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Clmrlcs.
Killcl,
Thermal physics.
Bibliography:p. index.
Includes
Statistical
!.
Herbert.1928-
536'.?
\302\260
I9B0
Copyright No pan of mechanical,
79-16677
by W.
bor.k
may
plioiographic,
H. Freeman and be reproduced
or electronic
copiedfor permission
in
Company by
any
process,or in
phonographic recording,nor
may
it be
system, transmitted, orotherwisc wriiicn or private use, without public from the publisher.
a retrieval
in
Pcimcti
this
of a
form
sioreti
ilie
United
State of
America
Twenty-first printing, 2000
9
Kroe/n Tillc.
II.
aullior.
O-7167-IO8S-9
ISBN
liic
joiiii
1930
QC3H.5.K52
I.
tiicrmodyn;miics.
About the Authors
Charles
at
has
Kiitel since
Berkeley
1951,
having in
work
undergraduate
solid
laught
physics
slate physics at the University of California been at [he Bell Laboratories.His previously was done at M.I.T. and at the Cavendish
His Ph.D. research was in theorclicai nuclear Professor Breit at the University of Wisconsin. physics with Gregory He has been awarded three Guggenheim fellowships, the Oliver Buckley Pme for Solid State Physics, and, for contributions to teaching, the Oersted Medal of the American Association of Physics Teachers. He is a member of \"he ;i id National of of Arts Science and of the American Academy Academy semio nSciences. His research has been in magnetism, resonance, magnetic and the statistical mechanics o f soiids. ductors, of Cambridge
Laboratory
Kroemcr
iferbcrt
is
liliy^ics.
I
a I'lt.D.
!c received
in Germany
with
Professor
liurhara.
at Santa
California
University.
a
Prom 1952 through
thesis
of Electrical
Engineering at the
His background
mid
in physics in I'J52 from on hot electron effects
I96S tie workedin
several
nre
training
the University in
lhc
semiconductor
then
of
University
in solid of
state
Gulling
transistor.
new research
labora-
of Stales. In I96S lie became in to UCStt at ltic University of Colorado; lie came [ilixirieul Engineering of semiconductors and technology 1976. His research has been in the physics and semiconductor devices, including transistors, negativehigh-frequency electron-hole mass effects in semiconductors, injection lasers,the Gunn effect, and semiconductor hetcrojunctions. drops,
laboratories
in
Germany
and
the
United
Professor
Preface
This book
an
gives
simple,
other
no
Probably
and
science
are
methods
the
applications.
physics. The subject is and the results have broad applicatheory is used more widely throughout of thermal
account
elementary
powerful,
physical
engineering.
We have written for undergraduate and for electrical engineering students
(not
but
original,
not
easily
physics
and
These
generally.
purposes have strong common bonds,most mcmls, gases, whether in semiconductors,
methods
of
students
notably
a concern
stars,
or ituclci.
accessible
elsewhere)
astronomy, for our
fields
Fermi
with
We develop that
are
well
to these fields. We wrote the book in the first place because we as compared to (hose were delighted by the clarity of the \"new\" methods we were taught when we were students ourselves. some because We have not emphasized several traditioual they topics, classical on statisnare no longer useful and some because their reliance cai mechanicswould make the course more difficult than we believe a course should be. Also, we have avoided the use of combinatorial first
suited
methods
where
they
are unnecessary.
Notation and units;
parallel. the
do
We
fundamental
not
use
We
generally
the calorie.
temperature
t by
use the SI and CGS systems in to The kclvin temperatureT is related
r =
kBT,
and
the
conventional
entropy
S is reialed lo the fundamental a by 5 = ka(j. The symbol log entropy will denote natural logarithm throughout, simply because In is less exlo Equation refers A8) of A8) expressive when set in type. The notafion of 3. the current chapter, but C.18) refers to Equation A8) Chapter with the assisfto course notes developed Hie bookis ihe successor (he ance of grants of California. Edward M. PurceSlconUniversity by from review of the to We benefited ideas the contributed first edition. many and Nh.-holns L. Richards, Paul second edition by Seymour Geller, Wheeler- Help was giveii by Ibrahim Adawi, Bernard Black, G. DomoK. A. Jackson, S. Justi, Peter Cameron kos, Margaret Geller, Hayne, Martin Ellen Leverenz, Bruce H. J. J. Klein, Kittel, Richard Kittler, McKellar,
F.
E.
O'Meara,
Norman
E. Phillips,
B. Roswclt Russell,T. M.
Preface
B.
Sanders.
An
by
in
added
was
the index.
treatment
elementary
atmosphere
Carol
thank
her help with
for
Wilde
Professor
Richard
John Wheatley, and Eyvind Verhoogen, for the Tung typed manuscript and Sari
John
Stoeckly, We
Wichmann.
1994
of the
on page
Muiier.
Bose-Eitistein condensationwas For instructors who have
solutions
manual
is
available
A
115, following an
page
on
added
to
adopted
via
effect in the Earth's
greenhouse
the
aioinic page
suggested
argument
gas experiments 223 in 2000.
on the
classroom
use, a
course
the freeman
atmo-
for
web site
(http:/Avhfreenian.
com/thermaiphysics).
Berkeley
and
Santa
Barbara
Charles Herbert
K'tttel
Kroemer
Note
to
the
Student
For minimum of the concepts the authors coverage presented in each chapier, recommend the following exercises.Chapter 2: 1,2, Chapter 3: 1,2, 3,4, 8, 6: 1,2,3,6,12, 5: 11; Chapter 4: 1,2,4, 5,6, 8; Chapter 6,8; 1,3,4, Chapter 9: 8: 14, 15;Chapter7: 2, 1, 2, 3, 5, 6, 7; 1, 2, 3; 7, 11; Chapter Chapter 12: 13: 3,4.5; 1,2, Chapter Chapter 10: 1,2,3; Chapter 11: 1,2,3;Chapter
3;
3,5,6,
3,7,8,10; Chapter 14: 1,3,4,5; Chapter
15:
2,3,4,6.
Contents
Guide
xiii
to Fundamental Definitions
General
xv
References
Introduction
Chapter
1 States
1
a Model
5
System
Eittropy and Temperature
Chapter 2 Chapter
of
3
Distribution
Boltzmann
27
and Hdmholtz
Free Energy 55
Chapter4
Thermal
Chapter 5
Chemical Potential and Gibbs Distribution
Chapter
Ideal
6
Chapter
8
il
Chapter
Binary
309
Mixtures
Cryogenics 333 Statistics
14
Kinetic Theory 389
Appendix B
261
275
Transformations
Semiconductor
A
353
423
Propagation
Some
7
225
13
Chapter15 Appendix
Phase
1 i
181
Gibbs Free Energy and ChemicalReactions
9
Chapter 12 Chapter
87
Distribution
Planck
151
Work
tHeaZahd
Chapter10 Chapter
Gas
Fermi and Bose Gases
Chapter 7 Chapter
and
Radiation
Integrals
Containing
TemperatureScales 445
Exponentials
439
Appendix
Appendix
C
Poisson
D
Pressure
AppendixE Index
465
Distribution
453
459
Negative
Temperature
460
Absolute
~
X
activity,
Accessible
Definitions
Fundamental
to
Guide
29
state
Boltzmann constant,
25
ka
Boltzmann
factor,
Boson
183
Chemical
119
/;
Classicalregime, n
74
\302\253
nQ
31
of systems
Enthalpy, H = U
+
246
pV
40
a
Entropy,
1S3
Fermion
Gibbs factor, exp[(NjiGibbs
free
Gibbs
or grand
Heat
61'
exp\302\243~~ \302\243/t)
potential,
Ensemble
139
exp(/i/t)
sum,
U
\342\200\224
\\a
\342\226\240%
138
+
p^
138
63
C
capacity,
68, 227
Heat, Q
free
HelmhoHz
Landau
G =
energy,
t)/i]
free energy
Multiplicity,g
F
energy,
\342\200\224 \342\200\224 U xa
function, FL
7
9
Orbital
Partition
function,
Z
61
t
298
Guide to Fundamental
concentration,
Quantum
Reversible
64
41 62
Thermal
average
Thermal
equilibrium W
hq
process
Temperature, t
Work,
Definitions
227
39
=
References
General
Thermodynamics A. B. Pippard,
of classical
Elements
1966. M. W.
R. H.
and
Zemansfcy
textbook, 6ih
anil
Heat
DiEEman,
ed., McGraw-Hill,
Cambridge University Press,
thermodynamics,
an intermediate
thermodynamics:
198!.
Afcchanics
Sitttisiical
U. K,
and M. Eisner, Statistical 1988. Agarwal mechanics, Wiicy, Dover PubticaHit), Statistical mechanics:principlesand selected applications, iions, 1987, cl956. C. Kittct, Elementary statistical applications physics, Wiicy, 1958. Parts 2 and 3 treat 1 has been expanded ioEo the Part to noise and to elemeniary transport Eheory.
T. L.
present Eext. R.
Kubo,
R, Kubo,
Statistical M.
mechanics,
North-Holland, 1990, cI965. Statistical physics !! (NanequHibrium),
N. Hashitsume,
Toda,
Springer,
1985.
L D.
Landau and E. M. Lifshitz,
Statistical
K. M. Lifshitz
and
L. P.
1985.
Scientific,
! (Equilibrium),
Springer,
1933,
tables
Mathematical
H. B. Dwight,
Tables
1961. A
3rd cd. by
physics,
1. Piiaevskii, Pcrgamon, 1980, part Ma, Statistical mechanics. World Shang-Keng M. Toda, R. Kubo, N. Saito,Statisticalphysics
and other
of integrals
mathematical data, 4ih
ed.,
MacmUton,
collection.
smati
useful
widely
Applications
Asirophysics
R. J. Taylor,
The
S. Weinbcrg,
The first
ed.,
Bainam
structure
their
stars:
three
Cooks,
minutes:
and
evotitiioit.
a modern
v:\\-w
1972.
Springer, of the
origin
of the
universe, new
1984.
Biophysics and macromolccules
T. L. Hill, Springer,
Cooperathity
1985.
theory
in biochemistry:
steady stale
and equilibrium
systems,
General Refer,
Cryogenicsand G. K.
low
J.
D. S.
and
Wilks
. .
Betis,
An
pa.
helium, 2nd
to liquid
introduction
physics, 3rd ed., Oxford
ed , Oxford
Univesity
1987.
Press,
Irreversible
thermodynamics
J. A. McLennan,Introduction 1989.
I.
in low-temperature
techniques
1987, ct979
Press,
University
physics
lempcrature
Experimental
White,
I. Stcngers,
and
Prigogine
Random
to
statistical
non-equilibrium
Order
out
of
mechanics, Prentice-Hall,
man's
chaos:
new dialog
with
nature.
1934.
House,
Kjnclic theory and transport phenomena S. G. Brush, The kind of motion we call heal, North-Holland, 1986, cI976. H. Smith and H. H. Jensen, Transport phenomena,Oxford University Press, Plasma
physics
I... Spitzer, Jr., Physical
in the
interstellar medium,
Touiouse, Introduction Wiley,
phenomena,
H. E.
and
Haasen,
Boundary
the
critical
and
critical
[ihenomena,
Oxford Uni-
1987.
Press,
affoys
Physical
metallurgy,
2nd ed.,
CambridgeUniversity
Press,
1986.
Superb.
value problems
and J. C. Jaeger, Conduction of heat H. S- Carslaw Press, sily 19S6,ci959.
Semiconductor
group and to
renormalizat'ton
1977.
Stanley, Introduction to phase transitions
University
Metais
to
197S.
Wiley,
.
.
P. PfeiHy and G.
P.
processes
transitions
Phase
19S9.
in solids,
2nd ed.,
Oxford Univer-
devices
Introduction to applied solidstate physics, t990. Plenum, 5th ed., Springer, 1991, Semiconductor K. Seeger, physics:an introduction, t981. S. M. Sze,Physics devices, 2nd ed., Wiley, of semiconductor
R.
Datven,
Solid
state
physics
C. Kittel, Introduction
to solid
state physics, 6th
ed.,
Wiley,
1986.
Referred
to ssISSR
Thermal
Introduction
Our approachto physics
to do
going
structure: in
in
thermal
the
this
differs from the tradition followed in beginning we provide this introduction 10set oul what we are that follow. We show the main lines of the logical all the physics comes from In order of lhcir the logic. physics
Therefore
courses.
chapters subject
in our are the entropy, the temporaiure, appearance,the leadingcltaracters story the Boltzmann the chemical factor, potential, the Gibbs factor, and the disiribu-
functions.
tion
The entropy A
closed
system
measures the number of quantum might
be
in
any
of these
states
quantum
to a
accessible
states and
system.
(we assume)with
statistical element, ihe fundamental logical o r states are inaccessible to the either accessible assumption, quantum and the system is cquaiiy likely to be in any one accessible slate as in system, olher slate. is defined accessible as Given accessible states, the cniropy any g = a lhtis defined will be a function of ihe energy U, lhe logg. The entropy V of the system, because theseparamnumber of particles N, and the volume as wirii. The parameters ciilcr enter the dctcrminaiion of y; other para meters may is a mathematical use of the logarithm convenience: it is easier to write 1010 than expA020), and it is morenatural to speak of a-y + o, lhan for two systems The
probability.
equal
fundamental
is that
ofg,3j.
When two systems, each of ttiey
may
transfer
energy;
energy,
specified
their total
individual energies are perhaps in the other, may increase on their
are brought
into thermal
energy remains constant,but A
lifted.
the
transfer
product
accessiblestates of the combined systems.
The
of energy
in
the one
coniact
comlraints direction,
or
g^g, that measures the tiumber of fundamental
assumption
biases
maximizes the the outcome in favor of that allocation of the total energy that and more likely. This statement is number of accessible states: more is better, is the the kernel of the law of increase of entropy, which general expression of
the second law
of
thermodynamics.
brought two systems into thermal contactso that they may transfer One ofthe encounter? system will energy. What is the most probableoutcome of the of the other, and meanwhile the lotal entropy gain energy at lhe expense two systems will increase. will reach a maximum for the entropy Eventually It is not the total difficult to show 2) that the maximum given energy. (Chapter We have
is atiained when
ihe same
for
quantity
in ihermai
system is equal to the value of This equality property for Iwo systems of the icmperat lire. Accordingly, expect for one
value
ihe
o((ca/cU}K_y the Second system.
coniaa is just the
we define the fundamental
property we
relation
by the
i
lemperaiure
1
U)
CUJ
The use of 1/r assuresthat needed.
complicated
directly
proportional
to t,
3. i.ti
a
c, be pjaeed in thermal reservoir. The loia! energy
.S\"
-
the
of
the Boltzmann constant.
i.fie :it cnorj>y
sinies,
the
have
fundamental
smali system
with
Uo
energy
assumpiion,
s to
energy
entropy a
g{U0 by
may
definition
the
be dropped.
This
may
~ e)
is Boitzmantt's
\342\200\224 slates
e, the lo
accessible
e)
B) exp[<*(f
be
a{Ua)
in a
expanded
~
o)]
Taylor series: =
\302\243{ca/tV0)
c(t/0)
- e/r ,
the temperature. Higher order terms in which Cancellation of the term exp[_a{U0)],
result.
have
wiil
of energy
A) of
energy (e) of the two state temperature
g{U0
and
Uo slate
- c)
and denominator of B) after
numerator
have
gWo)
^@)
:n
the ratio of tbe probability of finding [he 0 is the probability of finding it with energy
,
The reservoir
wiii
:iml
in
une
system Uial we eaH the systems is UQ; when the small
is in ihe stale of energy 0. the reservoir has energy in states it. When the small is the accessibleto sysiem g{U0)
it. By
tl
u large
system
reservoirwill
treau-d
factor
Dolt/nvnm
ibe
combined
\342\200\224 \302\243 and
is
T
temperature
kRa.
with
coniaci
energy
more com-
r; no
low
to
Kelvin
where kB is
Ayr,
entropy given by a very MiiipJe csampli? uf iwn Miiall sysicm wiih uiily
Now consider Uiajner
=
i
with
x
ihe
it will foilow'that 5 is
conventional
The
flow from high
will
energy
is
rclaiion
the
To show its system
in
substitution
use, we
thermal
calculate
contact
the
with a
expansion
occurs
of C),
the
C)
in the
leaves us
thermal
with
average
reservoir at tem-
t;
E)
imposed the normalization conditionon the
we have
where
of the
sum
prob-
probabilities:
+
P@)
The
derivation
in the most
The
as energy with the
as
eqtuti
the
transfer
the
to
For two sign
systems in
in G)
is chosen to
the
chemicai
high
The Gibbsfactorof
maximum with the transfer respect be of energy. Not only must %\342\226\240 {ca/cU)\\also be N refers to must equal, where
diffusive
ensure thai
ih'w
r,
<=
and
r2
of particle flow to low chemical
direction
potential
is Hio
coiutiiion
cttii:tiity
contact,
jt1
=
\\it.
The
as equilibrium
potential. and Bolizmann factor
of Ihe
an extension
5 is
Chapter
diffusive and
to
{uwn spevk's.Tin-*
and
thermal
is approachedIs from
(f it/jW)^,.
first
transfer
can
two systems in
For
a
of
4 as the
that
systems
be a
will
for Hie two systems, btil mmilKr of particles of ;i
average energy
in Chapter
this
theory is to
reservoir.
ttie
entropy as
well
of the
extension
thermal contact, of particles
F)
to find the
immediately
generalized
important
ns well
particles
- 1.
at temperature r, and we do of the Planck radiationlaw.
oscillator
harmonic step
be
can
argument
P{e)
is a treat systems ilia! can transfer particles.The simplestcx:unpie 1 parltcie with two states, one witli 0 panicles and 0 energy, and one with system The in at e. a reservoir and with temperature r and energy system is contact chemicalpotential;i. We extend C) for the reservoir entropy: ailows
us to
-
tf(f/0
t;HQ
- 1) =
~ By anaiogy
with
D),
we
-
0(r/o;A'o)
~
=\302\273
cxp[{/(
ratio ofihe
is readily
(S)
have
probability the system ts unoccupied, to the probability the system normalization
- l-{ca/dNo)
+ ^/r.
c/r
o(U0;NQ)
P{U)/P@,Q) for the
z{ca/dUo)
expressed
\302\243)/r]
is occupied
with
(9)
,
bv
J
particle
energy 0. The
at
z
energy
result (9)after
as
A0)
This particular result is known as the is used
particularly
the
theory
The classicaldistribution the limit
is just
The
of the
properties
The HeSmholtzfree
energy
for finding the entropy, eigenvalues
energy
thcnnodyiumic the
tcxl
illuminate
the
concerns meaning
Thermal
objects,
and
physics
of
and
the
derivation
in the
F=
low
of the ideal gas law-
occupancyPA,e) is much lessthan
V
the relation
\342\200\224 to
appears
{5F/cx)s
once we have
found
v
=
out
this
result
as an \342\200\224 a
offers
how
1:
6.
in Chapter
important computamethod
the easiest
to calculate
F
from
the
of 3). Other powerful tools for the calculation arc the in of functions reiftaiiuier of developed the text. Most that are useful in their own right and that illumiapplications and utility of the principal thermodynamic functions. connects the world of every iky objects, of astronomic;*!
(Chapter
chemical
and
atomic, and electronic systems. microscopic
used
ideal gas are developedfrom
because
computational function,
electron gas at
(Chapter 7).
function
of A0) when the
to describe the
and
function
distribution
Fermi-Dirac
of metals
high concentration
and
temperature
in
macroscopic.
biological It
unites
processes the two
with
the
world
of moiecular.
parts of our world,the
micro-
1
Chapter
of a Model
States
MODEL
BINARY
AVERAGE
Energy Example:
U
Function
16
System
Alloy
Sharpness
10
SYSTEMS
of States and the Multiplicity
Enumeration Binary
System
of the Multiplicity VALUES
of ihe
SUMMAItY
22
:\342\226\240
Binary Magnetic
Multiplicity
18
Function
Function
System for Harmonic
''
--.
Oscillators
-3 24
26
Bul
of a
!: States
Chapter
as a
although,
investigations development,
in
and becauseit
matter of history, statisticalmechanics owes it seems
thermodynamics,
both on account of the new
yields
quite
departments
ModelSystei.
outside
results
eminently and
elegance
and places
old truths
is the more
more different applicability.
upon me. It
kinds
is ihu
impressive the
only
mil never be overthrown,
the
the
greater
it relates,
of things
Therefore
independent
light
in
of thermodynamics,
J. A theory
to
principles,
a new
in
origin
of its
worthy
simplicity
its of an
physical within
theory
simplicity
and the more
deep impression
that
classical
of universal
the framework
W.
of
Cibbs
of its
premises, the
extendedits area of made
thermodynamics
content
which
I am
convinced
applicability of its basic
concepts. A.
Einstein
1: States
Chapter
Thermal
is the
physics
Mechanicstells
us
the
are three
heat. There
of
fruit
of work;
meaning new
in
quantities
ordinary mechanics:entropy, their definitions the in first
and mechanical
ofslatistica!
union
the
temperature,
principles.
thermal physics tells us the meaning of that do not thermal physics appear in and free energy. We shall motivate
deduce their consequences
chapters and
three
of a ModelSyster,
thereafter.
Our
for the developmentofthermal physics
of departure
point
stationary quantum statesofa the quantum states accessibleto a
of the
of
system
the
for
as the logarithm of the
is defined
entropy
The dependenceof the entropy
energy
of the
of states
number
of the system energy
concept
count
we can
the entropy
and the free
the temperature,
rhe entropy,
temperature.From
the
on
particles.
know
we
system,
is the
When
system,
(Chapter
2).
defines the temperawe find
the pressure,
the chemical potential,and
Fora as
such brevity
in
system
the
energy
we
usually
other therm odynamic properties of tlie system. all a stationary quantum state, all observable physicalproperties and the number of particlesare independentofthe time.For omit the word stationary; the quantum slates that we treat
are stationary exceptwhen 14-55. we discuss transport processes in Chapters The systems we discuss may be composedof a single more or, ofi^n, particle of interof many particles. The theory to handle genera! systems is developed but be can made in particles, powerful special problems interacting simplilicarions for
interactions
the
which
Each quantum said to
has
slate
may be neglected. a definite energy.
belong to the same energy
The
level.
Stales
with
multiplicity
energy level is the number of quantum states wiih very it is the number of quantum slates that is important tlte
of energy
number
states. Two states at not
as one
shall frequently
levels. We the
same
deal
must always
energy
levels
is taken
figure
of
hydrogen
at the
are shown
state of lowest
belonging to the sameenergy that
the
energies
arc
or degeneracy of an the some energy, nearly
physics, not over all quantum
in thermal
fti'th
sums
be counted as
two
slates,
level.
Let us look antic qiu.Umusuresam! eitcryy The simpler is hydrogen, with oik electron energy
identical
prolon
has
a spin
level
of
\\h
and
in
Figure
energy.
is in has
lewis
and !.!. Tlte
parentheses,
uOveia!
alomic
s>^u-ms.
one proron. The !u.\\-I>iiig The zero of energy in the of quantum stales number
figure we overlook orientations, parallel
in lite
two independent
of a Model
1; States
Chapter
SyMm
Hydrogen
Boron
Lithium
ant! boron. The lithium, lergy levelsof atomic hydrogen, votis, with t eV = 1.602 x tO\0211 erg. The numbers \302\243 stales the same energy, with no ac give ihe number of quantum parentheses having is taken forcouvei lakenofthe spinoflhe nucleus. The zero of energy in the figure ai die lowest energy slaie of each aiom. ei
Low-tying
energiesare
to the
or antiparaliel
of a
direction of an arbitrary external axis,such
magnetic field.To takeaccount
the values An
in electron
given
atom
of
the
multiplicities
of lithium
has three
electron interacts with the
of
shown
the
two
for atomic
electrons which
orientations
as
the
direction
we should
double
hydrogen, move
about
the
nucleus,and eachelectronalsointeracts
nucleus. with
Each ail
the
Chapter
I: Statesofa Model
Systet
IUpresc
30 Mill
diq
lip liciiy
4
3
1
4
2
2
4
2
3
3
25
20
J
i
i
4 3
2
3
2
2
2
3
1
1
2
1
\\
'
1
1
2
15
5 to 3
_ 1
Figure \302\253_,, jjj,,
1.2 n, ofa
1
and Energy levels, multiplicities, particle confined to a cube.
quanlum
numbers
other electrons.The energiesofthe levels of lithium shown in the figure are the collective energies of the entire system. The leveis shown for boron, which energy has five electrons, are also the energies of the entire system. The energy ofa system isthe total energy ofal! kinetic plus potential, particles, with
account
taken
of interactions
between particles.
system is a state ofall particles. Quantum called orbitals. The low-lying levels energy confined
to
a cube
of stde
are \302\243
shown
states
ofa
in Figure
ofa
A
state
quantum
system
one-pariicie
are
of mass M con-
single particle
1.2. We shall
of the
find
in
Chapter
3
Chapter
1: State* of a Model
free particlecan be characterized
of a
an orbital
that
numbers nf,
quantum
The multiplicities
System
The
n,.
nyi
by
three
integral
positive
is
energy
of the levelsare indicated
tn
the
orbitals
The three
figure.
wiih(HJl^>.fi.)equ.iltoD,l,l),(l,4,]),and{l,l,4)ai!l!aveii/ + the corresponding energy level has 3. multiplicity
+
nr2
=
n.1
18;
the
to know these!of values
siaie soTthejV states
in
any
assigned
the
of
arbitrary
such as
c is
s may
it is
A' particles,
of
\302\243S(N),where
energy Indices
system.
particle
convenient
theenergy
be
assigned
essemial
of the quantum lo the quantum
states should not
way, bul two different
be
index.
same
the
properties of a system
the statistical
To describe
It is a good idea to siart the properties of simple program by studying model which the energies for can be calculated A') e.vacily. We choose as a modela simple because the genera! statisiical propenies system our
systems
Ej{
binary
are believed to appiy equally This physical system. assumptionleadsto predictions
found
for
the
model
experiment. What as we go along.
statistical
general
are of
properties
to any
well
system
that
always
concern will
realistic
agree become
with clear
BINARY MODEL SYSTEMS
The
model
binary
separate
and
Attached
to
syslern
distinct
each
site
sites
is an
is illustrated fixed
in
N shown for convenience on a line1.3.
Figure
in space,
elementary magnet
titat
can
We assume
point
only
there are
up or
down,
tlie system means to To understand corresponding to magnetic an element of the no of count the slates.This requites magnelism: or no, red or blue, site of two states, labeled as system can be capable one or one. The sites are minus occupied or unoccupied,zeroor one, dtSFercni to in sties with numbers are supposed numbered, overlap moments
\302\261n>.
knowledge
any
yes
plus
and
not
of the sites as numbered parking spaces in space. You might even tltink vacant or 1A Cacti as in a car lot, Figiire parking spuce has two states, parking occupied by one car. the two slates of otlr objects, we may milure Whatever llic by desigreiic down. If (he magnet points arrows that can only point straight up or straight the up, we say thai ilie magnetic moinenr is -Hii.If the magnet down, points magnetic moment is -m.
physical
Model
Binary
123456789
10 Number of the
1.3
Ffgure moment
numbers
The \302\261m.
sire
each
Mode! system sites on a
at fixed
magnets
has
Syster
ils own
site
composedof 10elementary magncric line, each having shown arc aflachcd to ihc silcs;
magnet
assume
We
there
are no
magne'ic field.Each
moment may be oriented in magnetic up or down, so ihai there are 210disiincl of the 10 magnetic moments shown in the arrangements arc selectedin a random process, figure. If ihe arrangements two
ways,
(he probability is 1/210.
1.4
Figure
State
of finding
ofa
tile
parking
spaces. TiseO's denotespaces denote
vacant
shown
in Figure
spaces.
independent
probability
of arrangements
state of
site;
there
state of
2'v states.
10 numbered
with
particular
sites,
each
parking theO's
by a car;
occupied
This
moment
state is equivalent
nf the We may
bears be oriented
of which may
orientation
of the
the system h sjveitiedUy are
lot
shown
arrangement
to
that
(.3.
Now consider N different assume the values +\302\253i. Each number
particular
n
thai
moment
in two
ways
may
ofa!! other moments.
N moments is
The
2
x
2 x
2 x
\342\226\240 \342\226\240 \342\200\242 *=
2
a
with
total
2\\
A
the orient at ion of the moment oil c:k!i yiviiig for a single use ilio following simplettotation
the system of N sites:
nuimrr-
B)
of a
\342\226\240rl: States
Model Syst
four diflercnl Males of a elements numbered | and 2, vs here ench clctnetit can hsvc two conditions The element is a magnel which can be in condition f orcondiiion [. The
Figure
1,5
s> stem
of two
numbcr4hem to
We may are assumed to be arrangedin a definite order. ftom left to right, as we did in Figure1.3.According sequence the state {2}also can be wriitcn as
sites themselves
The
in
convention
this
C)
symbols B) and {3}denotethe same state of the system, the slate in the magnetic which moment on site 1 is +m; on site 2, the moment is -t-m; on site 3, the moment is -m; and so forth. It is not hard to convince yourself that distinct state of the system is every in N contained a symbolic of factors: product sets of
Both
D)
U)(U
The
is defined
ruie
multiplication
by
ti + till
liXti + li)
(Tt +
+
UU
The function D) on muitipltcation generatesa sum of 2*v the 2'v possible states. Each term is a product of N individual symbols,
denotes an
T1T3I3 For
one
with
magnetic
of
moment
elementary magnet on the line.Each term the system and is a simpleproductof the form
of
state
example.
of two elementary
to obtain the four
possible
states
(Ti + I1KT2+ is not
but
a state
The product on the it generates
for each
one
terms,
f\302\260r
t\\i
a system
The sum
E)
for each
symbol
independent
''\"
+
the states
of the
of Figure
side
of listingthe four of the
system.
(}x
+
li)by(t2
+ |j)
1.5:
Till
ii)
is a way
left-hand
magnets, we multiply
itTa
possible
+
F)
I1I2.
of the
states
system.
equation is calleda generatingfunction:
.
\342\226\240
\342\226\240 \342\226\240
.
Model
Binary
function
The generating
the
for
+
(Ti
of a
slates
system of
This expressionon multiplication generates21 = Three Two
magnets up:
Onemagnctup: None
The in
is
given
M
m
T1I2T3
lihti
tihli
IJ2I3
lilif3
lilils-
up:
be denoted
will
field. The
a magnetic
values
T1T2I3
totat magnetic moment of our
magnetic moment
states:
S different
T1T1T3
up:
magnets
three magnetsis
+ U)-
IjHTj
li)(?2+
Systems
model system
by Mt
value of M varies
from
of
which we will to --
Nm
N
each
magnets to
relate
Nm. The
of
the energy
set of possible
by
\302\253
Nm,
-
{N
2)m,
(N -
-
(N
4>n,
6)m,
\342\200\242 \342\200\242 -A'\302\273i\342\200\242,
G)
possible values of M is obtainedif we start with the state for which all = Nm) and reverseone at a time. We may reverse iV magnets magnets up (M to obtain llie ultimate state for which are down (A/ = - Nm). al! magnets There are N + ] possible of the total moment, whereas there are 2s values states.When N \302\273!, we have 2N \302\273N + 1. There are many more states than states ! 024 distributed values of [he total moment. !fW = 10,there are 210= For N many moment. 11 different values of the total large among magnetic the total moment ft/. of the the value different states of have same system may a given value of M. have in the next section how many states We will calculate
The
set of
are
Only
state
one
of a
system has the moment TTTT-
There arc N
is one
sue!
1
ways
state;
to form
another
a slate
with
-
M = Nm;that
magnet
is
(S)
-TTTT-
one
state
down:
mt
\342\200\242\342\226\240\342\226\240mt
tin
\342\226\240\342\226\240.\342\226\240tin,
AJ)
is \"
do)
1: Slates
Chapter
with one
slates
other
the
and
of a Mode!System magnet down are formedfrom
magnet. The states (9)
any single
of
Enumeration
moment
lot.il
have \302\243!0)
by
reversing
- 2w.
M = Nw
Function
the Multiplicity
and
Stales
and
(S)
spin as a shorthand for even number. We
It is convenient lo elementary magnet. need a mathematical expressionfor the \342\200\224 s magnets number of states with W, = {W + s magnets up and Nl = jN where sis an When we turn one from to Ihe down, integer. magnet up [he down + s s goes to jiV ~ 5 + I. I and orientation, {.V + 5 goes to jW ?N The difference (number up \342\200\224 from 2s to 2a \342\200\224 2. The number down) changes word
the
use
We
assume that
is an
N
difference
-
W,
ihc spin
is called
spin
excess of
right. The facior of 2 in prove to be convenient.
left to
from but
The
excess.
it
will
The productin
D)
may
be written
only
in
many
the order in
of the
magnets
have magnets
sites which
the
arrows
(ID
the 4 states in to be
+
in
\342\200\224
2,
a nuisance at this stage,
if-
from
are
a state
up or down, in a
appear
2,0, 0,
1.5 is
Figure
as
symbolically
drop the site labels {thesubscripts)
how
particular
25
appears
(I!}
(T \342\226\240.
We may
=
/V,
ft
D)
up or we
drop
we are
when
interested
down, and not the
labels
in
which
and neglect
given product, then E) becomes -
(t
II;
further,
(t +
We
find
(I
+
|)v for
I)' = Itt + arbitrary
iV
by
3ItJ + 3IJJ + jjj. the
binomial
expansion
A2)
We
may by
With
ihis result
t with
replacing
W, states
denote
\\N
expression{| +
|)'v
ivv =* y
becomes
tj-v+j
iA-+*
M*V\"J
A4)
of stittes having s magnets down. This class of $N + 5 magnets up and N, = i.V \342\200\224 = lias excess 2s and net raagneiic moment 2sm.Let us JV, spin JVj the number of states tn this class by g{N,s), for a system of N magnets:
coefficient =
but equivatem,
different,
a slightly
in
Function
Multiplicity
\342\200\224 s:
ihe symbolic
4-
The
exponents of x and y
write the
form
and the
of Stales
Enumeration
of the term
in
f
is the number
M*\"'
-
,n\342\202\254>T
(IS)
Thus
is written
A4)
as
I stJMT^l1\"\"
(I + i)'v= We
call g(N,s)
shall
llie same
of
value
ihe
field is applied to the spin s have
states
Note
tn
in a
system:
different values of the
of an energy level
for our
reason
5. The
deltnttion
a magnetic
of different
of
values
to the
field. Until
\302\260ur 9 is equal multiplicity we introduce a magneticfield, all
model system have the same energy, which the total number of states is given that A6)
of the from
of slates having when
emerges
field, stales
magnetic
energy, so that
a magnetic
ihe number
it is
function;
multiplicity
A6)
may
be
taken
as zero.
by
'
L
Examples related to g{h',s)for
coin, \"heads\" down.\"
could
stand
g{Nts) =
A'
~
for \"magnet
A
\\Q are
l)-v =
+
given
upland
in
\"tails\"
(H)
2-v
Figures could
1.6 and stattd
1.7. For
for \"magnet
a
Chapter t: Slatesofa Model
Figtorc 1.6 Number of 5 -f j- spins up and Values
of yf Npi)
tUc spin stales is
oixss
N
TTic values of the the
binomial
of distinct arrangements 5 ~ 5 spins down.
are for N - 10, when: 2.v K I. Tlic toul numtwt \\ -
9's
System
arc taken
h of
fro
coefficients.
I
-10
-8
I
-4
j
Spin
To illustrate that
the
the result, we consider sites, numbered from atom
provision
a
single
2
4
6
excess
2s
Alloy System
Binary
an
0
-2
-6
for
exact an
alternate
1 through
of chemical
species A
sites.
In brass,
vacant state
of the
of the two states
nature
on each site is irrelevant to
with N distinct alloy crystal 1.8. Each siteis occupiedby either chemical species B, with no provicopper and B zinc. In analogy to C),
system\342\200\224an
12 in Figure of or an atom A
alloy system
could
be
can be written
as
-
A8)
nry
Allay
Sya,m
\342\226\240= o
S
3
20
Number
Figure
were throw
1,7
An
experiment
NX) times.
thrown
10
23456789
01
of
heads
Was done in The number of
10 pennies
which
heads
in
each
was recorded.
0\302\25100 3
2
I
A
Fijutc
\302\251 5
chemical
0 6
10
0
7
S
II
012
1.8
A binary componenls
alloy syslcm of two A and 1!,whoseatoms
of a
state
distinct
Every
system on
binary alloy
in the
is contained
sites
N
symbolic product of N factors:
(A, + in
conventionally N
B2)(A3
+BN) ,
+ Bj)---(A.V
A9)
The Liverage composition of a binary is specified conalioy the chemical formula A1_1B1, which means thai out of a tola! by the number of A atoms is NA ~ A ~x)N and the number of B
to
analogy
of
4-
B1)(A3
atoms,
atoms is NB
D).
~
.v lies
.\\JV. Here
between Oand 1.
The symbolic
expression
is
of
g{i\\\\f)
B on
which
is identical
prediction
of
B' gives the
in A'v\"'
term
of N ~
A
\302\243 atoms
and
/
result A5) for
to the
the
spin
model
system, except
for notation.
Function
Multiplicity
experience that systems held at constanttemperature well-defined this stability of physical propertiesis a properties; The of thermal physics. stability follows as a consequenceof
have
exceedingly
bharp
function
that
the
from common
know
major the
of
or states
arrangements
possible
Sharpnessof ihe usually
The coefficient
N sites:
atoms
We
result A2).
to the
analogous
number
peak
in the
from
away
the steep
of and multiplicity function can show explicitly
the peak. We
large system, the function
defined
by
A5)
thai
us
allows
a very
sharply about
is peaked very
approximation that
variation for
to examine
the
jV \302\273 \302\253 We cannot (ook up these i and N. form of g(S,s) versus 5 when js| tables of factorials do not go above N = 100, and we values in tables: common may be interested in Af =^ 10'\302\260,of the order of the number of atoms in a soiid is clearly needed, felt. An approximation specimen big enoueli to be seen and
and a good oneis available. It
to work with to be understood
is convenient
logarithms
are
standard usage is In for ambiguity
whatever.
log
When
base
where
f-xccpl
log*/,
loy base*?,written c, but
it
is clearer
you confront a
very,
here
.-ilierwise as log.
to write very
specified, till The international
log when
there is
number
such
large
no
;is
Sharpnessof the 2iV, where
iV
is a
lG:o,it
simplification to look at the logarithmoft he
of both sidesof A5)
the logarithm
take
We
~
by virtue of ihe characteristicproperlyofthe logxy = the
With
logx + logy;
4-
\\N
number of magnetsup
and
of a
=
B2)
s)l ,
product:
~ logy.
log.v
in
N\\
\\N
B2)
down,
~
=\302\253
A',
5;
{23)
by use
B5)
a
of die
\302\273 1.
terms
\" \342\226\240 in the 1,A2iV) +
result
This
Stirling approximation, 1
ZjTi'Go
+
\342\226\240 \342\200\242 , \342\226\240]
s-t\\Q\\
argument
\\
B6)
N, large Appendix A. For sufficiently may be neglected in comparison with
is derived in
for N
B5)
logW,!.
+ IJ[12N)
BrtN)ti:iNNcxpl~N
B4)
as
appears
logiV,!-
s
which
.V!
the N.
of both sidesof B6)to obtain
the logarithm
take
of
the logarithm
evaluate
accordingto
We
-
log&N
logarithm
log(.\\-/_v)
logg(N,s) - logW! We
number.
notation
jV, =
for the
Function
obtain
lo
- logfrV + 5)! -
= logN!
\\ogcj(N,s)
Multiplicity
S
logN!
(N + |) log
2k +
I log
N
~
N.
B7)
Similarly
!ogjVt! s
I log
(JV, +
+
2?r
logN,! s iiog2* + After
- jV,;
BPJ
J)log.V,
- Nu
B9)
of B7),
rearrangement
log.Y!
+
(,Vi
JJlogN,
S
wlK'i'ciwcliaVcuscilW
Ito\302\243B!r/A')
= .V,
+
'.- f.V,
I-
.Vt.
\\
+
-V, +
i)logJV
- (.V, +
Nj)
,
CUf
\\VL-siil>ii;iLl{2S);iiKM2'J)froin{.!U)loobl:iiii
for B5):
tog
ilog(l/2;r,V) -
(.V,
+
iJlogty./.Y)
-
(,V, -f
i)Uui(.V,/.V). C1)
This
because
be simplified
may
logOV,/iV) = ~
by
log^l
4-
-iog2
+ Bs/,V)
the
expansion
!og(W,/A')
= logld
of
virtue
=
2s/jV)
-f-
2s/S)
- I2s2/N2)
x) =
logfl +
+ iog(i
-iog2
-
.v
C2) for x
\342\200\242 \342\226\240
jx2 +
valid
-,
\302\253 I.
Similarly,
On subsliiuiion in
C!) iogg
We write this
-log2 -
- 2s/N) =
-
B5/N)
Qs2/,\\'!).
C3)
obtain
we
+
s |logB/;E/V)
-
Wiog2
C4)
2s!/N.
result as
C5)
C6)
WnH)\"*?1.
Such
of values of sis
a distribution
of C5) number
over the range of states. Several useful \342\200\224 co to
The exact value of y(N,0)
caiicda
+ co
for s gives the correctvalue
integrals
is given
are
by A5)
with
The
distribution.
Gaussian
treated s =
2*for
in Appendix 0;
integral* the total
A.
C7\302\273
\342\200\242
The
replacement errors.
significanl
of a
slim
by
For example,
X s
=--
such as \302\243 by f{_. {. - \342\226\240)
an iniegrai, ihc rai io of
i{N2
+
N)
to
Ts./s
\302\273^o
is equal
io
1
+ A/N),
which approaches i
.)Js,
usually
'
as N
approaches
co.
= IN2
does
10
\\ 1.9
Figure (he
X.
\"
1
6
\\
\342\200\224
plotted
\\
4
cocfficicnis
iinearscaie. On this on
distinguish
The
approsimaiion (o
Tlie Gaussian
binomial
ihc
entire
scale
g{!OO,s) plotted on a ii is not possible io
drawilig range
the approximation of s is from
dashed lines ai\302\253 drawn from vaiuc or points at t/e or (he maximum 4- SO. The
1
n y
0
0
-10
-20
For N = 50, the in
C6) is
from
vylue
value
is 1.264
x 101*,from C7).The approximate
1.270 x 1014.Thedistribution plottedin Figure t.9 is centered = 0. When = to e~' ofthe s1 value is reduced the of g {N,
at s
a maximum
value.
maximum
of sE0,0)
That
is, when
s/N = A/2NI'2 ,
CS).
meaise\021 of g(N,0).The quantity is Ihus a reasonable A/2NI'1 width of the distribution. For N =s 1022, the fractional width is of the order of 10\"u. When N is very large, ihc distribution is exceeda in It is this sharp peakand the continued relative sense. defined, exceedingly sharply to a of the multiplicity function from the peak that will lead far sharp variation
the
measure
of g
value of
the
prediction well
defined.
fractional
properties of systems in thermal equilibrium We now consider one such property, the value of s1. mean
that
the physical
-
are
50
the y.
to
ChapterI: Stales
Modal System
of a
AVERAGE VALUES
The averagevalue,
mean
or
function
distribution
of a
vaiue, is defined
P(s)
function f(s) taken over a probability
as
= Z/<*m*),
normalized lo
is nol
we
and
g{N,s)/2s,
D0)
distribution A5) has lite property
have
If
unity.
\302\243J'($)
D1)
,
are equally
states
all
that
A7)
\302\253 2W
Zs(iV,5)
and
unity;
!.
\302\243?(*)-
The binomial
to
is normalized
function
distribution
Uic
that
provided
09)
probable, ihen P(s) ~
average of/(s) over this
t. Tfic
distribution
be
will
D2)
Consider tile function
C6),
we
and +
in
replace
f{s) = s2.In
D2) the sum
over
\302\243
the
led to
that
approximation
s by
an integral
\342\200\242 \342\226\240 \342\226\240
J
ds
between
C5) and ~- co
co.Then
- [2/nNI'3
(jV/2)J'J
J^Jxx^\"'*
= {2/7r,V)\"!{.V/2):i'J
,
(jt/4)\022
whence
The
quantity
spin
excess
<{2i);>
= iiV;
is the mean
=
,V.
D3)
square spin excess.The root mean
square
is
';J =>yfN
t
-
D4)
Sys
and the fractional
2s is
in
fluctuation
as
defined
D5)
The larger N
smaller is the fractional fluctuation. This means is, the that the central peak of the distribution function becomes more relatively sharply defined as the size of the system the size measured increase;, being by ihe number of sites N. For 10:oparticles, $F 10\0210,which is very small. the
of
Energy
System
Magnetic
Binary
The thermal properties of the model system when become relevant physically the elementary magnets arc placedin a magnetic for the energies of ihen ticid, the different states ate no longer all equal. If the energyof the system is specified, ihen only the suites having this ion occur. The \302\273f ink-net energy tn;ty energy ofa single magnetic moment m with a fixed external magnetic field B is \302\273 -m-B.
V
the potential energy of For the model system of
This is
orientations in
a uniform
the Ar
H6)
m
magnet
in
field 1J, the
magnetic
ihe
- MB ,
toial magnetic moment
2s/n.In this
or
Following
sections.
The value
^f (he
field ?.;
magnetic
dependence 2s
by
-2,
is,
spacing
for
energy
special
i..ucaka
by wrtitng
bsiween
Ae
=
the
by
inferact the
lite
example
moment
adjacent
by
tnodcl,
particular
only
but
is developedm
with
of s.
value
U{s). Reversing a
magnetic
difference
of
feature
ihat
moments
D7)
concreafe no difficulty. Furthermore, of (his model is constant, as in
of (he argument that
^ornrfcicly ridennined
lowers the total
by 2mB.The energy
is a
the generality
restrict
not
will
will
is
U
sec later (hat a
We shall
discrete.
levels
energy
adjacent
Constant
t.tO.
this feature ihe
between
is
spectrum
quast-conttnuous
ihe spacing Figure
of ihe encrcy U
of values
spcclnttn continuous
M for the
expression
tola! potentialenergy 2smB}=
m,- =Q.
using
two allowed
with
each
mngncts,
elementary
B.
field
the
exicnial
the
This funcitotutl
single moment lowers
~2m,
levels is
and
raises
the energy
dcnotcJ by
Ac,
where
DS)
Chopset
UM./mB
s(.)
+ 10'
1
0
10
2.30
+ 6
45
3.81
+ 4
120
4.79
+2
210
5.35
0
252
5.53
210
S.3S
+
\342\226\240
S
__2
+2
\342\226\240
log g(
120
-4
4.79
+ 3
-6
45
3.81
+ 4
-8
10
2.30
+5
-10
0
1
field S. The levels where 2s is ihe spin excess and \\N + s ==\342\226\240 5 + i isihe number of tip spins. The ititd energies UD muliiplictlics g(>) ^fe showti. Tor this
magnetic nmmersis
are labeled by
Example:
Multiplicity
system is
the
given
by Max Sludent
problem is given
The quantum
slates
problem is the
solvable
exactly
simple. The beginning do the
magnetic
s values,
oscillators. The problem of tlic function for harmonic for which an exact solution for the multiplicity problem
simplest
known. Another was originally
m m a
their
in
ofa
Chapter
harmonic
the
quantum
the oscillator. consider
a system
number of ways
number
The number
of N such in which
states
is
infinite,
oscillators,all
a given
total
derivation.
energy
The
excitation
of
modern
way
to
eigenvalues
D9)
or
zero,
and
the
the
is
solution entirely
sho) ,
s is a positive integer of
this
oscillator have the es =
where
for which the oscillator, is often felt to be not derivation
nol worry about 4 and is simple.
model
function
harmonic
Planck. The original need
binary
and to is the
angular
multiplicity
of each is one. Now
same frequency.
energy
We want
of
frequency to
find
the
can for
be distributed tlie
among
e^rher. pitcitv function fount! We begin the analysis by =
forwm'chff(i,\302\253)
1
problem of E3) below,
we
the
function
multiplicity
the same
as the
spin
g{N,n) mufti-
function for a single oscillator, here identical to m. To sojve the
numbers,
quantum
a function
need
the
is not
to tlie multiplicity
back
of
want
function
multiplicity
going
ail values
for
is, we
That
oscillators.
the
The oscillator
Af oscillators.
to represent or generate
ihe
scries
E1)
AS! Y,fl!!1 from
(S3),
but
^
^CfC
coS\302\260
not appear
t docs
' 's the
in
a temporary tool that result. The answer is final Jusl
will
help
us find
the result
(S2) provided we assume\\i\\
<
|. For
the problem
of JV
oscillators,
the
generating
function is
E3)
becausetlie
of w;iys n term in which the
number
number of onSctedwuys We observe
i\"
can
integer
;\\\\i\\Kai in the N-fold n c;m be foiuicJ as the
pftiJuct
sum
is picciscly ihe of iV non-iicg.nive
that
tj{N,n)
2) Thus
for the
\342\226\240 \342\200\242
(W
+
n
- 1).
E4)
system of oscillators,
ES) This
result
will be
needed
in
solving
a problem
in the
next
chapter.
1: States
Chapter
of a Mode!System
SUMMARY
1.
The
In
function for a
multiplicity
N, -
N't
limit
ihe
syslem of N
with
magnets
spin
excess
2s =
is
s/N
A'
with
\302\253 1,
\302\273 1,
we
have
the Gaussian
approximation
g[N,s) * {2/rlN)m2xexp{~2s2/\\').
2. Ifal!
of
states
the
mode!
spin system
are equally likely,
the
average
value
of
equal
to
2
52>
in the
3.
The
=
j''^JsstgtN,s)
p
Gaussian approximation.
fractional
of s2
fluctuation
is defined
as (s2yll2/N and
is
S/2N\022.
4. The
where
energy of the modelspin
in
is the
magnetic
syslem
in a
siaie of
moment of one spin
and
spin excess 2s is
B is
the magnetic
field.
2
Chapter
and Temperature
Entropy
11
ASSUMPTION
FUNDAMENTAL
PROBABILITY
3'\\
of ;in
Construction
Example:
3-
Ensemble
Most ProbableConfiguration
33
Spin Systems in
Two
Example:
Thermal Contact
3?
39
THERMALEQUILIBRIUM TEMPERATURE
-\342\226\240!
ENTROPY
41
Floiv
On Heat
45
of Entropy
increase
of
Law
Increase
Entropy
Example:
LAWS OF THERMODYNAMICS as
Entropy
-iS
50
a Logarithm
Example: Perpetual Motionof the
Second
Kind
50
SUMMARY
51
PROBLEMS
5:
1.
Entropy
and
52
Temperature
2. Paramagnetism
52
3. Quantum HarmonicOscillator
52
4.
5.
The Additivity
53
of \"Never\"
Meaning of
the
Entropy
for Two
6. Integrated
Spin Systems
54
Deviation
Note
we Jo
on problems: The iitil
cinplusi^e
53
melhoJ of fhis
problem
c
chapter
soKing dl
lliis
siu
Chapter
2; Entropy
and Temperatui
One shouldno! imagine will
mix,
the
one
contrary, \302\260
W10
will
years
recognize that
ff
we
thai
two
wish
there is
this
to find
of thermodynamics,
in
in a
gases
after a few days finds .., ilia!not
then again
0.1 liter
separate, until
a time
by any noticeable unmixing equivalent to practically
rational
we must
then
container,initially mix again,
unmixed,
and so forth.
long compared One may
enormously
On
to
the gases.
of
never. . . .
an a priori foundation for the seek mechanicaldefinitions of temperature
mechanics
principles
and
entropy.
J.
W.
Gibbs
between energy and temperaturemay are considerations. in statistical {Twosystems] by probability a transfer does increase the not probability. of energy The
genera}
connection
M. Planck
only
be established
equilibrium
when
We slart
this chapter
that enables us to a of average physical property system.We then consider in thermal equilibrium, the definition of entropy, and the definition of systems The of will as the taw second law of temperature. thermodynamics appear increase of entropy. This chapter is perhapsthe most abstract in the book. The chapters th;it follow wilt apply the concepts to physical problems. a
with
value
the
define
definition
'
FUNDAMENTAL ASSUMPTION fundamental
The
assumption of thermal of the quantum states
likely to be in any arc assumed to
be equally
states
accessible
states
A
closed
constant
system volume,
over
of
probability
\"
\342\226\240 -
of a
ttt;tt
a closed
accessible to it. All
probable\342\200\224there
accessible
other
physicsis
system
accessible
is equally quantum
to prefer
is no reason
some
states.
energy, a constant number of particles, values of all external parameters that may
will
have constant
and
constant
including gravitational, electric, and magneticfields. the A quantum state is accessible if its properties arc compatiblewith physical of the system: the energy of the stale must be in the range within specification which the energy of the system is specified, of particles must be and the number in the within which the number of parlictcs is specified. Wtlh range large systems we can never know either of theseexactly, \302\253 1 but it will suffice to have.SU/l/
influence
tmd&N/N Unusual
the system,
\302\253 I.
properties
of a
system
certain states to be accessible during
may
the
sometimes
time
make
the system
it
impossible
is under
for
observation.
at form of SiO2 are inaccessible low or starts with the that temperatures glassy amorphous fused form: in a low-tcmpcraturc to quartz in our lifetime of this type by commonsense. exclusions experiment. You will recognize many We treat are excluded all quantum states as accessible unless they by the the scale of the measurement of the time specification system (Figure2.1)and process.Statesthat are not accessible are said to have zero probability. Of course,it is possible to specify the configuration of a closedsystem to a If we specify that ihe are of no interest. point that its statistical properties as such
Fof example,the
states
of
the crystalline
in any observation silica will not convert
2; Enxropy
Chapter
and Temperature
I imtt
of
of ihe
spcMftcation
sjstcn
2, t A iwdy symbolic Ji:iKr;ihi: L-:idi solid s|x' slate of a closed sysn represents an accessible quantum fundymema! of statistical pliysics is tliat a assumption is equally likely to be in any of tlic quantum si; system accessible to it. \"Die empty circles represent some of thi that are not accessible because their do nc properties the specification of the system. vjfju -1 ,-/<_ IlovG
V\\uil-
system
is exactly
in a
stationary quaniurn state s, no statistical
is left
aspect
in
the problem.
PROBABILITY to be tn any likely Suppose we have a closed sysiem that we know is equally of they accessiblequantumstates.Let s be a genera] state label (a\302\273dnot one-half ihe in this slate is ihe spin excess). The probabtHty P(s) of finding sysiem
P(s) = if
the
fundamental
dosed, systems
state
5 is assumption.
accessible and P[s) = We
shall
on [/and on
A'.
0
panicle A),
consistent
otherwise,
be concerned
for which the energy V and wtH not be a constant as in P(s)
(t)
\\fg
but
taier
systems
that
ihe
fun-
are not
vary. For these have a functional dependence
number wilt
with
with
N may
Probubitiiy
sum
The
the
of
\302\243P(s)
the total
because
over alt
probability
probability
that
the
sysiem
_,
.
states is
is in
equal
always
to
unity,
is unity:
some state
.
B)
of tead to ihe definition of the averagevalue any physical properly. Suppose iliat the physical property X has the value when the is system in the state s. Here X might denote magnetic moment, X{s) of the energy, charge density near a point r, or any that square energy, property can be observed wlien the system is in a quaniumstate.Then of the the average observations of the quantity X taken over a system described by the probaThe
defined
probabilities
probabilities
by (I)
is
I'{s)
defines the average value of X. HereP(s] is the probability is are to the sysiem in the state s. The angular brackets used \342\226\240> <-\342\226\240
This
that
equation
average value.
For a dosed system,
the
average
of A' is
value
!'
'
>':
/\"i>-'-f
denote
i
-
D]
alt g
because
now
average
in D)
accessible slates are
likely,
equally
elementary exampleof we imagineg simitarsyslems, in is an
what
average; Such a group of systems
averageof
any
_1_
property.
one
constructed
over
properly
the
alike
ts
may
each
be
P(s) = an called
accessible
\\jg.
The
ensemble stale. quantum
catted an ensemble of syslems.The
is catted
group
with
ihc ensembleaverageof that
' \342\226\240' '\342\200\242\342\226\240>\342\226\240', .\342\226\240>.
alike. all consfrueled is composed of many systems, of the in one is a replica of the actual system Each system in the ensemble If there are g accessible stales, then quanium states accessible to the system. Each system for each stale there wilt be g systems in the ensemble, one system to the actual sysiem. in the ensemble is equivalentfor all practical purposes ensemble
An
Each and
in
sysiem
this
of systems
satisfies
sense
all
external
is \"jus! as
requirements
good\"as
the
actual
placed on the system.
Every
original system quantum
stale
Tit
'[!\".;\"*'\342\226\240\342\231\246
t
'';\"(:!'
*
-rrrt
\302\273'.\"*
t
t :\302\273\":;'\302\273''\342\226\240:
t';t
t: r;jt
:y
'[in
sT'\"\302\273
t
i
rn;
/.:Y:t
I
t
fit
t
t
rft
t
t
:t:;Y
i'YY 2.2
l;ij;ure
cuscmbJc
Tliis
of lOspiiw wiih
\302\253 iJirotijiJij
of an
Construction
represent a dosedsystem each frequent
in
a magnetic use
of s
of
field is
as a
Mi
;m
t
I ;l 2.v
tlie
cusciubl
as
in
is
ensemblefive spins,
implied
or
in
We conslruci each system with
-mB. |Do
stale index
2.2.
Figure
ensemble by one system in a We assume that the ensemble the fundameniat assumption.
in the
is represented
representsthe real system\342\200\224this
Example;
\\
t
sighificance.
stale,
quantum
t
Y.
[\\
ilic |0, so tliut Hi syML-ins,
=
accessible lo the actualsystem slaiionary
t
spin excess
sy>(cnis
t
t
icpresciiis
n!Litij]>]iciiy !/{.\\'.n) is yUU,4) cl]sv-]I]!iIc iiium ^inlnin rcpfcscutiiiive Uic vuriou^
t
\\
Y
Tlic
order in which listed has no
t
i
-t
t
-Sui/Jimd
etwujy
t
yi
t
-\342\200\242Tit
t
;i
t
'nrt
t
/[r;i
: t
t
Y
\342\200\242 \342\200\242
\"U.:'j\\
not
label.)
confuse
!he
Each system
in
spin
2.3
Figure
2s =
excess
use of
s
in
spin
an
ensctnMe
to
{.The energy of excess
represents one of the
with our
muliiptes
of
t tl'OutibiC
represents a system
enscmbie
The
2.3
Figure
i
at tin's
states
Tile 10 systems If the
have
2s
2s
have
and
2s
= 3; t
|Q sysiems has
system
have
2j =
7s
=
- 5.
- i.
- 5,a singie
ti1Ca\302\273embio.ThiS is
by tlic
I: 10
systems have 2s =
into
contact
-1;
.o!
function
multiplicity
m Figure 23 make up ttic cuwnibk. 5r ltn-'ii n sinylc sysitm tiiLiitueliefieid weic siidi lli^t 2.\\- \342\226\240=
in llie
- -3;
number ofsuch slatesis given
enetcy.Tlic
shown
energy
5 systems
\342\226\240\342\200\242! t
and spin excess 2i
A' = 5 and
Sy^m may represent ii ii
! t
.! I
5 spins
With
R\302\253ute2.4
\342\200\236
| f
=
N
with
t'T
f\\
jTIT:
tj
i\\
K.O[ij\\j}U\\iilI0
tomjiriscs
5
systems
Most ProbableConfiguration Let
two
systems
transferred
freely
2.5).The with
net
two
constant flow
from systems
ofenergy
of
and one in
be
\302\243t
to
tttc
contact
U =. Ut
brought
This
other.
form a
is called
so that energy (hermnl contact
larger closed
system
& ~
can be (Figure
Sx +
\302\243z
a + U2. from one system to another? The answerleadsto the concept The direction of energy flow js not simply a matter ofwhethef of the other, because the one system is greater than the energy
energy
of temperature.
the energy
5,
Whai
determines
whether
there
will be
r 2: Entropy
anil
Temperature
closed
Two
eomacl
in
The
ure
systems
u\\
in
of
Establishment
2.5
Figure
systems can be dificrcin
in
maximum
the
has
slates of two
systems when
in
contact order
thermal two
between
to define
spin
exchange of
and constitution. two systems. of
systems,
the energy.
&, and
lotal
energy
constant
of accessible states.
number
model systems contact.
A
[wo systems
energy is that for
t6tal
the
energy
between
contact
Micrmal
size
sbared in many ways between The most probabledivision
in
+ ut
Thermal conduclor allows
sulaliiin
accessible
ul
contact
thermal
system
s= +\342\226\240 f/;
We 1
then
and solve
first
and
2,
study in detail
in a niagoetic
The numbersof spins N
which the combined shall
We
can be
enumerate
what characterizes
the
the
the problem of thermal fieldwhich isintroduced maybe
u N2
different,
and
the values of the spin excess2s,, 2sz may be different rOr the two systems. All take have moment m, Tlie actual exchangeof energy spins magnetic might place the interface between via some weak (residual) couplingbetween near the spins the two systems. We assume states of the total system & can that the quantum be represented a state of accurately any by combination of any state of 3, with We but the of the excess are allowedto values constant, S2. keep N,, N2 spin change.Thespin excess of a state of ilie combinedsystem will be denoted by 2s, is dirccily where s = sx + sz. The energy of ihe combinedsystem proportional to the total spin excess: s2)
U{s)
The tola]
number of particlesisN
=
A'(
+ .Vj.
=
~2inBs.
E)
Most ProbableConfiguration the energy splittings betweenadjacentenergy leveh are equal so that the magneticenergy systems, given up by system I when one spin is reversedcan bo taken itp by the reversal of one spin of systctn 2 in the opposite sense.Any large physical system will have enough diverse modes of that
assume
We
to
both
in
7n\\B
The
is always possible. another energy exchange with system + is because is constant, but constant t he total st s2 energy systems are brought into thermal contact a redistributionis
two
the
when
s =
of
value
that
so
storage
energy
permitted
ihe
in
s3 and
of s,,
values
separate
the
in
thus
The multiplicity function of the combined g{N,s) product of the multiplicity functions of the individual the relation;
system
H2(Nj,s where
the
A.15).
The
To see
of s,
range
the
in
5Z by
F)
s,) ,
given by expressions of the form of < N2. if Nl -fiY, to %Nt,
gx, g2 are
functions
multiplicity
and
5[
systems
to the
related
is
&
Uz.
I/,,
energies
is from
summation
how F) comesabout,considerfirst
that
of the
configuration
combined systetn has
the first system has spin excess2sj and the second system for which A as the set of all states with excess is defined 2s2. configuration specified spin values of s( and sz. The first accessible states, each ofvvhich system hasg^N^s,) of the g1(N2,Si) accessible stales of ihe second any may occur togeiher with The total number of states in one configuration of the combined system system. of Sk functions the is given of the multiplicity product gl{N1,sl)g2{N1,Si) by ~ ~ s sit the product of the g's may be written as and &2. Because s2
This
Different
values
one
forms
product
term of of Uie
configurations
of j{.
suitesof
all
We sum ihe
sum we
hold
with
s, N,,
to obtain
ofs(
fixed s
states
N2 constant, as
the total
or fixed energy. We
numberof accessible and
characterized by
system are
combined
over a!!possiblevalues
configurations
where y{N,s) is the
the sum F).
part
of
the
combined
of
the
specification
thus
different
number of obi a in
F),
system. In the of therma!
contact.
The result F) is a sum
of
for some value maximum The configuration for which cmtfigurntiun;
ilie
number
of the
products
of s,, glg1
of states
to
sL,
say
is a in
it
gxiN1.Sl)g2[Nl.s
form G). be read
maximum
Such a product\\sill
us
\"st
hat\"
is called the
or
be
a
\"si caret\".
most probab!e
is
- 5,).
.
(8)
am! Temper
2: Entropy
Chapter
0
A*.
Thermal
of Ihe dependence of the repressaiion on [he division of ihe tola! energy multiplicity
Sclicmalic
2.6
Figure
coiifiguralioji
!f ihe
are
systems
and Sj.
two syslems.-S,
belwcen
equilibrium
large,
extremely sharp, as in
the maximum
with
to
respect
in Sj will
changes
be
of configurations will dominate the statistical of the combined system. The most properties alone will describe probable configuration many of these properties. Such a sharp of large maximum is a property of every realistic type system for which exact solutions are available; that it is a genera! property we postulate that'fluctuations of all largesystemsthe sharpness From properly it follows about the most probable configuration a sense ihat we will define in are small, The imporlani result follows l hat the values of the average physicalproperties
of
a
contact
in thermal
system
large
described by the properties nf which
configuration for
values
(used
Because
number
the
in either
of the
A relatively
2.6.
Figure
the
in
such
a replacement
(8).
and
the configura-
by an
the
equilibrium
average
average over
of a
average
values.
physical
only ihe most
example below we estimate ihe errorinvolved the error lo be negligible.
In the find
replace
F)
are accurately
is a maximum.Such thermal
may
system
configuration,
of these two senses)are called
sharp maximum, we
configuration
large
probable
of accessible st3tes
quantity over all accessibleconfigurations probable
another
with
most
small number
Most Prabahte
Example: tfiL-
sharpness
Tn-o sprn systems in thermal contact. We investigate for of the produci G) near the maximum (8) as follows.
functions for(j,(W,.s,)
Ilie muliiplicity
white ?i(Q)denotesjj,(:V,.G)
and
and
ffsf.Vj.Q). We
denotes
^@}
--^This
gives
product*
We
find
the
is,
properly
of
maximum
of statesaccessibleto is 2s,
and the
spin
excess
replace s,
-
of ihc first
- *,:
s
(jo)
-l^iiLj. system when
combined
the
by
spin syslem of product
the
spin
system is 2s,.
value of(tO}as a function of s, when the total spin esccss 25 ishd.l the energy of the combined is constant, it is convenient systems that the maximum of fog.m) occurs at the same \\atue of x as the From The calculation can be done eilher way. f !0|,
the maximum
constant; that to use
the number
combined system
of the
excess
modd
lUc
We form the ofiho form of|U5):
^olh
g^i-^J-
ConfiSui
>'{.y).
when
)
uy
be
if the
a minimuni, second derivative of
a maximum!
or the
z point of inflection. is negative,
function
^~-.
~^T-~
The
so
that
(II)
e\\tr\302\243nuim
is
curve
bends
the
downward. Ai
the
cxt
Nt, N2. and
where Equation
* The
product
s are
held
as s, is
constant
varied. The second derivative
A1) U
fund
ion of two
Gaussian
fund
ions
is always a
Gaussian.
c1/csi1o(
a
and is negative, so thai of the combined system
the
Tims the
is a maximum.
extremum
is thai for
which
most
configuration
probable
is satisfied:
A2)
t to interchange ofenergy when the fractional to the fractional spin excess of system 2. spin excess of system alt the accessible stales of ttic combined We prove itiat nearly satisfy or very systems then A3} of s, and sj at the maximum, nearly satisfy (U). if s, and 52 denote the values two
The
arc
systems
in
with
equilibrium
respect
1 is equal
o
of slates
number
iiic
find
most
the
in
we
configuration,
probjble
insert
A4}
in {9}
A5)
(-25I/N).
To
the
investigate
of
sharpness
of gigl
i
s, introduce &
that
such
-
the deviarion
measures
Hcre^
Square
to
j,,ij
s,
of su
+
<5;
sj
st from
their
in {9} and
A5) to obtain
deviation
from (H) ii
from
thai
\302\253
V.vi.so .'.\\'u
s,/jVj
As a numerical let Ni
=
values
- 5.
A6)
Su Sx at
the
maximum
o(g,gi.
equilibrium
so
of sialcs
45,^
2Sl
4s2S
slatesin !ii;it the the number number of of slates that
2d2
a configuration
is
l(N2J2
\302\253 ,V.
s2
number
the
/
know
=
fonn
we substitute
which
We
value of
a given
at
to
of ot \\
- 3) =
te.ffiU.e
~
~
;v~
the fraciionat devbiion from equilibrium is very smalt, example in wliidi = ^00, and the = !0:i and 5 10'2; ilia! is, <5/A'i = !O\0210. Tlien,2ii!,'iVi
Thermal
Equilibrium
product g,^i is reduced lo g\"\02100 = lO\0217* of its maximum value. This is an so that g,g, is truly a very sharply function of st. Tiic laigc reduction, peaked tlie fractional that deviation will be 10\"'\302\260or larger is found by integrating &
=
out lo a value
I0u
of the
probability
of (he order of Tiits
distribution.
of N, thereby is the subject of s or
by N x
including
ihc area
Problem 6. An
tO\"t14 =
extremely probability
{17} from
under
the
limit
upper
wings
to the
10~i!2,si ill very smalt. When two ihermal contact, the values ofsi, Sj thai occur most often will be very close to rare io (ind the values It is e.Uremely off,, S] for w hich ihe product g]g1 is a maximum. with values of*,, s, perceptiblydifferent from systems ?\342\200\236 st. to say that the probability a fractional What does it mean the system wjlh of ftitding deviation larger limn 5 .V, = \\Q~10 is Only \\0'!ii of the probability of finding the system as We mean that the system wilt never be found with a deviation as much ill equilibrium? in tO10, smalt as this deualion 1 part seems. We would have to sample !0IJ2similar systems lo have a reasonable chance of successin such an experiment. If we satnptc one system every is prctlv Usi sunk, we would 10'1 iuive to sample for 101J\"s. The age of the s, which is only 1O'Bs. Thereforewe say with great surely that lite deviation described wilt universe be observed. The estimate is rough, from never the message is correct. The quotation but Uoli/iitaun is relevum here. given at iUi beginning of this chapter We may expect to ob>mc substantial fractional deviations only in the properties of it of a small imiili in thermal contact with a targe system system or reservoir. The energy of 10 spins, in thermal contact with reservoir may undergo a large system, say a system on ihc lhat are largi! in a fractional fluclualions sense, as (lave beenobservedin experiments of small particles in suspension in liquids. The average energy of a small Brown tan motion be determined with a targe system can altvays accurately by observations system ill contact on one sronlt a! one time on a tatgc number of identical smart systems or by observations are in
systems
over
sysicm
result
period of lime.
a long
THERMAL
The
is given
probability
integrated
EQUILIBRIUM
for
of accessible
number
ihe
stales of two model spin
systems
in
in thermal contact, with generalized to any two systems constant toial energy U extension of the earlier arguBy direct U2. Ul +\342\226\240 is: the argument, g(S, V) of the combiited system multiplicity
may be
contact
thermal
summed
over all
values of
states of system 1 specified
by ihe
at
in
The
sum
o\\er
Jiicrgy
value of
ofaccesstblcstates
Us < V. Wzxz Ux. A
:nl configuraiioits
the
is lite
a configuration
gi
itU
%)
is the
number
of accessible
is of the combined system The number constants U,Nt, -V2.
coniigiiratioit with
(.\342\226\240',. together
g X{N
product gt{N],Ui)()i(Nz,U
~ Ut).
The largest term in the sum in (!S) governs the propertiesof the total system ill thermal equilibrium. For an extremuni tt is necessary that ihe differential* of be zero for an infinitesimal of g{N,U) energy: exchange
dg =
(-\342\200\224]
divide
We
+
g2t!Ul
by glgl
lV, +
9i(^~\\d
and use the
result dV2
=
~tiUx
dU2 -
to obtain the
0.
A9}
thermal
equilibrium conciiiion:
Lflfs we
which
may
write
as
B0b) We defitie the quanttiy
a, calledIhe entropy, a{N,U)
where
a is
the Greek
s
letter sigm:i.We
by
\\
now
write
B0)
in the iimii form
B1)
Ns is held
means
lhai
panial
dcrivalivc
wiih
consiam respect
in
ihe
to 0,
differttiljalion
is defined
as
o(g,{Nt,U,)
with
tespeel
lo U,.
Thai
is,
ili=
Teutperutute
is the
This
condition
Here
contact.
ant!
Afi
two
for
equilibrium
symbolize
not only the
us immediately
to the
/V2 may
on the
constraints
ail
thermal
for
systems
in
thermal
numbers of particles, but
systems.
TEMPERATURE
The last equality the
know systems
rule:
everyday are
leads
B2)
thermal
in
If T
of the
temperatures
two
equal:
be
must
denotes
inverse
the
equilibrium
= t2.
r,
This rule
concept of temperature.We
ttie
so that T
to B2),
equivalent ahsohite
B3) must be a function
in ketvin,
temperature
this function
of
V)s.
(ro/f
is simply
iSie
relationship
B4)
The proportionality L-on^mnt.
As
constant
determined
\302\273 U81
x
10~\"joulcs/ke1vin
\302\253 1.3X1
x
10\"Ulcrgs/kelvin.
discussionto Appendix B becausewe
temperature scale:we
called the
ilott/iiumu
experimentally, kn
We defer the
constant
;i universal
is
ku
define
the
fundamental
B5)
prefer
temperature
10 use
a more
natural
r by
B6)
This temperaturediffers
from
ihc
Kelvin
r
-
temperature
kBT.
|
by the
scale factor, kB: B7)
t has the dimensions Becausea isa pure number,thefundamental temperature unit of energy. We can use as a iemperature scale the energy scale,in whatever
2;
C/iapier
and
Entropy
Temperature
for the latter\342\200\224joule or erg. This procedure is much simpler than the introduction of the Kelvin scale in which the unit of temperature is selected so that the triple point of wmer is exactly 273.16K.Thctriple arbitrarily of water is the unique tcmperaiurc at which water, ice, and water vapor point may
be employed
coexist.
Historically,
the
to build
accurate thermometerseven
quantum states was measure
though
understood.
not
yet
temperatures
thermometers
with
age in
it was
possible of temperature to Even at present, it is still possible to in kelvin to a higher calibrated which
relation
the
wit h which tlie conversion factor kB iiself is known\342\200\224 million. of discussed in are per Questions praciical thermomeiry
precision
the
than
32 parts
about
as
from an
dates
scale
conventional
accuracy
B.
Appendix
permissible to take
the
of both
reciprocal
sides to
B8)
The iwo was
B6)
expressions
given as a
determined
U with
of
differentiation
V). The function of
of
definition different
respect lo a
with
Eemr^caiufe
is die
oilier
in lhermal
ej.pcrinicnts
same
in
The
q
r
other
itrot
V
\"What some
in
B6).
=
the entropy
= a{U,X).Hencei In B8). V(a,N),
but
cases, in
in
as a
\342\204\242 t(U,N).
mpties both
because
phy. we
id M
variab]
N consiant
variables.
independent
in
leaning,
diOcrcu
variables U
independent the same indcpendeiu
variables?\" arises frequently some variabtcs, and
have a slightly
the
has
B6)
from
and B8} of
function
it
is
however,
so that t
=
expressed
ihc independent we coniroi experiments are
variables.
ENTROPY
Tile quantity entropy
ts
as
defined
logg
introduced
was
as the
in B1)
logarithm of
system. As defined, the entropy the entropy i'is defined by
is a pure
the
as ihe
number
number.
entropy of the system:the of
slates
In classical
accessible
to the
thermodynamics
129)
Entropy
Figure 2.7 than
&
energy
if the
increase
U from
of a positive
system
total
the
condition if the
cooler
energy
body
established.This
U2+8U
of ihe value
words,
is
cm ropy.
increasing
Energy transfer
o,(final) + As a
>
cMXinai)
+ oa(iniiiai)
o^iniiia!)
consequence of B4), we
see
o are
5 and
that
connected by
scale
a
factor:
130) We of
wilt
call
S the
conventional
number of particlesin may 3)
the
on
depends
the
definition
on the of the entropy The of the system. entropy energy variables: the entropy of a gas(Chapter dependence
independent
of
thermal
... is
t
limited
by
the physical
physics
author
the
Thus
Britaiiuka,
Encyclopaedia
of entropy
fn
(he volume.
known.
not
the entropy,
the greater
on the
and
system
on additional
depend
in Uic early history was
entropy.
The more statesthat are accessible, a{N,U) we have indicated a functional
lih
the
ed.
fact
significance of the cm ropy
of the article on wrote:
A905),
that
\"The
it
does
not
in
thermodynamics
of
utility
correspond
ihe
the
conception
directly
to any
is merely a mathematical function but directly measurable property, of the definitionof absolutetemperature.\" now know v.hat absolute physical of the experiof the comparison properly the entropy measures, example physical
We
An
experimental
Chapter
and
determination
theorctic.il
of the
calcutiition
entropy
is
discussed
in
6.
Consider the
energy All
from
tot;il entropy changeAa 1 and
add
the same
when
we
remove
the
more probable flows from the wanner b when lhermat contact is an otamptc of the Saw of
wiii bt: in a
system
to
2 v.itt
-f a2
a,
entropy
of
amount
to system
t
combinedsystemsover ihe initiat
is higher
r,
temperature
the transfer
t2,
a positive
amount of energy to 2, ;is in
amount of Figure
2.7.
2:
Clmptcr
The
ihe quantiiy in
> r2
t,
p;ireniheses on ihe righ>hand
Loial change of entropy
so that the the
from
Tehtjter
change is
tolnl entropy
When
is
urn!
Entropy
system
the
with
is
higher
when
positive
to the
temperature
is positive,
side
the direction
system
of energy flow lower
the
with
temperature.
Example:
heal
with
fansitiariiy
This example makesuse
on heiitjlow. and specific heat.
increase
Entropy
of
ihc rentier's
previous
a 10-g specimen of copper at a temperatureof 350 K be placed in thermal contact identical specimenat a temporal tire of 290 K. Lei us find the quantiiy of energy ihe iwo specimens arc ptaced m contact and come Lo equilibrium at ihe transferred when final temperature Ttie of over hc;ii meiaitic ihe LempcraLure range 15:C specific copper Tf, to t007C is approxiinaieiy Lo a standard handbook. K~l, O.3S9Jg~! according The energy increase of the second specimen i> cmiai Lo the energy loss of ihc first) ihus the energy increase of ihz second is, in joules, specimen
(aj
Let
with an
AV
where
=
C.89J
ihe tempcraiures
- 290K)
K-'HTV
are
in
Tj
\302\253
|C5O
~
linat temperature
Ttie
kcMn.
- C.89JK-')C5OK
+
290JK
after
contact
Tf)
,
is
= 32OK.
Thus
At/,
=
\302\253
C.89JK~!)(~3OK)
-11.7 J ,
and
At/3 = -At/, (b)
What
is the
taken place, almost fraction of ihe final considered temperatures of
change of entropy immediately
after
of
the
initial
two
= U.7J. specimens
con'act?
when a transfer pf
Notice
that
this
transfer
0.1J has small
is a
contransfer transfer as calculated above.Becausethe energy at their initial temperawe may suppose the specimens are approximately of the firsi body is decreased by 350 and 290 K. The entropy
is small,
energy
Lan
Tile cnlropy of iiic
second
=
S2
iotal
Tile
increases
enlropy
AS, +
of the
= 3.45
,7~
of Enlnpy
by
x
by
10-4JK-' = 0.59
AS, = (-2.S6 + 3.45)x units the increase
In fundamental,
where
is increased
hody
of Imrrrmc
x
of entropy
1CTJJK~
*s
Botl/.ma\302\253n constant. This resuil mcaaS thai (lie number of accessible tjisihe - [email protected] 10l9>. two systems increases by (he factor exp{M
st;it
Law oflncrease of Eniropy We
can
ihc loial
thai
show
broughi If (he total energy V into
systems
arc
thermal
in
contact. ~
ctttrapy
two
jusi demonstrated this in U2 is consiant, the lotal multiplicity
V% +
systems a special after
;trc case. the
is
contact
thermal
when
increases
always
We have
ff(t/)\302\273=
^0,A/^A/
by A8). This expressioncontains the
term
t/,)
gi(EAo)i/i(^
C3)
,
~~
^to^
^or l^e
i-XiiiVd^
terms besides.Here ViQ is the the initial energy ofsyslem 2. Because initial energy ofsystem 1 and V l/lois increased all termsin C3) are positive numbers, ihe muitipliciiy is always by This is a proof of Ihe establishmenl of ihermal conlaci bclween two systems. taw of increase of entropy for a weli-definedoperalion. effect of conlact, the effect that slands out even after lakingthe Thesignificant of tcims in iiic summaof ihe multiplicity, is not just that Ihe number logarithm be very, summationis large, but that the largest single term in the summationmay very
niuitiplicuy
before
contact
and
many osher
~~
much larger than
the
initial
muilipiicity.
(Mi).,,
That
is,
= 9i(O,)gJiU
- 0.)
C4)
2:
Chapter
and Teniperatur
Entropy
with
Ut
= 0
parts and probable conftgL
the
presently
con figuration. rat ions
be very, very
Hie cm ropy
in or
found
increases or
muhiplicily
much larger than
ihe
moit
initial
essential
The
value
is lhat
effect
of
for which
Vl
the syslems
states
term
C5)
ihe product g^x is a cvoive
contacl
after
configurations lo their final configurations. implies thai evoluiion in this operation will final
configuratic
probable
Vl0).
9iiVtMU'~
Here 0s denotesthe
The entropy
probabilhy.
a{U) oflhe
ihe entropy
takes piao; between h dose 10 tlie most as the jysicm attains
of energy
will be
syucm
ofincreasirtg
reaches
eventually
may
* U. Exchange
anJ U,
maximum. iheir
initial
fundamental
The lake
always
from
place,
assumption with ali accessible
probable.
eqtutlly
The statement
C6)
fffjnil
statement of the law of increaseofcnlropy:the entropy when a constraint [ends to remain constant or to iucrease
of
is a
is removed.The operationofebtabitsliiiig removal of the constraint that Vu Ux
+
U,
need be
configuration
V2
each
contact
be constant;
u closed
to she
is equivalent
system
sjstem to ihe
after contact
only
constant.
The evolution of the takes
thermal
iniernul
combined
a certain
system
time. If
we
lowards separate
ihe
final
ihe two
thermal
equilibrium
systems before they
Add
energy
molecules
Decompose
Let a
2.9
Figure
(his cotifiguraiion,
reach intermediate
view
and
energies
the entropy
constraint,
called
Processes 2.9;
Operations
function
(he lime
that lend
the arguments
that follow.
as a
in
we
io increase
lend
thai
will
an
obtain
an intermediate of
the
lime
of evolution
in
of each
ofa
entropy
intermediate
up
syslcm.
configuration
with
io entropy. Ii is ihesefore meaningful tli.i' lias elapsed since removal of the
2.8.
Figure
lo increase the eniropy support
the
linear polymer curl
process
ofa will
system be
are
developed
shown in the
in
Figure
chapters
2:
Chapter
For a largesysiem* occur
never
/
and Temperature
Entropy
with another large sysiem)ihere will differences belween the actual value of the significant
thermal
(in
spontaneously
coniaci
value of the entropy of the most probableconfiguration of the system. We showed ihis for ilie model spin sysiem in the argument following A7); we used \"never\" in ihe sense of not once in ilie entire age of the 10's s. universe. We can only find a significant difference beiwcen Ihc actual entropy asid ihe
entropy
and ihc
entropy
of ihe
shortly which
most probable have
we
afier
implies
that
the nature
changed we
had
the
of
configuration
the system
prepared
system
macroscopic
of ihe contactbetween initially
in
two
some
very
systems,
special
way.
Special preparation couldconsistof lining parallel system up all the spins in one or to one another in the air of the room into the collectingall the molecules a small volume m one corner of the room. Such extreme by system formed situations artificial
never arise operations
m
naturally
on the
informed
left
systems
but
undisturbed,
arise from
system.
Consider ihc gas in a room: the gas in one half of the room might be prepared wjiti a low value of the average initially energy per molecule, while the gas in ihc other half of the room might be prepared with a higher value of the initially average energy per molecule. If the gas in the two halves is now allowed to interact of a partition, the gas molecules will come by removal very quickly' to a most probable configuration m which ihe molecules m both halves of the to room have the same average energy. Nothing else will ever be observed the We will observe to leave most never ihe sysiem configurahappen. probable configuration and
reappear
later
in the
even ihough the equations distinguish past and failure.
LAWS
OF
initial
specially
of motion
prepared
of physics
is true
configuration.This
are reversiblein
time
and
do not
THERMODYNAMICS
is studied as a nonslatisticaisubject,four posluiales of thermodynamics, in are introduced. Tiicse postulates are caiied ihe laws of thermal formuiaiion our statistical essence, these laws are containedwithin physics, bul it is useful to exhibit Ihem as separate slatemenis. a third If two sysleins are in thermal equilibrium with law. Zeroth sysiem, isa with olher. iaw be in each This must thermal consequence equilibrium Ihey When
1
The
thermodynamics
calculation
of Ihe lime
required
for
Ihe
process
is largely a
problem
in
hydrodyna
Laws
condition B0b)for
of the
(e\\oSgt\\
*
{~\342\204\242rk
in oilier
words,
=
r,
comact:
in thermal
equilibrium
of Thtentotiynamks
feioSg3\\
/cloggA
/cloggA
{-furl:
{imX=
{~7urk
= t3
t3 and r3
=
rj
imply
r2.
Heat is a form of energy. This law is no more than a slaicment of liie principle of conservationof energy.Chapter 8 discusses wliat form of energy First law.
heat is.
Second
entropy, applicablewhen is ill :l
of the
tluit is not
configuration
of ihe second law. We ilie law of increase of
of
law
(he equilibrium
consequence
will
in successive
instants of time.\"Tins is
wilh Eq.
called
lo a dosed systemis removed. The increase of entropy is: \"Ifa closedsystem
ihc enlropy of the
be lhat
have
iniernal
a constraim
used statement
commonly
statements
equivalent
many
statement, which we
the statisiicai
use
shali
are
There
law.
cotiliyitnilton.ilicmosi |>rubnble system will increase monoiotiic;ilty siaicineiil
a looser
ilian
I
he
one
we gave
C6} above.
The traditional of
is the Kelvin-Pianck formulation is for \"it to any cyclic process impossible of heal from a reservoir and the perextraction statement
thermodynamic
of second iaw
thermodynamics;
whose soie effect is the of performance an equivalent amount of work.\"An engine that vioiaies lhe second iaw by extractingthe energy of one heat reservoir is said to be performing motion ofthe second kind. We will see in Chapter 8 that the Kelvinperpeiual
occur
statement. Pianck formulationis a consequence oflhe statistical as the Third iaw. The entropy of a system approachesa constant value due zero. The ear! test of this statement temperature law, loNemst, is approaches at that ihe absolute zero the entropy difference disappears between all those
iaw follows
is
multiplicity
the
system
g@),
the
in
has
thermal
internal
definition
statistical
the
from
ground stateof
which are
a system
of
configurations
corresponding
zero. Glasses
is essentially
substantial,of in
real
life is
to
objection
the
order
that curves
must come in
flat
as
affirming
have a
of [he
The third
entropy, provided ihat the multiplicity. If lhe ground stale as t -* 0. entropy is o{0) = iogy@) is that does not appear to say much of the
a weii-defined
From a quantum point of view, the law not implicit in the definition of enlropy, provided, in its lowest se! of quantum at absolute states
would not be any
equilibrium.
frozen-in
number
of many
r approaches
reasonable
0.
the system is for zero. Except glasses, there that (j{0) is a small numberand c{0) for them o{0) can be and disorder, the third law tells us of atoms N. What plotted against x quantities physical however,
that
Chapter 2: Entropy as a
Entropy
and
Logarithm
Several
useful
stales itself.First,
the
ihe definition
from
follow
properties
number of
of the
rithm
1
Temperature
accessible states, hut two
of
entropy
of (he cmropy as the iogathe number of accessible systems is liie sum of lhe
of as
cad
independent
separateentropies.
Second, the
the
never meant to for
that
entirely
imply that
a discrete
tite
practical
purposes\342\200\224-to
We have
defined.
is
exactly, a circumstance
is known
energy
system
of energy
spectrum
ali
insensitive\342\200\224for
the energy of a closedsystem
with wiiich
6U
precision
is
entropy
eigenvalues would
of
number
the
make
We have depend erraticallyon ttie energy. simply not paid much attention io lhe precision,wlicthcr ii be determined by the uncertainly h, or determined otherwise. Define <0{U)as the number principle <5U 5{time) of accessible srates unit per energy range; O{U) can be a suilablesmoothed of V. Then y{U) ~ i>{UNU is lhe number accessible centered at average stales in the range SU al V. The cmropy is
accessible stales
a{U)
of
order
the system
as for
Typieally,
2V.
tola!
lhe
If
particle energy
then
A,
. Let N
logO(t/) of N
energy C(l')
. a{V) =*
~ IO20;A
=
*\302\273 0.69
\\
C7)
of states
number
total
the
order of N
is of the
times
same
be
of ihc
average
one-
will
2;7jVA. Thus
and 5U x
\302\273 10^'
10:o
C8)
+ log^C.
--'logNA
Nlog2
IO\021* erg;
a(U)
spins,
-t-
erg.
- 13-82-
2.3.
C9)
is dominated overwhelmof the entropy that the value effect on the the lhe precision dU is without perceptible overwhelmingly by is of stales of A' Tree particles result. 1 n the problem in a box, the number proporlhe + like a proportional io whence \\ogSU. Again U*dU, JVIogf something of units the term in N is dominant, a conclusion independentof even system used for lhe energy.
We
see
Example:
from
exam pie of N; value
this
iiuxion
Perpetual
o
Early lion
more energy
in.in
ii
absorbs.
in our
machine,
study
of
a machine
we came
physics
ilui
wjli
gi\\c
to
forth
motion of the second kind, as it is called, in Equally impossible is a perpetual machine which heat is exiracicd from part of the environment and deliveredto another of ihc part a heat in temperature environment the difference ihus established being usedto power we available for any purpose at no cost to us.In brief, engine that delivers mechanicalwork cannot tlic surrounding to ocean lo extract the cnefgy propel a ship by cooling necessary ocean 10 a transfer of energy from ihe low temperature propel ihe ship. The Spontaneous higher temperature boiler on the ship would decrease ihe total entropy of the combined and would thus be in violation of the law of increase of entropy. systems
SUMMARY
1. The
fundamental
2,
ifPfi)
is the
a cioscd
that a
probability
system is in
system is equally s, the
stale
the
to
be
average value
ofa
likely
is
X
quanliiy
is that
assumption
lite quantum slates accessible !Oit.
in any of
ensemble of alike.
3. An
systems is composedof very
4. The numberofaccessible states
the
of
combined
all constructed
systems,
many
1 and
systems
2 is
* where
5. The
s, entropy
conventional
+
= s.
s2
a(N,U)
=
entropy
S
6. The fundamental
with
the
conventional
7.
fundamental
entropy
kBa
connects
^' the
o.
temperature r is defined by 1/t
The relation r
S =
relation
Tiie
to$g{N,U).
,^ ^
s
= kgT connectsthe
[ca;cV)sx. fundamental
and
temperature
the con-
temperature.
law of increase of entropy states that tends to remain constant or lo increase
The
system is removed.
the when
entropy a constraint
ofa
closed sysicni intorn.il lo llic
2:
Chapter
and Temperature
Entropy
j
thermal equilibriumvalues
the physical of a system are properties accessible when the system is in contact with a large system or reservoir.If the first system also is large, the thermal of the states in equilibriumpropertiesare given accurately by consideration
8. The
as
defined
of
all states
over
averages
the most probableconfiguration
alone.
PROBLEMS
and temperature. SupposegW) tile number of particles, {a}Show
1. Entropy N is
and
is negative.Tliis
{c^a/rU2)*
2.
Find
Ptirtitttagiwtisni.
of
form
the
y( U)
=
wiiere
CUiNn, U
thai
*=
{Nt.
C is (b)
;t
constant
that
Show
;iciii;tlly applies hi an ideal gits. value
equilibrium
ui
ictn|KT;tit:rc
i of
ilie (Vac*
lional mngntrlizalion
M \\'m sysiem of
of the spin
is 2s.
excess
as given in
=
2/N
h' spinseachofmagnetic momem in in a magnelic field B. The the as Take lhe iogarlhilhmof lhc muliipliciiy g(N,s) eniropy
A.35):
c(s) =logg(Ar.O)for |s|
\302\253 iV.
Him:
Show
that
in
g{U) wtih
oQ
-
1og(/{N,0).
this
=
2s2/N
D0)
,
approximation
c0-
D1)
U2/2m2B2N,
Further, show thai
\\jx
~
-U/)\302\2732B2N, where
U denotes
, the lhermai averageenergy. oscillator, (a) Find the enlropy of a sel of N oscillators n. Use the muiiipiicily a fund ion of the loiai quanlum number \342\200\224 iV. and lhe make A.55) Sliding approximalion iogiV!= iViogiV \342\200\224 \302\273/itu the of JV 1 by iV. (b) Let U denote lhc tola!energy oscillators. the total energy at temperature x is the entropy as a(U,N).Show that
3. Quantum of frequency funciion
Replace
Express
hartitonic
m as
exp{/itu/r)
This
!S the
that
does
PSanck result; it us to
not require
is derived find
again
ihemuilipiicity
,42)
\342\200\224 1
tn Chapter function.
4 by a
powerful
method
used !ogl0 44 = 1.64345. a monkey-Hamlet {b} Show that the probability that we have
where
of the
universe
thereforezero
in
approximately
any
operational
of an
sense
the age of Hamlet is
be typed in
will
The
10\"'64316.
probability
event, so that
the
statement
original
one mudi less a library, book, beginning of this problem is nonsense: occur in the total literary produclion of the monkeys.
at the
never
will
5.
is
of
Addith-ity
A', =
ss +
as a
gig2
product
the
I012,
far
entropy
1012spins with
two spin
multiplicity
product
#sj72 is
(a) Compute (b)
s =
For
to
equal
1O10, by what
Yai9'it^i*5i)9i{^i's
two
jO\02174
s,
=
s, +
factor must ~
5i)\">
the
lOn and s =
you
\302\247've l^e
^ctor
form
to
value. Use
the
A7) may be
0.
(gijh),,,a!l
multiply
JV, =s
- ss), the
= s,.ForSj=
its peak
from
function:
multiplicity
of
systems
and g2{N2.s
g^x^i)
reduced by
the
for
gigz/{9-i9z)m3X
Given
is relatively sharply peakedat s(
functtonofsj
Gaussian approximationto useful.
systems. functions
fo make
it
nearest order of
the
magnitude.
' j.
Jeans,
htysteriota
utirerst,
Cambridge
Universily
Press,
1930, p. 4. The
slalenuill
is
attributed
to Huxley.
' For a rctaicd
Clarke in
malhematfco-iherary
2001.
explanations
sec'The
We arc gralcfut to the Population evidence. The cumulative as 2 x 10 s 3nd tl^c number njocti ksi than the'numbcr of
of the
average trretrme m an-seconds ii
study,
Libtary of Babel,\"
by ihe
fascinating Argentine
Revcttc for Bureau and to Dr. Rosier Reference number of man-secondsis 2 x iO10. if we take ihe oi numoc^ of lives as I ^ 10 , i he cumu*ai[\\c iaken in the problem. \342\200\242sccondi (t0\"> monkey
Cliapter2; Entropy (c)
How
and
Tcmperatur
iarge is the fractional
error
in
the
entropy
when
you
ignore this
factor?
6, Integrated
approximately the
is 10\"l0 \342\226\2405/JVj
use an
x
probability
or larger.
calculate
lhat gave ihe result A7), example the fractional deviation from equilibrium that = = You find it convenient to will IO2Z. iVj JV2 the
For
deviation.
Take
asymptotic expansion for
the
complementary
error
\302\273i,
xp(x2)
(\"\"e
x
1 +
small terms.
function.
When
3
Chapter
and
Distribution
Boltzmann
Helmholtz Free Energy
FACTOR
BOLTZMANN
58
61
Function
Partition
Example: Energy and Heat Capacityof a Definition; Reversible Process
Two
State
System
64
PRESSURE
6-1 ;>7
identity
Tlicrmodynaimc
Example: Minimum ParamagneticSystem
6S
ENERGY
KttEE
HELMHOLTZ
of
Property
the Free
70
7!
Maxwell Relation
Calculationof f IDEAL
GAS:
One Atom Example;
71
2
from
72
LOOK
FIRST
72
a Box
in Af
A
in a
Atoms
74
Box
76
Energy
Example:
of
Equipartition
Example:
Energy of ;i 69
Relations
Differential
62
77
Energy
7$
Entropy of Mixing
SUMMARY
SO
PROBLEMS
81
1. Free 2.
Energy of a Two State Sysiem Susceptibility
Magnetic
3. Free Energy of a Harmonic 4. Energy Fluctuations 5.
Effect
Overhauser
6. Rotation 7.
Zipper
O.ollaior
of DiatomicMolecules
Problem
S. Quantum
Concentration
85
Si S3 S3
S4 84
S5
S3
9.
auJ
Bol
ChapfcrS:
Partition
10. Elasticity
Function of
for Two
Polymers
11. One-Dimensional
Gas
lleliiiiioiiz
Systems
Free
Energy
Chapter3; BolRinattn
The
laws
statistical
We
are
of thermodynamics
mechanics,
able
of
to distinguish
which
on anotherfrom lhai which to specify cases of thermal
in
and
Distribution
lleimkoitz
may easily be obtainedfrom the principles are the incomplete they expression. Gibbi terms
mechanical
we call
mechanical
action and
the thermal in
the
narrower
Energy
of
action of one system sense . , . so as
cases of mechanicalaction. Glbbf
Free
this
In
of the
properties
physical
sysiem
iarge sysiem (ft,
called
system
as
particular,
in
of
reservoir.
The system
of
the values
calculate
Figure
3.1.
The
total
temperature. wiih equilibrium
a very
and the reservoirwill
energy Uo
sysiem is in a staleof energy
We
ihe
r because ihcy are in thermal contact. (ft + & is a closed sysiem, insuiaicd from
The iota!
if the
&
[he
temperature
influences,
us to
permit
a system as a ftinciion interest io us is in thermal
of
assume that the common
the principles that
we develop
chafer
=*
Ea, tltcn
U^
Uo
+
u!!
external
constant.
is \302\243/j
-
ihe
r,x is
a
have
energy
In
of
the reservoir.
Toial sysiem
.
Constant
J.I
energy
Vo
Rcprcs illation of a cioscd a! coniaci with a
irOtiiUhcn
BOLTZMANN A
5
central
problem
\302\253iilbe
proportional
to
in a the
loiat sjsieni
syst S.
iisioa
n decomposed
FACTOR of thermal
specific quantum Boitzmann factor.
10 find siaiu s of energy
physics is
the
probability t,.
This
ihe system is propor-
iltai
probability
be in ihe state s, ihe number When we specify that S should number of accessible Slates of the louil sysiem is reduced10the reservoir (H, ai ihe appropriate energy. That is, ihe number
of
accessible
states g\302\256t j
of
of ihe siaies
Figure 3.2 The change nsetvoit u&nsfcssenergy
fractional effec!
of
laigc TCMivoii
accessibleto (ft
the reservoir
of
Energy
of cnlropy t la
itie transfer
vil\\
luua
high
-
-
A)
for our
because
If the system
to \302\243,
Ihe
ratio
of Ihe
the
the
system
dependence
Multiplicity
of
at
Multiplicity
of
(ft
at
write
B)
on the
depend
is only in terms
are
situations
two
ow
if the reservoirsare very We
system
system is in
is^f/o
is
in
number
of
in Figure
3.2.
\342\200\224 The \302\243,.
-
as
\302\243,},
I al
stale
quantum
2 at
state
quantum
energy energy
direct consequence of what
assumption. The
about
that the
condition
the
is Uo
energy
.S ,
state of
the
specified
energy
energy Ei
is
Ihe
two multiplicities:
ts a
result
this
that
probability
probability
have
the reservoir \302\243\342\200\236
to the reservoir in
of the
The ratio
is
energy
stales accessible
This
present purposes we
the large,
of the
shown
constitution
temperature the
-
Uo
r.t
_
\342\200\224 \302\2432
[/q
have
we
ei)
\302\243a(^o
\342\200\224
called
B)
e2)
91s(^'o
the fundamental
3.3. Although questions of the reservoir, we shall seeihat
in Figure
reservoir.
of the
arc very,
multiplicities
very l.-irgcnumbers.
entropy of the reservoir: -
\302\273
E.)
-
ff\302\253(t/0
f
C)
when ihe
(he system.
The
on ihe reservoir ci\302\253iop>.
Bohzmnnn
Chapters:
Distribution and Helmholtz Free Energy'
Oi E
ergy yo
01
-', stale
En \302\253gy (/\342\200\236 -\302\253,
(/\342\200\236
-
',)
-
;\302\260
'='
slates
i
8
Ene
Fi\302\273urc3.3
has a,,(U0 id
The
State 2
tc I
Sta
Energy c.
gy\302\253>
system
in
t,},
-
(b) is in quantum c,) acccisiblc quanluin
(a),
slate
t, 2. The
slates,
(,\302\273(()\342\200\236
reservoir
in (a) and
(b)
D) the probability ratio for
two
the
1, 2
states
of the syslem is simply
E) Let
us
expand
in D)
the entropies
in
a Taylor
The Taylor seriesexpansion off[x)about
-
/(x0)
series
expansion
about
is
t)
0) where
1/t
=
(S^/cCV^
gives the
temperature. The partial derivative
is taken
Uo. The higher order termsin large reservoir.*
al energy an
infinitely
Therefore
defined
Acr^
-(\302\243,
ofvust
form exp(
the
of
term
It
utility.
h
ttie
to consider the
ealied the
exp(-e,/t)
=
in a
system in
:i
single
for
a!!
We see that
=
result
(II)
is one
average energy of the
= 1:
ZjZ
\302\243?(\302\243,)
U
probability
is V
useful
= (e)
oft'j, ~
convergence
\342\202\254)andnolg({/B
difficulties.
~
partition
p[Et)
and
is
function
(he
results
= X^fo).
of
the
Boltzmann
statistical
pro-
factor
e) because
[he cupansion
physics.
The
or
= T^logZ/ct).
=,
Zh^Zh!A
We expand
the Boltzmann factor
is unity. the sum of all probabilities
of the most
system
is over
system. The
the
5 of
states
A0}
,
5>p(~Ei/T)
summation
The
function.
partition
proportionality factor between the
gives
system
function
Z(r)
\342\200\242
the
Function
is helpfui
The
This result is
Boltzmann factor.
of Hie probability of finding of finding ihc probability
ratio
to
I
'
,as 3
known
is \342\200\224e/t)
the
gives
single: quantum state state 2. quantum
Partition
(8)
-\302\2432)/T.
expft/r)'
P{ez)
A
of
limit
E) and C) is
result of
final
in the
vanish
expansion
D) becomes
by
Affffl= The
liie
of ihc
'
A2)
tatter quanliiyimmcdiatcty
Battvnanti
ChapterS:
Helmholtz
and
Distribution
Free Energy
0.5
A
0.4
J-\342\200\224\342\200\224
Energy ystcm rgy
and
heat
as functions is plotteJ
capacity of a ofthe temperature
in units
V
J
ol t.
0.1 \342\226\240
u
0
The with
a
called for
(e)
refers to
energy
average
reservoir.
The
thermal
the
notation
average
in conformity
those statesof a
with
and not, as earlier,to the
can
that
system
\342\226\240 ) denotes <\342\226\240
exchange
an average
such
energy
value and is
average. In A2) the symbol U is used U will now refer to the system practice;
or ensembie common system
-f reservoir.
We trc^l a sy^lcm ofonc pitrtilicut cupticiiy of a two itatt systexn. 0 and of contact one one e. 11w of energy slates, energy partide is in iltcmut wiiti a reservoir at temjKrature t. We want of ihe lo find the energy and ttie heat cupaciiy as a tin iwo of i. The lit ion function for stales function the temperature of system pat
Exwiiplc? clc
Energy
uiid
wftli two
Z = \"fhe average
exp(-G/t) +
exp(~\302\243/r)
=
i
-f
A3)
expf-s/t).
energy is
This function is plotted in Figure 3.4. we shift iliczcro If of energy and lak^: W.c instead ofas 0 and e, the results appear different
energies
of the
two
as
states
-\\e
and +\\e,
ty. We have
\302\253p(-\302\243/2r)
=
2cosh(fi/2t)
,
A5)
Partition
Function
and
=
A6)
-jctanh(c/2c).
The heat capacityCv
of a
at conslant
syslem
by the
ihermodynamic
as
s x(iajdx)r ,
Cv which
is defined
volume
below
derived
C4a)
identity
A7a)
is
lo
equivalent
allcmate
the
definition
s (SU/dt)y.
Cr hold
We
because the values of the
V conslant
volume. From
and
A4)
In
of energy
dimensions
ri^i-immp
in ]Ilc
anomaly. For
i
for a syslem
at a
specified
-fEY
per kcivin.
A8a)
Tim
as VfSIST), or gU/dT),. ivliencc
siicctfic
pio! of Iicii]cnpiiciiy
\302\273 i, the
eXp(E/t)
A6).
C, is defined
unils
conventional
'
a
same resull follows from
The
arc calculaied
(!?b},
-
C
energy
A7b)
versus
heat
is drfmed \\\\\\
icmpcjiiiurc
us
itio
t'iiHirc3
deal
capacity
\"\342\226\240 is Ciiiic^l
per
^ jLtiOiiKy
A8a) becomes
heat capacuy
Cr^(E/2iJ. ihat
N'oiicc
lemperaiuie
Cy cc
r\"\":
is small
in in
ihis
as a-
exponen'.ial
-
0.
factor
A9)
high tempcraiufe limit. Iii ilie low teraperaiuie whh llic energy level spacing e. For i
comparison
Cy ^ The
exp[
\342\200\224rcJuces
r,'t}
unil
limil \302\253 e we
CO}
{c/lJ^Xpl-E/lX Cv rapidly
as r
decreases,
ihc have
because
exp[
~ 1/v) -> 0
Chapter}:
and Hdmhoki
Distribution
Boltzmann
Free
Energy
is reversible if carried out in such a to the close equilibrium condition. infmiiesimally Forexample, if the entropy is a function of the volume, any change of volume must be carried out so slowly ihat the entropy at any volume V is closely equal lo the equilibrium entropy is well defined at every the entropy Thus, a{V), of a reversible the of the change the and direction stage by process, reversing Reversible
Definition:
is always
to its
returned
be
will
system
A process
process,
ihe system
way ihai
reversible
In
condition.
initial
the
processes,
defined at ail times, in contrast to irreversible the processes,where usually we wilt not know what is going on during process. We cannot 10 systems apply the mathematical methods of thermal physics
condition of the system is welt
is undefined.
condition
whose
volume
A
the system
leaves
that
change
reversible exampleof an isentropic
same
state
process,
because
not
the entropy
change.
reversible processes,
change
the
number
of states
Any
process
in which
process.
and
process.
in
a special
an
If the
state
quantum
is an
system always remains in the any two stages of the pro(p. 31) of similar systems does
between
ensemble
the entropy
But reversible
we shaHhave
zero
be
will
same
the
in
change vanishes is
processes are
interest also in
to
limited
not
an
isentropic
reversible
isothermal
isentropic
pro-
processes.
PRESSURE
Consider a
system in
the
state
quantum
s of
energy e,. We
assume
e, to
be a
function of the volume of the syslem.The volume is decreased from V slowly V - AV of an external force. Let the volumechange take to by applicalron that the system remains in the same quantum state s place slowly sufTkrenliy the The \"same\" stale may be characterized by its throughout compression. of zeros in the wavefunclion. numbers (Figure 3.5) or by the number quantum volume The energy of the state s after the reversible change ts -
t\302\243V
eJ^V) -(dtJdV)bV
&V)~
a pressure p, applied normal done on the system by the pressure
Consider work
cube
volume
V
Kto
from
U(V
-
\342\200\224
Af)
AKappears
- U(V) .=*
+
\342\226\240\342\226\240-.
to all facesof a cube.
The
B1)
mechanical
(Figure 3.6) of the change of energy of the system:
in a contraction as the
At/
-
-(deJdVyLV.
.
B2)
\342\226\240\342\200\224-\342\200\224_
t.S
i.O
O.S
Volume,
relative scale
3.5 of energy on volume, for the energy levels of a free Figure Dependence -%- n_\\ -j- . t particle- connned 10a cube. Ttie curves arc labeled oy m ^\342\200\224 i^ as in Figure t.2. The niuhtpltctties ch; nge here g are also given. The volume ^s isolropier a cub1\" remains oc of ]tic stat1 & cube, i he criercy ranfie
represented in of ilie energy llu: av cragc
an
ranee
energy
ensemble
of systems
will
increase
in a revcrsibk
iiself is of no practical impottance. lhal is impotlaiU.
ti
is )he
Tigure
cha
3.6
compression
Volume of a
cliangc
cube.
- AI'
i
Chapter i;
Free
Ihtmhottz
of the system. Let A
the energy
V denotes
Here
and
Distribution
Batiziriann
Energy
be
face of ihe
of one
area
the
cube;then +
A{Ax
if ail
increments
=
AV
-
Ay
Az are
+
psA(&x
so that, on comparisonwith
=
AV
,
B3)
taken as
positive in
=
,
the
compres-
+
Ay
Az)
PiAV
B4)
B2),
=
-thjdv
stale
s.
P,
is the pressure on a system in We average B5) over all states
the
ensemble
the
of
B5)
to obtain
the average pressure
as p:
written
<\302\243>,usually
+ Az)
compression is
in the
done
wotk
compression. The
=s
Ax
V and
A
Ay
B6)
where U
m
The
.
of states
number
we have
in
described.
each remains
in
fhis
a is
entropy the We
ensemble hove
_st:ile
held constant in ihc tlcrivaiivi; is uuclianged
in ihe
in
reversible
the
of systems,
a collection
each
in
ihc
because
compression some
stme,
and
compression.
The result B6) correspondsto our
mechanic!
of the
picture
pressure on a
system lhat is maintained in some specific state. Appendix D discusses the result moredeeply.For applications we shall need also the later result E0) for on a system maintained at constant temperature. the pressure We look for other for the pressure. The numberof statesand thus expressions U and on V, for a fixed number of particles, so the entropy depend only on that
only
the
two
variables
U and
the
describe
V
system.
of
The differential
the entropy is da[U.V)
This gives the differential differential ch;i:v:_\342\226\240\342\226\240. JU and in such a dependency, way
B7)
Uu
change of dV. th;it
the
Assume the
two
entropy
Tor
arbitrary
independent
now that we select dV and dV interterms on the right-hand sideof B7}
Identity
Thermodyaamie
overall
The
cancel.
of dll
values
interdependent
entropy change
da
be
will
and dV by {W)a
and
zero. {&V}at
denote
If we the
entropy
these interchange
will
be zero:
B8)
B9)
) Bui
\302\273he ratio
(*5[/)\342\200\236/(<)F),
is
the
wilh respect to
V
at
o:
constant
s ldU/dV),.
(iV)J(SV), With
of U
Jerivative
partial
and
this
the dcfiniiion
l/i
C0)
s (So/5!/),.,Eq.B9)becomes
\302\273\342\226\240-<\302\243).-
By B6)
Therniodynaniic
pressure
and
to
equal
-ft
whence
Identity
Consider again the ihe
side of C1) is
the kft-h;ll)d
differential
the
B7)
of the
entropy; substitute the new
definition of i to obtain
Ttla
=
dU
-f pdV.
result for
Chapter3: Bohynaaa relation
useful
This
variable
will
appear
will be in
Free
and Helmbohi
Distribution
called the
E.38).
Energy
thcrtnodjnsmk identity,
A simple
with
transposition gives
dU = process of change of stateof the
If the actual
form
The
-
TVS
is
system
C4b)
pdV. reversible,
we can
and as the work done On identify xda as the heal addedto the system -pdV the The of increase is caused in mechanical work and part system. energy by in part as the transfer of energy be!ween by (he transfer of heal. Heat is defined two
brought
systems
into
thermal
HELMHOLTZ FREE The
contact (Chapter
8).
ENERGY
function
C5)
the Hclmhoitz free
is called
physics
at
constant
temperature
energy.
that
This
function
the energy
V
plays the plays
part in thermal
in ordinary
mechanical
no because processes, which arc always understoodto be at constant entropy, state are allowedThe free tells us how to bulance internal of energy changes and maximum entiic conflicting demands of a system for minimum energy The a Helnihoitz free will be a minimum for system S in thermal entropy. energy contactwith a reservoir (R, if the volume of the systemis constant. r and V. We that F is an cxtrcmum in equilibrium ;it constant first show 01 to &, reversible transfer from for infinitesimal By definition,
C6)
at constaot stant
volume.
so that temperature. But 1/t h {das/cUi)yi Therefore C6) becomes dFt
*=
0
C7)
,
the condition for F to be an extremum with at constant volume and temperature.We like F because the energy eigenvalues e, of the system (seep. 72). which is
dUL
respect
we can
to
all variations
calculate
it
from
Free
tfehuitoh:
We can show
Comment. i',R
+
\342\226\240 Then
Us
ihe tola!
is a minimum.
extiemum
the
that
- UtfffJtVrivji +
*
(\302\253*A'^),,v
L/j
=
a
Ft
w^ is ihc free energy of +
^
be a
The
hxti'iiplc:
with
o,
mus)
configuraiion.
the
-
\302\273
Fj thai
model
ffsiUj).
C3)
in
excess is 2s
=
respect
the
,
C9)
I. A't
-
with
when the constant r, V
respect
is in
system will
and we
is consiant;
. It follows from ihc mosi probable for any deparluce
to Us increase
Coilsklef tliC iyswin, parcmti^nctic = A1? -+- N^', Let N *uid down. Ni spins spins up is found in Tlie the SUHiiig approximmion W(. entropy
\\viUi
approximate
D0)
system.-Now
lo Vs at
system
vf the
property
of tliaptcr
tlicliclpofan
vviiii
of
the
is a maximum
equilibrium
minimum
free energy
Slltthnuiii
sy^tcfn
spin
i/t
- FJx ,
a^a^iV)
D0) lha)
==
becomes
C8)
rewll
is V
energy
that
We know
where
total
eniropy is
* e^U)
so thai
The
Energy
free
energy
tifa
N't
form of A.31}:
,4,, The
in a magnetic field Tile fice cncigy
energy
elementary magnet.
FJjaB)
&
V{s,m
-
Bis.- 2\\tttfl, funaion (to
w licrc
be called
m is ilic ldc
magnetic woniciilofan
lapdau
function
in
elemen-
Chapter
tO) is
io(s),or
D2) becomes of Ft{r,*.B) with respeel io s, this function cquat f because Thai free energy is a function F[i.B). is, Jt.<5>,BJ F(r,B), excess occurs when of FL with respect io the spin minimum Al
ihc
minimum
-
= 0
= -JmB +
tlog^4-|-
lo the of z
equilibrium
and B. The
D3)
Thus
in the
or,
on dividing
magnetic
and denominator
numerator
the
volume,
free energy of however,
The is
easier,
for
the
D5)
<2s>m/K =
exp(\302\273iB/r)
+
D6)
Hi\302\273wnh{mB/r).
can
be obtained the
from
by
D5) in
substituting
function
partition
for
one
\302\253
D7)
2cosh(mB/t).
expt-niB/r)
D2).
magnet:
F =
below. Multiply by N to obtain the result \342\200\224ilogZ as derived of Problem 2.) (The magnetization is derived more simply by the method
relation
N magneis.
exp(/\302\273B/r),
Ntanh(mi?/r).
ihe system in equilibrium to obtain F directly =
Z Now use
i
momeni If n is the number ofspins per unit volume. in thermal equilibrium in the magnetic field is
magnetizalion
M =
It
Is
excess
,U is the magnetic
The magnetization unii
by
ofihe spin
\302\273
<2s>
per
equilibrium value
thermal
B lhe
field
Differential Relations
The
is
off
differential
dF =
or,
use
with
of the
dU - xda -
thermodynamic
identity C4a),
dF = -adt for
oiit,
\342\200\224fdV
\342\226\240
which
D9)
These
relations are
The free for
an
energy
isvihcnwil
widely used. F
in
change
the
result
p
of volume;
=\302\273
-(rf/^K),
acts
as the eilecttvc
contrast this result witii
B6).
The
energy result
Calculation of Ffrom Z
may be written
as
' -
what
= U
of F
use
by
we
~ to. The two
pressure
~-(cU/dV)f
x(?a/3V)t
is dominant
in gases
\342\226\240
on
terms
the energy pressure and is dominant in most
call
may
\342\231\246
-($.
right-hand
side of E0)
represent
the entropy pressure.The energy solids and the entropy pressure
and in elastic polymerssuelias rubber is testimony of the importanceof the
{Problem
contribution entropy: Ihe natve from mechanics must tell that simple feeling everything -JUjdV about the pressure is seriously incompletefor a process at constant temperature, because the entropy can changein response to she volume change even if the of volume, as for an ideal gas at constant temperature. is independent energy 10). The
enlropy
Maxwell relation* We can now derive one of a group of called Maxwell relations.Form the cross-derivauvirscV/^l\" be equat 10 each other. It follows from that D9)
E!)
,
*{ep;eT)y
(ca/dV)t
relations tliermodynamic must Hx and ^FfcxcV,which
useful
a relation is not at all obvious. Other Maxwell will be derived later at that relations similar The of obtaining thermodyappropriate points, by arguments. methodology L Chem. 5964 namic relations is discussed A981). Phys. 75, by R. Gtlmore,
from Z
of F
Calculation
Because F = F
Wo show
-
U
thul
=
this
%a and
U +
equation
a =
-{cFjdi)Vt we have the or
i{SF/dt)Vt is satisfied,
F/x where
Z
is the
partition
fund ton.
differential
~x2c(F/z)Jcx
\302\253 U.
equation
E2S
by
- -log^ .
On substitution.
E.1)
Chapter3:
This
A2).
by
and Hchniipliz
Distribution
Bollzittnnn
the
F =
+ ar. is so
temperature are
log
g{t
=
Z)fct
We may
the
However,
low
that
the
only
limit logZ
that
In
occupied.
equafion E2). for Fjx to contain an
possible
appear
-riogZ
logg0 only
if
a ~
state
must
cj0 coincident
that
reduce to \\oggQ when the states at the lowest energy \302\2430
-* logg0~ W*.
s0
thai
a =
-cFjcr
\342\200\224
write the result as
factor
Boftzmann
the
entropy
a such
constant
additive
0.
Zand
E5)
-riogZ
differential
required
would
It
Energy
that
proves
F=
satisfies
Free
(II)
E6)
cxp{-f/t); probability of a
for the occupancy
quantum
s becomes
E7)
GAS:
IDEAL One atom
mass
free
M
free
in
A
FIRST
a box.
We
move
to wave
particle
LOOK
calculate
in a
equation
\\p{x,y,z)
cubical box -{/iJ/2.U)V:^
= Asia(nxitx
of
volume
== ei/i
We system
are
L)sm{iiyity}L)sin{n.iiz}L)
where ir,, nyt n. are any positive integers, as in not give independent orbitals,and a zero does values
function Zx of one atom of V ~ L1. The orbitals of the
the partition
1. Negative
Chapter
not give
E8)
, integers
do
a solution. The energy
are
neglect is
the spin
entirely
and all otherstructure
specified
by the
of
values of nlt
the nft
atom, n:.
so that
a state of
the
Ideal Cm:
The
the
Provided
we
function is ihe sum
partiiion
may
replace
the summations
2] = The notation a\" may
be
of adjacent
spacing
I
&
t!nx
First
Look
over the states E9);
energy values
is small in comparisonwiih
t,
by integrations: \342\200\224
*2{nx2
(/ji,cxp[
dnf
+
ny2 +
for convenience.
is introduced
Azji2/2A/12t
as ihe
written
I
A
'u2)].
F1)
The exponential
product of Ihrec factors
F2) in
of ihe
ierms
concentration
ji
~
XjV.
Here
F3!
is catted the
atom
in
a
quantum
cube
concentration.
of side
equal to
It is
the conccnlration
die thermal average
lie
associated with one Broglie
wavelength,
Here is a thermal which is a length to /i/M h'{Mi)ul. equal roughly will keep turning up in ihe thermal physics This concentration averagevelocity. of gases,in semiconductor and in the theory of chemical reactions. theory, For helium at at room lemperature, n. s= 2.5 x atmospheric pressure *3x lO^cnr3 and uQ = 0.8 x I02scrrT3. I(T6, which is very Thus, ii/iiq -
10 unity, so ihat helium is very dilute under normal conditions. compared Whenever n/nQ \302\2531 we say ihat the gas is in the ctassieai An ideal regime. gas is defined us a gas of noninteractingatoms in the classical regime. The thermal average energy of the alom in the box is, as in (]2),
small
F4)
becauseZj\"' exp{-
is (he
eJx)
log^i = so that for an
the system
probability
\342\200\224
terms
-f-
jlog(I/i)
is in
it. From
state
the
F2),
of t ,
independent
ideal gas of one atom
F5) If t
=*
known result for
The
thermal
:Iic Boltzmann constant, then per atom of an ideal gas. energy
kB is
where
kaT,
the
of'a
occupancy
average
free particle
=
U
lkHT,
the well-
orbital satisfies the
in-
equiility
an upper limit
sets
which
standard
at
atom
helium
regimeto
apply,
is always
positive
temporarily many
tiiitil
noniulcmaing
box, all atoms one
utoni
result.
identic!!, of ik
this
of 4 x
a free
must
the
of an
occupancy
and temperature.
concentration
occupancy
for
for
!0\026
be
\302\253 1.
We
note
that
orbital by a
For the classical
as defined \302\243\342\200\236
by E9)
atom.
meihod 10 deal with the problem of in Chapter 6 a powerful in a a box. We iitsi ireai an ideal gas of .V aioms atoms i\302\273 of the extension or different isotopes. This is a simple of different species correction factor that arises when ail atoms are We then discuss the major same of ihe b.unc specks. isotope ac
develop
identical
Ideal Gas; A
!\342\226\241
\342\226\241 I
1
2
Figure 3.7 An .V particle of.V bo\\es. The energy is
*
o
Figure
of different
Atoms
3.8
iV
on
in each
=
':,(!)
3.7), the
f-
sido includesevery
r.^2)
+
partition
is fhe
funciion
F6)
,
Zt{l)Z1B)---Zl(N)
indices of
a
boxes (Figure funclioiis:
distinct
!he right-hand
where 3, fi.... Cdenotethe orbital also gives lite pjiiiiioit function single box (Figure 3.S):
species in
partition
ZXha*\302\273
product
of free particles with one panicle for one particlein one box.
system
N limesthat
If we have cfne atom in each of product of ihc separateone aiom
because ihe
Look
First
of llic N
state
independent
,
F7)
\342\200\242\342\200\242\342\200\242r^N)
oUlotm
iV itottmiontciJng
in
ihc
suewssive
aiuiits
boxes. The
all ordilTcrctii
result
F6)
speciesin
a
for F7). If i!ic are iliesameas problem bseauic the energy eigenvalues ihe lotal partiiion funciion musses of all thsic dilTercni atoms hapjiened lo be lite same, would be Z,-\\ whsrc Zl is given by F2). When we consider ihe more common of N identical panicles in one box, we pfublem have to correct Zts~ because it overcounts the disiincl siaics of the ,V idutitic.tl parliclc
iliis
bang
the same
+ M*l - in a single bos, the siate lion numbers. For mo hbded particles \302\251 and ^(O) and ihe Male \302\243,(\342\200\242) combiti.tttons tmibi be counted i\302\253 + cfrtO|;irc tlisiinct Mates, and both Ij Hie lite function. Bui for hs'o tdetiitccil lHc st.iie of energy c, + \302\243* pitritclcs pmtiuon and only one cnlry is to be made in tfte si.ite sum in the p.irtiiioti idcmicai siate as c^ + \342\202\254\342\200\236 futiciion.
Chapter 3:
indices are alt onlj once
If tlicorbiial
entry
occur
slsould
a faclor
of ,V!, ;md the
Free
ami Hdmhohz
Distribution
BoHzmann
ihe
correct
llic cniry wilt occur S< limes iti Zts, whereas are identical. Thus, Z,v ovcrcounisthe Stales by is function for N identical particles
each
diffrfem, if
Energy
panicles
partition
F8) \342\226\240mz'\"-*]<\302\273>**\"
classical regime.Here nQ There isaslep in ihe argitmcni
in ihe diticrcnt
oEotla's-
It
i\302\243 no
=
Simple
|
from F3). {\\(zi2nk3)yl whcii we assume iliat ail jV occupied orbiuis rnallei lo cvaltsaie o^rccily {lieerror tmrouuecci
arc
always
by
but laicr %sfg v,'il\\ cotifj^ni by another rnctliod the validiiy of FSJ in ^pproxitnaiiorij tlassicai regime n
The energy of
Energy,
{he ideal gas
from
follows
{he
N particle
ih^S (lie gas. the
partition
funciion by use of{l2):
l/ = T3^l0gZ.v/(}T}=^T , with
consistent
{65} for
F =
With
{he
earlier
tionlogN!
c=
result NlogN
F =
From
the free
gas of
N
aioms.
one panicle.
The
free
energy
is
- -i]ogZi'v + ilogN!.
-ilogZy
G0)
= (Mr/2n/!2K'i2KandtheStirlingappfoxima-
Zt = naV \342\200\224we
have
N,
-tNIog[{.Ui/2n/i2K'2K]
+ rNtogN
energy we can calculateihe entropy The follows from {49}: pressure
p -
F9)
-BF/3K), =
I
PK
=
Nr,
NxjV
and
,
-
xN.
the pressure
{71}
of the
ideal
{72}
G3)
IdtalGas:
the iJcal gas law.In conventional
is called
which
1>V =
The entropy follows a =
from
=
-{CF/?z)r
first
A
Look
units,
G-1)
NkuT.
D9};
+
W!og[{Afr/2\302\253/r}3';F]
-
\\N
+ S , G5}
NlogN
G6)
the
for
equation
the entropy
method
that
h
a direct
through
The energy
argument.
{7!}and {76}we
NjV,
of a
entropy
involves
6 by
Chapter
particle
a quantum
\342\200\224
n
involves
result
The
result is known as the Sackur-Tctrodc This monatomtc ideal gas.It agreeswith experiment. the term ideal gas hq, so even for the classical shall derive these results We again tn concept.
concentration
(he
with
V
have
not
does
explicitly
{69}also follows
from
involve the U
*=
F
N\\
or
identical
xa; with use
+
of
= \\Nr.
The energy U = |j\\'r from F9} is ascribed to a contriof energy. each \"degree of freedom\" of each panicle, where the number of degrees of freedom is tlic number of dimensions of the spsce in which ttie slonis move: 3 in Itus In ihe classical focm of siatisiicat the function contains example. mechanics, the partition kinetic energy of the particles in an iniegral over the momentum p,. components pt, p,.
Example! contribution
Equipanlihn
i* from
For one free
particle
(
Jjjexp[~
a result
to Ft).
similar
average energy may The
result
The limits of integration
be
ts generalized
in the
of degree2 in thermal averagekinetic energy is homogeneous hamiltonian
is
average potential
homogeneous energy
by use
calculated
are
fat \302\261ao
momentum
2
with
component.
The thermal
in
a posilion
component,
will
the
wilt
coordinate
that coordinate
of the system limit of the if the Further, \302\243r.
Kamiltonian
the
with thai momentum
associated of degree
each
{77}
of A2) and is equal wfr.
classical theory. Whenever
a canonical
associated
,
pI1)/2Mx']dpIdpfilp.
also
classical be
component, theihermal be Jr. The resull thus
!
3 Vibt
-
^\342\200\224\302\273
2 lion
T,\302\273s
I
0
50
25
250
75100
500 1000 2500 5000 K
Temperature.
Heat capacity at constant volume of one molecule of Hj in The vertical scale is in fundamental units; !o obtain a value in conventional from the three units, multiply by kB. The contribution transnational at high temperature degrees of freedom is j; the contribution from the two rotational degrees of freedom is 1; and the contribution from ihe potential and kineiic in the motion energy of the vibrattonal limit is I. The classicallimits are attained wlicn high icinpciaturc i \302\273 relevant energy level separations.-.
Figure the
3.9
gas phase.
applies to ihe At
arrangementsof the
and
temperatures
high
Example:
for
harmonic
oscillator
harmonic
in the classical limit. The quantum diatomic roiator are derived in Problems the dais teal limits as in Figure are attained, oscillator
for the
Entropy ofttuxia*. In Chapter 8 in a solid made up of ,V
A and
number
1 we
-
(
calculated
the
A arid t
atoms
number
for the
results
3 and
har-
6, respectively.
3.9,
of possible
atoms B. We
found
arrangein A.20)
of arrangements:
A/i
G3)
The crtiopy associated c{A',0 = and
is piotied
in Fi^me
with
these
arrangements
logflMf) 3.10for
jV
~ 20.
is
bg,V! This
!og{iV
contribution
r)!
to the
~ logti
G9)
,
total entropy
of ;tn
alloy
itfcalGas;
A First
lj>ak
i
\\
4\342\200\224-~
4
vj
7
At_x
composition
Alloy
0.S
0.6
0-4
0.2
0
1.0
Br
as a function of ora random binary alloy Figure 3.10 Nftxing entropy ihe proportions or the constituent atoms A and B. The curve plotted for a total of 20atoms. We see that this entropy is a was calculated maximum when A and B are present in equal proportions 0.5), (x \302\273 anci trie entropy is zero For pure A or pure H.
system iscalled the by use of the Sliding a(N,i)
entropy
.viih
x
result
G9)
be put
may
a more
in
u
convenient
approximation:
=* NlagN
- N -
=
- (N
NlogN
= -(,V
The
of mining.
-
(N
-
i)\\ag{N -
!)\\og(N
- t)Iog(l -
//N)
- i) + t)~~
N
-
t ~
rlogt
+
r
l\\agl
l Iog(f/jV)
,
= t/N,
- -v)Iog(I -
-v)
.vlog.v].
treated as a random of an n'loy A^^B, (homogives the entropy of mi\\iug 11. solmx^i. TIic in detail in Chapter (homogeneous} soiici problem is J^dopjd of a mixture of A condition We ask i Is the homogeneous solid solution ihe equilibrium of and B atoms, or is the equilibrium a two-phase ofcrysi2l!::i*s system, sudi as a miMure of the science answer is the basis of much pure A and crystallitesof pure B?The complete of metallurgy: the answer will depend on the temperature and on the imcniioiTiic ii;;;raction case iluit tltc iiue faction energies bciw.vn energies t/M, f/EB,and UAa. In the special This
result
A., BB, and AB neighbor pairs are all equal, ihe lower free energy than Hie corresponding mb-lure free energy of the solid solution A [ -^B, is
A
F =
which
we
must
Fo
-
\302\273
is
of
the mixture always
in
There is a any
other
tendency
very small xN is ihe
if a
evert
A atoms.
sutroundiug
Let this
proportion -\\ number of B
\302\253 t of
strong
The
(81)
,
xlogx]
a
have
a minimum
atoms.Tlic
.vj
the
The entropy of mixing solid soiution has ihe lower to x.
of any
eniropy
mixing
-
(SO) is
dissolve in atom and ihe
B to
element
between
exists
etlergy
a B
a positive
quantity.
energy is
xi\\U,
a
approximately
(83)
~.\\N\\ogx
+ txiog.v)
N(sU
If
wlk-re
,
(84)
= 0 ,
{85}
when
BFfBx
x This shows there is a
that
-
(82)
repulsive energy be denoted by U, B aioms is present, the Iota! repulsive
Fix)
has
A
small proportion
repulsive
a =
which
proportion
positive\342\200\224so
at least a very
for
A,
+
will
elements.
pure
case.
this special
element
a)
solution
- Fo
+ xF0
x)F0
in the arc
entropies
positive\342\200\224all
free energy
- .\\)Iog{l -
A't[A
-
A
B crystals
and
A
ofc/yitatlilci of the
with
compare
F = Tor
+
Fo
tct(.v)
solid
homogeneous
natural
\302\273
+
N(U
=
impurity
+ t)
xiog-v
(86)
exp(~!}exp(-t//T}. content
in all crystals.
SUMMARY
1. The factor =
?{\302\243j)
.
is the
probabiHty of finding
a system
expf-e./tJ/Z in a
slate s of energy c, when
the
system
comaei
is in thermal
particles
the
in
2. The partition
The
3.
is
function
is given by
pressure
~{cUldV)a = x$gIc\\')v.
p =
4.
free
Hejmltoltz
The
is defined
energy
held at
a system
for
equilibrium
a
6.
ss -tlogZ.Tltisrcsulitsvcry }\342\226\240
p
-i?F/dT)y;
such as p
constant i, ttscru!
in
~
\302\253 \302\273a.The
N/V
-
o =
JVt;
state at all
of spin
atoms
zero,
(nQVf/N\\ ,
concentration
quantum
is reversible
process
equilibrium
and ofquuniiiics
incilcuhitiansorf
from F.
o derived
and
pV
A
mittimum
is a
~{8F/dV)r
ZH =
8.
H
xa.
V.
\302\273
7. For an ideal monatomicgas of N
u
as F ^ U ~ '
-
5.
if
r. The numberof
constant.
is assumed
system
ai temperature
reservoir
a large
with
s (A/r/2n/t2K/I.
nQ
+
W[log(iic/H)
=
Q
5];
Further,
fW.
if the sysiem remainsinfinitesimally times during ihe process.
close
to
for
the
ihe
PROBLEMS
/.
Free
energy as a
function
of
a two of
t
state systenu (a) Find of a system with two
fro; ettcrgy, find energy the system. Tttc entropy is plotted in c. (b)
in
of
energy
2.
function
moments
of
for
ihe
tnagiteliAUiou
temperature
in
a magnetic
and
M
magnetic
one
states,
for
end
function
pitriiiion tltu
for
io x
susceptibility
field for the another
at energy
the energy
itnd
free
0 and one entropy
3.11.
Figure
field. The result
mutanli^jiB/i}, as derived in {46}by
expression
expressions
00 Use ihe
susceptibility.
Magnate
expression
From ihe
an
model system
the
method.
of
magnetization
Here
find nn exaci s ilM/tlB as a
n
magnetic
is M ~ the is particle
Si
Chapter
and
Distribution
Bolizmann
OR 2
tjelinholt;
Free
Energy
-
0.6
0.4
A/
/
0.2
/
]
ft
0
Figure
3.12
i. oscillator
/Vcc has
as a function loui macaeiicmoment is the momeni a U.icarfunciio mB/x r the manicnl Sends 10 salurale.
ofmfl'r.
Notice thai
31 lou
of HiiJ/r,
bul
hiS
Ai
high
contciiiraitot!. Tlie result express thul the
of ihe
Plot
the result
is x
sust;eptibiltty energy of an
is
plottcJ
as a function
infinite
a
only
in Hgure 3.12. (b) Find and the parameicr.v
series of
ihe
ofr
\342\200\224 in the j\302\253\302\273V
harmonic
2.0
1.5
1.0
0.5
oscillator.
ttniit
s
free
energy
M/tuu,
(c)
and Show
\302\253 /\302\273B r.
A one-dt'mcnsian;il
equally spacedenergy
states,
with
liurmonic =
\302\243,
sho,
oscilwhere
s is a
of frequency
positive imegeror zero, the
chosen
have
Wu
Enin
3.13
Figure
osciliuSor
and
frequency of she oscillaior. for a 0. (;s) Show that
tlic classical
to is
of energy
the
at
s =
state
the free energy is
oscillator
harmonic
zero
(S7)
Note
at
that
high
such ihal
temperatures
of the logarithm to obtain F^i
!og(/itu/t).
x
\302\273 fitu
(b)
we
From
may expand
the argument
(87) show thai the
entropy
($8}
The 4.
entropy
Energy
is
shown
in Figure
fluctuations.
wiiis a
reservoir- Show that
syslcin
is
3.13 and the heat
Consider a systemof fixed the
mean
square
capacity in volume
fluctuation
Figure
ta thermal
3.1-i.
contact
ia she esiemyof lhc
Z the conventtooai symbol for
Here
to
U is
relate
3:
Chapter
S\302\273re3.S4 r
harmonic
ifiiontal
wilh
unncal mlcln iiit iluc
capacity oscillator of
T
flE.
Cv
is k^own
mpcratures
'as
Energy
u>.
^
The
of i/fiw, whkh is calkd rhc
is
\302\253herc0,
\\n the high
Icniperamte.
- kB.ot
frequency
units
in
Free
IIcIihUoUz
Versus u-mperatufe
Heal
scale is
mJ
Distrihut
Bohzmnn
1
temperature miiis. This V'^Iv:;. i\\i low
in run
\\\\\\t
c^issic^I
C(- decreases
y
csponeniially.
1.0
0.5
r
t
not fluctuate
attiiude
otltcr
Any
v,\\
value
would
of a system.The energy do^s
Thus
\\shcn
is
the system
be inconsibletit
in
thermal
with
contact
of the
definition
our
a reservoir.
with
temperature
may fiuetijiitc, but the temperature Sorr.c workers ltot. do not acllterc to a rigorous definition of temperature. Landau arid Ltfshttz etve Hie result of
a system
such
\302\253AtJ>
=> t2/Cv
(90)
,
this should be viewed as just another form of (89) with At becomes AV;CV. We know that AU = Cy At, whence (90) <(Ak which is our result (89).
but
5.
effect.
Suppose
one
whenever increase positive
system
- r=C,.,
suitable external mechanicalor to add ae the energy of 'he heat reservoir arrangement the reservoir passes to the system the quantum of energy e. The net of er.eray of the reservoir is (a \342\200\224 Here a ts some numerical factor, l}e. or negative. Show that the effective Boltzmantifactor for this abnormal
Ovcrhaussr
electrical
equal to
that
by a
can
is given by
This reasoninggives the statistical basts of the Ovcrhattsereffect whereby the be thermal nuclear in field can enhanced above the a magnetic polarization Such a condition polarization. equilibrium requires the active supply of energy to the system from an external source. The system is not in equilibrium, but ts be to in a state. Cf. A. W. Rev. 411 said 92, Overhauser, Phys. A953). steady 6.
Rotation
considered
only
of diatomic molecules. the translationa! energy
In our of the
first
look
particles.
at But
the ideal molecules
gas we can
con-
rotate,
energy. The rotational motion is quantised; and the energy molecule arc or the form
with kinetic of a diatomic
= jlj
ft;}
+
posilive integer includingzero:/ = level is
of e;ich
for the rotational states
of
stiitcs,not over all le\\cls\342\200\224this
0, I,2,.. ..
a difference,
makes
The
multiplicity
function
Zk(t) Z is a sum over all jb) Evaluate ZH(x) approxi-
the
partition
that
Remeniber
molecule.
one
(92}
l)\302\243o
is any
where;
levels
the sum to an integral, (c) Do the same for after the second term,(d) Give expressions for the energy V and the heat capacity C, as functions tn both limits. Observe oft, that the rotational contribution to the heat capacity of a diatomicmolecule
approximately for
r
r
truncating
\302\253 sOl
by
\302\273
by
e0,
1
of L'(t) nnd
C{x),showing
7. Zipper pruhh'W. with
units,
in conventional
{or,
approaches
closed
converting
the sum
the A
citefgy
h:is N
zipper
u siate
0 and
r
and
links; each link
has n
state
ii is
which
in
for
behaviors
-\302\273 ro
limiting
open
with
that
can
the
r -*
0.
and
1}
are
it is
tti which t:. We
energy
zipper only im/jp from the left end, mmiher s can only open if all links to jhe left {1,2,.. .,s tn the (a) Sliow ihat the parution fuiicliotl can be summed
however,
the behavior
Sketch
r \302\273 r.0.lc)
A-,,)\\Uv?n
require,
site link
that
open,
already
form
193}
e \302\273 the average number of i, find of model lhe ofnvo-siraiided unwinding very simplified C. Kittel, Arner. J. Physics 37, 917A969).
{b) In the
5,
iirnii
Quantum
L;
in the
when
one
Consider
concentration.
rai ion in
concern
lhe
effect is n
=
ground orbtial. There
lhis zero-poini quantum
kineiic
open links.The modelis
particle
DNA
confined to a
cube of side
of
1/L3. Find the kineiic energy will be a value ofthceoticentraiioii is equal
energy
to the
a
molecules\342\200\224see
i
he
particle
for which
temperature r. (At
this
of unity; ihe ration the occupancy of the lowestorbitalis of the order Show thai lowest orbital always has a higheroccupancyilian any oilier orbiial.) the concentration nQ thus defined is equal to the quantum eoticcntraitonnQ defined by {63), wtthtti a facior of ihe orderof utitiy. concern
9.
Partition
function
Z(l 4-2) of two
for
independent
two
thai
Show
systems.
I and
sysiems
temperature t is equal 10ihe produci
of
ihe
2
in
thermal
paniUon
the
partition contaci
at a
funciions
of the
function common
separate
systems:
Z(I +
2)
=
Z(I)ZB).
(94)
ChaptcrS:
10.
identity for
a One-dimensional
is
system
dU ~ fdi
zda = when
Energy
The lhermodyiiamic
of potythevs.
Elasticity
free
Heimhaltz
and
Distribution
Boltzntann
(95)
the external forceexertedon the line and witSi C2) we form the derivative to analogy
/ is
line. By
ill is
the extension of the
find
.-Ha-
The direction of
the
force
is
to the
opposite
conventional direction of the
pressure. We
a polymeric
consider
of N
chain
equally likely to be directedto the right of arrangementsthat give a head-ioiail
=
\302\253 JV
\\s\\
show
of
=
/
/>, with
each
link
(a)Show ihat the number 2js|p
is
' -T-
(kN
(b) For
to the left,
length
+ q(N,s)
q(N,-s)
links each of length
and
\342\200\224~
4- 5)!
-
(\302\243N
(97)
s)\\
that
(98)
(c) Show
that
the
/ is
at extension
force
/ = h/Np1.
(99)
because the temperature. The forcearises wants to cur! up: the entropy is higher in a random coil than iti ati polymer uncoiled a rubber band makesit contract; a warming configuration. Warming of rubber is discussed steel wire makes tt expand. The by theory elasticity H. M. James and E. Guilt, Journal of ChemicalPhysics II, 455 A943); Journal of Polymer Science4, 153A949); seealsoL. R. G. Treloar, Physics of rubber elasticity, Oxford, 1955. II. Ouc'dimcmionaigas. Consideran idea! gas of A' particles, each of mass at temperaline oflcnath L. Find the entropy confined to a one-dimcsisional M,
The
force
temperaturer.
The
is proportional
pti5tides
to the
have spin
zero.
4
Chapter
and
Radiation
Thermal
Planck Distribution
PLANCK
DISTRIBUTION
PLANCK
LAW and
Emission
FUNCTION
AND STEFAN-EOLTZMANN
96
Law
KirchhorT
Absorption:
LAW
Estimation of SurfaceTemperature Black
Cosmic
Example:
97
Background
Body
Radiation
93
ELECTRICAL NOISE
PHONONSIN SOLIDS:
104
Modes
Phonon
Numberof
102
THEORY
DEBYE
109 Thermal Photons
Number of Surface
no 110
Ike Sun
of
Temperature
Average Temperature of the interior of the Age
111
Sun
of the Sun
111
Surface Temperature of tlte Free Energy of a Photon Heat
111
112
Gas
112
Shields
Heat Capai
ty
HeatCapa'
ty of Solids in
HeatCapai;ty HeatCapai
of
Photons
High Temperature
Limit
and Phonons
Flux
Energy
113
113
113
113
114
Radiant Object
and
114
Occupancy
Isentropic
Expansion
Reflective
Heat
SUPPLEMENT:
113
Space
tribution of Radiant
D: of a
Entropy
ol\" intergalactic
tuations in a Solid at Low Temperatures 4He at Low Temperatures ty of Liquid
Flu>
Energy
Angular
112
Ga: in One Dimension
Photon
image
111
Earth
Radiation
of Thermal
Pressure
Shield
of Photon and
GREENHOUSE
Gas
Kirchhoff's
114
Law
EFFECT
115
Chapter's: Thermal
[We v,
//
Radio
the distribution of consider] U is viewed as divisiblewithout
distributionsare possible.We of the finite
whole
equal
gives the element
N oscillators
U among
energy
then
limit,
an infinite this
however\342\200\224wUl
of frequency
number of is the
essential point
an entirely determined nimiber of natural constant h = 6.55 x fO~21
made up of
and we make use of the
erg-sec.This constant when oscillators
consider
calculation\342\200\224Vas
parts,
the
multiplied
by
of energy e in
the common ergs
frequency
....
M. Planck
v
of
the
Planck
Fu
Distribution
FUNCTION
PLANCK
DISTRIBUTION
The Planck
distribution describesthe
of
spectrum
radiation
the electromagnetic
a cavity. Approximately, it describes within the emission equilibrium of the Sun or of meta! h eated a torch. The Planck distribuspectrum welding by distributionwas the first appHcation of quantum thermal physics. Thermal electroradiation is often caHed black body radiation. The Planckdistribution electromagnetic also describes the thermal energy spectrum of lattice vibrations m an clastic in
thermal
solid.
The word
\"mode\"characterizes a particularoscillation amplitude pattern low = 2nf as the frequency
the cavity or in the solid. We shall refer always the radiation. The characteristic feature of the of oscillation hio.
energy
of frequency w The
energy
of \302\243,
may
be
the
state
excited
with
radiationproblemis only
of the
in units
s quanta
that
is
U)
e,
where s is zero or any
integer
(Figure
4.!). We omit the
zero point
\\hai.
energy
These
of
positive
of
a mode
quantum of
mode
the
in
in
harmonic as the energies ofa quantum to, but there is a difference between the concepts.A
energies
frequency
are the same
Figure
4.1
represents
to s
oscillator harmonic
Stales of an oscillator a mode of frequency
photons in
the
mode.
that tu of
an
(-haptcr
4-
modes a and magnetic
b, of
Distribution
and Planck
Radiation
Thtftjml
and \302\253>t. The &>\342\200\236
frequency
field is suggested
in
the
figures
occupancy ofeach mode.
amplitude for one pliotor
oscillator is a iocasizedosciiiator.whereas the an electromagneticcavity mode is distributed
and
electric
of
magnetic
energy the: interior of
throughout
the
(Figure problems the energy eigenvaluesare integral of ho, and this is the reason for the similarity in the thermal physics of multiples the two problems. The used to describe an excitationis different; s for language the oscillator is called the quantum number, and s for the quantized electromode is called the number of photons in the mode. electromagnetic We calculate the thermal first average of the number of photonsin a mode, a reservoir at a temperature when these photo ns are in thermal equilibrium with t. The partitionfunction is the sum over the states A): C.10)
both
For
4.2).
cavity
B)
}{~shia/r}. This
sum
is of
ihc form
the infinite scries
may
TV, be
with
.\\-
s
and
summed
exp{-/i
Because
has the value 1/A
.\\-
is smaller
than
1,
-- x), whence
C) 1
-exp(-tou/T)
Planck Lax and
The
the
that
probability
state s of energy slim
is in the
system
Slefan-BohimannLt*
is
by the
given
factor:
Bohzntann
P(s)
The thermalaverage
of
value
exp(
\342\200\224
shci/x)
s is
E)
With
ftej/r, the summation on the
>' =
the
has
side
rmht-hand
form:
-cxpt-v)/
From C)
and E) we
find
I
-exp(->.)'
F)
This is the Planck distribution
photons(Figure
4.3)
number of with
energy
PLANCK The thermal
in
for
function
a singie
mode of
AND
average
frequency w. Equally,
phonons in the mode.The result in the form of (!). LAW
thermal
the
lo
applies
any
it is
number of the average
kind of
wave
field
LAW
STEFAN-BOLTZMANN
average energy in the modeis
\342\200\224 )
1*
G)
4: Thermal Radiation
Chapter
Disiribttlio
Planck
and
+ *(\302\253)
as a function Figure 4.3 Planck distribution ofihe reduced temperature i./rw. Here is Hie thermal average of the of rmniber photons in the mods: of frequency en. A plot of where O(o)> + i is also given, $ is the effective 7ciopoint occupancy of ihc mode; the dashed line is i!ie classical asymptote.Noie that we
0.5
/
/ A
The
high
be
may
exp{frfcj/t)
t
limit
temperature
0.5
as
approximated
often
is
\302\273 ha)
lna/r 4-
1 4-
limit. Here
the classical
called
* \342\200\242 whence \342\226\240,
the
classical
average energy is ^ <\302\243>
There mode
n
is an has
own
conducting cavity
in
frequency the
form
wn.
of a
For
(8)
modes
of electromagnetic
number
infinite
its
T.
radiation
within
cube of edgeL, there
is
cavity.
any
Each
a perfectly of modes of the
within
confined
a set
form
Ex
=
ExOitn
wtcos(fiJji.v/L)sin(iiyjij'/L)sin(fi.Jiz/L)
ID Et
Here Ex, Er and are \302\243;0
the
independent,because
sin(fl=Tiz/L)
,
(9a)
,
(9b)
= E-0 Ex
are
(9c) the
three electric
field components, and
The three
corresponding amplitudes. the field must be divergence-free:
components are
\302\243lQ,Ey0 not
and
indepen-
A0)
When we insert {9}into A0}
and
+
\302\243,0\",
the
field vectors must
rhar the
states
This
nx,
components
4- E:Qn:
E^nr
ny
transversely polarized
be
condition
the
find
-. Eo \342\226\240 0n \302\273
A1)
vector
to the
perpendicular
the electromagnetic field
>l, so that
and
Law
Slcfan-Bolwtann
factors, we
ail common
drop
and
Law
Planck
field.The polarization direction
is defined
in
the
n
as the
with
is a
cavity
direction
of Eo.
For a
given
n,,
triplet
directions,
polarization
can choose two so rhat rhere are mo distinct nft
n. we
On substitution of (9) in the wave
c the
wilh
velocity
of light, we
if we \";\342\226\240
\"y
>h<
4-
are of the
The total energy of the
iij
independent
by an integral
indices.
That
n;2)
mode
the
is,
over
we set
+
,,/
=
photons in the
= w3L2. in terms
A3) of Hie triplet
of integers
+
A4)
, rta\302\273)\302\273'i
A5)
mrc/L. is, from
cavity
G),
integers alone will the sum over nx, modes of tlic form (9). We replace the volume clementditx dny dnx in the space of the mode
The sum is over the triplet of ny,
+
Hya
form
w.
all
iriplei
define
the frequencies
describe
each
.(V
'.
trJJ.
\302\253) of
= (rtj[2 \342\200\236
then
perpendicular for
find
cWnJ
This determinesthe frequency
modes
equation
i:y
V'-v1
mutually
integers nx,
ny,
n..
Positive
Chapte
where the factor We
involved. two
|
now
the sura
the positive
only
or integral by
a factor
of the electromagnetic
polamations
octant of the Spaceis of 2 because Ihere are field (two independent
Thus
modes).
cavity
because
arises (\302\243K
multiply
independent
setsof
=h
1, = ji ft
hu)n
Jo
with
A5) for
over
a dimensionless
Standard \302\253\342\200\236.
dnn*
r
(nVic/L)
\302\273\302\260
is to
practice
We set
variable.
\342\200\224
A8)
ex
transform
x =
the definite integral to one and
nhcn/LT,
A8)
becomes
A'J)
integral has the value z*/l5; it
The definite
such as Dwight
in the
(cited
general
is
in good
found
references}. Tlie
standard
energy per unit
the
volume
lional !olhe of
law
fourth
V =
L1. The
power
oflhe
result that lemperalure
is
B0)
\\Shs
with
volume
tables
the
radiant
is known
energy
density
is propor-
as theStefan-Boltzmann
radiation.
we B0) into the spectral decompose applications of this theory as the energy per unil of the radiation.The is defined density spectraldensity We find \302\273u from and is denoted as \302\253\342\200\236,. can volume per unit frequency range, in terms of w: (IS) resvritlen
For
many
B1)
U/V
so
that
the
spectral
density
is
B2)
Planck
Law
andSufan-Boltzmam
Law
A
\\
1.2
1.0
/
\342\200\224
/
/
0.6
1
0.4
0.2
\\
/
\\ \342\200\224 \342\200\224
/
0/
~ l)willi.v = bttf/t. T\\\\h runciion h involved in the Planck radiation law for llic of a black body may spectral density uw. flic temperature be found from ilie frequency tjmil ai which the radiant Figure-1.4
Ploiof.vJ/(c*
is a maximum, energy density per This frequency is directly proportional
unit
ffequency
so ihe
range.
tempera sure.
of distribution This result is the Planck radiation the frequency law; It gives thermal radialion (Figure4.4).Quantum here. theory began the relation The entropy of the thermal photonscan be found from A34a) ~ at constant volume:da from B0), dUfr, whence
Thus
the entropy is
B3)
The constantofintegration
is zero,
from
C.55) and the
relation belsvecnF and
a.
4: Thermal
Chapter
gy
flux density
area
and
length
is of the order of the of equal to the velocity
factor is equalto \302\243; the
The geometrical
The
by
for
result
final
use of
gy
v
y
The
Distribution
Planck
and
Radiation
the
radiant
the energy density
B0) for
light
the
is (he
time.
of
unit
of
Thus,
subject of Problem 15.
The
is often written
result
as B6)
\302\253
aB s= b2V/60AV
has the
x
5.670
vahie
10~8
W m~2
(Here cts is not the entropy.) A
as
a
black
A small
body.
body
the
on
10\"*
K~* or 5.670x that
a cavity
radiates whose
walls are rate walls
in
is said to radiate
thermal
given of
K~\".
s~'
cm\022
erg
at this rate
as a blackbody at the of the physical constitutionof the
is independent only
in
B6a)
T will radiate
at temperature
depends
hole
unit
is
flux
U/V.
Jv
times
derivation
energy
p
a column
in
contained
energy
equilibrium in
B6).
the cavity
The
rate
and de-
temperature.
Emission and
Law Absorption; Ktrchhoff to the ability of the The of a surface to emit radiationis proportional ability surface to absorb radiation. We demonstrate this relation, first for a black body
or biack
surface
is defined to
incident
upon
biack if
the
and,
second,
be blacktn it
hole
for a
surface
with
arbitrary
properties.
An object
electromagnetic radiation a ho!e in a cavity is in that range is absorbed. By this definition small incident the hole will is enough that radiation through a given
frequency
range if all
t
reflect
times from the cavity through the hole.
enough
in the
be absorbed
lo
walls
cavity
with
back
loss
negligible
oj Sutj
The radiant
a black surfaceat temperature x is from a small hole a in equal density Jv cavity al the same temperature. To prove this, let us close the hole wilh the black in thermal the thermal average surface,hereaftercalledthe object, equilibrium fiux from the black object to the interiorof the be equal, must energy cavity but opposite, to the thermal average energy flux from the cavity to the black to the
energy
flux
radiant energy
Jv from
density
emitted
flux
object. We
the
prove
following:
fraction a of the
non-black object
If a
radiation incidentupon
at
temperature
t
absorbs
a
by the emitted by a black body at the same and e the emisstviiy, where the flux emitted by the object is e times it,
the
radiation
flux emitted
object will be a ttnies the radiation flux temperature. Let a denote the absorptivity cmissi\\ity is defined so that the radiation the fiux emitted Theobjectmust emit by a black body at the sametemperature. at the same rate as it absorbs that if equilibrium is to be mainiamed. H follows a is law. For the special case of a perfectreflector, a~e. This is the Kirchhoir
whence e is zero. A perfect docs not radiate. reflector The argumentscanbe generalized to apply to the radiation at any frequency,
zero,
of
Estimation
Surface
Temperature
of a hot body such as a star is emission takes from the Ihe maximum of radiant energy frequency is depends on whether we look at the this (see Figure 4.4). What frequency place For if,,,, the fiux range. range or per unit wavelength per unit frequency energy the Planck the maximum is given from energy density per unit frequency range,
One
way
to
estimate
the
surface
temperature
at which
law,
Eq. B2),
as
\342\226\240\342\200\242 0
,
3
- 3exp(-x)
= x.
This
be solved
may
equation
and Planck
Radiation
Thermal
4:
Chapter
DiVnfmrion
numerically. The rool is
kca^JkgT = xm
^
2.82
,
B?)
Figure 4.4.
as in
is that the Example:Cosmicblack body background radiation. A major recentdiscovery universe accessible to us is filicd with cudutian like a thai of black approximately body Tor big bang at 2.9K. Tlic existence evidence of lliis radiation [Figure 4.5) is important assume ltw! tli^ unhorse is expanding and cooling wiitl liliic. cosmotogiol modelswhich
so
h;id coo
universe irucructs
This
Most
lines.
mancr.
Tftcrcafier ihe radiaiion evotved with \302\273mcin a very siinpte way: llic plioion gas of 2.9 K. Tlic pfioion gas will al constani cniropy io a icmpcraiure by expansion at consent ihc expastsion during cniropy if Ihe frequency of each mode is towered universe wilfa llic nuoi^er of pljotons it\\ c>icti mooe Kept con^JunlL We show in ^joj that ihe entropy is constant if lhe number of pholons in each mode is consiant\342\200\224the
cooled
was
remain
of
;uid itie bhiirk body radiaiion were in Itiaiipl cqtiitilirium. IJy ihe lime ted lo 300A K,! tie m;iiUv Mas primarily in she farm of atomic liydrogen. with bLjck body r^diLitJi^Ji si the fic^ucOci^b of jlic liydro^jcti &fH:ctriJ only itie of llie biack body r;idi;ition thus was cffctiutty docouptoti from clergy
nimicr
Ihc
ihaf
itic
tnc
below
lhe
After inlo
ihe eniropy. Urn evoluiion
determine
occupancies
decoupling
and
stars,
galaxies,
Electromagnelicradiaiion, is superimposedon the
black
cosmic
As an important exampleof the
sponMucousthermal called
H. Nyquisi.*
noise,
H. N!jquisi,p!:js
lical jijsus,
ts
4.6. We
\342\226\240
Wiley,
botiy radiation.
shall
R !
sec
across
that
by J.
property of lo llic
proportional
dimension, we
in one
law
Planck
were discovered
The characteristic
square noisevoltage by Figure
atoms (which are organized lhan before decoupling. complicated radiated by the maiter since lhe decoupling heavier
inlo
more
tn voltage
fluctuations
are
tnaucr
NOISE
ELECTRICAL
which
of
ciuSl clouds) was as slarliuM, such
a resistor.
B. Johnsonand
Johnson
value of the
is also
noise
consider the
These fluctuations, explained
resistance
by
ihc mean*
is that R,
as shown
directly proportional to tlie tem-
[
Microwave
|
Interstellar CN
Q
measuremeni
IR
2.9
K Black
body-
Frequency (cm\"!) 4.5
Figure
me surememsof lhe spcclrum Observations nf the flux were made with
Experimental
body radiation.
speclrum ofinlcrslettac
oflhe
infrared
a bulbon-borne
wiih
;
s
Caurlesy of P. L. Richards.
pcrature r and tlie band'-vidili
edge of Tlie voltage
A/of wave
cLvironuignetic
tlicorcm
Nyquist
generated
uccded in
any
gives
by a resisior estimate
of
neiir
lhe peak,
ai frequencies
the
circuit.
propagation
a quanlhative. in
thermai
Hie iinnting
cosmic
of the microwave
W
heicr
unJ vvere rneusi abo\\c lhe peak.
(Tliis secifon presumes a knov.1at Hk inicrtncdiiitc level.)
expression
equilibrium.
signal-io-uoise
The
for ihe llicrma!noise theorem
is lhcr^fore
ratio of an experirr.cnial
and Planck
Radiation
Thermal
4:
Chapter
Zl ~
\342\200\242 Carbon
filamen
+ Advance wire
xCuSO, m H,O
vNaCi
RcMblancu
Mil
in
component,
kinds of conductors,including
0.5
0.4
0.3
0.2
0.1
0
H..O
in
Afiei
electrolytes.
J.B.Johnson.
In ihe
apparatus.
square
original a
across
voltage
the
form
resistor
states
theorem
Nyqutst
of resistance
in
ft
ihe mean
that
thermal
at
equilibrium
temperature t is given by
B8) where A/
is
the
frequency*
bandwidth
the
which
within
circuit
the
of Figure
4.7, the
power deliveredto an '
al
which
'
In
this
maich
section
(R' = the word
(R + Rf
fluctuations
voltage
are measured; all frequency components ouisidethe given We show below that the ihermal noise power per unit frequency a to where resistor a the facior4 enters matched loadis x; by arbitrary
are
range
range
it does resistive
ignored. delivered
because load
'
in
R' is
B9)
R) is <1\">/4J!.
frequency
refers
to cycles
per
unit
time,
and not
to
radians
per
unit till
Noise generator
4.7
Figure
load R'. The
which this
hi
power enables
oftliermal noise that
Consider characteristic
as in
impedance
is a maximum wiih condition the ioad At
supply.
us to
the
limit
voiiagc
be
absorbed
circuit is maintained
power
said
at
= R.
when R'
lo the
to bo matched
- (Yi)f4R. The
filter
bandwidth under [lie bandwidth 10 whicii the mean
fluciuaiion
applies.
line Figure 4,8 a losslesstransmission ~ R terminated at eachend Zc by
wiihoui
with
to a
frequency
is.
ihal
to R'
respcel is
&
match,
line is matched at eachend, in will
resistance ft
delivers
cilrrenl
consideration; square
ciicuii for a
Equivalent
a generatot
sense
the
reficciion
in
that the
ail energy appropriate
of lenglh a resistance
L and
charac-
R. Thus
the
traveling down the line resisiance.
The
entire
t.
temperature
line is essentiallyan electromagnetic sysiemin one dimension. We follow ihe argument given above the distribution Tor ofphoions in thermal but now in a space of one dimension instead of three dimensions. equilibrium, has two photon modes (one propagating in eachdirection} Thetransmission line = 2nn/L of frequency in the from A5), so shat freihere are iwo modes 2nfa A transmission
frequency range
C0)
Sf~c'lL,
where c' is the
propagation
velocity
on the
line. Each
mode has energy C1)
exp(ftwAJ
Figure 4.8
Transmission
Hue
derivation of ihc Nyquist aclerislic line
has
ihe
heir
The
liieorcm.
impedance 7,c of ihe [ ra us mission vaiuc R. According lo ihc of Iransrnission iines, [he matched to the line ivlicn
theorem
mdamcniai erminai
L with
oficiijjtli
resistors
resistance
are
same value R.
has ihc
in
is r. It
in
the
that
follows
hw
limit
classical
the
Planck distribution.
to the
according
equilibrium,
with circuits
on the
energy
\302\253 z
so
line
in
the
are
We
energy per mode A/ is
range
frequency
concerned
usually
the thermal
that
C2)
The rale at which
line in one
off the
comes
energy
direction is C3)
The powercoming
off
the
al
line
end is
one
that end; there areno reflections
impedance
R at
is matched
to the line.In thermal
line at the same rate, or elseitstemperature to the load is
9 = but
V
~
2R1,
itt
temperature(hermomctry,
dc current
when
no
when
a dc
iA/
the power input
C4) used
been
{Figure 4.9)
where
in low it
is
tempera-
{not
con-
more
than t. Johnson noise is the noiseacross (V1} discussed is flowing. Additional noise here)
resistor
appears
current flows.
PHONONS IN SOLIDS;
DEBYE
THEORY
calculate the spectraldistribution of this distribution for a continuous solidand to consider So I
energy to the
,
The result has
regions
temperature
impedance
a
measure
to
convenient
B8) is obtained.
so that
\302\273
Thus
rise.
terminal
the
terminal
must emit
load
would
R
the
when
the
equilibrium
all absorbed in
decided to
approximationto
the
actual
distribution.
The
the
possible
as a
fvee vibrations
good enough of a lattice must,
sonic spectrum
j in
Solids:
Dcbye Theory
square noise \\ o'uge flucluations observed cxperimcn::i))y from a 3 jiO resistorin ihe mixing chamber of a dilution as a function of magnetic refrigerator 4.9
Figure
Mean
icmpcralurc indicated by tlidrmometer.
After
R.
C Wheailey, J. 533 A972).
and
J.
a CMN'
R. GiiTarJ, Low
powder
li.
Tcnir
Physics
100
T
of course,deviatefront
its soon its t!ie wavelength becomes comparable to . .. The only thing which had to be lione was lo to she fact that every solid ofjunta dimensions numfrc'r contains adjust ajiuite atoms and a At low has mint her vibrations.... of therefore L'uoiujh finite of free and ttt perfect analogy to the radiation htw temperatures, of StefanBoltzmann ..., the vibrational energy contentof it solid will be proportional
t/ie
disittuees
this
of the atoms.
P. Dcbye
The energy of an elastic wave electro
elastic ;is for
wave in a cavity
magnetic
clastic wave
is calleda of
wave
in
oj is
is quantized
just as the energy
is quantized.The quantum thermal
The
phovwn.
frequency
a solid
average
number
of
energy
of an
of pitonons in
Planck distribution function,
given by the
an
of
an
just
photons: 1
We assume
that
t!ie
frequency
ofan
elastic
wjive is independent
C5}
of theamiMttmle
and heat capacityofiheelastic be carried of the resiiks obtained for photons waves in solids. Several may th:tt the velocities of ail over to plionons. The resultsare simple if we assume elasticwaves are equal\342\200\224independent of frequency, direction of propagaiion, but it helps Thisassumption is not and directionof polarization. very accurate,
ofthe elastic
sixain.
We
want
to find
the energy
A-
Webb,
6,
the general trend of the observed results in many with a solids, of computation. Therearetwo important of the experimental results: the heat capacity features of a nonmctallic solid varies as tJ at low temperatures, and at high temperatures the heat capacity is independentof the temperature. In metals there is an extra for
account
minimum
contribution Number
the
from
conduction
There is no limit to the number of but the number of elastic modesin with
each
3A?.
An
wave
elastic
in
Chapter
7.
Modes
of Plionon
of N atoms,
treated
electrons,
three
possible a finite
modes
in
a cavity,
If the solid consists
lite total number of modesrs
of freedom,
degrees
has three
electromagnetic is bounded.
solid
possible po! matrons, a
two
transverse
and one
polarizations of an electromagnetic of the atoms is perpendicular displacement a wave the displacein wave; longitudinal displacement is paraiicl over all to the propagation direction. The sum of a quantity modes 3, may be written as, includingthe factor
longitudinal,
in
two possible
to the
contrast
wave.In a transverseclasticwave the to the propagationdirectionof the
| JW of A7). Here n as for photons. We
by extension exactly
elasticmodes
ts
to
equal
C6)
of the triplet of integersnxt i\\yt iu, ;iralI such that the total number of
in terms
is defined
to
want
<*\302\253(\342\200\242*\342\200\242) ,
find
3JV:
C7)
In the number
photon problem there was no cor espondinglimitationon the total of modes. It is customary to write D, after Debye, for nraaI. Then C7)
becomes in\302\273D3
The thermalenergy
of
the
=
3;V;
phonons
nD =
is, from
FW/7!I'3.
A6),
C8)
,\\umbcr
by C6)
or,
ofPho
and CS),
D0) with
analogy
By
place
=
V
where limit
X
of
usually
=
of A9),
the evaluation
{'in2twf2L)(xL/n!n-)-i
For
ji/irii/Lr.
the
velocity
of sound
v
written
in
L3 we
.
J*V*
D1)
\342\200\224~~^
write the volume V.
with
Here,
C8),
the upper
is
integration
written as -
,vfl
where6 is
the
called
Dcbye
0/7
= fcsO/'t .
D3)
temperature:
0 =
The
with
velocity of light r,
of the
(hv/kB)Fn2N/VI \\
D4)
special interest at low temperaturessuch than that T \302\253 0. Here the limit .xD on the integral is much umly, and .vo larger 4,4 is little contrithat there be We note from may Figure replacedby infinity. = we have to 10.For the definite the integrand out beyond x contribution integral result
for
D1)
is of
the energy
f\"^
Jo as earlier.
Thus
the energy in the
proportionalto
T4\302\273 The
heat
__\302\243_-?-
exp.x-
D5)
,
15
1
low temperaturelimitis
capacity
is, for
i
\302\253 kB0
or
T
\302\253 0,
D7a)
4: Thermal
Chapter
Radiation ami
Distribution
Planck
-
17.78
T
A
-\342\226\240\342\226\240 \342\200\224
a E 13.33
A
i
A
^
4.44
Y \\A
0
3.99 in
r3,
Figure 4.10 show
TJ10
these
from
Low tempcralure
heat
Ihc cxcettent daia
is 92
agreement K. Courlesyof
capaciiy whh
!tie
5.32
K3
of solid Debyc
and
L.1 Fincgold
argon, pioiicd against The value of 0 N. E. Phiiiips.
T1 law.
In conventional units,
D7b)
This
h known
result
as the
Debye T1 !aw.*Ex
plotted in Figure 4.10.Representative temperature
plotted
Problem given in
are
given
in Fig-jrc
in Table
4.1. The
13.297 A912):
values
calculated variation of
4.11. The high temperaturelimit functions
tiiermodymmiic
4.2
and
14,65 A9K).
are
plotted
in
for argon are of the Debye tem-
results
menta!
experimental
II. Several related TabL*
peri
Figure
4.12.
T\302\273 0
for
Cv
is
versus
the
a Debyc
T/6
is
of solid are
subject
Aiimitr
Lu
of
210
Jjg
7.
UJ
1
X
UJ
U
EQ
pT
UJ
<
S
e
Z
(J
a.
Photon
Modes
4: Thermal RaJhtion
Chapter
ami
Dim
Planck
25
-
i ^\342\200\224^\342\200\224
20 -
/ Heal
Figure-5.11
according10ihc
capacity
solid, The
approximation.
Dcbye
vertical scale is in
Cv of a
J mol~'
K\"'.
The
Iiori^onla! scaieis ihc temperature to the Debye temperature 0.The normalized TJ law is below 0.10.The of ihc region value al high values of 7\"/1? is asymptotic 24.943
Jmor1
L\342\200\224
K\021.
/
0.2
0
'able
4.2
Values of
C,, S, U,
ai
id F on the S =
k^o
Debye
0.6
0.4
ihc ory.
0.8
in unlls J
U,0
moI\021
1.0
K\"
IT
Cv
0
24.943
0.1
24.93
90.70
X 2402
- 666.8
0.2
24.89
73.43
115.6
-251
0.3
24.83 24.75 24.63
63.34
74.2
56.21
53.5
50.70
41.16
24.50
46.22
32.9
27.1
0.4
0.5
0.6
CO
0.7
24.34
0.8
24.16
0.9
23.96
1.0
23.74
42.46 39.22 36.38 33.87
1.5 2 3
22.35
24.49
20.59
18.30
16.53
10.71
4 5
12.55
6 7
9.20 6.23
-87
-60.3
-44.1
-33.5
-26.2 -209
16.82
-17.0:
9.1 5.5 2.36
6.51
1.13
4.08
0.58
2.64
0.323
0.1S7
0.114
2.53
10
1.891
0.643
0.048
15
0.576
0.192
0.0096
9
-137
19.5
1.22 0.874
3.45
170
22.8
1.77
4.76
8
1.2
0.073
-7.2:
-3.6.
-1.2
-0.4!
-0.2
-0.1
-0.0
-0.0
-0.0
-0.0
-0.0
Summary
i
20 0
10
Q
-10
-20
-30 1.5
1.0
0.5
\024\302\2600
0
FiKiire^.12
Energy
solid, according
of tlic
solid
to tiit
free energy
t/and
theory.
DcbyC
f
s=
I/-
Tlic Dcbye
roof
a
temperature
is 0.
SUMMARY
Planck distributionfunction
1. The
for
the thermal
2. The
is
average number of
photons in a cavity
mode
Stefan-Boltzmannlaw is
for the
radiant energy density
V
15/iV
in
a cavity
'
at temperature
t.
of
frequency
a
3.
Planck
The
and Planck
Radiuiwn
Thermal
4:
Chapter
Distribution
law is
radiation
h
\302\2533
r2c3
the radiation
for
energy per
unit
volume
The
low
Debye
is,
in
where
limit of
temperature
conventional
i
unit
per
4. The flux density of radiant energy is Ju Boitzmann constantn1ks*/(i0hici.
5.
\342\200\224
exp(/io)/t)
range
\342\200\224
of frequency. aB is
where
oBTA,
the Stefan-
the heat capacity of a dielectricsolid
uniis,
the Debye
temperature 0 s (hvjkB)Fn2$fvy'\\
PROBLEMS
1.
Number
of
thermal
at temporal
equilibrium
Show
photons.
ure
a cavity
in
x
N
that the number of volume V is
of photons XXs\")
= 2.404n~2K(t/Ac)s.
in
D8)
i-rom B3) the entropy is a = Dn2F/45)(r//icK, believed that the total number of photonsin the
whence
=s
3.602.
It
is
universe is 109 larger than the both are of total numberof titiclcons(protons,neutrons). Because entropies the the order of the respective of number (sec Eq. 3.76), photons particles of the universe, provide the dominantcontributionto the entropy although the particles dominau- ., c total energy. believe that the entropy of the We so that Mie entropy of the universeis approxipllotonsis essentially constant, approximately
7. Surface density
with
constant
at
temperature ofthi! Sun. The value from the Sun normal (o the
the Earth
constant of the wavelengths
and
time,
Earth.
referred
The
observed
value
of
the
incident
integrated
total rays
radiant energy Hilx is called the solar
over all emissionwave-
mean Eanh-Sim disiance is:
to the
solar
constant
=
0.136
J s
D9)
(a) Show thai the total rate (b) From this result and
10\"12J s~' cm'2 K\"\"\\ Sun treated asa black x
theSunasl.5
thai
body
is T
Sun is 4
X
10\026 J S\"
constant
Stefan-Boltzmann
.the
show
10!3 cm
generation of the
of energy
*.
5.67 x
the effective temperature of the surface of the ~ 6000 K. Take the distanceofthe Earth from
and the radius of the Sun as 7
x
10l\302\260cm.
of the Sun, (a) Estimate by a dimenof magnitude of the gravitationalselfof the Sun, with AiQ = 2 x !033g and RQ => 7 x 1010 cm. The gravienergy gravitational G is 6.6 x iQ\"8 dyne cm2g~2,Theself-energy constant be negative will referred to atoms at rest at infinite ihe total thermal Assume ihat (b) separation, kineticenergy of the in the Sun is equal to \342\200\224 atoms limes the gravitational \302\243 theorem of mechanics. Estimate the average energy. This is the resultof the virial temperature of ihe Sun. Take the number of particles as 1 x IG*7.This csUmaie too low a temperature, somewhat because the density of tlie Sun is far gives from \"The range in central temperature for different uniform. stars, excluding of of those for which Saw matier the only composed degenerate perfect gases and does not hold (white those which have excessively sniall average dwarfs) and is between densities(giants 1.5 and 3.0 x iO7 degrees.\" supergiants), B. Lynds, and H. Pillans,Elementary Oxford, 1959.) (O. Siruve, astronomy, 3.
of t/te interior temperature or otherwise (he order
Average
dimensional
argument
4. Age
radiates
of the Sun. Suppose4 at
energy
for
on
radiation,
of hydrogen
x
Find
the present time, (a) the rough assumptions
(atomic weight
the
the
of
he
Sun. It is
(AAf)c\\
temperature
Surface
us much
renijiates ihe
that
1.5
(a) to
books
converted
estimate the
universe
by
that
Peebles
is about
and by
Calculate the iemperatureofl hesurface Earth. it ihat a black body in thermal as equilibrium assumpiion from the Sun. Assume also thermal radiationas it receives of the
over ihe day-night Earth is ;it a constant jemporiittini 5S00K; RQ = 7 x 10locm; and ihe I2arih-Sundislanwof
of the
face 7\"o
ihe
and
4.0026)
-
IOlJcm.
x
6.
sin
Use
cycle.
the
on
Earth,
in
conversion
ihe
has been
(b) Use
available
of the Sun
believed ihat [lie age of the
109 years. (A good discussion is given died in the Wcinberg, generalreferences.)
5.
energy
hydrogen
original
= \302\243
10 x
ofthe
total
L0Q7S)to helium(atomicweight
the
expectancy
i
which the Sun
total rate at
that the energy sourceis
stops when 10percentof to helium. Use Einstein relation
the reaction life
is the
lQ;6Js~!
Pressure
(a)
of thermal
variation.
Show for
p = -(cUfcV),
a photon
gas thai:
- -^s/iiiluij/ilV) ,
E0)
and Planck
l Radiat
where s; is the
in the
of photons
number
(b)
Oiildbnti,
=
dojjfilV
p =
(c)
moJej; -mjyV;
E1)
U/iV.
E2)
to 3 x (energy density). with the kinetic pressure of a (d) Compare pressure radiation of t mote cm'3 characteristic of the Sun. gas of H atomsat a concentration At what the two pressures equal? The average are temperature (roughly) of the Sun is believed to be near2 x tOT K. The concentration is temperature at where the the highly nonuniform and rises to near rOQmoiecm.\023 center, kinetic pressure is considerablyhigher the radiation than pressure. radiation
the
Thus
is equal
pressure
of thermal
the
7. Free
energyof a p/iot
photon gas is given
where
ihe
the parlhion
that
Show
(a)
gas.
function of a
by
is over
product
direclly from
on
the modes \302\273. (b)
free energy
Helmholtz
The
is found
as
E3)
E4)
F-T][tog[t-exp(-AuiA/T)].
Transform
the sum to an
integral; integrateby F =
5. Heatshields.
to a between
black
the
constant
at
plane
two
black
A
planes
is allowedto come and T,, and show presence of this
to that
net
energy where
is inserted
temperature
energy flux density is the principle of
Tu is parallel
at temperature
- T*),
plane
state
a steady the
plane
T(. The net black
third
find
E5)
is Jv = aB{Tf
used in B6). A
to
-n
(nonreflective)
temperature
parls
flux
aB
density
is the
in
vacuum
be-
Stefan-Boltzmann
between the other two and Tm. Find Tm in terms of 7'B is cut in half because of the
the heat shield and is widely used reduce radiant heat transfer. Comment:The result for N independent heat shields floating in temperature between the pianes TM and T, is that the - T,4)/(N + !)is Jy = ciT^ net energy flux density plane.
This
to
9.
L on transmission line of length wave waves satisfy the onc-dimcnsional electromagnetic equation =s c2E/ct2l where E is an electricfield Find the heat v2d2Ef3x1 componentof the on thermal at the w hen in capacity photons line, equilibrium temperature
which
Photon
gas
in one
dimension.
Consider a
r. The emiiiicrulioiiof modes lake ilic soiulions siundmg waves
ihe usual way for zero amplitude ai
in
proceeds
with
as
line.
10.
Hcitt
radiaiion
by
hcai capacity of mailer lo lhal of radiaiionis ~-
of ilic
ralio
ilic
thai
in a
atoms
by
Sntcrgalactic space is believedto be concentration =laionim~\\ The space is al 2.9 K, from the Primitive Fireball. Show
space.
of iniergalaciic
capacity
occupied hydrogen alsooccupied thermal
one dimension: cacli cud of ilic
10'9.
limit. Show lhal in ihe iimil of solidsin high temperature a solid towards die limit of capacity Cy \342\200\224\342\226\240 goes 3A'\302\243B, in conventional units. To obtain higher accuracy when T is only moderately can be expanded as a power seriesin 1/T, of larger than 0, the heat capacity
Heat capacity
//. T
heal
liie
\302\273 0
the form
E6}
Determine the first nonvanishing - 0 and T comparing with temperature
Dcbye
in the
lemperature at which the photon contribution equal to the phononcontributionevaluated 13.
Energy fluctuations
atoms is
in
from
the
in
ihermal
contact
with
3 to
show
at
ai low
region
in which
a heat the
that
IF is given by
inserting
dtelcciricsolid
a
with
the
cm ~3. Estimate
heat
the
to
a solid
in
temperature
Chapter
your result by
sum. Check
4.2.
of photons andpltonotts. Considera 10\" atoms equal to 100K atid with
Heat capacity
/2.
term Table
capacity
would be
1 K.
temperatures. ihe Dcbye
Consider a solid of N
Ti
law
is
vaiid.
The
solid
fluctuations reservoir. Usethe resultson energy root mean square fractional energy fluctuation
E7)
(
T =
that
Suppose
a side;
$F
then
order of uniiy 14.
Heal
longitudinal
p=
0.02.
in liquid
waves
(a)
0.145gcm~3.
transverse
no
are
gram
capacity
per
value
=\342\226\240 0.0204
Calculate
on the x
T\\tn
= 200K;and N
1O1S for
*
the fractional particle of volume
!0\025 K
of liquid
capacity
sound
At
a dielectric
for
s\"'. There
CK
I0~2K;0
the Debyc
waves
0.01cm on is of
energy
ihe
cm3.
*He at low temperatures. 4He at temperatures bctaw sound
in
fluctuation 1
a particle
in the
The 0.6
of longituvelocity K is 2.383 x 104cm
liquid. The
is
density
temperature, (b) Calculate
the
heat
with the experimental and compare K*1. The T3 Jg\021 dependence of the experimental
Dcbye theory
value
thai
suggests
the
Physica
Distribution
most
Hit;
due
32,
625
to f
excitations
important
value
experimental are
experimems
Kramers,
are
phonons
below 0.6 K. Note that
liquid.The H. C
and Pfanek
Radiation
Thermal
4:
Chapter
in liquid
4He
expressed per gram of C. G. Niek-Hakkenberg,and
has been
J. Wiebes,
!957}.
distribution that the spectra! of radiant energy flux, (a) Show of the radiant energy flux that arrives in the solid angle i!Q is 0 is the angle the normal to the unit area makes with fuucos0*
15.
Angular
density
16.
Image An on a
radiant object. Let black object of area Ao. of a
a
lens an
Use
the hole in
image
a
cavity
argument
equilibrium
of
area
to relate the
QQare the solidanglessubtendedby the the hole This general property of object. It is also true when focusingsystemsis easily derivedfrom geometrical optics. is diffraction the that all Make important. approximation rays are nearly
product
to
AuQtl
axial
{al!
parallel
Qtl and
where
AaQ0
from the
as viewed
leas
from
and
small).
angles
17. Entropy and occupancy. We
chapter that the entropy of the time because the number with body the frequency in of each mode has not changed with time, although each mode has decreasedas the wavelength with the expansion has increased of the universe. Establish the implied between and occonnection entropy that for one mode of frequencyw the entropy cupattcy ofthe modes,by showing is a function of the photon occttpancy<<(s) only;
cosmic of photons
=
to start
from
a
is convenient
It
18.
haxiropic
expansion
not changed
l)log<5 + !) the
and
the
Consider
of photon gas.
temperature
a cube
-
ES)
function.
partition
thermal equilibrium radiation in cavity volume increase;the radiation expansion,
in this
argtted
has
radiation
black
the
of volume
pressure
V
performs
of the radiation wit] in such is constant
drop.
gas at
of
temperature
photons
of the r. Let the
work during the expanFrom (he result for the
an expatision. (a) Assume that was the radiation from cosmic black-body decoupled temperature was the radius of of the mutter when both were at 3000K..What temperature to now? If the radtus has increasedlinearly the universe at that time, compared with at wltat fraction of the present time, age of the universe did the decoupling take place?(b) Show that the work done by the photons during the expansion
entropy we know
of lite
the
The
iV1'3
titat
subscripts
i and
/ refer to
the
initial
and
final
siatcs.
19, Reflective heat shieldand Kircbhoff's Consider (aw. material of absorptivity it, e-mtssjvtty e, and rcllecttvjty r
suspended between and parallel with temperaturesru und t,. Show that the net flux
black sheetsis (I
also
often
are
dewars
film called
Mylar
by the
and in clouds,
alumtntzed
an
of
the wanning of the surface of of water, of an infrared absorbentlayer
describes
interposition
and of carbondioxidein
the Earth. The water may a
such
e =
GREENHOUSE EFFECT
the
as
contribute
much
layer,
the
90 percent
as vapor
and
of the warming
of the surface of
temperature
Earth
the
the Sun
between
atmosphere
effect. Absent
between the
intermediate sheet is helium I; r - 0. Liquid
by many, perhaps 100,layers Superinsulation.
Effect
Greenhouse
caused
=
a
with
at tempera-
when tiie
density
means
8, which
sheet of matethe sheet be
maintained radiation thermal
of
density
flux
the
times
\342\200\224 a. Let
insulated
SUPPLEMENT:
The
r)
in Probfem
as
black
-
a plane \\
sfieets
black
two
~
is
Earth
the
flux the by the requirement of energybalancebetween incident on the Earth and the flux of reradiation from the to the fourth power of the temperaEarth; the reradiationflux is proportional of Problem temperatureof the Earth, as in D.26). This energy balance is the subject the where 4.5 and leads to the result Ts \342\200\224 temperature 7'\302\243is {RsI^seV^Ts, of the Sun and DSE of the Earth and Tsis that of the Sun; here/fjis the radius
determined of solar
primarily
radiation
is the Sun-Earthdistance.
The Sun is
that
of
result
\342\200\224
much hotter than the Earth, but
subtended by
the
Sun)
of roughly
factor
is TE
problem
the solar
reduces
280
the
K, assuming geometry
T,=\302\273
(the
FC The
5800
smali
solid angle
iiux density incidentat ihe E;irth
by
(i/20)*.
a
that the atmosphere is a perfectgreenhouse, radiation that that transmits al! of the visible layer falls oa it from the Sun, but absorbs and re-emits a!! the radiation (which lies idealize the problem in the from the surface of the Earth. We may infrared), of the infrared layer portion of the by neglecting the absorption by the the solar lies almost incident solarradiation,because entirely at spectrum from 4.4. The layer will emit enerry flux Figure higher frequencies,as evident Oux will balance and the the suiur i!ux 1$, Su UiJi tL up IL down; upward ** ft Is- The net downward flux will be the sum of the solarflux Is and the incident The latter increases the net thermal Oux flux lL down from the layer. at the surface of the Earth. Thus We
defined
as
assume
as
an
an
example
absorbent
lEt^h + h-ns. where
is
l\302\243g
the
thermal
Oux from
the Earth in the
E9)
presenceof
the
perfect
effect.
greenhouse
=
the
that
greeahouse
flux
as
varies
T4, the
new temperature
is
Earth
T\302\243s
so
the thennal
Because
of the surfaceof the
\302\253= 2\302\273/\302\253rfi A.19)
warming
of the
280
K ~
333 K,
Earth is 333 K \342\200\224 280
F0) K =
53 K
for
this extreme example.*
\342\200\242
For
end
detailed discussions i. T. Houghton
1992:
see Climale change and
ct aJ, editors.
Climate
change
.
1992, Cambridge
U.P., 1990
5
Chapter
and
Potential
Chemical
Gibbs Distribution
CHEMICAL
OF
DEFINITION
Example: Chemical Potential of the Idea! Internal and Total Chemical Potential Example:
Barometric
of
Variation
119
POTENTIAL
122 with Altitude
Pressure
Magnetic Example: ChemicalPotentialof Mobile \"'\"
in a
Field
Magnetic
Example;
Batteries
Chemical
Potential
120
Gas
Panicles 127
'
\342\226\240
129
131
and Entropy
Thermodymtmic Identity
Numberof
133
134
GIBBS SUM
AND
FACTOR
GIBBS
125
139
Particles
140
Energy
Example:
Occupancy
Example:
Impurity
140
Zero or One !omz;ition
Atom
in a
Semiconductor
143
SUMMARY
144
PROBLEMS
145
145
1. Centrifuge
2. Moleculesin the Earth's Atmosphere 3. Potential Energy of Gas in a Gravitational 4. Active Transport
5.
Magnetic
6.
Gibbs
7.
States
8.
Carbon
Monoxide
146
146 146
Poisoning
of O2 in a MagneticField
147
147
Fluctuations
Concentration
Ascent
145
Sum for a Two LevelSysSem of Positive and Negative Ionization
11. Equivalent 12.
145
Concentration
9. Adsorption 10.
145 145
Field
Definition of ChemicalPotential
of Sap in
148
148
Trees
13. Isentroptc
148
Expansion
14.
Multiple
Binding
148
of O2
15. ExternalChemical
Potential
....
149
\342\226\240al Potential
(H. We found earlier thai reservoir <2t, ihe Helmholtz
and
Gibbs
Tor
a
Distribution
free
&
will
assume
x
and
wiiii
number equilibrium
with
a minimum,
subject free
energy
h\\
to N
+
=
of the
F2
jV,
- t/( + t/2 +
jV2
the minimum
value
restraints
on the applies pp
oiher
-
N,,.V\302\273
T(ff,
This
+
in
result
diiTusive
equilib-
ihi5 sysicms
between
A)
)
Because N is constant,She
= constant.
combined system is
(R.
a
with
equilibrium
of pparticles.
^2, the
F=
Heimholtz
thermal
in
paiii^lc diaiiibution S, t!io toial Hchulioltz free energy .md
between
makes
of
energy
compatible wiih the commontemperature such as the volume and the system, y, equally to ihc combined\302\243,+ S: in equilibrium
S
system
single
a
minimum
with
respect
to
of
Definition
System
Chemical
Potential
.Sj
t*\"Ener\302\260y
exchange\"\"}
conlacl with each oihcr Example of Iwo systems,Sx ;mdX., in ihcrmal and wiih a large reservoir Of, forming a closed total system- I)y opening the take, contact wliilc remaining at the common S{ and ^j can be brouglit in tlillusive transfer for a net panicle tcmpcraiurc r. Ttie arrows at tlm valve liave been drawn lo S2. from \302\243j Figure
5.1
variations
= -5N2.
5Nt
tlF with
K2, also
K[,
Hutt
at
the
minimum,
- (cFi/cN^^lNi
held constant. (/F
so
At
=
With
= 0, + lcF2/dN2)sc!N2 (!Nt
=
~-tlN2,
WG
- (SF2/cNl),yNl
[(cfj/t'iV,),
B)
have
- 0,
C)
cqutlibriuni
- (cF3/cN2)t.
((Tj/oV,),
(\342\226\240\30
DEFINITION OF CHEMICALPOTENTIAL We
define
the
chemical
potential I
as (/;0 I
E)
Chapter 5:
ChemicalPotential
where ji is
Distribution
Gibhs
and
ihe Greek letter niu.
Then
'-
/'I
for ditTusive when diVt 5j, ihe value of dN \\
ihe condition expresses elF will be negative thai from
to
5,
free
energy
is
we sec from C) are transferred panicles tlN2 is positive. Thus the
If/(,
equilibrium. is
negative:
When
negative,
and
> ;i2,
^i to &2; ilitit
flow from
us particles
decreases
t'l
the system of high chemicalpotentialto the system potential. The strict definition of /i is in terms of a difference are not divisible; derivative, because particles
particles
of
low
chemical
and
not
a deriva-
from
)
=
F(t,V,N)
- F{x.V,N-
(low
's.
F)
1).
between systems in which the contact, rcguhucs important fully transfer. are Two ihat can exchange both energy and particles energy systems in combined thermal and ditTusive equilibrium when their temperaturesand chemical are eqtial: i[ = t2;/i; = ji%. potentials of A difference in chemical potential acts as a driving forcefor the transfer force for the transfer particles just as a differencein temperatureacisas a driving of energy. has its own chemical potential. If several chemicalspeciesare present, each The
chemical
and
regulates
potential
it is
as
the particle transfer as the temperature,
For species j, G)
the
wherein
for the
Example: of
Ihe
differentiation!
TTumbcrs
of
all
particles
are held
constant except
species j.
Chemical
monatomic
potential
of the
Ideal gas. In
C.70}
showed
that
ihe
free
ideal gas is
/logZi
-
logN!] ,
(8)
of
Definition
is the
for a single
function
partition
Chemical
Potential
panicle. From (8),
A0)
If
which
approximation for
the Stirling
use
we
factorial,we
tlutt v,e
assume
and
A\"!
can differentiate the
fold
approaches
log
iV
=
log
for
large
A\"
+
+
[N
llcttcc the
of \\.
values
- 1 = togtf
-1)--
chemical
+
\342\200\224
.
potential
(II)
of the
ideal
gas is =
,<
or,
by
- log.V)
-rUogZ,
(9),
A2a)
where
n
\342\200\224
NjV
is the
defined
concentration
F{N
-
From
approximution.
system
volume
(S) we obtain By use
separately.
The chemicalpotential
composedof
electrons
of
=
ths
definition
=
ji
concentrationof
/t
we expect intuitively: lower concentration.
the
i)froin.{6)as
A2). The result depends on the the
na
=
(A/r/2nfi2K
2 ts
the
quantum
by C.63).
Ifweuseji = F{N) -
Stirling
of particles ami
concentration
-
-i[log2, particles,
ideal
Ttog(p/tiiti).
gas
we do
af/j,
not
logjV],
not need to which
on their total
law p
=
nr
we
can
agtces
use the with
numheror on write A2} as A2b)
This is what concent mi ion of particles increases. chemical from to to lower higher particles higher potential, on of an the concentration ideal gas dependence Figure 5.2 shows the boiling temperature or of helium atoms, for two temperatures,
increases
as the
flow from
Chapter 5: Chemical
Potential
The concentration dependenceof/*,
Figure 5.2
composedof dectroio.jrhelium regime
with n
\302\253
nQ,
a ^as
at 4.2
atoms,
must have a value
of -
of
least
at
^i
concentrations apprccwhly le\302\253lluin semiconductors. For gasesit is always lypical
of an ideal gas K..To be in the classical
o! r,
units
in
K and 300
oulv for
satisfied range
Distribution
andGibbs
t. For
electrons this as in ihe normal
is
in metals, satisfied under
those
conditions.
of
and
liquid
pressure, 4.2 K, lave ncgaiivcchemical
ill atmospheric
licimin
tnolccubr
jtascs aiwjjs
condiiioas: at classical concernrations
such
ihnt
and
room
unJer
potentials
nfnQ
300 K. Atomic tciilii^blc plijiically soe from A2) iliat /i is
icmpcraturc,
\302\253 !,
we
ncgaiive.
Internal
and Total
The best equilibrium
way
in
Chemical Potential
to
understand
Ihe
presence
the
of a
chemical
potential
potential step that
acts
is to discuss diffusive on
Ihe
particles.
This
Internal and
Potential
Chemical
Total
Figure 5.3
A
tiie voltage polarity shown, of positive parlic'es energy in
be
the
potential charge
!o J2. The potential energy particles would be loweredin 3, respect
has wide
problem
13.
in Chapter
discussed same
application and includesthe We
\342\200\224 n
Sx und
systems,
junction al
&->t
the
and capable of exchanging particles,but not yel in diffusive We assume that initially > /(j, and we denote the iniiial nonfix
equilibriumchemical
difTcrence
potential
the potential
iliat
above
Af[(im'tial)
way to
by
- ^i-
= ;/2
A/j(iimial)
Now
a
let
such energy be establishedbetweenthe two systems, of each in is raised energy by exactly panicle sysiem Si its ininal value, if Ihe particles carry one a charge qt simple
in potential
difference
I'
p
temperature
equilibrium.
A
hvo
consider
again
semiconductor
establish this potentialstepisto apply
the
between
two
a voltage
systems
lhat
such
A3)
the
with
shown
polarity
also can
in Figure
serve asa potentialdifference:
of mass
M
by
the gravitational
the
height
/i, we
by
sfcp
in
difference
A
when
we
raise
potential
gravitational
a system
of particles each
establish a potential differenceMgh,
where
g is
acceleration.
Once a potentialstepispresent, this
5.3.
is included
in (he
the
potential
energy
of ihe
patrides produced
energy U and in the free energy F of the
keep the free
system.
step raises the energy system *= free of Sx by /Vs A;i(mitial) energy AV/ A!7 relative to its initial value. In to the ihe language of energy states, energy of each suite of .Si the potential has been added. The \"insertion of the b^rier potential energy ,Yt A/i(initial) ilie to male the chemical of Hveificd by (B) mises potential $x by A/i(!il!ti;il). of ^, c\302\253i\302\273:il to ih:lt fin::!i.-hciiiiL:;il oi\\S,: puk-ntuil If in
Mgurc
5.3 we
/affinal)
= ;(,(initial) -f
of
S2 fiNed, ihe
[,..(initial) -
established
with
i, would be raised by
system
two
between
step
potential
systems of chargedpaniclescan
/i,(inilial)] A4)
qAY
q > wild
0
of negative with
cespect
was
barrier
the
When
/^(initial)
chemical
difference
equilibrium.
is equivalent to a true potential energy; the between two systems is equal to the potential will bring the two systemsinto diffusive equilibrium.
that
barrier
potential
gA
potential chemical
in
barrier
the
the two
brings
/(i(initial)
The
pj was held fixed.Thus systems into diffusive
inserted,
\342\200\224
a feeling for the physical effect of the chemical potenfor the measurement of chemicalpotentialdifferences two between To measure ft2 - /<,, we establish a potential step systems. that can transfer betweentwo systems particles, and we determine the step at which the net particle transfer vanishes. height The absolute differences of chemical Only potential have a physical meaning. value of the chemical potential dependson the zero of the potential energy scale. The idealgas result A2) depends on the choice of the zero of energyof a free as equal to the zero of the kinetic energy. particle When total chemical external potential steps arc present, we can expressthe gives us the basis
statement
This
it forms
potential, and
of a
potential
system as the sum of two = iv
/'
Here
/jtM
is the
internal
be present electrical,
Pi
the
is
if
chemical potential* external
the
magnetic,
\342\200\224ca'i
Hi
=
definedas the were zero.
potential
A5)
+ft,,;\342\226\240\342\200\236,
per particle in the
energy
potential
parts:
external potential,and
chemical
The term /iCM
gravitational, etc. in origin. The
be expressed
total
the potential \342\226\240
Gibbs
measures
called
differences
further
chemical
in
fi.
potential
mechanical,
-A
A6) chemica!
potential
those of
the
working
semiconductors,
chemical
words
qualifier.
potential
may be
called the
electrochemical
barriers of interest are electrostatic. Although n the
be
equilibriumcondition
and internal external Unfortunately, the distinctionbetween sometimes is not made in the literature.Somewriters, particularly and with charged particles in the fieldsof electrochemistry when use often mean the internal chemical potential they
The
nmy
jiial
would
as Apeil =
potential without a
that
potential
and .
ihe intrinsic it,*, '
if
potential the
that potcnsjal. He recognized
term
electro-
a voltmeter
mea-
Chemical Potential
and Total
Internal
SysicmB)
5.4
Figure
A model
of ihe variation
aimcispliericpressure\\>.iih
altitude:
of gas at different gravitaiiona! field, jn iherma! \\olumcs
heights
and
con wet.
System (!)
chemicalpotential is clear unambiguous, The use of \"chemical potential.\"
we
and
without
potential\"'
avoided
tit
situations
in
which
shall
any confusion about
an
tisc \"total adjective
chemical should
be
tis meaningcouldoccur.
The of the simplest example pressure whhahhutte. external is itte equilibrium sysiems in difTcrcm potentials bciwccn lo be isothermal. assumed layers ;lt different llcigliis of ihe Canh'satmosphere, Tim rcul uimosplicfc is in imperfect equilibrium: ii is'cunstaiiily upsa by Hitititoroloeicut temperature processes, faoih in the form of macroscopic air movemems and of strong We may from clouii fonn;uion, ;inii becauso of heal input from liiil ground. grudiunis make ;in ;ippro.\\imaic the dilL-iem air layers as model of the aimosphcrc by KtMliiia ^iih each oilier, in diHerem syslemsof idea! gases in ilierilia! and diftusive eiiuilihrium
of barometric
Example;
Variation
diffusive
equilibrium
exieroul
\\c\\c\\, (Figure 5.4}.If we place ihe zero of the poieinul energy ai ground is ihe particle massand g ihe energy per moleculeai heighi Ji is Afyfi,whcrc.\\f acceleialion. The internal chemical potential of ihe parlidesis given by A2}.
poicnlials
the poiemial gravitational
The loial
bciwccn
diemical poiemial is
{17} In
equilibrium,
this
musi be
independent of {he
Tlog[i.(/t)/H0] and
ihc
conceniralion
\302\253(/t).alhcijjlU
+
A/y/i
hcidil.
-
Thus
rlog[ii@)/nQ]
,
h satisfies
(IS)
i
in
di!
Chapter 5.-Chemical
and
Potential
Dhtnliution
Gibls
1.0
1
0.5 \\
0.2 \\
o.i
V
i 5.5
figure v-iih
atmosphere
The
of atmospheric The crosses represent
Decrease
ahiiude.
concerting
as
on rocket
sampled
corresponding lo a
pressure Ihc average
0.0S
fliyhis.
a slope
lias
Vine
siraight
2
icmpcraiure T =
K.
227
h
X
\\
\342\200\224
20
0-10
Heighi,
The pressure
aullilude
li
of an ideal gas is proportional
lo
llieconcenlration;
M
40
30 in
50
km
Ihe pressure
therefore
is
p(/i)
=.
= p@)exp(-/i/'iJ-
p@)exp(-.V9li/t)
A9)
conation Ti i*i\\ os ^Jtc ilc^czitjciicc 01 liic pressiifcon uli^iLide of a single chemical Al ilie cliaracierisiic Mght hc = anuoipliere species. ihe e~l =3 0.37. Hie decreases ihe fraciion To esiimaie t/A/g .-.unospi^er-c pressure fay characteristic fietght, consider an isoihermal nioiccuies atTuosphere composed of nitrogen wiLliiinioiecular is 48 x 10\" \"gill. At a teiliper.lweigh! of 28.Tliema^orail .V^ niolecuie iureof :90KiIic value of r = kBT is 4.0 x 10\" l4crg. \\W\\nj = 9K0L1HS\021. the d1.1r.1tand will icrisiic iieiijlii 5 mites. He, Li^Uiermolecules, Jit is S.5tm, H, approximately &s
Thl^
in an
itio
fatthcr dp,
t\\ictul
11UC.U1-.C
pressure
ll.o
UanhS
iiut
these
Ivavc
LiiiiiO^pticrc
Fiyuie 5.5 is :i loyatiifunic on rocket flights. The data
behavior. iaken
uuromoLrtc
isoihermat
hrgeiy i>
not
;n;tiir.ittly
jjloi potnis
cscipctl from ilic oi\"
f;i!i
pressure
near
iiiiiinspUufc:
see
iwitlicniiiit, ni/i)ft;is ;i iitoic tfatj bciv^un 10 and a siraighi
fine, suggesting
Probicm
1.
a'ttipliaital
4QKiloiMcttrs, roughfy
iso-
vtl and
Tlie straight = 1000:1,
behavior.
thermal range
jj{/ij):pl/i,J
line connecting the data points over an altitude range from ft,
total Chemical Pot
of Figure 5.5 spans a pressure 2km io A, = 43km. New,
=
from A9),
so that of
the
of the
slope
at temperature
which leads to
is Mg/x,
line
wish !hc point k ~
eurve
observed
tltc
T ~
0, p(li),'p{0)
\302\253 227
x,'kB
\302\253
I, is
caused
K. The
non-intersection
by the higher
tempera-
aliiiudes.
lower
one species of gas. In atomic the compercent, A'j. 21 pet Oz. and 0.9 pet Ar; oilier constiiuems account for of tlie aimosphcre may be content pci each. The water vapor at T ~ 300K B7\"C}_ to 15 a relative 100 of appreciable: corresponds pet humidity pet The carbon dioxide concentration varies about a nominal value of 0.03 pet. In an H,O. be in equilibrium wftli iisclf. The conideal static Uolhenrtal atmosphere eadi gas would of each would fall off with a separaie Boltzmann factorof the form concentration cxp( -\\/y/i/t},
Tlic atmosphere
M the
with
consistsof
dry air at ica iess than 0.1
of
composition
appropriate
p^insiTrii^itic
T *}\\
tiff
than
more is
fevel
TS pet
molecular mass. Becauseof
tlie
JiiTerences
in mass, ihe
difTcrcnt
nk*c
u\\ /iifrprcnl
magnetic field. Consider m. For simplicity suppose f or antiparalld [ io an applied magnetic field B. is Then the poiential energy of a f particle is - iufl, and the potential energy of a 1 particle We may treat the particles as belonging to the two distinct chemical specieslabelled 4-ii]\302\243i. t and i, one vhh external chemical potential ;in,(|J = -mB and the other with jjm,U) = \"iB. of an clement or as two The particles 1 are as distinguishable as Uvo difTerent isotopes f and we speak of f and 1 as distinct species in equilibrium with each other. different elements; with concentrations of the particles viewed as ideal chemical The internal gases potentials Example: Chemical potential of mobile a system of .V identical particles each each moment is directed cither parallel
n, and
iij
If
particles
a magnetic
in a
moment
are
ill
iltcrti:iM\302\273\302\253:lK-|k-lJ
it. mtht
magnetic
with
vary
out
are
potentials
!t\\:nivs
iintwi'iiittKlfovcrtbtf
\\\\w \\ wlumc
in unlcr
out
i|ic volume irigmo
5.6).(Tlie
dent
of posiiion,
if there
is free
total diffusion
to
ut:tintaii)
potential of a species is eoitstam iiiJc within the volume,) Becausethe particles
chemical of
vo!ui\302\273c^iltcsyslciH.llici-.-:-.vt.ij.H Mi ,i unal clicnnt.it i- \342\226\240\342\226\240
j cotiMjisl
Potential and
5: Chemical
Chapter
Dhtribut
G'thhs
---
\342\200\224\342\200\224\"
Iff'
ion (i, 5.6
Figure pari
in
1GIU
of a gas of magnetic ofihe chemical potential values of the magnetic field concentration, at several = 2 x iO7 cm\021 for I! ~ 0, ihcn ai a point \302\253here B \342\226\240= 20
Dependence
on ihe
ides
Ifn
intensity.
x
kilofiauasl2iesb)ihccona:nlraiionwillbe2
species in
have
equilibrium
equal
desired
\302\253,{B)
where
ti@}
is tiie
concentrationat n{B)
n(B)
The result
magnetic magnetic colloidal
Held
such
as
-
= consiant B3) are easily
total concentration a poini at magnetic
\302\253
n,{B)
=
+
n^B) = ii[ 4field
intensity-
The
at
nt
a point
of
to be: ,
where
the
field
B =
B4)
0. The total
S is
* n{0)M
form of ihc
but is applicable to
fine
+
exp{-\302\273)B/T)];
+
4-
^-
panicles
magnetic
fujictionai
B3)
^\302\253@)exp{-)nB/i)
= Jn@)[exp{jHB/T)
ji,(B)
n{0)cosh{mB/T)
shows [he tendency
;i,01(|). fay substitution
seen
in{0)exp(mB/t);
orientauons, solution. Such
\342\200\242 \342\200\242
-Y
to concentrate
result
fecromagncfic
is not
limited particles
B5)
in
regions
to atoms
of high wi[h
in suspension
two in
3
in the siudy suspensions are used the laboratory of superconductorsand fhe domain strucfure of ferromagnetic materials.In ace used to tcsf for fine structural cracks in high strength the suspensions sfcel, When fhese arc coated a furbine blades and aircraft with gear. landing ferromagnetic
flux strucfure engineering,
of {22)and
solutions
!09cm\023.
diemioil poientiais,
lUl) The
\342\200\242
ciir3
particles
in
of
f
he magneiic
Internal
and
suspension
fields
intense
ttie
field,
placed in a magnetic at the edges of the
flic
and Total ChemicalPotential
becomes
concentration
particle
enhanced at
crack.
discussion we added to /<\342\200\236, the internal chemical potential of ihe were ideal would be particles gas atoms, /iml particles. given by A2). Tlie logarithmic for /iinl is not restricted to id^al gases,but form of the conditions is a consequence iliat the do not interact and their concentration low. that ii sufficiently Hence, particles A2) applies to macroscopic particles as well as to atoms that satisfy these assumptions. The only is the s;tlne of the quantum difference concentration n,,. We can thereforewrite the preceding
In
the
If
=
/'iM
where the constant
t log
\342\200\224 does t log \302\273,,) (\342\226\240=
Oneof
Example; Batteries.
it
not
-f
constanl
,
on
depend
B6} of the
concentration
the
puctielcs.
vivid examples of chemicalpotentials and potential hi the familial lead-acidbattery the negative electrode consists of metallic lead, Pb, and the positive electrode is a layer of reddish-brownlead ari: ininwisiid in ditulod on a Pb substrate. The eltclroiics sutfurie oxide. acid, PbOj, which ions is into H* ions and ionUai H.SCU, SG4~~ (Figure 5.7}. (protons) partially steps
in
of
electrochemical
the
w
the
both
process
discharge
the positive
most
the
battery,
electrode arc
Pb +
Because of B7a)the negative chemical potential ji(S04\"\"\") inside
than
two
the
the
PbO.
reactions;'
SO4\"\" -+ PbSO.,+
2e~;
B7a)
electrical currents If the battery electrode
the electrolyte
(sec Figure
5.7b).
B7b)
2H2O.
\"
acts as a sink the sulfafe ions at
electrode of
+
PbSO4
for
SO*~
the
surface
ions,
the internal
keeping
of the negative because of B7b) the
Similarly,
electrode positive
chemical ions, keeping the internal potential /((H1\") of 'tie ions lower at the surface of ihe positive electrodethan inside the electrolyte. The the tons towards the electrodes, and they drive the potential gradients drive
acts as
hydrogen chemical
and
exactly the
a
sink
for
the
during
potential
correct
H+
in
not
the
net
electrode,
negative
are depleted
from
thereby charging both.
the
positive
As
a result,
the interfaces, electrode-electrolyte steps of to stop to equalize the chemical potential steps and the If an external the chemical reactions from proceeding further.
steps develop at
magnitude
given are
electrons
connected,
diffusion of ions, which stops current is permitted to flow, the is of because electrolyte negligible, reactions
process.
discharge
terminals arc
accumulate
electrochennca!
The
PbSO4, via
+ HjSO- + 2e\"-+
+ 2H+
PbO2
\342\200\242
electrode and
of the negative sulfate,
electrode:
Positive
electrode
Ie;td
to
electrode:
Negative
lower
Pb
metallic
tlie
converted
reactions.
resume.
reactions a negligible
The
Electron fiow directly concentration in
electron
actual reaction steps
through the
electrolyte.
are more complicated.
the
clcclrodc (\342\226\240f)
PbO,
i
l>b
I
T
T (b).
5.7
5-igurc
parlialiy two
H
(ajThekjJ-aciJ ioni/cd
T
ions
U^SOj.
plus one
PbSO4 + 2HjO,Lii before the
and
H*
and
ihe chemical
barrier.
bailcry coniists of a Pb anj a PbO2 One SOI\" ion coii\\crfs one Pb atom
im-ionizcd HjSO^ molecule
isiirning
Iwo
ciccirons.
development ofimcfiia!
one
convert
(b) The poltnltai
reaction, (c) The eicctroiuticpoieuliai
cicctrotic into
PbSOt
PbOj
ilnmclsed
moltrcul^ imo
eiectrociiemicaipokntials for barriers y(.\\)
thai after
in
+ 2e~;
slop the dilTuston the formation
\"
SOI
of the
and
Potential
Chemical
Entropy
take place, because now an process the reactions opposile lo B7a,b) Eliai generates elect rostalk [wicin b I steps ;H ihc surfiigc t'f Use ei^mMc of such mugnituilti us lo reverse ihc s!ytl of Hie (toUi!)ciieiiik;ii poiciiliiit gradients, Iwik-u the diction ami of ion How, U'c denote of the negative and by A K_ and A f. the difTcrenccsin dectcostaitc polcmial
During
I he
charging
cMornai voiUigc
is ;i}>[i!ied
posilncclcclrodcsrelative
to
negative charges,diffusion
will
=
2gAK_
of
H+ ions wi!i
the
stop
two polemics forces);
(electromotive
ions
curry
two
BSaJ
A/i(SO4\"\.")
\302\253
B8b)
A/i(//').
are called
Al'+
and
AK_
suifute
when
+ ^AK+
The
the
when
slop
\342\200\224
DifTusion
ciectroiyie. Because
ihc common
ha!f-cc!i potentials or
half-eeil
EMF's
their magnitudes are known: AV_
=
Aft
-0.4voll;
electroslalic polcnliai difference as required to stop the diffusion reaclion.
Tiic lolai
AK =
AK+
developed
\302\273
+1.6volt.
across
one
full
cell
of
ihc bniicry,
is -
AK_
\302\253 2.0voli.
B9)
the nccaIl drives ihc electrons from opcn-drcuitioitage of EMFof the battery. lo the positiveicnr.ma!,when ihc iwo are connected. in tha electroljle. We have ignored free electrons The polenlial steps tend to drive electrons from the negative elecirodesinto the electrolyte, and from the cleciroiyie into ihe is so small as to be the Such an current is but electrode. electron magnitude present, posilive is many the the concentration elcclrons in because of electrolyte practically negligible, is ions. The only effective electron (low path less than lhal of the orders of magnitude ihe electrodes. the external connection between through
This is ihc
live lenninal
Chemical
Potential
and Entropy
of we defined the chemicalpotentialas a dertvathe an alternate relation, needed later: energy. Here we detivc
In E)
the
Hclmlioltz
free
C0) This expressesihe raliopi/x 1/r
was
defined
in Chapier
as
2.
a derivative
of ihe
entropy, similar to tlte way
Clmpler5; Chemical To derive
U, V,
C0), considerthe entropy
N.
and
Cibbs Distribution
and
Potential
a [unction
as
of the
independent variables
differential
The
+
cU
^
C1)
differential change ofthe entropy Tor arbitrary, independent = and rfJV. Let dV 0 consideration. for the under
After
the differential
division
by {5N)r,
C2)
The ralio {Sa)J(SN)lis (fo/5N),,and
volunie. With
the
definition
original
is (dU/dN),,
D[/),/(JW),
of 1/t,
all at
conslant
we have
C3)
dNj,, This expresses derivative
at
a
By
the
original
of the
definition
\023 and on
comparison with
oTderivatives at constantt
U in terms
constant
chemical potential, C4)
~
[H,
w., we
C3)
obtain
C5)
The two expressionsE)
same
quantity
ft. The
and
difference
C5)
represent
between
two
ways to express the the following.In E), F is a
different
them is
Table 5.1
of liic cm
partial derivatives
a, U, and F
F, witii
energy
relations
of
Summary
expressing cr, liic
ropy
as
given
am! Entropy
Potential
Chemical
ti-.c temperature
free
U, and liic
energy
of their
functions
itatur.il
variables
independent
o(U.V,N)
, is independent
(IU\\
variable
\\ia)rtl
y,,
function of
a
of
function
C5)
its natural independent variables the
variables.
same
yields n as a function of
and C5), but expressedin
is to
a third
find
we
In C1)
U,
terms
x,
N.
V,
The
assumed a =
quantity
fi
is
that n appears as that
so
a(U,V,N),
the
in both
same
The object
variables.
of different
N, so
V, and
E)
of Problem1i
for p:
relation
C6)
and in
derive
10 we
Chapter
compilesexpressions
Tor
have their
i,
a relation for p asa function p, and ^ as derivatives
of
t, p,
and N.
of o, U, and
F.
Table 5.1 forms
All
uses. identity.
Tlicvmodynamic dynamic
identity
particles
is allowed
given
We
can
in C.34a)
lo change. As
generalize
lo include in
the statement
of the Ihermo-
systems in which the
number
of
C1),
C7)
By use
C0)
for
of the definition B.26)of i/r, the fi/i, we write da as .
da =
dU/x
+
relation
pdV/x
C.32)
- pdN/x.
for p/r,
and the
relation
C8)
This
Potential and Gibbs
Chemical
5:
Chapter
lo give
be rearranged
may
Distribution
-
dV = xJs
statement of the
broader
is a
which
to developin
,
;u|jv
C9)
I
we were
than
identity
ihermodynamic
able
3.
Chapter
GIBBS SUM
AND
FACTOR
GIBBS
+
pdV
The Boltzmannfactor,
in
derived
3, allows
Chapter
us lo
probability that a system will be in a state of energy Et (o the ii\\ thermal system will he in a state of energy t2, for a system
of the
ratio
the
give
the
probability
a
with
contact
reservoir at temperature t:
P(z2)
is
This
contact
of the BoHxmann of that
much
argument retraces
a very
We consider
presented
large body
number N0.The
is
body
&, in thermal and diffusive
the system
particles and energy.Thecontact .
potentialof
the
system
iV
the
reservoir
particles,
are
has Nu C/o
Chapter
that
to those
energy
(Figure
e.
Uo and
diffusive
constant particle
obtain
reservoir
large
may exchange and the chemical
5-8), They
the temperature
oftiie reservoir.When
- M particles; when
\342\200\224 To
factor
Gibbs and
thermal
3.
parts, the very
contact assures
equal
the reservoir has energy
in
of two
composed
system
in
j and chemical potential p. The arguconstant
with
The
mechanics.
lo a
factor
at temperature
reservoir
a
with
known resuh of statistical
the best
perhajtls
is thegenecauzauon
ihe
the
system
the statistical
system
lias
energy
has
e,
properties of the
+ on identical copies of the system accessible slate of the combination. What reservoir, copy quantum will be round to conis the in a that ihc observation given system probability N particles contain and to be in a stale s of energy \302\243j? number of panicles. stale s is a state of a system The some having specified The energye](iV) is the energy of the state s of the A'-particIc sometimes system; When can we wriie ihe energy of a we write only e,, if ihe meaning is clear. A' particles ihe energy of one pur tide in an orbital as -V times system having
system,
make
we
observations
one
in ihat
orbital'.'
the the
particles
Tor
Only may
as before
each
so interactions between Ihe particles are neglected, be treated as independentofeachother.
when
iit thermal and diffusive conlacl wiill of energy anti of panicles. Tile total system S is tnsulaicd from the external world, so thai
A system
Figure
5.8
a large
reservoir
is in and the probability the system has particles of accessible to number a particular stale s. This probabilityis proportional is exactly stales of the reservoir when the stateof specified. That sysiem
Let P[NtEs}denote
jY
that
the
the
of \302\243, the state the number is, ir we specify the number of accessibleslatesofO?:
\302\273((H+
The factor ! reminds stale,
The
y(
us
stales
the Uo - e,. Because
stalesof of accessible
of
that
the
system the
S)
=
gift)
+
^
P{N,i:J
in
is proportional
reservoir.
NtU0
-
is just
{41)
looking at ilic sysiem S reservoir have No - N particles
-
(H
x 1.
we are
probability
of
states
oraccessible
zt).
a single
and
spiciixd
Ikivc
energy
to Ihe number
Potential and Gibbs Diitrihufit
5: Chemical
Chapter
fit Particles
i\\'o
Energy
UQ
-
Panicles
,V, \342\200\224
Ft
Enemy
-
Uo
Energy
Panicles
-
,VU
Particles
A\\
Energy ,
r,
(b)
(a)
dirTusive
and
to
it. In
stales number
{b] the
system
is in
quantum state
2, and
with the
contact
the
system. In (a) tile
has g{N0
reservoir
- A'j,
of
Uo
to it. Because we have specifiedthe exact state of the system, the ofsiulcs accessible tool states accessible 10 01 + i k just the number
accessible
Here g refers to the rcsetvoiraloneand in and on the energy of the reservoir the We can express D2) as a ratio of two
state 1 and the
lhat
other
the
number
one
that
\302\243;
of particles
reservoir. probabilities,
is in state
the system
on
depends
-
total
the system
is in
2:
g(N0 --JV,,[/0 -
e,)
D3)
j
where
g refers
to the
By definition of the
state of the reservoir. The
in Figure
5.9.
entropy
=
s(N0,y0)
so that the
is shown
situation
probability ratio hi
D3)
,
ejpOfNo.f,,)]
may
be written
D4)
as
D5)
>ffel)
=
\" eXp[\"(N\302\253
N\"y\302\260
D6)
Act is the
Here,
eniropy difference:
Ac h a(N0 ~ The reservoir is very
-
NUUO
approximated
and
\302\243 that
-
N2,U0
D7)
e2).
system, and Ac may be tn a series expansion in
lerms
order
firsl
to the
relate
-
with the
in comparison
large
quite accurately by the
the [wo quantities N
- a[N0
e,)
system. The
entropy of the reservoir
becomes
-
N,U0
For A
defined
~i)
by D7)
D8) we have,
to the
first
in
order
iVE
\342\200\224
N2
in e,
und
\342\200\224
c->t
E0a)
original defmilion of the temperature. This is written the system will have the same temperature. Also,
by our bin
K
for
the
reservoir,
(?e'
E0b)
by C0).
The entropy difference
D9)
is
(N,
Here
Aa
refers
to the
- Nt)n
(t, -
reservoir, but Nit A'^, \302\243lt t]
result of statistical mechanicsis found
on
refer
combining
E1) system. The and E1):
to the D6)
central
E2)
5;
Chapter
and Gibbs Distribution
Potential
Chemical
ratio of two exponential factors, c;ichof the form A term of this is The Gibbs factor form calleda Gibbsfuctor. cxp[(A'/j e)/t], is proportional to the probability is in a state 5 of energy e, that the system and number of particles YV. N. The result was first who J. Gibbs, by given referred to it as the grand canonical distribution.
The
the
is
probability
-
The sum of Gibbs factors,taken
of particles,is normalizing absoluteprobabilities:
that
facior
the
of the
states
overall
converts
. This is called the Gibbs sum, or the grand sum, The sum is to be carried out over all states
particles:
the
of
-
exp[(A'/i
or the
ASN.
abbreviation
the
defines
this
system for al! numbers relative probabilities to
grand pan
it
for all
system
E3}
E],VJ). r].
function.
ion
numbers of
written e, as eJ(:V,
We have
to
of part icles .V. That is, stale on the number hamiitonian. The state s(N) of ihe exact A'-particie = 0 must be N term included; if we assign its energy as zero, then ihe first in ^ w'-' be 1. term The absoluteprobability the will be. found in a state Nlt e, is that system Gibbs facior divided by lite Gibbs sum: by ihe given
the
emphasize
of the
dependence
c,|.v,is ihe energy
of
the
E4} chemical temperaturet result our cenlral ratio of any two P's is consistent ihe correct factors. relative probabilities E2} gives
This
applies
a system
lo
lhat is ai
and
E2} for
with
Thus
and
of
jV2,\302\2432.Thesum
of tlte
system
ihe
probabilities
of all
stales
for
al!
p.iiiictiljrl)-
lite Gibbs
stales
the
for
A',,
of par*
numbers
s,
icles
is unity:
f by ihe
ji. The
poieinial
definition
IicIpM.
of 0-. Thus
The
nuriliod
E4)
us,cil
gives
there
the
correci
to ikme
iibsuime
the Potion
-
1
\342\226\240 155)
probability.*
dimibuiion
impends
on
I
6744s
and Gibbs
Factor
m diffusive and thermal contact Average v;i!ucs over ihe sybiems are easily found. If X{N,s) ts ilie value ihe sysiem of -V when and is in ihe slate then ihe thermal particles quantum s, average of .V s N and all ts
Sum
a
with
lias N
reservoir
all
over
{56)
Number
the
of
The
particles,
is in diffusive
system
number of particlesin
ihe
contact
with
system
is
a
been multiplied can beobtaftied
liie
by
from
of N.
definition
in ihe
term
each
because
average of the
thermal
The
reservoir.
value
appropriaie the
,
in the system can vary
of panicles
number
To obtain tile numerator,
lo E6).
according
thermal averages.
to calculate
result
this
use
shiill
We
Gibbs sum
has
More convenient forms of
:
of#
whence
<,V>
Tile J
by
tiicrma!
number
average
use
direct
of E9).
will
We often
be
written
l-ll
we
E9)
= r^-1 is easily found from the
of particles
When no
thermal average . Wlicn
for
-
confusion arises, we
speak
of the
later
for
interchangeably
,V
or
shall
write
Gibbs sum jV for
the
occupancy of an orbital,
<.V>.
eniploy the handy notaiion
F0) where
k
see from
is called
the absolute
A2) that for an
activity. Here /. is Ihe Greek letter
ideal gas ). is
directly
propcrliona!
to ihe
lambda.
We
concentration.
Chapter5: Chcutkul
and Gibbs
Potential
Distribution
The Gibbs sum is wriiien as
F1} '
A
A5S
ensemble average numberof particlesE7)
and ihe
is
F2}
is useful, because in
This relaiior. finding
many
actual
problems
come out
will make
ihat
value
ihe
we determine
/. by
equal to the given number of
panicles. thermal
The
Energy,
v -
of the
energy
average
^i
-
system is
F3)
where we ha\\elemporarilyintroducedthe notation /J write
V
E9} and
so that
F3} may -
A
Chapter 3
in
terms
Example:Occupancy occupiedby may
be
5.10). A
zero
occupied single
usually
be
lo give
combined
F5}
toi
more widely used in partition function Z-
that is of the
zero
molecules
by one heme
We shall
;r-\302\253
expression
simpler
1/t.
that
Observe <\302\243>.
for
=
or one.
A red-blooded
or by one
molecule is the
red colorof meal.Ife is ihe energy
in
the
of an
example of heme
group,
a
syslcm
obtained
in
may be be vacant or
ihat
which may
than one Oz molecule(Figure which is responsible for the myogiobin, prolein to Oa at rest at adsorbed molecule of O2 relative
Oz molecule\342\200\224and
group occurs
was
calculations
never
by more
5.10
Figure
where c is
the Gibbs sum is
ihen
distance,
I
\302\243~
musl be added io remove
If energy
sum arises from
in ihe
Theseare itjc
only
occupancy zero; ihe We have
possibiliiies.
from, tile heme,
atom
the
F6)
+;.cxp(-\302\243/r).
lerm
The term t wilt be negative. arises from single occupancy. where Mb denoies present,
t
Aexpf-c/i)
Mb 4- O2 or MbO2
17 000. myoglobin.a protein of molecular weighi ihe conceniraiion versus of oxygen occupancy Experimental results for Ihe fractional arc show a in Figure 5.t t. We compare ihe observed oxygen saturation curves ofmyoglobin and in Figure 5-12. Hemoglobin is ihe oxygen-carrying component of blood, hemoglobin ti is made up of four molecular wiih ihe single sirand of identical each slrand nearly strands, ihe classic myogiobin,and each capable of binding a single oxygen molecule, Hisiorically, work on ihe adsorpiionof oxygen by hemoglobin was done by Chilian Bohr, the father of Niels Bohr.The oxygen saturation curve for hemoglobin (Hb) lias a slower rise at low because ihe binding of a single O, Io a moleculeof Hb is tower Ihan for pressures, energy Mb. A! higher lhat is concave of ihe Hb curve has a region pressures upwards, oxygen because trie binding the first is adsorbed. increases after O2 energy per Oj
The Oj thai
molecules
chemical
the
on
are
hemes
by
the
A
e
exp(/i/i}.
in
potentials
the Oj
find
ihe
the
value
liquid, so
surrounding
and
in
solution:
= ?.{O2)
/(MbO2)
/i(O2);
From Chapter 3 we
in
ihe myoglobin
on
\302\273
of /
in
F7) of the
icrnis
gas pressure
retalion A =
We
with
equilibrium
are equal
of Oj
/i(MbO2)
where
assume
Oj
adsorbed
by
a hemc
O;
rctal
10 an Oi al infinite from ihe site. sepatuiion If energy musi be supplied10detacri llie O3 from ihe heme, then c will be negative.
D
infinite
of an
Adsorpiion
ihe energy of an
ihe ideal gas resuit
_
+
F3)
p/t\302\253Q.
sotulion.
conslant
At
temperature
/(Oj)
is
p. O^
is found
Ac\302\273pl-t/i) I
-
io C), in
applies
direclly proporiiomUto tjic pressure Mb occupiedby The fraction/of
n/nQ
/.exp(-\302\243/t)
=
from {66}to
be
1 ' r'exp(\302\243/t)+
1
Fg)
Figure 5.11 The molecule with
reaction o( u
(Mb)
nvyoglobm
may
oxygen
(he adsorptionof a molecule of on ihe large mjoglobin molecule. results
Gj
at a
as
site
Tlie
isotherm
3 Langmujr
follow
be viewed
<{tiitc
molecule can accutaiely. Each myoglobin adsorb one Oj molecule.These curves show (he fraction of myogtobin with adsorbed O. as a function of ihe partial pressure of Oj. are for human myoglobin in The curves solution. is found in niuidcs; it is Mvogiobin for the color of slKtfc. After A. responsible Rosst-Fanellt and E. Antoiiini. Archives of and Btoihciittsiry Biophysics77, 47\302\253 (tySK).
Concentration
curves
Saturation
Fi\302\273ure5-12
lo nijoglobin
bound
of O.
and
iKmoglobtn (Nib) (J-ib) in wmer. The partiat molccuks in soluiion pressure of 0. is plotted as the Iiorizonl;i! Tiie
;t\\is.
the
vertical
molecules
a.\\is gives
of Mb
0, mokcuk, or the
which
ihe fraction of lus
one
bound
suarnis of lib which have one bouiid Qt innlcutle. h;is a much Uirgcr change in Hunioglabm cotitem in ihe pressure range bensecu oxygen ttw nuerks and the veins. Ihis circumstance facilitates ihe aciion of ihe heart, viewed as :i The
pump.
curve
for myoglobin
pioitual lonn (or the The
MbO..
of the
ft action
curve for
reaction
has the Mb
lienio^lobin
+ O2 Itits
\342\200\224\342\200\242
;t
forrn because of ifsicracironsbenveen molecules bound to ilic four si rands of ihe niolccule.The dr^Atntjis after J. S. Fruton
uiiiCicm O,
lib and
S. Siminondi,
1961.
Gvm-J
bioJtaniMry.
Wiley,
of O3,
rcbt
as
[he same
is
which
subsiiiute F8)
in
the
with
in
7. We
Chapter
to obtain
F9|
} 4- I
or.
derived
function
distribution
Fcrmi-Dirac
Gtbbs Sut.
and
Factor
G'tbbi
G0}
+
\302\273iurexp(E/r}
p
\302\256 jiqtcxp(\302\243/i).
p0
' +P
Po
whac
G1}
is
!>0
wiih respect
is constant
easts on the surfacesof
Example;Impurity
oikn
ionizatian
in a
as impurities
in
the atoms exchange
Lei / be the
bound
is accessible.Therefore one
an
with
When
^
the
with
to
large
and
electrons
ionizalion
can be
electron
semiconductor.
of the impurity energy an impurity atom;
the
system
fanned
with lite
& has
by
the
attached
as this stale; the stales of \342\226\240$ are summarized
the electron
rest
semiconductor. We suppose
atom.
that one, but only one. i of the electron spin
either orientation t or three allowed states\342\200\224one
with
ionizalion
by
band
systems S in thermal of the semiconductor:
small
are
atoms
reservoir
energy
electron
conduction
the
In
chemical
numerous
may iose an
crystal-
with
has zero
Suite
result
the adsorptioiiof
without
attached
spin f, and one atom is ionized. We choose tiic electrons, the impurity other two stales thereforefiave ihc common energy c =
electron
The
temperature.
of
Atoms
a semiconductor
the semiconductor
of
diffusive equilibrium
and
the
to dcicribc
used
when
as an idea! gas. The impurity
be treated
on
but depends
solids.
atom
elements when present to the conduction band may
to prcsfi re,
Langmuir adsorption is ndtrm
as the
known
an
electron
zero
an
electron,
with spin ofenergy
\342\200\224 /. The
].
of &
accessible
below.
number
Description
Electron
detached
Electron alladicd, s
1-kcifoti ijjnjLlicJ. The
Gibbs
sum is given
s
by
G2)
The
tliat
probability
&
is ionized
tliat
probability
0) is
=
\302\253
P(ioaized)
The
\302\273
((V
P(O.O)
$ is
neutral
t
is just
\342\200\224.
G3)
is
(im-ionized)
= /'(It,-/)
P(neutral) which
-\342\200\224-1-
\342\200\224 =
+ P(U,~/) ,
G4)
- PtO.O).
SUMMARY
1. The chemicalpotential found from
/*
equilibrium
if jtx
is
v
{cUJcN)a
\342\200\224
i{cojvN}v_v.
parts, external ittid in an external particle
of two of
a
The internal part is ofthermalorigin; Tlog(u/iiy),where is the concentration it
3.
The
Gibbs
have
and
^{Mi'lnh2)*'2
nQ
at chemical
a system
that
N particles
and be
state
a quantum
in
sum
5.
taken
over
The absolute
6. The
The
field of force.
-
gas ;t(int) is the quan-
potential ;i and temperaturei s of energy ts.
4, TheGibbs
is
mooaiomic
internal.
factor
gives the probability will
ideal
an
for
concentration.
bo
also difl'usive
= }tz.
2. Tl^c chemicaipotentialis made up external part is the potential energy
quantum
= {tF/tN). v and may Two systems are in
as h{t,V,N)
defined
\342\200\224
for all numbers
all states
activity
X
is
thermal averagenumber
defined of
of
by
X
particles
~
particles.
expQi/r). is
kg
\302\243
PROBLEMS
1.
velocity
angular
j[(r) on the for
as
an expression
i. Find
temperature
H
A circular cylinder of radius to. The cylinder contains an
Centrifuge.
radialdisiance
<\342\226\240 from
an ideal
for
the
axis
the ions
about
rotaies
R
ideal gas of atomsof niass
lhe
dependence
axis,
in terms
with
SI
al
the concentraiion of *i\302\2430)on the axis. Take of
gas.
2. Moleculeshi die Earth'satmosphere. If a is the concentration of mofcculcs at the surfaceof the Earth, Al lhe mass of a molecule, and g lhe gravitational at lhe surface, show that acceleration at constant tlic total number temperature of molecules in the atmosphereis =
N
wifh
r iiuMstirtiJ
The
integral
ttf (he
diverges
aimosphere cannoi be in are always escapingfrom 3.
I'ntuiitiul
each of mass
at
tipjwr
flic
limit, so
ihe
R is
here
l-yrih;
the railing of
ihe liarili. and the
be bounded
cannot
N
that
Molecules,
equilibrium.
em-i'ity of gas
M
of
ihc center
from
f75)
-
4nn(K)Cxp(-A/yR/t).Jjti
light molecules,
particularly
atmosphere. fa
t in
temperature
a uniform
of moms
a column
Consider
Jield.
gravitational
gravitational
field
average average potential energy per atom.Thethermal is of Find the total heat density height. capacity independent total heat capacityis the sum of contributions from the kinetic the potential energy. Take the zero of the gravitational energy ~ 0 of the column. = 0 to h ~ oo. h h from Integrate
thermal
the
Find
y.
kinetic
energy
per atom. The
energy and at lhe
from
bouom
transport. The concentrationof potassiumK+ ions in lhe internal of !04 of a a fresh water alga) may exceed by a factor sap plant cell(for example, the concentration of K+ ions in the pond water in which the cell is growing. The chemical of the K* tons is higher in the sap because their conpotential at concentration it is higher there. Estimate the difference in chemicalpotential 300 K. and show that it is equivalent ihe cell wall. to a voltage of 0.24V across are Take ;i as for an ideal gas. Because the values of the chemicalpoiemials The ihe ions in the cell in are not in diffusive and the different, equilibrium. pond to the passive impermeable leakage of ions plant cell membrane is highly ihrough tt. Importantquestionsin cell physics include ihese: How is ihe high concentrationofionsbuilt up within the cell? How is metabolic energy applied 10 energize the aciive ion transpon? 4. Active
5. drawn.
If
T
= 300K,
particles contain to
shown?
Determine
concentration.
Afaguctic
how many give
.\"-\342\226\240..\342\226\240
a
magnetic
the
ratio
m/x for
Bohr magnetonsfiB concentration
=
effect
which Figure5.6is the
eh/Zinc
ivould
of the
magnitude
whh
unoccupied
that system, (a) Considera system one in either by particle
a two level
sum for
Gibbs
6,
energy
one of energy zeroand oneofenergy
= i +
3-
time.
in the
c include
ttuitn
Notice
+
X
excludes the possibility
Our assumption
that
s. Show
the Gibbs
be un-
may
or occupied
zero
of two
sum for
ihis
states,
G6)
;.exp(-E/r). of
one
is
system
state at
in each
particle
sum a term forN
\302\253= Oasa
particular
system of a variable numberof particles, Show that the thermal (b) average occupancy of the system
the same state
of a
is
G7)
(c)
Show that
(d)
Find
an
(e) Allow one
3.
particle =
for the
expression
thermal average energy
the possibility that the orbitalat at the same time; show that
I +
1 +
average occupancy of the state at energye is
the thermal
+
Aexp(-\302\243/i)
Because ^- can befactored
as
uitd 7, Statesof negative diogen atoms;suppose each tliat
atom
can
Stale
Number
of
-JA
0
-if)
ion
conUiiion that the will
involve
5,).,
and
replaces
the
O2
lu luihuning. ad^oibed on hanoglohin
tl-.e clTect, consider a model for which vacant or may be occupied either with energy
eh
by one molecule
CO. Let
J3
1
U of
per mom
electrons
be unity.
r.
mtmoxitle
Cmtitm
2
number
Livci;t^^
hy-
Energy
Posiiivc ion
The condition X.
electrons
i
E\\cii<.-d
FioU rlic
a lattice of fixed exist in four states:
Ground
Negative
G9)
+ Aexp(-e/i)].
/.)[!
Consider
iotuzation.
posttiw
each by
in effect two independent systems.
we have
shown,
+
A
system.
be occupied
ate may
0 and
A2exp(-\302\253/t>-\302\253
the
of
A'
catbon monoxide (lib) mulcculcs in each energy fixed
adsorption ea home
poisoning the CO ilic
Wuoil.
To hhnw
a heme may one molecule O2 or
by sites
site on
be in equilibrium
be with with
Froblcn
CO in the gas phases at concentrations such that the are activities = 1 x SO\025 and 1 x 10\021, all at body 3TC. X(OZ) X(CO) tempcraiure factors, the system t\" the absence Neglect any spin muhiplicity (a) First consider of CO. Evaluate \302\243A such that 90 percent of the Hb sites are occupiedby O2. the answer in eV Now admit the CO under the specified Express per O2. (b) conditions. Find Sgsuch that only 10 percent of the Hbsitesare occupied by O2. and
O2
field. Suppose that at most one O2 can be Problem and that when /.{O2)~ iO\025 we have 8), group (see 90 percent of the hemes occupiedby Oj. Consider O2 as having a spin of i and a magneticmomentof i (ts.How strong a magnetic field is needed to chance T = 300 K.? (The Gibbssumin the limit of zero the adsorption by 1 percental field will differ from that muhiof Problem 8 because there the spin magnetic the was of bound slate neglected.) muhiplicity
9. Adsorption to
bound
ofO2
in
a magnetic
a heme
10. Concentration fluctuations. contact
in diffusive
system
We have
a reservoir.
with
is not
of particles
number
The
constant
a
in
seen that
\342\200\242
re:
<,v> from E9).
(a) Show lhat
(81)
=LLi.
3- ^
The
deviation
mean-square
= <(N
<(ANJ>
(b) Show
-
<.V\302\2732>
that this may
be
of N from
((AWI) =
is the diffusive
fi
xvi:
;u>rly
iliis
n-iilt
-
<,V>:;
= tc<,V>/c>
in llio
i.K'iil u:is to liiul
(83) liial
square fractional fluctuation in the population with a reservoir. If is of the order of contact
mean
=
by
as
<(A.V)'>
In t'liaptcr
defined
is
- 2 + (N}'
(N>)
written
of an \\02Q
atoms,
idea! gas then
in
ihe
Chapter 5:
ChemicalPotential
and
Distribution
Gibbs
fluctuation is exceedingly smalt. In is it cannot panicles well defined even though
fractional
such a system be rigorously
contacr wirh ihe reservoir. When is allowed is
diffusive
11.
of chemical
definition
Equivalent
defined
E) as (cF/cN)tiy.
by
An
fi
ihat this relation, which
Prove
the definition {5}that
resultsC1)
and
C5).
have
we
potential moreoften
a function
as
Schindler,
was
5.1 is (S5)
(cU/cN)ay.
to
Gtbbs
It will be convenient for treating {5}as the definition In
two-fold.
of the
to make use of the and
of/i,
we need
practice,
to
ji, is equivalent
define
adopted.
consequence, are
a mathematical
in Table
listed
expression
was used by
reasons
Our
*=
H.
The chemical potential
potential*
equivalent
the number of conslatil because low, this relation weight of large
temperature tihanasa
(85) as
the chemical of
funciion
the
a particle is added to a system a process in which entropy a while the is constant is more natural of the temperature system process kept one in which the entropy is kept constant;Adding a particle to a system at shan a finite temperature tends to increase its entropy unless we can keep eachsystem of the ensemble in a definite, although new, quantum state.There is no natural defmtiton the E) or F), laboratory process by which this can be done.Hcnee in which the chemical is as the in free expressed energy per change potential added particle under conditions of constant temperature, is operationally the ~ We point out that will not give U (85) simpler. /tN on integration,because is a function of N; compare with H{N,c,V) {9.13}.
a. Operationally,
12.
of sap in trees. Fitsd iree under the assumption
Ascent
in a
uppermost 0.9. The concentration of
in air
are
leaves
maximum
the
lhal
the
containing
water may rise of water and Ihe pool
to which
height stand
rools
in a
waier vapor at a
relative humidity
r
=
is 25\"C.If the relative humidity is r, the actual concentraair at the leavesis where n0 is the vapor uppermost rii0, in ihe saturated air lhal slands immediatelyaboveIhe poo!of
temperature
in the
water
concentration water.
13.
(a) Show
expansion,
hentropic
as a function only of show ihat xVin is constant in an
expressed
the
thai Ihe entropy of an orbital
expansion
gas
(b) From
occupancies,
isentropic
idea!
of an
can
be
this result
idea! monatomic
gas.
14. infinite
of
binding
Multiple
molecules.
Assume distance.
. solution), (a) What
Oi.
is
the
molecule can bind four O2 at rest relalive to Oaat O2,
A hemoglobin
lhal e is the energy the Lei / denote probability
bound
ofeach
absolute
activity
that one
exp(ji/T}
of
the
free Oi (in
and only one O2 is adsorbedon a
ts the
Sketch
moiecule?
hemoglobin
probabiiity
four
Uiui
the result qualitatively as a functionof/.,(b) and only four O2 arc adsorbed? Sketch this
What result
also.
15.
chemical potential.
External
N atoms
of mass M
potentialat value
of the
the
surface
in
volume of
tola! chemical
Consider a
t.
temperature
with
chemical the (a) Prove carefully atid honestly that
V. Let
the earth,
at
system
potential for
}i[Q) denote the value of the the
identical
system
when
translated
to altitude h is
p(h) = where applicable
g is
y
Mgh
,
the acceleration of gravity, is this result different from (b) Why to the barometric equation of an isothermalatmosphere?
that
6
Chapter
Ideal Gas
DISTRIBUTION
BOSE-EINSTEIN
153
FUNCTION
DISTRIBUTION
FERMI-DIRAC
157
FUNCTION
CLASSICALLIMIT Potential
Chemicai
Free
160 161
163
Energy
Pressure
16-\",
Energy
164
Entropy
165
Heat Capacity
165
Testsof Example:Experimental Chemtcai Potential of Ideal Gas
Sackur-Tetrode
the
with
Example: Spin Entropy Reversible
in
Isothermal
Zero
internal
167
Equation Degrees
\342\226\240
Field
Magnetic
169
of Freedom
J70
17'i
Expansion
Reversible Expansion at ConstantEntropy into a Vacuum Sudden Expansion
IT.t i \025
SUMMARY
176
PROBLEMS
177
1. Derivative
of Fenni-DiracFunction
2.
of Filled
3.
Symmetry
177
and Vacant Orbilals for
Function
Distribution
177
Double
Occupancy
Relativistic Particles 4. Energy of Gasof Extreme for an 5. Integration of the Thcrmodynamic Identity 6. Entropy of Mining 7. Relation of Pressure and Energy Densily 8. Time for a Large Fluctuation Gas of Atoms with Internal Degree of Freedom
10. Isentropic
Relations
11. Convcelive \\2.
Ideal
Gas
of
Ideal
isentropic Equilibrium
iit Two
Piitu'itsiuns
13. Gibbs Sum for Ideal Gas 14. Idcai Gas Calculations 15.
Diesel
Engine
Compression
Gas of
the
177
Statistics
Atmosphere
177 Ideal
Gas
177 17S i
7S
17S 17') 179 J~0
1^' 1SU ISO
ISO
Chapter 6: IdealGai
gas of noninteracting atomsin the limit of low concentration. The limit is defined below in terms of the thermal average value of the number ideal
The
is a
gas
of particleslh;it
distribution the orbital.
function,
An orbital is a stale of [he terns is widely usedparticularly
Schrodingcr equation for by chemists.
are weak,Ihc orbitalmodel allows of the Schi'6'tlingcr equation of a
!fthc
one
only
to approximate of A' particles system
particle.
sin
exact
quantum
in
terms
of an
by assigning the
This
between particles
interactions
us
we construct
thai
state
approximate quantum
as /(ej./i),
designated
usually
average occupancy is calledthe where t is ihc energy of
The thermal
orbital.
an
occupy
N
panicles
state
approxi-
lo
orbituls,
orbital a solution of a one-particleSchroOinger equation.There are an infinite of number orbitals available for term The usually occupancy. \"orbital\" is used even when there is no analogy lo a classical orbit or to a Bohr orbit.Theorbitalmodel gives an exact solution of ihe N-particlc problem if there are no interactions between the particles. only with each
It is a fundamental result of quantum of which mechanics derivation (ihe lead us into would thai ail species of particles fail two distinct astray here) fermions with is and bosons. a Fermion, classes, Any particle half-integrai spin with There are no intermeand any particle zero or integral spin is it boson. the an intermediate classes. same rule: atomof Hie is follow Composite particles
composed
an
of
each of spin j, An
aloin
of 4He
odd
so
of
number
must
Hie
that
has one
particles\342\200\2242
have
2 protons,
electrons,
spin and
half-integral
more neutron, so ihcreare
an
even
I neutron-
must be a fermion. number
of panicles
of spin |-, and 4Hemust be a boson. The fermion or boson nature of the particlespeciesthat make up a manyon the states of the system. body system has a profound and important effect as applied to the orbital model of noninleracting The resultsof quantum theory 1.
as
appear
particles
An
can
orbital
be occupied
species, An orbital
The second
rule
of
occupancies
of any
by any integral
number of bosons of the same
zero.
including
2.
rules:
occupancy
can be is a
need
occupied by
statement
not be
0 or!
of the
fermion
same species.
of the
Pault exclusionprinciple.Thermalaverages
but integral or half-integral,
individual system must conformto one or
the
the other
orbital rule.
occupancies
Distribution
Fa-mt-Dirac
The
different
two
rules give rise to two stun over all integral
occupancy
there
each orbital:
boson
a
is
Function
sums
Gibbs
different
values of the
for
occu-
orbital
= 0 or N = ! N fermion sum in which Different only. Gibbs sums lead to different quantum distributionfunctions for the /{e.t./i) thermal average occupancy. Ifconditions are such \302\253 it will notmaitcr that/ 1, whether the occupancies N = 2, 3,... are excluded or are allowed. Thus when <:< I the and fermion functions must be similar. Thislimit boson distribution / in which the orbital with unity is c;ii!cd the occupancy is small in comparison
there
and
A',
occupancy
is a
classical regime. now
We
occupancy
treat
the Fcrmi-Dimc
of an
orbital by
the
fcrmtonsand
for the thermal average occupancyof an
equivalence of the to treat
two
of famioit
particlesis
bosoti
and absolutely
in the
functions
the properties of a
gas in
Bose-Einstein orbital
opposite
thermal
the
for
average
function distribution bosotis. We show the
occupancy, and
we
go
on
treat the properties limit, where the nature of the of the gas.
Chapter
the properties
for
by
limit of low
limil.-ln
this
in the
g;ises cruchi
function
distribution
7 we
FERMI-DIRAC DISTRIBUTIONFUNCTION We
a system
consider
fermion. The
single orbital that
of a
composed
be
may
occupied
by a
is placed and diffusive contact with a reservoir, in thermal A 6.i and 6.2. real Figures system may consist of a large numberNo of call it the system. but it is very helpful to focuson one orbitaland fcrmions, Our are of as the reservoir. All other orbitalsofthe real system problem thought thus out. An is to find the thermal averageoccupancyof the orbital singled No orbital can be occupiedby zero or by one fermion. other occupancyis be taken will allowed Pault exclusion by the principle. The energy ofthe system to be zero if the orbital is occupied The energy is c if the orbiial is unoccupied. one fermion. by 5 we have The Gibbs sum now is simple: from (he definition in Chapter system
as in
\302\243=
The term 1 comes from the = 0. The term -Uxp(-~-\302\243/r) \302\243
so
that
of the
N
*= 1 and
1 +
configuration comes
the energy
orbital is the ratio of the
(I)
;.exp(~f./r).
when
A' = 0 and energy occupancy the orbital is occupied by one fermion, with
is e. The thermalaveragevalue term
in
(he Gibbs
sum
with
of N
=
the
occupancy
1 to
the entire
Gibbs sum:
B)
Ideal Gm
6:
Chapter
Occupied
,VQl
Figure
Wo)
6.1
fermion. energy
e
=
log
G,,) \302\243<*,\342\200\236
We consider as the The system is in thermal of the occupied orbital
spin orientation and considered as forming
confined to
unoccupiedand
-
JVD
We
be
might
the
Uu
if the
reservoir
ihe
will
contain
system is occupied
JVO by
one
average oecupancythe
5 that
Chapter
write B) in
the
a! most
by
one
at temperature of a definite states may be if the
fcrmions
t. The
sysiem is
fermion.
conventional an
symbol orbital
that
/(e)
of energy
e: C)
<.V(\302\253)>.
/. = cxp(/i/i), where
is the
p
chemical
potential.
form
standard
-,0/r]
\"exp[(\302\243
+
D) 1\"
as the Fenm-Dirac distributionfunction.* D) Equation of energy e. The value Ihe average number of fermions in a single orbital is known
result
Tills
gives
O2
be occupied
may
thermal averagenumberof particlesin
from
We may
UJ _(|\302\243^-t(J^
encigy of a free electron Other allowed quantum
fie) s Recaii
*) = a{A'u.
-
kinetic
The reservoir
for the
introduce
I,
volume-
a fixed
fermions
denotes the
_
system a single otbhai that and diflusixc contact v-uh
reservoir.
the i
o(/V0
119261,
iicovcrcd. general
and The
itaicmenl
P. A.
M. Dirac, Proceedings
of
i!:c Royal
Sociciy of London AI12. 661[
ihe new quantum meclianits is concerned with paper by Dirac of lln: form assumed b>- iht P^ulj principle on ihis iheory.
a'ld
cont
Functlor
Distribution
Fertni-Dhac
System
Figure
6.2
shown
here.
(a) Tlic obvious method of viewing The energy levels each refer to an
Sclirodinger
parliclc
equation. Ttic
loial
of noninicracling a system orbital thai is a solution of of
energy
panicles a
single-
is
the sysiem
of panicles in the orbital n of energy \302\253\342\200\236. For fcrmtons .V. = 0 ihan treai n orbital as the to simpler (a), and equally valid, single The system in this scheme may a of energytn. All other orbitals be itic orbital system. is jVoca. whert are viewed as ihe reservoir. The lotal energy of this o\302\273e-orbital system is the number one orbital as ihe of panicles in the orbital, This device of using Nn only weakly with each system works becauseihe particles are supposedto interact oiher. If we think of the fennion sysiem associated with the orbital n, these arc two dsher the system h;is 0 panicles and energy 0, or the system lias t pariki-possibilities: the Gibbs sum consists of only and enesgy \302\243\342\200\236, {wo terms: Thus, where
JVB
or 1.(b)
is ihc
number
much
Ii is
3- ~ I Tltc
firsi
from the orbital
arises
scrm
+ Acxp{~ejti =
occupancy JVH
0, aad
the second
tcnti
arises
;Vq = 1.
from
function distribution lies between zero and one.The Fermi-Dirac is plotted in Figure 6.3. In the field of solid siaie physics the chemical potential <* is often called the Th\302\243 on the temperature. Fermi level. Tlie chemicalpotentialusually depends is often written as ef; ihai of ft at zero temperature is, value
of/always
Mi
We call *
En
ilie
cF
the
Fermi
semiconduUor
ihen called she
Fennt
energy,
liseraiure level.
- 0) & not
fi(Q)
to be
ihe symbol
cF
=
E)
Ef.
confused* uiih
is often
used for
the
;( a! any
Fermi
lempfniurt.-.
level
which
a:-d ef i
6; Ideal Gus
Chapter
I
1
1
N
\\
A'
\\ 1
........
6.3
Figure
of the
Plot
Fermi-Dkac
r. Tlie value
the temperature
function
distribution
Fermi
higher
energy energy
/t(t). the
are
occupied
are
unoccupied.
chemicalpotential departs If there is an orbilalof orbital
is exactly
by
energy
half-filled,
in
electrons
Consider
ihe
in a
a system
- p in
of
unils
metal,
ji
1
might
of many independent
orbitals of energy below exactly one fermion each, and ail orbitals At nonzero temperatures the value of the t =
temperature
from
/i
versus e
/(e)
6
5
4
3
2
the fraction
which are occupiedwhen the system is heated from absolute zero, fcrmions are to the shaded region at c/ji > 1.Forconduction lo 50 000 K. correspond
!he of
I
of orbitals at a given energy tlfe system in thermal equilibrium. When is from the shaded region at E/p < transferred
gives
of/(c)
is !he temperaturedependent orbiials, as in Figure6.4.At
\342\226\240~m.
-4 -3 -2 -1 0 e \342\200\224 of t ji, in units
-5
_6
-7
Ihe Fermi equal
to
sense
of
0, ali
energy, as we will see the chemical potential a thermal average:
7.
in Chapter
(e
*=
//),
the
F)
Orbitalsof are
lower
less than
energy
are
more
ihan half-filled,
and orbilals of higher
energy
half-filled.
We shall discussthe physical of consequences in Chapter 7. Righi now we go on to discussthe
the
Fermi-Dirac
distribution
distribution
function
of non-
-I
Figure think
6.-1 of
interacting bosons, and in the
bosons
and
ihe ideal
establish
we
then
convenient
do
with
pictorial
gas law for boih fcrmions
FUNCTION
DISTRIBUTION
spin. The occupancyrule of bosons, so lhat for bosons can be occupiedby any number is an bosons have essentially different than fermions. Systems of bosons quality can have rather different physical properties than systems of fermions. Atoms of 4Hc are bosons;atoms of 3He are fermions. The remarkable superfluid < of the 2.17 low (T K) phase ofliquid heliumcan be properties temperature of a boson gas. There is a sudden increasein the to the attributed properties In and in ihe heat this temperaturefluidity conductivity of liquid JHe below to 2.17 K was found of 4He below experiments viscosity by Kapitza ihe flow be iess than IO~7 of the viscosity of the liquid above 2.17K. Photons(the quanta of the electromagnetic field) and phonons (the quanta number is of elastic waves in solids) can be considered10be bosonswhose is a
boson
A
not of
an We
of the one
m
the
consider
all
it
is
as we
osciiiator,
thermal
as
but
conserved,
in
by
particle wtjh thai an orbital
and
Figure
of the
of photons
to think
and phonons as excitations
did in Chapter 4.
distribution
contact
Let \302\243 denote
function with the
bosons for a system of noninteracting a reservoir. We assume the bosonsare of a single orbital when occupied energy
is N\302\253, there arc N particles in ihe orbital,the energy ireat one orbital as the system and view all other orbitals
when
6.5. We
value
integral
simpler
diffusive
same species. particle;
an
way to
composed of independent not interact with eutli oilier 3 common tcservoir.
limit.
appropriate
BOSE-EINSTEIN
iutecacl
A
a system
orbitalslliat but
function
Distribution
Chapter 6: IdealGo:
as pan ofIhe The Gibbs
reservoir.
sum
taken
for
the orbital
of particles may
number
arbitrary
Any
be in i : orbital
is
G)
The
tola! numberof paniclesin the combined and reservoir. However, the reservoir may be arbitrarily large, so that system N may run from zero to infinity. form. The series G) may be summed in dosed Let x s Aexp(~E/t);then upper
limit
on
N should
be the
i
i
\342\200\224 A\"
provided
inequality; Gibbs
(8) ;.cxp(-\302\243/t)
< 1. In
all applications,/,exp(\342\200\224e/i)will satisfy this otherwise the number of bosonsin the system would not be bounded.
that
kcx.p{~t/z)
The thermal averageof !henumber
the
i
-
sum by use
of
in the
panicles
orbital is
found
from
ofE.62):
-(\342\226\240)/<] L\342\204\242-___
__
A0)
Distribution
Bosc-\302\243insicin
Funclio
and Fenni-Dirac Comparison of Bose-Einsicin for funclions. Tlie classical regimeis aiiafoeii \342\200\224 \302\273 where ihe two distributions become (e r, /i) nearly idciuit We shall see in Ciiapier 7 that in the degenerate regime ai low
Figure 6.6 dislribuiion
chemical
ihe
temperature
potenijai
postnve, and changes 10 negative
ft
at
for
high
a FD
distribution
is
lemperaiure.
Et differs Bosc-Efnstcindistribution function, malhcmaticaHy tnsicad of + i -1 from the Fermi-Dirac distribution function having only by can have in the denominator.Thechange very significant physical consequences, 7. The as we shall sec in Chapter two distribution funclions are comparedin the two distri6.6. The ideal gas represents the limit e \342\200\224 r in which /i \302\273 Figure distribution are The choiceof ihe functions approximately equal, as discussed below. in any e is always arbitrary. The particularchoicemade zero of ihe energy
This
defines the
wiU affect
problem
ihe
difference e
\342\200\224 has
is discussed
iuriher
in
the
classical
A gas orbital
is
/;
in
is much
\\a\\uc
to be B0)
less than
of
dieniic.il
the
independent
/i,
potential
buE
the
of ihe
vaU:e
of the choice of the zeroofe.This
point
below.
regime
when
the average
number of atomsin
one. The averageorbitaloccupancy
for
a gas
each
at room
temperature and atmosphericpressureisof the order of only 10\026, safely in the classical Differences between fcrmions (half-integral spin) and bosons regime.
6: Ideal
Chapter
and
Gas
Uau iuiu
liicq
reg
imcs
Class
of
parlic
Ic
Regime
___
T!Krma!
it
o CCup.iticy
i*: ny
\342\200\224 \342\200\224\"~
III
Boson
Aluays
Fcfmioii
Close
10 bul
Boson
Orbital
of
Quantum
much
less
lhan one
less ihan one. energy lias
lowesi
an occupancy much
greater than
one.
so that for occupancies of the order of one or more, in the dassieai is lhe their are identical. The regime quantum regiir.e equilibrium properties These characteristic features are summarized opposite of the classical regime.
arise
oniy
6.L
Table
in
LIMIT
CLASSICAL
as a systemof free nomuteracting particlesin ttie classical \"Free\" means confined in a box with no restrictions or e>uernaiforces
ideal
An
gas
regime.
acting
is defined
the
within
We develop
box.
the properties
of an idea! gas with
the
use
the ideal gas powerful method of the Gibbssum.in Chapter3 we treated use the the of buE identical by partition function, panicle problem encountered there was resolved clear. by a method whose validity was not perfectly The Fermi-Dirac and Bosc-Einstein distribution functions in the classical limit in an orbital to the identical iead result for the average number of atoms Write /(e) for the average occupancy of an orbital at energy e ts the e. Here of it is not the energy of a system energy of an orbital occupiedby one particle;
of the
N
particles.
functions
where
The Fermi-Dirac
(FD) and Bosc-Einstcin(BE)
func-
distribution
are
lhe plus
sign is
for
the
FD
distribution.In orderthal/(c) must have in
this
classical
;
be
distribution much
smaller
and th;m
the minus sign for the BE unity for ail orbitals, we
regime
exp[(e-//)/r]\302\273l.,-
.
.
.
A2)
of ail
of (tt).
orhiiat of energy r.
/. hs
wiih
is satisfied we may neglect Then for either fermionsor bosons,the
this inequality
e. When
all
for
denominator
exp(/j/r).
function. !t is functions when
the
always
be a
occupancy
average
/(i.)
*
The
limiting
-
cxp[{/i
result
the
of
f
A3)
Fcrmi-Dirac /(s)
occupancy
average
dfstnbulion and Bosc-Einstein distribution the classical
is called
is very
small
in
with
comparison
although called classical,is still a result mechanics: we shall find that the expression the quantum constant /i.'Any theory which, contains
Equation A3), unity. described by quantum involves
the
in
is
limit
the
+!
term
the
for
particles
for
A
or
/.i
h cannot
classical theory.
thermal
entropy,
ideal gas. There aremany
= Aexp(-r/i) to of
topics
study
classical of
the
the
importance:
chemical potential, heat capacity, the prcssure-volumc-temperaiure distribution of atomic velocities.To obtain results from distribution function, we need first to find the chemical potential ihc concentration of atoms. and the
relation,
terms
of the
properties
function /(r)
distribution
classical
the
use
We
the in
Chemical Potential
The chemical of total number
is
potential
of atoms
the
that the
the condition
from
found
equals the
number of atomsknown
be the sum over all orbitalsof the
This number must
distribution
thermal average to
be
present.
function/(eJ:
N =
A4)
start with a monatornic gas where s is the indexof an orbitalof energy es. We N we include atoms of identical of zero spin, and later spin and molecular of atoms is the sum of the averagenumber The number mocksofmotion. total
of atoms
in
each
orbital.
We useA3)
rn
A4)
to obtain
AS)
JV-/yexp(-\302\243,/T).
To evaluate is
just
the
this
partition
sum,
observe
function
that
the summation
Zx for a
single
free
over free particle orbitals atom
in
volume
V, whence
Chapter 6: Idea!Gils In
=
shown that Zt
3 it was
Chapter
where
HqV,
nQ
s
rs the
(Mrjlnh2K11
quantum concentration.Thus A'
=
;.Z,
number
of the
terms
in
=
density
is
n;nQ
\302\253I.
out to
A7) j
volume quantum of the ideal potential
=\302\273 T
l/nQ. in the
monatomic
(IS)
,
tog(n/iiQ)
result
The
may
he written
give
-
i[)ogN
see
as
log
-
V
(IS) for
atoms
\\\\;i\\e
spin
spin ;:iuliipiidiy^S is JoublcJ; nQ will an
added
lo sevcra!
be subject
can
examples.
i
h3\\c
as the concentrationincreasesand
(lie chemica!polentiaf
If the zero of ihe energy scale is shifted by energy of an orbila! falls ill e0 = A instead
wifl
A9)
the temperature
Comment: The simple expression modifications.We mention four
(b) If lite
+ *SogBn/i2/A-/)]-
Jiogt
potential increases increases.
the chemical
that
decreases
(a)
A6)
the
chemical
The
,
i?/hq
,
obtained in another way.
wiih E.12a)
\302\253
We
nftiQ
in
atoms
P in agreement
=
exp(/i/T)
of
=
W/i?QK
Finally,
NJV.
\302\253=
number
to the
is equal
classicalregime gas is
=
n
/
which
;. =
/.,,QK;
S, the
=.
A
an energy of at
\\
ferm -rlog2.
(he
zero
of the
in ihe
sum in(iS) and
by
Tile tfTctt
kinetic
B0)
il isdiJublcdLlhevalucof
t\\eryw[icre
be replaced
that
+ Tlog(n/nQ).
number of orbiiafs
4- I. For s.pin
A so
e0 = 0, then
2\302\273G,
of
Shi: spin
Ihe
i\302\273 mufliplicd
iheparlition nght-hand
by ihe
fund ion Z^ side of{18)
on She enSropy is
Srcalcd
below. fc)
the &3s is not fTiOnatonuc, moiion wifl vibraiional If
t^e enter
ihe
iiitern^J
partition
sS^ies associated with function, and tfic chemical
energy
ro^itionaj
potential
-ind
wilt
free
Energy
- t log Zjni, per D8) below, where 2ial is the of partition function the internal degrees of freedomof one molecule. K the gas is nonideal, the resuK for /t may be considerably more complicated;sec 10 tk for to j g;is of interChapter relatively simple van der Waals approximation
have
Free
an added
term
Energy
chemical
The
is related to
potential
free
the
= n ,
(CF/dhr)t_y
according to Chapter 5. From F{N,t,V)
where so
the integrand
by
energy
B1)
this,
=
=
is found
+
B2) \342\200\242\342\226\240\342\226\240],
Mow Jt/.vlog.v
in A9).
brackets
in
[log
tJ^N
j*JN}j{N,t,V}
= xlog x
\342\200\224
x,
[hat
F
- iVi[logW -
-
I
F
=
Tile free energy increaseswith
Comment: Thus, from
The
integral
-
Wr[iog(n/nQ)
concentration
should
in B2)
- Jlogr + JlogBn/iVA')]
fog V
be
strictly
.
C3)
B-t)
1].
and
decreases
a sum,
because
with
A'
is a
temperature.
discrete
variable.
E.6),
-
f(N,r,l')
which differs from Ihe
integral
f. log.V
only
in ihc
= Iog(l x 2
lenn
x
3
in
log
x
B5)
.
Ji(-V)
f
,V in
A9), for
= iog/V!
\342\226\240 \342\226\240\342\226\240
x
N)
,
B6)
ChapM6:UtaICai the
wHcfg
A1 \302\2433\\e
inic^EiiI
\342\200\224 _V in
log A'
B3).
log,V! ~ may be
used, and now B5) is ihc
Bui for
MogN -
she Stifling
N
targe
N
approximation
,
B7)
as B3).
same
Pressure
The pressure
is relatedto the
With B3)
F we
for
free
by C.49):
energy
ha
Nt/V;
which
is the
pV
idea! gas law, as derived
in
Chapter
F
s
=
Nz
,
B9)
3.
Energy
energy U is found
The thermal
+
U-F
With B3)
for F
we
ta-F~
from
U ~
=
t(cFi<\"t)|.,\302\273
-t:
C0)
'
CD
have
2t
so that for
or
iff,
an ideal gas
C2) The
factor
| arises
from
the
exponent
of x
in
because
\302\253Q
the
gas
is in three
|
or two dimensions, the factor would be of 1, The average kinetic energy of translational motionin the classical respectively. of freedom of an atom. limit Is equal to ^c or %kBT translaiional degree per The principleofequipartitionofenergy of freedom was discussed degrees among
dimensions;
if
in Chapter 3.
iiq
were
in one
Heat
A
has rotational
molecule
polyatomic
and degrees of freedom,
the
Capacity
average
relational degree of freedomis%r when ihe lemperature is high in comparison with the energy differencesbelwcenlhe roialiorta! |e\\els energy of ihe molecule. The rolaliona! energy is kinetic.A linear molecule has two molecule has degrees of rolationa! freedom which can be exciled;a nonlinear tliree degrees of rotalional freedom. of each
energy
Enlropy
The
is related
enlropy
to the free
=
a
From B3)
F
for
This is idenlica! lhiil
so
\\o^n0/n)
for the
equation
C.76).
In the
The result C4) is known
is positive.
absolute
historically and is
of an ideal gas:
result
earlier
our
with
\302\253 1,
[33)
-(oFJdT)v,K.
the entropy
have
we
energy by
of
entropy
classical regime h/uq as
lhe
idea! gas. It is
a monatomic
h
Siickur-Tflrode
imporiani
in of chemical reactions. Even thermodynamics though the equation containsh, the result was inferred from experiments on in chemical rcaclions iong before the v;ipor pressureand on equilibrium it was a greal challenge to quantum-mechanicalbasiswas fuliy understood,
theoretical unsuccessful
the
essential
physicists
to
attempts
to do
explain
the
Sackur-Tetrode
so were made in
the
equation,
early
years
and
many
un-
of this century. We
shall encounter applicationsof the result in later chapters. The entropy of the idealgas is directlyproportional to the number of particles N if their concentration n is constant, as we see from C4). When two identical are side by side, each system having gases at identicalconditions placed entropy JV that connects the is doubled. Jf a valve ffj, the total entropy is 2al because is unchanged. We see that the entropy scalesas systems is opened,the entropy the
of the
size
concentration-
valve is
at constant system: the entropy islinear in the numberofparticles, increases when the If the gases are not identical,the entropy
opened (Problem6).
Heat
Capacity
The
heat
capacity
at constant
volume is defined
in
Chapter
3 as
C5)
Chapter6: IdealGos
for
the expression
wlien
derivative
the
calculate
can
We
is
nQ
from the entropy C4) of an
directly
idea!
gas
out:
wn'tien
IE
From this,
idea!
an
for
gas
C6)
or Cy
=
to perform the work
be added
must
units.
conventional
capacity at constant pressure is largerihan
The heat heat
in
%NkB
gas againstthe constant lhe thermodynamicidentity
discussed
p, as
pressure
=
xda
to
needed
because C\302\245
expand in
detail
in
additional
the volume of 8. We Chapter
the use
-f pdV to obtain
dU
/eu
C7)
p
only on Hie temperature,so tliat ldU/i)z)p will have the same value \342\226\240dSidU/di}^, whieh Cy by the argument of C.17b). isjust = N. Thus lhe ideal By gas law V C7) Nt/p, so that the terra p(dVldt\\ becomes
The
of
energy
ideal
an
gas depends
Q. + N
C, = in
fundamental
or
uniis,
C, in
capacities
have
as R,
We
units.
conventional in
the
two
=
notice
systems
+
C,
again of units.
C8b)
Nk,
the different dimensions that heat
For one mole,NkB
is
usually
written
called the gas constant.
The resultsC8a,b)
are
decrees
C8a)
of freedom
of a
for
written
an
ideal gas
without spin or otherinlernal
molecule. For an alom Cy
c,
= In + n
= |,v
\342\200\224
:N,
so
that
C8c)
Heat
units, or
in fundamental
Cp units. The
in conventional
Example;
from
found
\302\273
C8d)
$NkB
ratio CpjCv is written as 7,
of the
tests
Experimental
are oflen
entropy
Capacity
Sackur-TeiroJc
gamma.
equation. Experimental values of Ct by numerical integration
values
experiment!
letter
Greek
the
ihe
of
of C7)
C9)
Here
ihe
denotes
at the lowest
entropy
temperature attained
in
rhe
measurements
suggests rhat o@) may be set equal ro icta unless fhcre are mullipliritics not removed at the lowest temperature attained. We can calculate the entropy of a monatomic: ideal gas fay use of rhe Sackur-Teirodc and pressure may be equation C4). The value thus calculated at a selected temperature value of the entropy of ihc gas. The experimental value compared with the experimental is found ihe follow ing confributions: by summing
of C,,. The fhird
Enlropy increase on
1.
2.
of thermodynamics
law
solid
healing
Entropy
increase
in ihe
.1.
Entropy
increase
on hearing
4.
Lntropy
increase
in ihe
5.
Enlropy
change
absolufe
from
transformation (discussedin
solid-to-liquid liquid
zero to the melling
from
niching
[mini to ihe
point. 10).
Chapter
boiling point.
liquid-to-gas transformation.
on heating
gas from
the
point
boiling
to ihe
selected temperature and
pressure. There
may
further
of experimental
We enlropy
I.
be a
give delails of the is given in terms
lo C4)
corrctiion
slight
and theoretical
values have
comparison
for
for ihe
now
been
nonidcalilyoflhe gas.Comparisons out for many gases,and
carried
afler ihe measurements
neon,
oflhe conventionalentropy
S
very
of Clusius.The
=s kua.
M.55 K The heat capacity of ihe solid was measured from 12.3K to the melting point The heat capacily of ihe solid below I2.3K was under one atmosphereof pressure. zero of the me^iuceslimaicd by a Debyc law (Chapier 4) extrapolation to absolute found of ihe at is menlsaboie IJJK.Theeniiopy solid the melting point bj nuir.e.'tcal of UlTiCp/T) 10be inlcgralion 5lolid
\342\200\242
A classic
ihe American
=,
vtudy is \"The heal rapacity of Chemical
Society
i4.29Jmor'K~t.
oxygen
51.2300 A939).
from
12 K to
its
boding
point
,ind
its hiat
of
Chapter 6:
idcutGa: 1
able
xaluaot wide
uropy ullli c
Enlropy
r.lp,lnK
Gas
Nl-
212
Ar
87.29
Kr
119.93
sul'rce:
hc.it
associated enlropy
The
3.
point
to mc!l
required
inpul
The
of melling
heat capacilv of the of 27.2 K under one
heat
The
mol\021.
associaied
The cxpertmenlal sphere adds up to sK3i
The
=
^.oiid
calculafed
value
+
14506
labics, 6lh
cd.,
Voi
was
liquid
the
measured
atmosphereof
pressure.
K is observed
lo be 335
J mol\"'.
The
from
+
&SUvAi of neon
entropy
=
the
poim
inching
[o ihe
increase was
The entropy
found
boiling
10 be
- a.SSJmol-'K-1.
of neon
oflheeniropy
ASmelling
value of
ai 24.55
solid
the
is
5g\302\273,
from ihe
96.45
12924
at to vaporize ihe liquid enlropy of vaporization is
required
input
96.40
129.75
BonMvin
ASIJ4Uid
4.
Iculalo
Ca
I
144.56
Lumhh
From
The
E Nperimenia
'K\"
Jmol\"
in
4, pp. 394-399.
Pan
. 2.
boilin
iiosph
96.45
+
gas a! 27.2K
af a
pressure
same condifions
Jmor'K\021 agreemenl
lo be 1761J
ofoneafmo-
\302\253 96.40Jmor'K\021.
ASViPQliMlion
under ihe
Sackur-Tefrodeequation. The excellent
was observed
27.2K
is
, with
fhe experimental value led to the Sackurfind it verified by 6.2. in Table
gives us confidencein the basis of the entire theoreticalappamlus that Tctrodc equation. The rcsuh <31) coutd Unrstly l[:ivc been guessed; to observation is a real experience. Results for argon and krypton are given
of IdealGas
Chemical Potential
Infernal
with
of FreeJoin
Degrees
Chemical Potential of Ideal Gaswith Internal
of Freedom
Degrees
We consider now an idealgas of identical polyaiomic has rotational and vibrational degreesof freedom
tional degrees of freedom. Thetotal energy
\302\253 \302\243 en
to
motion of the
problem
rotational
In
classical
oscillator
is the
the transla-
sum of two
regime
of Problem '3.6. sum for the 1
terms
in
ofthe orbital is assumed to to correspond occupancies states Gibbs sum associated n
greater
the
be
+
degrees of freedomand molecule.
treated
r.n
vibrational
earlier. The rota-
n is
orbital
;.exp(-\302\243fl/t)
The
.
D1)
because the averageoccupancy \302\253 I. That is, we neglect the terms in 3- which than In the presence of internal energy unity. with the orbital n becomes
of ),
powers
higher
D0)
,
problem
the Gibbs
\302\243\302\253
where
eifli
center of massofthe
harmonic
the subject
was
energy the
the
ts
+
and vibrational
rotational
the
to
refers
einl
the translations!
energy
to
addition
in
the molecule
parts,
independent
where
e of
molecule
Each
molecules.
are omitted
D2)
D3)
The
is
summation
just
the partition
function of the
internal states: (\024)
which
is
~tlogZ;ol.
related
to the
From D3)
internal free energy of the
one
molecule
by
/\"in,
=
the Gibbssum is
D5)
Gas
6: Idea!
Chapter
The probability
the state of in
X
the
to
the
that
of
The classical regimewas of
Several
tlie
(a)
the molecule,
sum 3-:
Gibbs
analogous to
n is occupied, irrespective of is given by the ratio of the term
orbita!
transtationat
motion
internal
the
for
A3)
\302\253 i. The /(\302\243\342\200\236)
ease, but
monatomic
derived for the
results
the
as
earlier
defined
XZinl
now
D6) is entirely the role of A.
result plays
monatomie ideal gas are different
for
ideal gas:
polyatomic
Equation
is replaced
for,*
A7)
by
D7}
with
defined
as before
exactly
ideal gas
monatomic
(We shall
of atoms with
zero
always use ne as defined spin.)
Because
X
=
for
e\\p(/i/i)
the
we
have
p. =
(b)
The free
~ r[log(ri/nQ)
energy is increasedby,
for
IogZiol].
D8)
JV molecules.
D9)
(c)
Thecniropy
is increasedby
E0)
The former
Example:
Spin
represcnl
boih
resuU U
enmtpy
\342\200\224
jh'z
10 die
energy alone.
iranslalional
ntasnrtic field. Consider an and nuclear spins. TIic iniernal
in zero
electronic
applies
of Spin /, funciion partition atom
where /
may
associaied
Reversible Isothermal Expansion
the
with
is
alone
spin
ZiM = this
the number
being
of independent
{2/ + t) ,
spin siaics.The
the
entropy
spin
=
E0).
TheefTec!
of the spin
Isothermal
The to the
is the
x !0Jcm\\
gas
the
What The
initial
of D8): E4)
I)].
pV
at an initial volume of
of 4He
at constant
The temperature is
temperature constant
maintained
reversible
the
expansion
\342\200\224 S'x
we
see
final
until
by system
the
thermal ai
any
is equal
temperature
the final pressure
that
is
pressure.
increase of entropy an an of ideal at entropy gas
is the
a(V) =
expansion?
constant A'log
V
= (ix
4-
constant
on
volume
as E5)
,
l0\{0.69?)")
entropy is larger at the largervolume, states in the largervolume than in
same temperature.
depends
temperature
= jVIog2
Notice that the more accessible
+
slowly
expand
initial temperature. From
one-half
with the help
is found
after expansion? is twice the initial volume; the
pressure volume
final
E3)
logBi
atoms
1022
x
contact with a large reservoir.In a instant is in its most probableconfiguration. What
1) .
Expansion
!03cm3 at 300K. Let the is 2
E2)
entropy on the chemicalpotential
Consider as a model example1 volume
1) ,
+
logB/
}i = T[Iog(n/ii0)-
Reversible
free energy is
is
\302\273,.,
by
to the
contribution
spin
Fial = -tlogB/+ and
E1)
*=
0.069
because
the smaller
x
the
1033.
E6)
system
h.TS
volume at the
Work
6.7
Figure
is done
by
(he
gas in
an isothermal
expansion.Herethe gas docs work by raising llic weights. Under isothermal conditions pi' is constant for an ulcal gas, so thai the pressure mus! be reduced to affow !hc volume to llio load of expand. The pressure is reducedby removing weights a
work is done by
How much When
Figure
at
little
the
gas
the
ht the
gas
isothermally,
expands
work done
The
6.7.
a time.
expansion ? it
work
does
from
directly as4.!4 x
30serg=
W as
the work
negative of the work done by
=
\302\253
DJ.4J){0.693)
The assumption that the process that a knowledge of Vat every stage define
the work
Thus
43.4J.
done on the
E7),
Ni!og2
We
in
E7)
Ntlog2.
H\\NxjV)dV
pistonis,
as
on the piston when the volumeis doubledis
=
We evaluate/Vt
a piston,
against
is reversibleentersin determines
p at
done on the gas by the
w=
2S.7J.
gas
on the
E8) E7)
when
we assume
every stage of the expansion.
external
piston. From
-jpdV= -28.7J.
agencies.
This
is the
E8), E9)
ruble
What is the
in ihe changeof energy expansion? of an idea! monatomicgas is U
The energy an
at constant
expansion
decreases
temperature. which is the work
by Afilog2,
*=
mt Entropy
Expai
and
jNx
not change
does
However, the Helmhoitz free
in
energy
done.The connectionis discussed in
S.
Chapter
flowed inio the gasfrom the reservoir? that ihe energy ofihc idea!gas remained the constant when of energy it is necessary thai a gas did work on the pision. By conservation (low of energy in ihe form of heat into the gas occur from ihe reservoir through the walls of the container.The quantity heat added to be of the must Q gas but be in because equal, opposite sign, to the work done by the piston, How
much heal
Q +
seen
have
We
=
W
0. Thus
Q = result
the
from
Reversible We
the
at Constant
Expansion
expands
insulated container.
at constant temperature. Supposeinstead from I x 103cmJ to 2 x 10Jcm3in an in-
reversibly
No heal
to
flow
is constant in a system
The entropy
processis
Entropy
an expansion
above
gas
F0)
E9).
considered
that
28.7 J ,
out
carried
or
isolated
(slowly).
reversibly
is permitted, so that Q = 0. the reservoir if the expansion from of entropy A process without a cluvnge the gas
from
is called
adiabaticprocess.
has
is
an isentroplc process or an the specific meaning that there we
simplicity,
stick
shall
term
The
no
heat
transfer
in the
\"adiabalic\"
process. For
wilh\"isentropic.\"
What is the temperatureof the gas after expansion ? The entropy of an ideal monatomicgas depends
on
volume
the
and ihe
temperature as
a{z,V)so that the entropy
constant
remains
at
constant
eniropy '
Tl3'2f, for
an
ideal
monatomic
gas.
+ constant)
logK
,
F1)
if
t3/2K =
constant;
Iogt3/2K=
In an expansion
+
N{iogT3;I
from
V1
to
V%
constant.
F2)
have
we
= rivlV1
F3) .
.
Chapter6: IdealGas
-
idea! gas
the
use
We V
law pV =
forms. We
alternate
two
obtain
to
Nz
insert
F3) and cancelN on both sidestoobtain
Nxjp into
F4) r =
insert
we
Similarly,
to obtain
pV/N in F3)
F5) BothF4)
and
hold
F5)
for
an iscnlropic
monatomic gas.
an ideal gas of motion {rotations, vibrations). We obtain
to
10
Problem
internal
with
molecules
for a
only
It is ihe subjectof
of
degrees
these
generalize
results for
process
F7)
t.'\"'\"\"'V. =r2\"\"-
Here'/ s
is
Cp/Cy
the
ratio
heat capacities at constant pressureand
of the
volume.
constant
With
Tx
- 300K and
=
Vs/V2
find from
|- we
F3):
\302\253 189
Tt\\k is
ihe
is cooled
in
final
the
expansion
Tt
Expansionat constant What
is
ihe
change
the
after
temperature
hi energy
300
-
K
is
entropy
in ihe
an
\302\243/,=
189 K
at constant
entropy. The gas
- !!IK. method
important
G0)
of refrigeration.
expansion?
The energy change is calculatedfrom ideal monatomicgas U2
F9)
by
process
- Tt =
expansion
K.
Cy{x2
the
-
temperature
r2 -
change
r,) ,
G0). For
an
G1)
fi
or, in
conventional
Expansion
into
units,
ut =ivA-B{ra- r,)
a2-
x
\302\273i(l
x
=\302\273 -2.3
x
10IZ)(j.38
JO\0216ergK\"!){-IHK)
\302\273 -231
108erg
G2)
decreases in an expansion at constant entropy. The work done by to the decrease in of equal energy the gas, which is Ul - U2 = 23 J.
The energy the
is
gas
Sudden Expansion inlo a the
Let
Vacuum
suddenly into a vacuum
gas expand
a Hn:t!vohmicof 2 liters.Thisis an is opened
a hole
When
in
the
excellent
an
from
of an
cxiimpfe
to process.
of I liter
irreversible
ihc expansion,
to permit
partition
volume
initial
the first
moms
and strike ihc oppositewall. If no heal How through the is no way for the atoms to lose their kinetic permitted, energy. The flow may be turbulent (irreversible), of tht! with different subsequent parts values of the energy between gas at different density. Irreversible energy flow will the assume the regions eventually equalize conditions throughout gas. We whole process occurs rapidly enough so that the walls. flows in through no heat rush
the hole
through
walls is
there
Hois1much work is done in the expansion ? No meansof doing external is provided, work is not
work
Zero
work
the
is
What
is zero the
so that the
work done is zero-
of all irreversibleprocesses,but for expansion into a vacuum. a characteristic
necessarily
after expansion?
temperature
No work is doneand no heat is added in the expansion: W = 0, Q ~ 0, and ~ = 0. Because the energy is unchanged,the temperature of the ideal U2 t/j is The in the because of a real gas unchanged. process energy gas may change the atoms are moved farther affects their interaction which energy. apart, is
What
the
change
The increase
of entropy
the
in
?
expansion
of entropy when the volumeis
at constant
doubled
temperature
is given by E6): Au
w
a2
into a Expansion into a vacuum
For
most
the expansion probable
(equilibrium)
~
a,
vacuum
is
- Nlog2 = 2
not
0.069
x
1023.
G3)
= 0.
a reversible
configuration
process: the
system is not
in
the
at every stage of the expansion. Only
'
Gas
Idea!
ChapterS:
Reversible ] isothermal
t-
y 0
V
expansion j
I-
A'rtog-
-.Vtlog-^
Nlog^
Reversible
] ]
isentropic expansion
-J.Vt,I-(\302\243) L. \\vi/
J
Irreversible
j
\021
into
expansion
y 0
V
vacuum
Wlog
\342\200\224
'
J
before removal of the partition and the final conare most probable configuraiions. Al intermediate configuration equilibration the distribution in concentration and kinetic energy of atoms between stages the two regions into which the divided does is not system correspond to an distribution. The central results of these calculationsare sumequilibrium the
initial
configuraiion
after
6.3.
in Table
summarized
LAW
GAS
(a) /(e) =
LEADING TO THE
STEPS
SUMMARY:
GAS
MONATOMIC
SPINLESS
FOR
IDEAL
Aexp(-\302\243/r)
of an
Occupancy
limit
classical
(b)
(c)
X
=\342\200\224-\342\200\224-\342\200\224~~~
Given
i
x
e*P(-\302\243nA)
= -\342\200\224-177x73
^
Energy
'j
=
X!CXP(\342\200\224\302\243JT) \"
this
N,
-[n the
dassical of a
Ik
'/ii j
n2
exp(\342\200\224e/t)
Transformation
equation
determines
limit.
free particle n
number
quantum
volume (d)
of/(\302\243)
the
in
\302\253 1.
N
=
En
orbital
orbital of
in a
cube of
V.
of the
sum to
an
integral.
(e)
X =
N/nQV
Result of the integration subsitution
(f)
nQ
= {Mi/27rfi2K/2
(d) after
in (b).
Definitionof the qaantam
. concentration.
=
H
(g)
T!og(ll,'!!Q)
(h) F
=
(i) p
- -icFfcV)t,N-
J(/,V/i(.V,t,K)
=
-
1]
NiDog(tr/HU)
Nt/K
PROBLEMS
Show
Derivative of Fermi-Dirac function. Fermi level e = ji has the value Dt)~'. 7.
steeperthe slopeof 2.
Symmetry as
appears
the
Thus
+
f{fi
of 'filled and vacant i5). Show that
e = /i
Let
orbitah.
probability an orbital6 below known as a hole.
to the
ihc
at
evaluated
the temperature, the
Fermi
the
that
ap-
/(c)
G4)
is equal is orbital
is occupied
is vacant.
level
so
+ 5,
fin + 5) = 1 -'/(/< - 3). an orbital 5 above the Fermslevel
that
probability
\342\200\224cf/vc
function.
Fermi-Dirac
the
that
the lower
Thus
vacant
A
sometimes
3.
Distribution
of
values and
siaiist'tcs. Let us imaginea new an are 0, I, and 2. The of orbital occupancies lo be 0, e, the energy associated with are assumed these occupancies
It,
which
double
for
function
in
mechanics
occupancy
the allowed
respectively.
(a) Derive
an expression for
system composed of
ensemble is
orbital
this
at temperature now to the
reservoir
the
in
t and chemical usual quantum
average
thermal
(N>, when the
occupancy
contact
and diffusive
a
with
potential/t.
mechanics, and derive an expression of an energy level which is doubly deaverage occupancy the e. If both orbitals are identical degenerate; that is, two orbhals have energy Return
(b)
for
the
ensemble
occupied
the
4. Energy particles
have
total
is 2e.
energy
of gas of extreme p such
momenta
relativistk
that
pc
particles. \302\273 Me2,
where
Extreme M
is the
relativistic
parti-
rest mass of the
/. = particle.Thede Broglie h/p Tor the quantum wavelength continues to apply. ideal that the mean energy per particle ofan extreme nonrclativistic the (An interesting gas is 3t ift S pc, in contrastto ir problem. in Notes on Thermoby E. Fermi variety of relativisticproblems discussed of wul Suiiisiks, Chicago Press, 1966,paperback.) University relation
relativistic
Show
Tor
are
ttynumU-s
5,
Integration
modynamic
of
identity
the
tlwrntodynanric
at constant
identity
for an
ideal gas. Fromthe
number of particleswe
thcr-
have
+
^. X
G5,
Gas
6; Ideal
Chapter
Show by integration thai
6.
if
Relation
pressure
+
N log
a system
,
at
G6)
V.
and
of
a system
that
+
jV
of N atomsof type
diffusive
after
V
atoms B at
A is placed same the temperature of type
the total entropy
is reached
equilibrium
known
in
{A
The difference
is established.
contact
and
pressure
of
(a) Show
density,
energy
contact
in thermal
a system
in
logT
results
the
in
has
ihc Gibbs paradox.
been called
7.
Cv
2. The
diffusive
when
entropy
a =
entropy is
as [he entropy eniropy increase2N log2is ihe atoms are identical s B), show that there is no increase
by 2N log
increased
ofmixing.
gas the
Suppose
with
contact
and volume. Showthat is
ideal
of mixing.
Entropy diffusive
in
an
a constant, independentof t
crj is
where
for
reservoir
a heat
with
ihat
ihe
average
pres-
is given by
G7)
where the sumis over all states
of
the
(b) Show
system,
for a gas offree particles
that
;
result of the boundary conditionsof the whether es refersto a stateofJV noninteracting that for a gas of free npnrelativistic particles as a
G8)
,
-r77
problem.
particles
The
result
or to an
hoids equally
orbital (c) Show
p= WjlV , thermal
is the
U
where
average energy of
to the classical
regime;
long as
nonrelativistic.
8.
they
Time for
cases
in
a 0.1
are
it
holds
liter
container
wiH
We
unmix
the system. This result Tor fermion
equally
a large fluctuation.
G9)
Boitzmann
quoted
only
in a
is
and boson
to ihe
lime enormously
not
limited
particles, as
effect that
two
long compan^
problem: we let a gas of atomsof \"'\"'years. of 1 aim, volume liter at 300 K and a pressure a container of ofOJ occupy and we ask how long it will be before the atoms assume a configuration in all are in one-half of the container.. which initial accessible to the system in this of states (a) Estimatethe number
to 10\" JHe
condition.
We
shall
investigate
a related
(b) The gas is compressed isothermallyto a volume stales are accessiblenow? the
For
(c)
system of
number
for which
states
number of states
ofthe
frequency
(e)
Estimale
are
with
the of
one-half
in
9. Gasof atomicgas, energy
l-'nut ttic (a)
system
iutermil
has two internal energystales,one
the atom
There ;ire H aiomsin
chemical puk'iiljai; (h)
an
tdcal
of ideal
in an
gas
isentropic
gas.
an ideal man-
Consider
of freedom.
degree
which
for
[he oiher.
relations
hentropic
of
total number
is Ihe
of the
live
volume
V at
(c)
entropy;
energy;
(o) heat capacity;it constant pressure.
10-
'
as a crude estimate system changes. number of years you would expectto wait before all atoms the volume, starting from ihc equilibriumconfiguration.
one
but
A above
volume
We use this
a year?
in
the slate
which
m'th
atoms
in the
what
10lC)s\"',
volume
the
of
one-half
are anywhere
atoms
an atom is %
in the
atoms
all
are in
all atoms
the value of the ratio
estimate
container,
the
which
for
rate of
collision
the
If
(d)
collisionsof
liter
0.1
the
in
How many
liter.
0.05
of
an
r.
temperature
(t|> pic.^uic;
(a) Show that the differential
changes for
process satisfy
(SO)
where
= these relations apply even CpjCv\\ of freedom, (b)The isentropicand isothermal degrees y
BB
Show
for
that
given by
c
-
ideal gas of Here p //.
the
at
constant
Bs =*
~V{cp!cV)a;
idea!
gas Ba
(Ba/p)\022; ihcre of
molecules
mass
= yp; B,
is very M
iittie
= p. heat
we have
is the massdensity.
Convcaire
of
an
=
iscntropk
troposphere\342\200\224is
atmosphere\342\200\224the
entropy,
equilibrium of the
not
constant
The
if
bulk
internai are defined as have
molecules
the
moduli
(81)
-V(dp/dV)t. of
velocity
transfer
sound
in a sound
p = pi/M,
so
that
c
in a gas is wave. For an =
(yr/'A/I''
.
atmosphere. The iower iO-15km often in a convccijve sieady state
temperature.
\\\\\\
such
equilibrium
p\\\"'
is
Use the condition of mechanical independent of altitude, where CplCr. field to: (a)Show that dTjdz = constant, equilibriumin a uniform gravitational where r is the altilude. This quantity, important in meteorology,is calledthe was relation that adiabatic pressure dry lapse rate. (Do not use ihe barometric derived in Chapter 5 for an isothermal (b) Estimate JT/i/r, in atmosphere.) = mass density. CC per km, Take y 7/5. (c) Show that p x p1,where p is the = -,\342\226\240
Chapter6: IdealGas gradient is greater than the isentrapic temperature be unstabic with respect to convection. may
actual
the
If
atmosphere
12. Idealgas ttro monatomicgas in
in
=
A
Gibbi
13.
Chapter
is 3in
an
=
(b) Find
sum for ideal sas. 3, show that the
{a)
V in
diffusive
With
coniact
P(N} =
Hie
expression
a. The the
(c)
X P(N) = 1
r.
Zs ^
of
help
ideal there
probability
gas
of
arc
;V
from
(i>QV)sf,\\\\
atoms
identical atoms
,
is just ihe Poisson distribution function thermal average number of atoms in ihe XVnQ,
temperature is
in
Hie gas
wiih a reservoir is
which
previously as
confined to a square of area for the energy V of the gas.
N atonis
Gibbs sum for an
exp(;.iiQK).(b) Show that
volume
an
the entropy
for
expression
with
the
potential of an ideal
ihe chemical
Find
(a)
dimensions,
spin is zero,
L2. The
(c) Find
dimensions, two
gradient,
Confirm
that
and
(82)
Here
is the
we have
evaluated
C).
{Appendix
volume,
which
P[N) above satisfies NP{N) \302\243
=
.
ideal monatomic of an gas ai gas calculations. Consider one mole 300K and 1atm. First,let the gas expand and to twice isothermally revcrstbly an let the initial Hits be followed by volume; second, iseniropic expansionfrom io four limes ihe initial volume, (a) How much heat is added io iwice (in joules) at ihe end of the gas in each of these two processes? (b) Whai is ihe temperature is replaced the second process?Supposethe first process by an irreversible a (c) What expansioninto a vacuum, to total volume twice the initial volume, in joules per kelvin? is ihe increase of emropy in ihe irreversible expansion,
14, Ideal
15, Diesel
engine
in
engine
so
which,
compression.
fuel is
compressed
highly
initial temperature of maximum y
diesel
engine
is an
internal combustion
sprayed into the cylinders after ihe air chargehas been !o tgntte it has attained a temperature sufficient the that
i sen tropically from an that the air in the cylindersis compressed If ratio is what is the the 27\302\260C C00 15, compression K). the is healed \302\260C to which air in the by compression? temperature
fuel. Assume
Take
A
=
1.4.
7
Chapter
and Bose Gases
Fermi
183
GAS
FERMI
Fermi Gas in Three Dimensions
Slale of
Ground Density
of Electron Gas
Heal Capacity
FermiGas
IS9 194
Metals
in
Stars
Dwarf
White
185 1S6
Simcs
of
Nuclear Mailer
BOSON
of
Spacing
199
199
Near Absolute Zero
Chemical Potential Example:
19S
CONDENSATION
EINSTEIN
AND
GAS
196
'
and Second
Lowest
Lowest Orbilals
of FreeAtoms Orbital
201
Versus
Occupancy
202
Temperature
205
Einstein CondensationTemperature
207
\"He
Liquid
Quasiparticles
and
212
*He
Superfluidity,
Phases of
Superfluid
210
of Helium
Relations
Phase
217
3He
SUMMARY
217
PROBLEMS
218
1. Density 2. Energy
3.
of Orbitals in One and Two of Relativists Fermi Gas and
Pressure
Versus
Potential
Mass-Radius
for
Relationship
Heat
Energy,
Gas
Boson
219
Fermi Gas
219
Temperature
White
7. Photon Condensation
8. 9.
218 219
as a Fermi Gas
5. Liquid'He
6.
of Degenerate
Entropy
4. Chemical
218
Dimensions
Capacity, in One
and
Entropy
Dwarfs
219
of Degenerate Boson Gas
221 221 222
Dimension
Stars 10. Relativistic White Dwarf !!. Fluctuationsin a FermiGas
222
12.
222
Fluctuations
13. Chemical
14.
Two
Orbital
in
222
a Bose.Gas
222
Potentia! VersusConcentration Boson
System
.
223
It is
a fundamental result of quantum
and molecules,areeither
regime
the
which
in
They behave
bosons.
or
fermions
is small
concentration
all particles,
that
theory
in
including
atoms
alike in the classical with
comparison
the
quantum
concentration, (i
Whenever n >
the
iiq
gas
s
\302\253
ifQ
to be
is said
A)
(Mt/2tt/ijK'2. the
in
regime
quantum
is called a
and
quaniutn gas. properties between a quantum gas of ions a gas in the classical lemi and one of bosons is dramatic, and boih are unlike A low has a high kineticenergy, low heal regime. Fermi gas or liquid capacity, a high and exerts magnetic susceptibility, low interparticle collisionrate, at absolute zero, A Bose gas or liquid has a pressure on the container, even \342\200\224 of concentration in the ground orbital, and these particles high panicles
Bose
the
called
in physical
difference
The
a superfluid,
as
act
condensate-\342\200\224may
with
practically
zero
viscosity.
For many
systems the concentration
important variable.The
is
n
obtains
regime
quantum
fixed,
is the temperature when the temperature
and
the is
below
t0 s n ~
by the
condiiion
is often
said to
h
be a degenerate gas*.
realized
defined
was
by
Nernst
iiq. A
that
B)
{2nh2/M)n213,
gas
in
theentropy
the
quantum
regime
with r
\302\253 ro
of a classical gas divergesas logr
both fermion boson and theory removes the difficulty: -+ a the i so that 0, gases approach unique ground stateas entropy goes to zero.We say that ilie entropy is squeezed out on cooliuga quantum (see gas 3 and S). Problems in the classical regime{Chapter6) the thermal number of particles average as
t -*
in
an
0. Quantum
orbital
of energy
\302\243 h
given
by
Wilh Ihe result for^
appropriateto
this
regime,
-
J{z)
with
the usual
choice of the origin of eat zero
The
form
assures
D)
is any
fcrniion
A
spin. occupancy
low
0 or
of
I,
it is
temperatures
by ihe
=
/
zero
the classicalregime. a half-iniegra!
composite\342\200\224with
low-iying orbiials will
all orbitals
energy below
<
0
with
are
there
which
just
fermitm
one
have
eF will be
e <
occu-
orbital
these limits. Ai
between
anywhere
is always
orbital
Pauli exclusion principle to an occupancy
average
lowest orbital.
of the
energy
picture of
original or
dear that many
orbital. At absolute I. Here ef is the
in each with
an
with
D)
occupancy of any
panicle\342\200\224elementary
is limited
fermion
A
our
with
\302\253!,consistent
the
for
the average
us that
,
(n/HG)exp(-\302\243/i)
occupied orbitals
enough
system. This energy is calledihe = Oiit r = 0. As t increases the Fermi will orbiusls t-iierfiy. Abo\\cr-fall have/ a high energy mil, as in Figure distribution function will develop 7.3. Bosons have integral or zero spin. They or composite; be elementary may if composite, they must be made up of an even number of elementary particles if these have spin \\, for there is no way to arrive at an integer from un odd of half-integers. The Pauli principle does not appiy to bosons, so there immber is no limit on the ground ihe occupancy zero of any orbital. At absolute in the orbital of lowest orbital\342\200\224the energy\342\200\224is occupied by all the particles the is increased the ioscsits As lowest orbital temperature populasystem. single
to hold the
number
population only
contain
each excited
and
slowly, a
of
orbital
orbital\342\200\224any
small number
relaiively
to the
assigned
ofparticles
of particles.
We
higher
energy\342\204\242will
this
discuss
shall
feature, carefully. Above r = r0 the ground orbital losesits special becomes much like that of any excited orbital. occupancy low-lying
Fermi
is called
gas Fermi
ihe
with
energy.
energy lower than orbitals of higher wlicrt
regime
the
ihe
in
will
Chapter
The
compared
The
most
striking
A
vacant.
entirely Fermi with
gas the
is said
Fermi
is occupied
orbital
An
to be
energy,
as
no udegenerate in
the
classical
6.
of tlie theory of degenerateFermi gases the wliiie dwarf stars; liquid 3He; and of 3 fermion gas is the high kinetic property
unportam applications conduction electrons in metals, matter.
be almost
one fermton. ishigli
temperature
treated
degeuerate when the temperature is low in comparison the orbitals of When the inequality i \302\253 e^ is satisfied and the be almost Fermi energy ef will entirely occupied,
energy
it contains
when
fully
its
GAS
FERMI A
point ;md
include
nuclear energy
level f, for
Fermi
16 eleclrons;in
60 \342\226\240|
gTOUnd
\"a
slalc
Hie
the
louesi eight levels
(!6orbilals) are occupied
(a)
(a) The to a fine of
7.1
Figure
confined
spin up and one for Orbitals above the
energiesof spin
down,
n = i,2 10for an etecfron level correspondsto two orbitafs, one for The ground siaicof a system of t6 electrons. orbhals
the
L. Each
lenglh
(b)
shaded region
in the
vacant
are
Slate.
ground
that it Is necessary ground state of the system at absolutezero.Suppose to accommodate N nomnteracting electronstn a length L in one dimension. What orbitals will be occupied in the ground state of the N electron system?
of the
In a one-dtmenstonal is a positive form
the
crystal
integer
stn{fJ7ix/L)
number ms
If the (i
=
other
Any
filling
we
and
3, 4
1, 2,
fill
higher
tire
with
in,
orbitals
in Figure
then
7.1.
electrons in
the
ground
and
fiHed,
a higher
from
starting with
the
in
are \302\261\302\243
gives
arrangement orbitals
The orbitais that are filled shown
=
by
supplemented
it,
the
spin
quantum
spin down.
up or
spin
8 electrons,
has
system
empty. state
\342\200\224 for \302\261 j-
free electron orbitalof
of a
number
quantum
it until
ground
=
1
all
the orbttais
state
the orbitais
of higher n are
energy. To constructthe at
the
botiom,
N electrons
with
and we
ground
continue
are accommodated.
state of a systemof 16electrons
are
Ground Sttite of Fermi Ground Slate of
Fermi Gas in
in Thtee
G&s
Dimensions
Dimensions
Three
be a eube of sideL and volume V = L3. The orbilais have the of C-58) and their energy is given by C.59). The is form the Fermi energy Ef of the highest filled orbiiai at absolutezero;it is determined energy by the that the N in the stale Iioid requirement system ground electrons, with each filled wiih one electron up to the energy orbital the system
Let
E) <5)
Here
the
radius
filled
and
is
tiF
separates
of a
sphere (tn the space empty orbitals. For the
orbitais must be filled
to
up
nF determined
of the system
factor| in
n
because
arises
the
n, = {lNfn)ll\\
F)
has two possiblespin orientations.The nx, ny, nz in the positive octant of the sphere
triplets
only
to be
are
space
ihai \302\273r)
tty,
an electron
because
2 arises
factor
nx,
N electrons
hold
by
y
The
integers to
counted. The volumeof the
sphere
We may
is 4nn//3.
then write E) as
This
Fermi
the
relates
so-called \"Fermi total
The
energy
=
[/\342\200\236
with
En
with
F),
=
2
have
e, \302\243
=
2 x
conversion
=
N/V
n. The
| x
In (8)
471
\\^r
eln n2
eB *= --1
j j
P'
(in
\302\273a ,
(S)
and (9),n is an integerand isnot N/V.Consistent
let
21 in the
electron concentration
temperature\"tf is definedas t> s ef. of the system in the ground state is
(h2/2m)(n}t/LJ. we
to the
energy
(9)
(\342\226\240\342\226\240\342\226\240)-> 2(fcX4n)
of (he sum into an
JdrtirV-)
integral.
Integration
ground state kineticenergy:
l0m\\L
of
(8)
gives
the total
0 10
40
30
20
60
50 in
Volume,
Total ground stale energy
7,2
Figure
elecirons,
80
70
90 100
cm3
Uo
of one
mole of
volume.
versus
and is f of The average kinetjcenergy is UJN per particle the Fermienergy N the energy increasesas the volume decreases cF. At constant so the to the that Fermi a contribution (Figure 7.2), energy gives repulsive in most metals and in white dwarf and neutron stars binding of any material; it is the most tends important repulsive interaction. That is, the Fermienergy the volume. It is balanced in metals to increase the Coulomb iiltraction by between decuoos and ionsand in she stars by gniviimicurjS attraction. using
and F).
E)
Density of States
Thermal
averages
rr denotes
where
the
n; and
orbita!
distribution
orbital
function, energy
for
independent
the
quantum
problems
particle
orbital;
XR
is
the
have the
value
form
of the
quantity
A'
in
f(t:a>T,!i.)is the thermal averageoccupancy,calledthe disof the orbital n. We often express <.Y) as an integral over the
\302\243. Then
A1)
becomes
A2)
Density of States
where
transformed to an integral by
has been
orbitais
over
sum
the
the
sub-
substitution
(\342\226\240\342\226\240\342\226\240)-> *>(eH' \342\226\240\342\226\240)\342\226\240
X
A3)
Jrfs
of energy betweent and t 4- dt. The cailed the density of slates, although it is more always accurate to call it the density of a of orbitais because it refers Jo the solutions one particleproblemand not to the states of the N particlesystem. Consideran example of Ihe calculalion of \302\251(e). We see from G) thai the number N of free electronorbitaisofenergy iess or equal lo some e is than Here
of orbitais
number
is the
quantity 'D(e)is nearly
N{e)
=>
V. Take
for volume
A4)
,
(V/in2)BM/h2)il2til2
the iogarithm of both sides:
logN and take differentials
of
The quantity dN =
log
s=
N and
(iN/lsjdt is
A5)
+ constant,
flogs
loge:
the
number
of
of energy
orbifals
between c
and e + (/e,so that ~dNltlt
C(e)
is
the
counted
write
density
of
\302\251(e)as
a function
two spin orientations of an electronhave this derivation because they were counted in F). We of e
been
can
alone because
JV(\302\243)A
from A4). ThenC 7)
A7)
ZN{t)llt
The
orbitais.
throughout
=
=
(V/3n2){2m/h2)y'2F.in
,
(IS)
becomes
A9i
Chapter
7.3
Figure
of energy,
dimensions. densiiy
and BaseGm
Density of orbitals as a function for a free electron gas in three The dashed curve represents the of occupied orbilals ai a finile
/(eVD(e)
temperature,
bm
comparison
wiih
the
7: Fermi
occupied
is small
such thai r
in
cF. The shaded area rcprcsetus orbiiah ai absolute zero.
Energy, e
When
*D(e)
total
function (Figure 6.3), the densiiy of orbilals (Figure 7.3).The becomes of occupied
multiplied
orbilals number
*\"\342\226\240
the disiribulion
B0}
described Fcrmi-Dirac distributionfunction of probkms where we know the total number patiides, from ihal the total of calculated number requiring particles value. The total kinetic energy of ihe electrons is correct
where
is Ihe
f{t)
V
If Jhe
sysiem is which
above
in
they
the
ground
are vacani.
in
we B0)
Chapier
determine be
equal
=
6. In }t
by
to
Ehe
B1)
stale,
all orbitais
are filled up to
the
energy
\302\24
The number ofekcirons is equalto
B2 and
ihe
energy
is
B3)
Heat Capacityof Electron
Heat
Gas
of Eleciron
Capacity
a quanliiaiive for the expression of electrons in three dimensions. The
derive
We
gas
impressive accomplishment ideal monalomic gas the much
lower
agreement
wiih
heal capacity of a degeneraie
calcufaiionis
heat capacity is \302\247W,bui The calculation that
for
The increase
results.
theexperimemal
of N electronswhen
0 10
from
healed
i is
(he
in a
denoted
meial
~
an very
agree-
of a
energy AU
by
im-
For
excellent
gives
total
most
gas.
elecirons
follows
in
Fermi the
perhaps
of the theory of (he degenerateFermi
are found.
values
Gas
U{x)
system \342\200\224
L'(Q),
whence
f{c)
is (he
energy
range.
Here unit
-
=
AU
JjVcrfWO:)
Fermi-Dirac fimetion, and O(c) is We multiply the identity
N \"
B4)
J0\"(E).
of orbilals
number
the
~
B5*
J^W
Jo\342\204\242
per
by tf to obtain
+
{jo use
We
The
first
electrons
fo rewrite
AU
=
from
- tr\\n^y0{c)+
Je(b on
integral
B6)
B4) as
B6)
JJ
- f/dttfiiz).
ifjAttrfteW
side of
the right-hand
ef to the
bring
contributionsto !he
energy
ihe are
eiecirons positive.
~
W)
fW&iz)-
B7) gives the energy neededlo take
orbitals of energy \302\243 >
ihe energy neededto
- $0
fc'M*r
The
the
and \302\243f,
to ef
second
from orbifals product
integral
gives
below ef. Both
f(c)'D{c)de in the
firsi
of electrons elevated to orbitals in ihe energy is the number range ihe in the is an energy e.The factor [1 \342\200\224 second integral probability /(\302\243}] thai an elecironhas been removed from an orbilal c. The function A (/is plotted versus function 7.5 we ihe distribution Fermi-Dirac in Figure 7.4. In Figure plot of the Fernii for six values of the lemperalure. The electronconcenlration \302\243, was laken such that tfjkB ~ 50000K, characteristic ihe of conduction gas electrons in a meial.
integral
dc at
The heat respect
to r.
capacity
of
the
electron
gas is
found on
The only temperature-dependentterns
in
differentiating
B7)
is /(e),
AU
with
whence we
Figure
7.4
of the energy of dependence The fermian gas in three dimensions. in normalized form as AU/NeT,
Temperature
a noninteracting is plotted
energy
uhere
N is the
is ploued
number of decirons.The temperature
\302\243\342\226\240
S; 05 ^
as xjtF.
0.4
4
3 e/A'b, Figure
Fermi-Dirac
7.5
graph as
the
was energy
calculated
at which/
in uniis
7
of 104
K
for lempcraiures, in ihreedimensions.Th gas The chemical of temperature. with the help of Eq.B0)and may be read
disiribuiion
= J.
ai
function
TF = \302\243F/kB= 50000 K. The resuhs apply number of particlesis constani, independent
each temperature
6
5
0.6
various
10 a
Courtesy of
B. Feldman.
Heat
-Region of UcMsieraic
quantum
Capacity of
ElectronGas
gas
/
/
Rc\302\253ion
\342\200\224-^
of classic-,
\\
Figure
gas of
7.6 Piol of ihe chemical poiemiai /i versus noninteraaing fcrmions in ihrcc c!iincisions.
can group
units of
ihc
plotting,
ji
i arc
and
temperature
T
for
For convenience
a in
0.763cf.
terms to obtain
B8) At
(he
that the approximation
derivative df/dx to
nseiats x/eF < 0.01,and we see from Figure 7.5 only at energies near er. It is a goodapproxiof orbitais
of interest in
lures
tempera
the
evaluate
is
large
density
integral:
Ccl S
Examinationof suggests
that
potential ji constant
the
when t in
cF. We
the
graphs \302\253 we \302\243f
Fermi-Dtrac
B9) hi
Figures
ignore
the
7.7 of the variation of/i with x temperature dependence of thechemica! 7.6 and
distribution
function
and
replace ;i
by
the
have then:
C0)
7;
Chapter
thai
;i is
calculated for the
Gas
of she
free electron Fermi and three dimensions. In common a= 0.01 at room (empcraiure, so
potential
gasesin one meials t/nF
and Bost
with temperature
V&riaiiors
1.1
Figure chemical
Ftrmi
ji, for
lo cF.Thesecurves expansions of the
closely equal from series
number of particles in
ihe
were integral
system.
We set
x a follows
and
it
We
may
from
safely
B9) and
C0)
replace the
(e.- e,)/t , \"
that
lower limit
integrand is already negligibleat such that cF/x -~ f 00 temperatures
by
x = -ef/r or
more.
The
inicgta!
is not
demciuary,
but
may
p,
ondilTeremhuion
of bsjih
sides uiih
rcspcci
be cvafua^d
-V
lo ihc
if
are
we
The integral*
+ iI
\342\200\242
because
-co
~
3
from ihe it2
parameter a.
n
the factor concerned
becomes
Heat Capacity we have for
whence
she heat capacityof an
r
when
gas,
\302\253
xf,
I
C4)
units,
C5)
Ctl of orbitajs at the
the density
that
found
We
-
\302\253(\302\243,)
electron
free
for
a
if
is not
For r
gas,
the temperature \302\253 xF
Gas
'
- WUEfU.
Cel
In conventional
electron
Electron
of
the
with
xF
s
eF. Do
06)
not be deceivedby
gas, hut
for r
is degenerate;
gas
= 3iV/2tf
3N/lcf
ofthe Fermi
Fermi energy i,
only
\302\273
if
(he
reference
a convenient
is in the
gas
notation
the
rF:
point.
classical regime.
Thus C4) becomes
C7) in
units
conventional
there
is an
extra factor kBl
so
that
C8)
C\\,
where
A'flTF
s
sF. Again,
TF is
but not an actual temperature,
only
a reference
point.
We can give a
physical explanationof the
form
of
the
result
{37). When the
those electrons in states within specimen is heatedfrom absolutezero,chiefly because an energy range r ofthe Fermi level are excited the FD thermally, over a region distribution function is affected of the order of r in width, illusisofthe order illustrated electrons 7.3 and 7.5. Thus the number ofexcited by Figures r. The of A'i/\302\243F, and each of these has its energy increasedapproximately by as total electronic thermal energy is thereforeofthe order of Uci Ni2/eF. Thus the electronic contributionto the heat capacityis given by
C9)
A'r/rF
is directly
which with
the
experimental
proportional to t, results.
in
agreement
with
the exact
result C4)
and
Chapter 7:
Bos
and
Fctmi
Fermi energy parameters for
Cuiculaied
Table l.\\
free
eleciroi
Comiuciion
Fermi
eieclron
concemralion
NiV.
of
1.1
3.S
K Rb
1.34
0.85
2t
2.4
1.0S
0.79
1.8
0.56
2.1
Cs Cu
0.73
1.5
8.50
1.56
Ag
5.76
1.38
Au
5.90
1.39
7.0 5.5 5.5
the
4.7
and copper, silver,and
64 6.4
electron per
valence
one
have
gold
the conduction
becomes
electron
valence
the
electron in
the
metal.
of conduction electrons is equa! to the concentration the be evaluated eilher from and ihc atomic densiiy may
concentration which
lattice dimensions. weight or from the crystal If the conduction electronsact as a free eF
1.8 8.2
In Metals
atoms,
energy
5.5 x 10J 3.7
2.5
10\"
[. k,
inK
4.6
atom, and Thus
1.3 x
x 10\"
T., =
in eV \302\243F,
Na
alkali metals
The
s~'
SV, in cm
cm'3
in
iempcratur
energy
Li
Gas
Fermi
Velocity
Fermi
gas,
the value
of she Fermi
from G):
be calculated
may
fermion
ef = {hl12m)Cn2uI!\\
Valuesof
vF at
velocity kinetic
the
given in Table
ef are
of
and
n
surface is also given equal to ef:
is
-
^ttuy3
where eFjkB
T
nt is for
\302\253 TF
the mass of the in
it is
defined so that
HO
. \302\243f
of (he
Fermi \\emperatureTF ~
the order of 5 x l04K,so that (he derivation of C5) is an excellent approxima(ion
The heat capacity of Sow
\\ab!e;
electron
are of
the
assumption at
room
and below.
temperature
the sum of
(he
tn
values
The
electron.
metals
ordinary used
7.1 and in Figure 7.8.The
the Fermi
energy
D0)
an
temperatures
many
electronic
the sum
constant
at
metals
and
contribution
a lattice
volume
may be written
as
vibration contribution. At
has the form Cv
~
yi
+ At3
D2)
Na
a functio Rb
monovaienl mclals. The siraighl line is dra whh iiin !brEf ^ 5.835 x 10\023'n1/J ergs,
J
Eleciron
Figure 7.9 TK
After
where
y
from C7),
capacity
Lien and N.
E. PhiJHps,
A are
Ii is
helpful
10\302\260
in cm~3
concentration,
values Phys.
characteristic
constants
for polnssium, plolled as 133, AI37O A964}.
to
display
as a plot
t
and
of the material. Here y
is dominant
the experimental
discussed
at sufficiently
values of the
in
s
Chapter
jn2N/iF
4, The
iow temperatures.
heat capacity for a given
of Cvjx versus t2: Cy/t
~
y + Ax2
The intercept the points should lieon a straight line. the value of y. Such a plot is shownfor potassium in 7.9. Figure of y are given in Tables 7.2 and 7.3.
for then
C/Tv
Rev.
and the lattice vibration term-4i3was
electronic term is linear in material
5
Expcrimcnlalheal
W. H.
and
2
10\"
5
D3) at t Observed
\342\200\224 0
gives
values
7.2
Table heat
of monovalent
capacities
y
eteciron eSecironic
and free
Experiment!
metals e electron},
Cexp),
1
mJmol\"'K\"
y/y0
Li
1.63
0.75
2.17
Na
1.38
1.14
1.21
K
2.08
1.23
Rb
2.41
1.69
1.22
Cu
3.20 0.695
197 2.36
0.50
1.39
Ag
0.646
0.65
Au
0.729
0.65
Cs
1.35
1.00
1.13
The values of \342\226\240/ nud yo arc in i iUlU u: Oluilcsy of N. li. 1'hillim.
oni:
Table
7.3
values of
ExpenmenUt
Li
Be
1.63
0.17
Na
mb
declronic heat capaciiy consiain
y of
mcials
I Al
1.38 1.3 K Ca 2.08 2.9 Rb
Sr
V
Cr
9.26
t.40 9.2
2r
Nb
Mo
2.80
7.79
2.0
_
Sc
Ti
107
3.35
Y
10.2
\021
Mr
Cs
Ba
La
Ht
T*
W
2.7
10.
2.16
5.9
1.3
value of y
Cu 0.695
Ni
4.73
Tc
3.6
sOTE;Thc
N
Si
P
Gc
As
1.35
2.41
[3.20
C
7.02
Zn
Cra
0.64
0.596
In
Rh
P
Ae
a
4.9
9.42
0.646
0.638 1.69
Re
lr
Pt
Au
He
2.3
3.1
6.8
0.729
0.19
Tl
1.79 1.47
Sn
Sb
1.78
0.11
jpb
Bi
2.98 o.oos
is in ,s
fuiniihed
White Dwarf White
dwarf
E
by R
Phillips
and
N. Pear
Stars have
slars
masses
comparable
to
that
of
the
mass and
The
Sun.
radius of Itic Sun arc Q
*
2.0
x 10!3
g;
Ko = 7.0 x
dwarfs are very small, perhaps of the which is a normal star, is of the Sun, density of water on the Earth. The densitiesof white dwarfs The
radii
of white
10'\302\260cm.
O.Oi order
are
of the
that of
D4)
1
gem\023,
exceedingly
Sun. The like
that
high, of
the
White
are
107gem\023.
oflO'1 to
order
into
ionized
entirely
degenerategas,as Besselobserved
that
its predicted
2.0 x
3O33
estimated
as 2
radiant
energy
in
shown
below.
the
of the
path
dwarf
white
and the electron gas is a In 1844
discovered.
to be
star Siritis oscillatedslightly
a straight
about
x 109cm by a flux,
comparison of the
the
using
surface
of thermal
properties
temperature
ts
the
and
radiant energy developed
4.
Chapter
mass and
The
and
dwarfs
prevalent in white
invisible companion.The companion, SiriusB,was discovered position by Clark in 1862.ThemassofSiri us B was determined on the orbits. The radius of Sirius B g by measurements
lincasifithadan
to be
free electrons,
nuclei
of Sinus was the first
The companion
near
be
will
the densities
under
Atoms
Stars
Dwarf
radius of Sirius B
to
lc;id
the mean
density
D5)
This extraordinarilyhigh following words; \"Apart reason
particular
times
100
atoms at a
x
2
10~6A3peratom. order of 0.01 A, as
molecule
electrons form
graviiational
In
the
temperature
A
the no
dwarfs
white
has a mean
density
a volume
a
10\" atoms
The average
2
x
per
atom
equal
\\o
iO~30cni3pcratorn
mol\021)
nearest-neighbor separation is ihen
of
of 0.74 A in a separation compared with the internuclcar atomic the of hydrogen. Under conditions of such high density ionized are electrons The are no longer attachedto individual nuclei. is held togetherby in dwarfs an electron matter the white gas. The which is the binding forcein all stars. attraction, the interior of white dwarf stars* the electron gas is degenerate; The Fermi is much less than the Fermi energy energy of au ef.
cr *
discussion \302\243ood
WinHon,
No. 2
density of 106gcni~3have
electron gas at a concentration of 1
'
Other
suspicion.\"
Maanen
=s
(I06molcm\023){6
and
Van
named
~
A
the
incredibility
the calculation with
that
by Eddington in 1926 in of the result, there was
appraised
higher.
Hydrogen
or
the
from
to view
densities;
have higher
was
density
1973.
as (h2/2i}i){3n2nI13
of while
dwaif siais
is
x
0.5
given
is given
103Oelcetronscm~3
x
10~6erg
by W. K
as i
x 10s eV
Rose, Aitropkvsics,
Hotf.
by ,
R
D6)
characierisiic
Liquid
energy of
Fermi
7.4
\"able
of mailer
Particles
3Hc
atoms
Metal dwarf stars
While
degenerate fsmiion
g
values)
Tf, in
K
0.3
demons
5
x
electrons
3
\302\253 !09
Nuclear
matter
nuctcons
Ncujron
stars
neutrons
tQ1
3 x \302\2730u 3 x 10!I
about 10' higher than in a typieal metal. The Fermi temperature zFikB of the m the interior electrons is =s 3 x 109K, as in Table 7,4. The actual temperature of a while dwarf be order I0\021 K. The is believed of the of electron to gas in the interior of a white dwarf is highly because the thermal energy is much degenerate
lower
Fermi
the
than
energy.
energies in the relativistie regime?This question arises because our nonrelativistic of the Fermi gas has used the theory expression the an kinetic electronof momentum The energy p2/2m for p. equienergy of equivalence of the rest mass of an electronis the electron
Are
=
\302\243Q
This
will
x
\302\273 A
of
is
energy
effects
me2
* 1.
x I010cins-!):
10\"\"g)C
D7)
order as the Fermi energy D6).Thus relativistic densities but not dominant. At higher the Fermi gas
same
the
be significant,
is reiativistic.
NuelearMatter We
which
state
the
consider
matter
nuclear
of matter
within nuclei.
is composed
qualiiaiively. We
estimate
here
of a
contains
A nucleons
that
nucleus
R
Accordingto volume
goes
b nuclear
this
relation
as R3,
the
gas, at
fermton
degenerate
of
least
energy of
x Al!\\
x ltT13cm)
average
protons
tbenucleongas;The radius is given by the empirical relation
the Fermi
A.3
a
a
form
The neutronsand
volume
which is proportional to
per particle A.
The
D8) is constant,for the of nucleons
concentration
matter is ?
S
0.11
x 1039cm-3
,
D9)
Potential
Chemical
about
103 times higher than
Star.Neutrons the neutronsneed and
concentration of
one
protons not
not
the
Fermi
equal
Absolute
in a
white dwarf
particles. The Fermi energy of energy of the protons. The concentra-
identical
are
oilier, but not
Or the
of nudeons
concentration
the
Zero
Near
both, enters the
familiar
relation
E0)
\302\243,=\342\204\242C>r'\302\273)M
let simplicity number of neutrons. For
us suppose that the number of protonsis equal Theit
from D9)
as obtained
a cm\023 \302\253\342\200\236\302\253\342\200\236\342\200\236(,\302\253
on dividing
\302\243C.17 \302\243f
x
x 1039
0.05
*
'W>n>
2. The
by
Fermi
energy is Mev.
kinetic energy of a particlein a degenerate Fermi gas is is energy, so that in nuclear matter the averagekineticenergy
The average Fermi
the
{51}
,
x 10\024erg \302\253 27
10\023>1/3a;0.43
to
E2)
-J
of
ihe
16 Mev
per nucleon.
BOSON A very
GAS
remarkable
CONDENSATION
EINSTEIN
AND
effect occurs in
transitiontemperature, of particles the system will below
a
gas
of nonintcmcting
a substantial
which
fraction
bosons at a certain of the total number
called occupy the single orbital of lowestenergy, the orbital of second lowest the other orbital. Any ground orbital, including the will be occupied by a relatively same temperature at negligible energy, be number of particles. The total occupancy of all orbitalswill always equal to the specified number of particlesin the system. effect ts The ground-orbital called the Einstein condensation. stale be nothing surprising to us in this result for the ground would There occ ccupaney if it were valid only below I(T14K.. This temperature is comparable in a system whh orbitals the energy spacing between the lowestand next lowest of volume 1 cm3,as we show below. But the Eitistein condensation temperature of for a gas of fictitious noninteracting helium atoms at the observeddensity most familiar is the is very much higher, about 3K. Helium helium liquid example of Einstein condensationin action. in
Potential
Chemical
The key
to
the
of a boson
system
Near Absolute
Einstein at
Zero
condensation low
temperatures.
is the
behavior of the chemicalpotential potential is responsible
The chemical
of a large population of particlesin the ground a orbital. system composedof a hrgc number N of nonintcractmg is at absolute zero all particlesoccupythe the bosons.When lowestsystem and orbital the system is in the state of minimum energy. It is ceriainiy energy should be in the orbital of lowest noi surprising dim at i ~ 0 ail particles We can show orbila! that a substantial fraction remains in the ground energy. at low, although experimentally obtainable,leniperaturcs. our energy on If we scale, then put the energy of the ground orbital at zero from the Bose-Einstein distribution function for
the
stabilization
apparent
We consider
<53)
the occupancy
e obtain
i\342\226\240-+ 0 the
When
Here
we
have
dial
know
the total
made
.v, which
in the
use
ground orbital at e =
of the
occupancy
of particles
number
of the
0as
ground orbital becomesequal to the
total
system, so ihal
of the
*s series expansion cxp{\342\200\224 x)
hji/x, must be smallin
wiih
comparison
number of particles N could not
be
large.
I
~
unity,
From
x +
\342\226\240 \342\200\242 \342\200\242. We
for otherwise
this result
we
find
E5)
asT
ForN = 1022at
-.0.
T = IK,
we
have
/i
-1.4 x lO'38 erg.We
s
noic
that
fromE5)
E6)
as i
~+
0.
The
chemical
energy than the ground b\302\243 non-nesative.
potential
in a
boson system must
state orbiral, in order
tirar
rhc
occupancy
always
be
of every
lower
in
orbital
Cxampte;Spacingoflonat
of an atom free to move
where
n,,
n>4
are \302\273,
integers.
positive
of free
('
of volume
a cube
in
ttrbiiuh
lowest
second
and
orbital
=\302\273 /-1
The energy e(I 11)of
the
energy
eB1 i)
of one of
set of
the
atoms. The
Zcr
of
energy
an
is
orbital
lowest
ihe
is
E8)
1) ,
+ 1 +
ind
Near Absolute
Potential
Chemical
next lowest orbitais is
E9)
+ 1+1). W
energy of
excitation
lowest
The
is
atom
the
As =*\302\243BH)e(III)
lfA/{4tie}
- 6.6 x lO'^gandi, = = C){8.4
Ae
In
is extremely
spiining
ijtK,
iO\0211
closer to
/i]/r}
is
\342\200\224
.
F0)
cm,
= 2.4S x
NT30erg.
F1)
- 1.S0x KT1*K. small,
tiiRtcuU to
it is
conceive that
it can play unimportant accessible temperatures such as reasonably I mK {55} gives )i ^ -1.4 x temperature zero of energy.Titus /i is orbital ES) as the than is the nexl lowest orbital E9), and cxp{[t(J 11)dominates the disso that t(Ill) ap{[t[2li}~n\\h}t
and
ntucit
the
orbjia!
ground
closer
io
than
l
is
function.
distribution
Boltnnann
factor
exp{~A\302\243/i)
exp(-1.8 which is essentially unity. would first excited orbit;il gi\\cs
x
physical problem even at the lovvest is I0\021 K. However, at the whicli = \\0:i aiouts, referred to the for N erg
much
The
3
(-)
x IO-3!)(9,S6)
in a
part 1
Ae/*8
units,
temperature This
I
=*
an
entirely
By D)
vuluti
1
mK.
x 10-\") we would
is
s I expeel that
1.S
x
even
I0\0211
,
*
\302\253cilte
if/t
F2)
occupancy
of tlie
the order However, ihe nosc-Eiiliit-'itidis.tr ibutioii ofhii;il: of ihe first exched for the occupancy
only be of
dilfctcni
at
of 1.
Ae
because
\302\273 p..
of
the occupation
Thus
the
orbiial
exciied
fust
ai
1
mK
i:
F4)
so thai
the
5 x 10\" 13t
of
which
is very
is [datively
temperatures
simple
low
called
from
thai
are
in this
small. We see thai
the
occupancy
particles much
very
F2).
lower
in
*
is
is quiie in
left
as
orbiial, in ihe
5
exciied
be expeciedat
distribution
ihe greatest part of tlie population The particles in ihe ground temperatures. the Bose-Einsicin condensate. The atoms
the atoms
orbital is/.iV of the first
than would
The Bose-Einstein
in which
situation are
factor
Boltzmann
cienily
the N
fraction
first
long
condensate
as
iheir
low
oibiialai
from the
sight
strange;
the ground
=
10i0v 10ir
x
ii
favors
a
orbiiai at sufi'lnumber is \302\2731,
act quiie
differently
stales.
excited
do we understand the existence of the condensate? Suppose ihe aioms were for holding distribution (Chapter 4), which makes no provision by ihe Planck of photons consiani the loial number of particles; instead, the thermal average number increases wiih temperature ai i\\as found restricted ihe 4.1. If lav. s of in Problem the nature of photons loiiil *iumbcr to *i vliIuc $i we wotild suy thitt the i^rousid orbital of ii\\c plioion the difference No = S* ~ N(r) between the number aiioued and ihe number gas contained excited. The ,V0 noncxciK'd tjic would be described as condensco1 into thermally photons but becomes zero at such ;ill ,V a that orbital, A'o ground essentially photons temperature i, arc excited. There is no actual on the totiil number of photons; constraint however, there is a constrain! on the total number A' of material bosons,sucli as MIe atoms, in n sysiem. of the condensation into the ground The diiTcrctKc This consiniint is tht' origin oibiul. between the Planck distribution and the Bosc-Einstein h the laner will lliat distribution of so conserve the tot;tl number none\\ciicd (hat particles, independent of icmperaiurc, are really in the ground atoms ilalc condensate. How
governed
Orbital Occupancy We
fora and
saw
in A9)
VersusTemperature
that the
number of free particle orbitalsper unit
particle of spin zero.The total exciied orbitals is given by the
of atoms
number
sum
of the
have
for
/@,t),
integral
in
F6)
the
ground
F6) the
separated
the
in
occupancies of all orbitals:
N . We
ofheliutn-4
is
range
energy
number gives
sum over
n
of atoms
in
the
number
into the
two
parts.
ground
of atoms
Here N0(t) has been written orbital at temperature t. The
NJ,i) in all excitedorbitals,with
\\
\\ \\r
= 0.5
t
/(\342\200\242.
-\342\200\224\342\200\224 \342\200\224\342\226\240\342\200\224__
o!
7-10 Plot of Ihe boson distribution particles present to ensure ). a I. The integral Figure
slates arc
gives
condensed
the
funciion
wilh sufficient two temperatures, of times the density distribution the rest ofthe particles present orbitals; on value of No is loo large to be shown for
ofthe
number N. of particles in exciied into the ground slate orbital. The
Hie plot.
The /(e,i) as the Bose-Einstetndistributionfunction. integral gives only the number of atoms in excited orbttals and excludes the atoms in the ground = because 0. the function is zero at e To count the atoms correctly orbital, D(t) we must count separately the occupancyNa of the orbital with e = 0. Although in a gas of of No may be very the value large only a singleorbitalis involved, bosons. We shall call NQ the number of atoms in the condensed phase and Nt the number of atoms in the normal phase. The whole secret of the result wluch follows
is that
in energy
at low
to the
temperatures the chemicalpotentialp is very
ground state orbital than
ground state orbital.ThisCloseness the
population
of
the system
into the
of
p to
the
first
excited
the ground
much
orbital
closer
is to the
orbital loads most of
ground orbital (Figure7.10).
7:
Chapter
Fermi
and Base
Gases
The Bose-Einstein distribution function
when
written
e - 0 is
NM=V^~, where
as
in
E4),
in
all
excited
or, with
x
will
X
orbitals
s
on
depend
for
the
at
orbilal
F7)
the temperature
x. The numberof particles
as tm:
increases
e/z.
Nolice the facfor
zil2 which
of Nc. dependence of in Ihe si ale low sufficiently ground temperatures particles a very large number. EqualionF7) tellsus that / must be very close to wiUbe / is very accurately constant, becausea macwhenever 1. Then unity No is \302\273 to be for the validity value of forces closeto unity. The condition /. macroscopic ;V0 of the calculation is that No \302\273I, and it is not required that Ne\302\253 N, When to small g s\302\273 i in the integrand, deviathe value of the integrandis insensitive -' deviations of a from 1 tn F8), although not in F7). 1, so that we can set /. Ihc
gives
The value of the integral*in
The
temperature
the number
At
infiniic
= U1 10 >\342\226\240
gi
F8)
is, when
). =
1,
it ton
Thus ihe numberof atomsin
ts
states
excited
Temperature
\\.IO6VB\\H\\3;:i
..
G0)
where
=
/iQ
N to
(Af r/2rr/i2)a/2
is again the quantum
obtain the fractionof atomsin =
N./N
The
X
value
We
concentration.
divide
Nf by
orbitals:
excited
2.6l2nqV/N
=
G1)
2.6l2na,'n.
which fed to G!) is valid as long as a large \\/N are in the ground state. A!I particles have to be in some in an excited orbital or in the ground The number in orbital=
1
or
1
\342\200\224
of atoms
number
either
orbital,
to small excited Orbitais is relatively insensitive changes in X. but the rest of this we must take /. tlie particles have to be in the ground orbital.To assure very close to 1 as long as NQ is a large number. Even 103 is a large numberfor Yet witliin ihc occupancyof an orbital. Ar/rE = 10\"fi of the transition,where is > !015 a loins is defined orbilal r\302\243 by {72) below, [he occupancy of the ground at the concentration of liquid 4He. Thus our argument is highly cm\023 accurate
at
Ar/r\302\243
10~5.
Condensation
Einstein We
=
Ihe
define
Tcmperaiure condensation
Einstein
number of aionis in
which the
atoms.
That
is,
A^frJ
\302\253 N.
temperature* states
excited
Above
ilie r\302\243
as
i\302\243
for
(cmperaiure
total number of of the ground orbital is is macroscopic. From G0)
is equal
occupancy
(he
to (he
not a macroscopicnumber;below n the occupancy A7 for Ne we find for the condensation temperature witli
G2)
M \\2.6I2k Now
Gt)
rimy
be written
as
G3)
wlicrc
is
v;irtes ;is
ii:2 a I
jV
value
of
tlic
tola!
Atademic
of atoms. below
Ictuperiitures
T^foralomsof
in,
number 4Hc
The numberofatoms
if;, as
sliown in Figure
in
excited
orNuils
7.11. Tlic uik'iilaled
is a=3 K.
dcr W'issenschaficn,
Berlin, Siuunesbcricliie 152-1,261;1925.3.
aid
Fermi
7:
Chapter
Hose G:
1.0 \"n
/
0.8 \\
t
Superfluid
component'
\\
/ ormal
flui
\\
omponen
/ y
of the of
ihc
i.o
0.8
gas: tempcraiuce dependence in ihe ground orbiial and of aloms in all exciled orbilals. We boson
Condensed
Figure7\302\273tl
\\|
0,6
0.4
0.2
0
proportion
No/N
pfoponion
NJN
ofaioms
as normal and superftuid have labeledihe two components wilh the cusiomary description of liquid !o agree helium. arc intended !o be zero at The slopes of all Hucc curves
The
number
of particles No
that
note
We
N may
the
in
=
have
phase
said
or the
that
be
the particles in
in the
ground
temperature
VM
is
G4)
even
slightly
the ground orbitalbelowt in
is given
k'elvin
the
molar
volume
in cm3
For liquid helium Vu~ 27.6cm3
mol\021
mol\021
and
less
than te
a
orbital, as we see in Figure7.11.
M)
where
from {73):
form
superfluid phase.
The condensation
= 0.
{x/x\302\243?ir\\.
be of the order of 102\\Forx
large numberof particleswill We
-
- Ne = N[l
N
is found
orbital
ground
x
and
M
by the
the
condensed
numerical relation G5)
.
M
= 4;
is the
molecular
weight.
thus TE = 3.1K.
Liquid4
Liquid
He
4He
tempera! ure of 3 K is suggesiivelycloselo j he actual tempera Jure a! whicha transitionlo a new stale of matter is observed to Jakeplace in helium liquid (Figure 7.12). We believe that in liquid 4Hebelow2.17K there is a condensation of a substantial fractionof iheatomsof 4He inlo the ground orbiial of jhe system. This is different from the condensation in coordinate of a gas to a liquid. Evidently ihe iit the condensation j'nicrspace that occurs of4Heat 4.2 K under a pressure of alomic forces that lead to ihe liquefaction one atmosphereare too weak to destroy the major effects of the boson concondensation at 2.17 K. In this respect tlte liquid behavesas a gas. The condensaof bosons. condensation into the ground orbital is certainly connected with the properties
The calculated
of 2.17K
2.5
2.0 \"\302\253,
s
t.o
y
X
0.5
0
1.6 1.8 2.0 u;
Tempera!
Heat capacity
Figure 7.12
peak near
2.17
K is
of
liquid
evidence of an
*He. The
vanishingly
viscosity
by rate small,
above
of flow
through
sharp transition
important
nature of the liquid. The viscosity above the transition temperature is typical liquids, whereas the viscosity belowtlic in the
determined
of
the liquid of normal
merely becauseof et al.
as
transition
narrow
slits is
at least I06 times smaller than the transition. The transition is
called a lambda transition of the graph. After Kccsom
2.6
2.4
2.2
tlte
the
often shape
Chapter It Fermi
is normally not
condensation
The
Gases
Base
twd
may act as bosons,as in
in metals. A
different
superconductivity
type
of transition
been observed
has
properties
permitted for
the
in
but
fennions,
of electron
to complex
of fermions
pairs (Cooper
phases
with
pairs)
superflutd
Atoms of
3mK.
below
3He
liquid
pairs
3He have
of 5Hc atomsact as bosons. can several of liquid as a in support of our view helium give arguments of this is a drastic A t first oversimplification sight gas noninteracling particles. of fite problem, bur there are some important features of liquid helium for are fermions,
\\ and
spin
but pairs
We
the
which
is correct.
view
{a) The molar volume
absolute zero is 3.1 timesthe volume [hat we calculate from the known interactions of helium atoms.Titeinteraction between forces of helium atoms are we!!known aud pairs experimentally and from these forces by standard methods of solid theoreticaily, elementary state physics we can calculate the equilibrium volume of a static lattice of to be 9 cm3 helium atoms. In a typical volume calculation we find the molar
mo!~', as the
atoms
helium
tn
structure
expanded
distances.
appreciable
responsible for
the
liquid
4He at
mo!\"l. Thus the kineticmotion leads to an exhas a large elTccton ihe liquid siale and which the aloms to a certain extentcan move freely over motion We can say ihat ihe quantum zero-point is
with
compared
of
of
the
27.6 cm3
observed
volume.
of ifie inoiar
expansion
ofliquid helium in ihe normal state are not very (b) The transport properties a ihosc of normal classical gas. In particular,the ratio of the different from thermal conductivity K to the product ofthe \\iscosity heat times the tj capacity per unit mass has the values ~_
JC__ JJCV
[3.2,
These vaiues are quite closeto
ihose
The
values
Table
temperature\342\200\224see selvesin
gas
the
at the
liqirid
14.3-
are with
same denstly.
in
an
at
2.SK
at
4.0 K
|16,
for
observed
order
Normal liquids
of
gases
at
room
the transport coefficientslliem-
of magnitude act
normal
quite
of those
calculated
for
the
differently.
are relalively weak, and I be liquid does not exist (c) The forces in [he liquid above the critical temperaiureof 5.2K, which is itie maximum boiling pouit ten limes stronger in Ihe observed. The binding vvotiki be perhaps energy :i si:uic lull ice, hul the expansion of ilte molar of equilibriumconfijjuralion volumeby the quanlum motion of the atoms is responsiblefor tile zero-poml The value of ihe critical value. reduction in the binding energy to the observed
io binding energy. iemperaliireis dirccliyproportional (d) The ikjuid is slablc at absolutezero pressures 25 aim the solid is more stable. the
ai
muter
25atnt;
nbuvc
'He
Liquid
r
1 as
g I
T, Fi\302\253iire7.13
Comparison
uuder
4He
iiqujd
onset of high
gravify
fluidify
in
K-
of rales of flow through a fine
and B.M.
3He and
of liquid hole.
Abraham,
fhc suddcr
Noiice
in \"He. After
or superfluidity
B. Weinsiock,
Osborne,
'lie
r'Hz
\\
D.
Pliys.
W.
Rev.75,
9S8 A949).
The new stale of ntallcr
2.17
has
K.
quhe
inio
asionishing
which
liquid
4He enters
The viscosity
properlies.
when cooled beiow
as measuredin
a
flow
is zero (Figure 7J3), and the [hernia! conduciiviiy is a supervery high. We say thai liquid 4He below Ihe [ransilion icmperalure fluid. we More denole *He below ihe transiUon lempcralurc precisely, liquid as liquid and He U, and we say llial liquid He II is a mixlurc of normal Huid of Hie helium Tlte normal fluid component consisfs Supcriiuid coniponcnls. of consisis aloms excited in thermally orbiials, and the superfluid component life helium atoms condensed into the orbital. It is known lhat liie ground tat:e in Ihe in liquid 4He does not radioacljve boson 6He in solution part in 3Hc in solution supijrflow of the latter; neither, of course,doesthe fermion 4llc kike part in ihe super/low. We speak of liquid \"Mle as liquid ik i. ;ihovc liie iransilion leniper^lnrc Thereis no supcrdnid in Ikjuid lie 1, for here the grama! oihii;il coiliponcnt of uKiyuiiude as ihe oct.'up:mt;y is ucgti^ihlc, order ot-vupiuicy being uf l\\ic suim:
experiment*
fliii.ls
ofiliirftciil
Bic.isufi;
is essenlially
visttiMlics,
the .ivct.igc of
K'ii,
sums i)f 'be
c\\fKfinn:liii
aierjgt*
iiicisuic lluidiiy.
ifJiraif(Jj;t;
i imnily.
.t\302\273Juliet
l'vjil'Ijjjk'j
Kjurc
7.14
The mcUing
curve of liquid
and
helium (*Hc). and the transition curve between itie two forms oftiquid He I helium, and He II. The liquid He ll form exhibits solid
sispcrliow
ptopciiics
as a
consequence of Uic
condensarionof aroms into the ground orblial of the s; stem. Note ihat licl'mm h a liquid at absolute zero ai pressures below 25aim. The tiquid-\\apor
this gcaph
boiling
curve is
as ii would
pressuic line.After 19,626E950).
C.
in
included
not
[he zcco
wjih
ractgc
Phys. lev.
A. Swenson,
of any other
in which
temperature
liquid
seen. The
we have
as
orbital,
low-iying
He { and
({ exisl are shown
regions of pressureand in
7,14.
Figure
The development of superftuid propertiesis no[ an automatic consequence A dvanced of [he Einstein condensation of aloms into the ground orbira!. interaction
aloms
among
aloms
[he
existence of some form (almost form) of interany [hat leads [o the development of superfluid properties in in the ground orbital.
that il is [he
show
calculations
cal-
condensed
Phase Relationsof Helium
The
phase
of
diagram
4He
was shown
can be followed
from
any appearance
of the solid. Al
called He I, makes II.
He
A temperature
Ihe
a
transit
ion
called
the/
in
Figure
poinl of 5.2 K
crilical
ihe
transilion
to the
The
7.14. down
to
discussion of solid heli 1967, pp. 85-95. Solid 'He
An tnlercsring
August
poinl is the triple point al which
m
by B. Bertram and in three crysral structures
is given
:xisis
curve without
liquid, form wilh superfluid properties,called
who first solidified and vapor coexist.Keesom, the solid* did not existbeiowa pressureof 25alm.Another
can,
zero
Ihe normal
temperature
liquid He ii,
\342\200\242
iiquid-vapor
absoiule
hciium, triple
R. A.
liquid
He
point
exists
Guyed Scientific Am ro rhc condili
according
(,
found that
;
'He
of Helium
/ Solid
/
\342\226\240i
5
Liquid
/
40
Gas
7.15
Figure
Pliase
kelvin and (b) in
slopeshown a
in (a)
'he
lo
for
liquid
In rhc
on ihc phase boundary liie liquid, and we
liquid io
[he
has
soiid
have io add
it. Superfluid
solidify
(a) in
3He,
region of negaiive
ihan
eniropy
higher
heal
diagrams
miKikeivin.
properiics
JHc. The appear in(b) in ihc A and B phases of liquid A phase is double\342\200\224in a magnetic field [lie phase divides imo hvo componenls wiih opposiie nuclear magnciic momenis.
at
K:
1.743
He I and regions
here the
He II.The
of existence
The of4He.
Figure
negaiive
two
7.15
slope
triple
of He of
phasediagram
ihe
soiid is in equilibriumwiih
3He
exhibiis
of ihe
poinis
are connecicd
II and He I. differs
in a
ihe
iwo
by a
liquid
modifications.
line !ha!separatesihe
remarkable way from the phasediagram of ihe fermion nature of 3He.Note
the importance
coexistence curve at low temperatures.As
explained
in
negative slope means thai
10, the
Chapter
ISlower lhan !heentropy
the
purposes
many
!hc
ground orbila! by when
only
I
laboratory reference frame\342\200\224as
Ilie
when
supcrfluid for the
energy,
superfluidhas energy to tlic
relative
a velocity
is set
superfltiid
of the
atoms
exeitation
superfluid is given
of the
of mass
center
he
No
energy. The
no excitation
has
definition
il behaves as
of liquid helium
component
superfluid
were
it
liquid phase
phase.
a vacuum, as ifi! were not thereat all.The are condensed into the groundorbila!and have no if
of! he
entropy
and Superfluidity, 4He
Qunsipariieies
For
solid
the
of
inlo
to
relative
flow
the laboratory.
component of Na
The condensed
as
flow
the
in tile
cxcitatiosis
ereate
not
does
atoms make transitions bctwecilthe
Such iransitions might in
irregularities
the
The
transitions,
loss
from
can
occur.
of the
wall
the
relationshipof
and the
superfluidity
excitations
free
a
E
between the energy e and show
that
superfluidity
of helium
atoms
cause of energy loss and
=
the
is
flow
!!. If
resistanceless
not
the
involves
He
in liquid
orbilais of freeatoms,with
so
is,
as no
long
orbitals.
with
irregu-
tube through which the helium atoms are Rowing.
the moving fluid,
The criterion for
viseosity so long
!ho excited
and
orbital
ground
occur, are a
if they
with zero
superfluid\342\200\224that
by collisions
caused
be
flow
will
atoms
collisions
such
were
orbitals
excited
the
if
relationreally like the
Momentum
and
energy
momentum
of
relation
parliele
= -_(/,/,-)*
l\\.ivi
Mo
momentum
G6)
or /ik
Here would not be expected.
of an atom, k =
then
we
can
But
^/wavelength.
atoms the low energy the bceause of the existence of interactions between but are longitudinal sound excitations do not resemblefree particle excitations, waves, longitudinalphonons(Chapter4).After all, it is not unreasonable that a in any liquid, even though we have longitudinal sound wave should propagate no A
of
has
language
many
of superliquids,
experience
previous
grown
These
atoms.
up to
describe the low-lyingexciiedstatesofa
slates are called elementary
excitations
articles. particie aspect the states are called the elementary excitations of liquid He II. shall
Longitudinal
(jiiasip
We
menial evidence superfluidity.
This
for
condition
this,
but
will
first
show
elementary excitations leads to the
we derive
us superfluid
wiiy
give
and
phonons
the clear-cut
a necessary conditionfor the
phonon-tfke
behavior
of liquid
system
in their
nature
He II.
are
experisuper-
of the
Quaupanhlts andSuptrJluMty,
V down
velocity
He
Body of mass Mo a cylinder th;it at absolute zero. 7. [$
Figure \342\226\240\342\200\242a
'lie
It
a steel ball or a neutron, of with V mass down a column of liquid helium aJ rest at falling Mo velocity absolute zero,so that initially are excited, if the no excitations elementary Jhere will be a damping nioliou of the body generates excitations, elementary forceon Jhe body, in order to generate an elementary excitation of energy ck alid momentum hk, we must satisfy the law of conservationof energy: We consider
in
Figure
7.16
\\M0V2
where
V is
Furthermore,
a body,
*=
perhaps
+
|A/0F'2
the velocity of the body after creationof the elementary we must sarisfy the law of conservation of momentum =
Ma\\\"
conservation taws cannot always direction of Jhe excitation created in
the
hk.
+
The two if
G7)
, \302\243k
be the
satisfied process
excitation.
(IS) at the
same time even
is unrestricted.
To show
7; Fermi
Chapter
and Base Gases
this we rewrite G8)as A/0V
and lake the
On multiplication
by
-
Af0V
0
we
l/2Af0
G9)
from
+ h2k2
2A/0/iV-k
/V
|A-/K2
subtract
l
square of boihsides: M02V2
We
-
G7) to
= MQ2V'2.
have
k +
\\M0V'\\
G9)
obtain '
1
\342\226\240
(SO) 2mTq
V for this lowest value of the magnitudeof the velocity which will occur when the direction of k value equation can be satisfied.The lowest is parallel to that of V. This critical velocity is given by
There is a
-h2k2 Vs
The conditionis a become
very
large,
little
-
minimum
simpler
of-
(81) if we let the
to express
mass Mo of the body
for then
(82)
A
body
with
moving
in the liquid,so that to
be zero.
A
body
a lower the
velocity than
motion
moving
will be with
Vc
will
not
be able
to create
resistance less. The viscosity
higher
velocity
will encounter
excitations will
appear
resistance
and
Quusipartfctes
*lle
Superfluidity,
from below. The slopeof this line is equal to the critical velocity, if as for the esciiation of a free atom, [he straight line has zero h2k2/2M,
the curve ck
s=
slope
crilical
the
and
Free atoms:
The
=
Vs
of
energy
is zero:
velocity
a low
minimum
phonon
energy
frequencyregionofsound is equal to of sound velocity product of vs
Phonons:
The
wavevector
the
times
Vc
He
liquid
the
t'a, or
the
to [he
in
where
waves
= 0.
oUik/2M
product
II is
(83} =
\302\243k
tui>k =
of wavelcnglh
tusk in [lie and frequency
where the circular frequency mk k. Now the critical velocity is
= minimum
is
equal
= v,.
offtr^/hk
(84)
velocity of sound if (84} is valid for ;ill wavcveciors, which it is not in liquid helium ii. The observed criticalflow arc indeed velocities nonzero, but considerably lower than the velocity of sound and lower than the solid straight line in Figure 7.i 7, presumably beeause usually the plot of Ek versus lik may turn downward at very high hk. The actual spectrum of elementary excitationsin liquid helium II has been of slow neutrons. determined by the observations on the inelastic scattering 7.17. The solid straight line is the The experimentalresultsareshown in Figure Landau for the range of wavevectots coveredby the neutron critical velocity and for this line the critieal velocity is experiments, critical
Vc is
velocity
Vc
A and k0
where
ions
Charged
\302\253
&/hk0
~
5 x
arc identified on the figure. of helium in solution in of
conditions
experimental
to the
equal
pressure
and
I03 cm
closely conditions
motion
the
of
temperature
Sueh vortex rings are transverse
appear
longitudinal
the
(85)
,
modes
covered
have been observed*to move * near 5 x 103 cm s\" veloeity
modes by
eertain experi-
11 under
have a limiting drift value of (85).Underother the ions is limi(ed at a lower velocity
vortex rings. in
helium
liquid
free partieles and to to the calculated equal
like
almost
s\021
Figure
condi-
experimental
the
by
of
motion
ercation
and
of
do not
7.17.
is more (84) for a neeessary condition for the critieal velocity that demonstrates general than the calculation we have given.Our calculation II if He at zero a body will move whhout resistance through liquid absolute at the velocity V of the body is less than the critieal velocity Vc. However, Our
* L.
Meyer
result
and
F. Reif, Phys.
Rev.' 123,727
U96t|;
G. W.
Rajfiekt, Ph)s. Rev.
Lcllcrs
16,934
A966).
in
WavevccioT,
Energy ck versus in liquid helium ai
7.17
Figure
exciiaiioHS
of 50s
units
cm\021
wavevecior i of elementary t.JJK. The paraboliccurve
represents the iheoreiicaiiycalculated helium aioms ai absolutezero. Tlic open circles lhe otigin
from
free the
3.0
2-0
1.0
and
energy
momentum
curve has beendrawn rising
wfili
from
iineariy
of the
exciutions.
measured
rising
for
curve
lo correspond A smoosh
Hie broken curve she tiieorcsiealphonon brynch m s\"'. The solid straight line
ihe poinls.
through
lhe origin is
of sound of 237 gives uniii: Ttie line imxnin jh\302\253 vdocily, gi\\e$ appropri^se in ihcse expcrimenls. After ofij/A\" over ihe region of k covered Hcnshaw Rev. t2l. 1266A960and A. D. B. Woods. Phis.
a vclodfy
the crmcai mum D.G.
femperutures above wili be a normal fluid
exciied.The
of
the
body.
fluid
The
supcrflow
component
ihrough a may
remain
aspect fine
appears first
lube
behind
the
in in
lhe
m
that are
thermally resistance to the motion in
experiments
which
side of a container. The container
component leaks out without resistance.The derivation
there
temperature,
excitations
source of
is the
component
Einstein
the
below
of demcnlary
component
normal
liquid flows out fluid
absolute zero, but
while we
have
the
the
normai
superfluid given of the
critical velocity also holds for
the velocity of the superis the niass of the fluid. Excitations A/o interaction between the flow of the liquid
this
V as
with
situation,
the tube;
walls of
the
to
relative
fluid
would be createdabove the V( by and any mechanical irregularity in the walls. of
Phases
Supcrfluid
Three
phases of liquid
superfluid to
contrast
3He
liquid
3He
transition
4He\342\200\224with
of electrons
slate
surface form a
[he Fermi
known*
be
to
In
a
in
Uvo
so
of particlesin
type of bound siaie
as
known
a diatomic like molecule, pair is qualitatively is much larger than the average iniereleciron mlerparticle spacing in liquid 3He.
but\342\200\224in
a few milhkelvin. similar to the super-
qualitatively
where pairs
in metals,
7.15b),
(Figure
of only
temperatures
The superfluid phases are beiieved superconducting
are
pair. Such a
the radius of the
but spacing
in
a metal
near
orbiials
a Cooper
molecule
or the
average
[he two electrons [hat form a Cooper pair are 3He the siates the staie. In of nonmagnetic superfiuid liquid (singlet) spin aioms [hat form a pair are in the triplet spin states of the two JHe nuclei, metallic
superconductivity
ihree
thai
orientations
M,
1, 0,
and
-1, or mixturesof these
both
the
magnetic
magnetic
\302\243nd
been confirmed.
have
properties
The
states.
three
have been explored experimentally, and
superfluids superfiuid
corresponding to spin orienta-
are possible,
supcrfiuids
magnetic ~
SUMMARY
1. Comparedto a classicalgas,a
energy, is zero
in
ground
the
The
ground
state of a free
lota!
kinezic
For
elementary
energy
C
Wll
roif^s,
f
Ph
lias high kinetic
heat capacity. The entropy of The energy of the highest filled of spin j is particle gas of ferniions
in
the
ut\\ ground
see j. C. U heat icy, 311 J. R. Hook, ii97j|;aritt PU)sicsfluiic!tn25. \342\200\242
temperature
slate.
' ~ 2,
at low
gas
low
and
pressure,
high
Fermi
the Fermigas orbiial
\342\226\240 v
stale
)
is
ly/b. Physics Today, February p. i For Bulletin 29. 5i3A97SJ. i%sici R \\d Ph
in
the
Chapter 7: .3. The
Cases
Base
and
Fermi
density of orbitats at
is
r,f
~
\"D{Cf)
A. The
heat capacity of an
at r
gas
is
\302\253 i>
units.
in fundamental
5. For a Boscgas at r
fi. The
electron
3,V/2c^.
<
rE
of atoms
fraction
the
temperature of a fj;iS
Einstein condensation
N
__2nftV
excited
in
of
is
orbitiiis
bosons
nuniiitenicthig
Y>
PROBLEMS
i.
of
Density
of orbiiafs of a
hi one
orh'ttah free
electron
and
O,(e) where i.
of area
dimensions,
two
(a)
2
=\302\273
(t,/rc)t2*rt/Aa\302\243I
is the length of the line,(b)
the density
(86)
,
in two
thai
Show
that
Show
is
dimension
in une
dimensions,
for a square
A,
\302\251i(e)
=
^tiii/Tcft2
(87)
,
independent o(e.
2. where
of
Energy
in is
refarivistic
the rest
Fentu
For p is [he momentum. is of [he (nh/L), multiplied limit, (a)
of
Show [hat
in
energy of a gas ofN electronsis given zr
this
+
+ it/
by (n/
form
with
an
energy
is given by e volume V = i.3 the
energy
a cube
in
electrons
nonrelativistic
For electrons
gas.
mass of the electron,the
n:2)ll2>
relativistic
extreme
exactly
e
\302\273 me1,
= pc, where momentum as
limit the
for
the
Fermi
by
\302\273
AncCii/n)\025
,
(88)
Problems
where
u
=
the total
(b) Show that
N/V.
=
(/\342\200\236
The general
problem is and entropy
3. Pressure electron
In a
gas
Ireaied
the
in
by
the
(S9)
Fermi
gas.
furPhysik47,542({928).
(a) Show
uniform
that
Notice
\"/.
a -*
Chemical
-a
verms
potential
where
different,
the integral the behavior
electron
Fermi
temperature.
is upward
*D, is
given
in
a
for
downward in three dimensions (Figure are
a
that
Fermi
gas
has
orbital or
to
in the
i!s energy
(b) Kind
l/f/2|J.
region
r
\302\253
t>.
Oast-0.
of p. versus t
curvature
of
Use entropy
for
expression
gas is
J,Vef.
decrease of the volume of a cube every The of is raised; to XjL1 energy an orbital proportional tin
stale uf ihc
ground
exerts a pressure
stale
ground
of
F.Jutlner,Zeilsduifl
degenerate
of
ener\302\273y
Explain graphically why the initial gas in one dimension and The C,(e) and *D3(e) Him: curves useful So set up t. It wiii be found
fermion
7.7).
Problem
of particles, and to the number N, of the integrand betweenzerotemperature
for
consider from and
a finite
the
graphs
temperature.
5. Liquid 3He as a Fermigas. The atom 3He has spin 1 = \\ and is a fermion. 7.1 the Fermi (a) Calculate as in Table sphere parameters vF, ef, and TF for 3He at absolute zero, viewed as a gas of nan interacting fermions. The density of the liquid at low temperatures is 0.081 g cm\" \\ (b) Calculatethe heat capacity = T \302\253TF and compare as observed with the experimental value 2.89NfcBT Cr W. Reese, and J. C. Wheatley, Rev. for T < 0.1K by A. C. Anderson, Fhys. 7.18. of of 495 see also Excellent the 130, properties A963); Figure surveys liquid 3He are given by J. Wilks, Properties of liquid and solid helium,Oxford, 1967,and by J. C. Whealley, \"Dilute solutions of JHe in \"Heat low temThe Journal of Physics 36, 181-210A968). American temperatures,\" principles of refrigerators 12 on on 3He-*He mixtures are reviewed based in Chapter to 0.01 K. down cryogenics; such refrigerators produce steady temperatures in
acting
continuously
6. Mass-radius
operation.
relationship for
M and radiusR.Let the
are nondegenerate. Show self-energyis -GM2jR, (a)
where
density
is
constant
within'the
white
of mass protons order of magnitude of the gravitational the gravitational constant. (If the mass of radius R, the exact energy is potential dwarfs.
be degenerate
electrons
the
that G
sphere
is
a white dwarf Consider the but nonrelativistic;
Chapter 7: Fermi
and
Sose
Gases
5.0
SO
20 Temperature,
Figure
3He in
7.18
Heat
liquid
*Hc
the
fegion
theoretical
T. Thus for
pure 3Heis slight slope. The curve
taken for
gas in the
a Fermi
of C/T
cufves
The curve for the
percent solution of
3He and of a 5 capacity of liquid The quansisy plotSed on she vcrSica!
she horizontal axis is
200
$00
in K
at constant
at constant the solution
axis
is C/T,
and
degenerate temperature
volume
pressure, of Hie in
are which
liquid
horizontal. accounts
4He
fof
indicates
at low 3He sis solution acts as a Fermi gas; she degenerafe region over to the at temperature goes higher temperature. nondegene/ate region The Sine through she experimental possHsfor the solution solid is drawn if the hrTf - 0.331K, which agrees with the calculation for free atoms effective mass is taken as 2.38times the mass of an atom of 3He.Curves J. C. Wlicatley, Amer. J. Physics36 A968). after
thai she
-3GM2f5R). (b)Showthat in the
electrons
w
m is
here
lhai (;ss
ground
she
gravitational
by ihe
if
order
of magnitude
of the
kiiicsic esiergy of the
mass of a proton, (c) Show the same orderof magnitude and kitsctic energies are of virsal theo/esiiofsnechanics), Mll3R ~ 10;ogW2cm.{d)Iflhe
she mass
required
the
state is
of an clcclrosiand
Mit
is
the
Figure
Eimtcin
mass is equal to dwarf? gas
H ss
(e) of
believed thai Show
neuUons.
she value of the Express
she
Sun
she
of
that
Heal
gas ut
capacity
of
a
lQ33g), what is the density of the while
ofa colddegenerate pulsars ate starscomposed neutron star Mll3R =s 10!7 g\023ciyi. What is neutron star with a mass equal so that of the Sim?
tn km.
resuls
7. Photon condensation. Considera science in which she universe ftclton The number number of photonsN is constant,at a concentration of I0Iocm\"s. excited of thermally we assume is given by the result of Problem4.1, photons = is Ne 2.404 Kt3/*2/] V. Find the criticaltemperature in K below which which \342\200\224 will < JV. The excess N of lowest be in mode the jVe frequency; Nr photon there is a large the excess might be described as a photon condensatein which concentration of photonsin the lowest mode. In reality there is no such principle that loui! the number of photons be constant, hence there is no photon
A1.
sions
heat
Energy,
op.idiy,
;mii
to ;i volume
cn[[opy V.
ami entropy
capacity,
;is ;i function Put
of tcmpcf.ifun; in of ;i the
y;is
del'milc
o\\
of degenerate bosongas. the
in
region
dim
ex prcs-
th for the cne-igy. hc;il of hpiil zero confiiial bosons be not ion less form; it need x <
,V noniiilcMlcling
imegnil
Find
ens
The calculated heat capacity above and below r\302\243is evaluated. was in Figure 7.12. The shown Figure 7.19. The experimental curve
;tn
cortilam volume.
tot a
that
radius for
B x
7.19
shown
in
dilTcrcnce
7:
Chapter
Gases
and Base
Fermi
between the two curves is marked;It is ascribedlo the between the atoms,
9. Bosongasin dimensional
bosons, and show
gas of noninteracting
not converge. This result suggeststlial a not form in one dimension. Take/. ~ I for
really be treated by
of
white dwarf
Relativhtk
10.
means
of rest mass m such that the
great
p
the
the
that
ground
(The
on a finite
The de
momentum.
is the
does
integral
does
should
problem
each dwarfs are
A' electrons white
kinetic
relativislic
extreme
have
electrons
onc-
line.)
Consider a Fcrnii gas of radius R. Conditions in certain of
a
for
condensaie
stale
calculation.
over orbitats
a sum
majority
energies e = pc, where
boson
stars. of
a sphere
in
for Nt{x)
the integral
Calculate
dimension.
one
interactions
of
effect
Broglie relationremains
A1 electrons of the ground stale kineticenergy on the assumption that pc for ali elecrrons. Treat the sphere as a cube of equal volume, viria! theorem argumentto predictthe (a) Use the standard of N. Assume value that the whole star is ionizedhydrogen, the but neglect of the kinetic energy of the protons comparedto that electrons, (b) Estimate the value of N. A careful treatment leads not to a single by Chandrasekhar value of N, but lo a limil above which a stable while dwarf cannot see exist: D. D. Claylon, Principlesof stellarevolution and McGraw-Hill, nucleosynthesis, M1973. 1968, p. 161; Harwii,Astrophysicat concepts, Wi!ey, ).
~
Inhjp.
2 gives the
Problem
e =
11. Fluctuations
in a
Fcwri gas.
Show
of a
orbisa!
a single
for
fermion
system
that
- A -
<(ANJ>
if
is \342\226\240GAO
fluciuation
of a
single
orbiial
of
a boson
If
as
(I
enough
deep
N ~ in
+
below
the
the Fermi
(I!)
system, fhen from
<(ANJ> =
(91)
,
in thai orbiial. Notice that
number of fermions average vanishes for orbitals with energies = I. By definition, AN s thai the
energy so 12. Fluctuationsin a Basegas.
average occupancy
E.83) show iiiat
(92)
are if fhe occupancy is large, with \302\273I, ihe fractional fluctuafions of the order ofuntty; <(ANJ>/2\302\2531, so mat the actual fluctuations can Thus
be enormous.It been said that \"bosons of thrs text has an elementarydiscussion of
travel
has
13, Chemical
the
potential versusconcentration*
potential versus the number of
for
flocks.\"
fluctuations
Sketch
(a)
particles
in
a
boson
The
first
edition
of photons.
carefully gas in
the chemical volume
V
at
both classical and quantum
icmpcraiure
r. Include
for a syssem
of fermions.
14.
at
orbitals
with
/x, and the
of
boson
orbilai
Two
(he
Consider
systent.
a system of
0 and e.
particle energies
single
temperature is r.
r
Find
such,
fegimes. (b) Du
The
chemical
the fhefma!
that
is
twice
rE
spin
same
zero, is
potential
average population
the population of the orbital JV 3> | and make what approximations are reasonable. If the atoms in a gas have integral spin (counting the sum and nuclear spins), they can form a boson condensate when the below the Einstein condensation temperature te given G2): by orbital
lowest
the
of
bosons
A'
the
e.
at
of
Assume electronic is cooled
gas
~ BTTh2iM){Ni2.6\\2VJ/\\
For atoms in the vapor phase she Einstein'condensation is very temperature In Sow because the number densities are very low\"; A995) early successful and elsewhere. Such experiMIT, experiments were carried out at Boulder, are of which mark the exciting forefront the experiments, extraordinarily complex, field. A on BEC and is literature on gas large quantum experiments theory the Web.
set of experiments (MIT) started a beam of sodium atoms with an of SO14 oven at 600K at a N/V cm\023. Whas concentration exiting happens next is the result of a number of clever with laser beams directed on tricks one pint or another of the beam of atoms. First file atoms are slowed by one laser bcum from an exit of 800 in jt1 to about 30 in s~'. This is velocity slow eiioug.lifor !0!\" ntonis to be trapped within a magneto-optical trap. of Fusthcr tricks, including evaporation, reducedthe the temperature gas to 2 /nK, the uhraiow condensate was formed. The at whieli the rE temperature at rE was again !014atoms/cmJ. concentration The atoms in the condensed are in tlie ground orbital and expand phase states released the move once from Giiiy slowly trap. The atoms in excited The out of their of the relatively steady-state positions. positions rapidly a asoms can be recorded as a funcsion laser beam. of sime after release, using One
The
law, G3). she
of
number
sudden
decreased condcusase; orbitals.
With
asoms this
appearance
in excited sechnique
of
orbitals
is
in
with
agreement
good
she t3
signasure of Bose'Einssein condensation is is a sharp peak of atoms as the semperature the
through rE. The peak comes from she wings of she line from light
lighs
scattered
scattered
by by
atoms
atoms
in
in
excited
the
8
Chapter
and Work
Heat
OF HEAT
DEFINITION
AND
2-7
WORK
CONVERSION
ENGINES:
HEAT
TRANSFER:
ENTROPY
AND
ENERGY
OF HEAT INTO WORK
2IS
OirnOl
2-S
Inequality
Sources of
252
Irrevcrsibiliiy
233
Refrigeraiors Heat
and
Condiu'oners
Air
255
Pumps
2?6
Carnot Cycle
Example:Carnol
an
for
Cycle
Idea!
Energy Conversion and the Second Palh Dependence of Heat and Work
of Thermodynamics
242
Example: Sudden Expansion of an
Gas
245
PRESSURE
245
Ideal
AT CONSTANT
AND WORK
CONSTANT
OR
TEMPERATURE
245
Work
Isothermal
IsobaricHeal
245
Work
and
and Fuel
Elecirolysis
Example:
237 240 240
Work
Irreversible
HEAT
Gas Law
Cells
Chemical
247 250
Work
Example:
Chemical
Magnetic
Work
Work
and
for an
Ideal Gas
25! 2--
Superconductors
SUMMARY
257
I'UOBLEMS
2?
1.
Heat
2.
Absorption
).
I'lwicm
:::
Pump
4. I leal
2!:<
ReiYigerator
Carnol
6. Room Air
25S
Hnginc
Engine\342\200\224Kel'rigcralor
5. Thermal
7
Pollution Condilioner
Cascade
25S
258 258
7. Light
Bulb in a
8. Gcotlierrnal
259
Refrigerator
259
Energy
9.
Cooling
Solid to
of Nonmetallic
10. Irreversible
Expansion
of
a Fermi
T= 0 Gas
259 259
and
Energy
AND
ENERGY
and
of
to a The
system by thermalcontactwith a system by a change in the external
The most the
civilizationis
combustion
physical
conversion
of heat
device to convert heat of the steam enginegave Energy
most
because
The
fundamental
da
transfer
it
reservoir
~ dU/x
is accompanied
in external
of
is a
limitations
the
understanding
com-
of the
much
to
rise
Consider
2. This
of Chapter
made
was
development of thermodynamics. central applications of thermalphysics electrical energy is generated from heat. difference between heal and work is the difference in the
the
which
with
modern energy-intensive civiliza-
into work. The IndustrialRevolution
to work.The problem of the
one
remains
transfer.
entropy
in. a
The internal engine, which converts heal to work. to dominate man as much as It serves seems him,
which
conversion
the
magnetic field, electric field,or heal from work will be clear
volume,
process
important
engine,
transfer
that describe
para meters
energy conversion processes.
by the steam
possible
reservoir.
transfer
is the
Work
we distinguish
reason
The
potential.
we discuss
when
include
may
parameters
gravitational
energy transfer.Heat is the
forms of
different
two
are
to a
system.
TRANSFER:
ENTROPY
work
of energy energy
Work
OF HEAT AND WORK
DEFINITION
Heat
of Heat and
Definition
Transfer:
Entropy
a reservoir the energy transfer dtl from is/in thermal contacl at tcmperaiure
energy transfer is what
we
transfer- Work, as
parameters\342\200\224such
t; an
ihe energy transfer, accordingto the
accompanies by entropy
to a
the
position
argument
and we see
as heal,
above
defined
being energytransfer by of a
piston\342\200\224does
not
system
entropy
a change
transfer
any
from when only to come system.There is no placefor entropy work is performedor transferred. we must be careful; the toial energy of two systems brought into However, contactis conserved,but their total entropy is not necessarily conserved and increase. The iransfer between two systems in ihermal contactis may entropy wcl! of one system increasesby as much as the entropy defined only if theenlropy
entropy
of
the
lo the
other
processes such
we
heat
processes
in
the
that
constant.Later
are
Let
decreases.
will
which
flow example
us restrict
combined generalize
the total
in Chapter
2.
ourselves
of entropy the discussion
entropy
for
the
present
the interacting to irreversible
to
reversible
systems remains processes which
of the two systems increases,as in
the
CkapterS:Heat We
Work
and
a quantitative expressionto the distinction between heat and dU be the energy of a system during a reversible change process;da and t is ihe icmperitture. We define entropy change, can give
work. Let
is
the
as the heatreceived by
of
the
xda
s
i!Q
process. By the
in the
system
(I)
principleof conservation
energy,
=
dV
which says thai the and
energy
ihe System
t1W =
is (be work performed designatingheat and For da
below.
\342\200\224 we
HEAT
of
~ zda
dV
reversible
the
dW rather
0) process.
reasons for
Our
ihan dQ and
d\\V
roles
energy
arc
explained
for dV = xda, pure heat.
pure work;
sysictn
Then
CONVERSION
INTO WORK
of the
because
consequences
work doneon the
reservoir.
the
=
(tQ
in
by iftjand
Catnotinequality. Heat processes
dU ~
B)
partly by
from
the system
have
0,
ENGINES:
HEAT
OF
on work
+ dQ ,
is caused
change
by heal added to
partly
aw
different
Have
work
and
difference in
in
entropy
conversion
conse-
two
Consider
transfer.
difference:
the
(a) All types
of work are
each other, becausethe
is zero.
transfer
entropy
ideal
An
electrical
electricalresistance,is device ideal electrical work. generator of work mechanical work into electricalwork. Because forms to are each convertible, they thermodynamically equivalent (o denotes mechanical work. The term equivalent particular,
mechanical friction or electrical work into mechanical without
and into
work
mechanical
into
convertible
freely
a
motor, convert
to
converts
An
are
all
other
work
freely
and,
all
types
in
of
work.
(b) heat
Work
cannot
be
can
be
completely
but docs
the beat,
work from
completely
heal
necessarily
converted to work. The entropy
be pennittedto ultimately
be
removed
pile
up
from
inside
work.
the
with
strip the entropy removed
the
the device.
device
from
from
way
to
do
system
with
A device that
generates
that
has been
the
the converted
indefinitely;
The only
inverse is not irue:
Entropy entersthe
into work.
converted
not leave the system
must
into heat, but the
converted
heat
input heat
cannot
this eniropy must ultithis is to provide more
Heal\302\243itgines:
of Heat
Conversion
into Work
Entropy
Figure 8.1 Eniropy and energy reversible devicegencralmg work must equal ihc entropy inflow.
as waste
heat
is given
that only
and outpul eittropy
ail
the A
there
input prohibition
all
the
heat
input
.it which
the
be ejected
need
of the input heat. Only
the entropy
heal can be convened must
thai
iransfer
heat
reversible
by the temperature
part of the
;t\\\\ay
cany
\\\\ca.t.
heat, at a tempcrat ure lowerthan
BecauseiiQ/ifc = r, entropy
in any
from
operating
continuously The
outflow
entropy
converted to work, and 10 ejectthe
the amount
than
heat
input
flow
be some
output
to
work.
To
excess
input
input heal (Figure 8.1). of one unit accompanying It follows is transferred. heat of the
at the the
lower temperatureio bctv^^.i
difference
prevent
input
of
ihc accumulation
heat; therefore h is impossibleto convert
heal to work! against
unlimited
entropy
accuniLiiaiion
tn
a
mean entropy cannot accumulate temporarily, that provided removed. Many practical energy-conversion devicesoperatein
device it
does
not
is ultimately
cycles,
and
the
contained
entropy
deviceis calleda heat The entropy contained a value
is
There the
stroke
intake
the
What
in each
not
is at a
cylinder
minimum near the beginning
of
pile up indefinitely.
of the
fraction
The
engine.
Such a cyclic is an example:
and a maximum near the beginningof the exhaust stroke. of the entropy eontent to which the devicereturns cyclically;
does
entropy
with time. periodically internal co'rnDiisTion engine
varies
device
the
in
input heat Qh taken in convened into work.?
one
during
cyele
at ihe
fixed higher
The input entropy associatedwith QjTh. inpui confusing signs, we define in this discusdiscussion all is into heat, and entropy flows as positive whether the flow or energy, out of the system, rather than ihe usual convention following according to which a flow is positive into the system and negative out of the system. If Q, is the waste heat leaving the system fixed ;j{ the lower per cycle temperature t,. = Qjxy In a reversibleprocessthis output the output entropy per eyclc is as
temperatureth heal
the
is equal
entropy
be
can
is ak
\342\200\224
To
avoid
to the input entropy:
Q,fa = QJik ,
Qi
=
14)
E)
ir,/rh)Qh.
generated during one eyeleof a reversibleprocessis the the heat added and the waste heat extracted:
The work between
W
ratio of
The
ealled
the
=
Qh
- Q, = [1 -
the work generatedto the
Carnot
=
QA
(
-\302\261\342\200\224>-
F)
Qh.
in the
added
heat
difference
reversible process
is
efficiency:
G)
is named in
This quantity
a remarkable feat: the Carnot*s
form of
derivation
of
concept
entropy
by some 15
preceded
derived
it in
1824. It
was
not yet been invented,and the recognitionthat heat is years had
a
energy.
The Carnot efficiency efficiency
honor of SadiCarnot,who
ij
=
W/Gh,
is
the
the output
highest
value of the
possible
work per unit
of
input
heat,
energy conversion in any
cyclic heat
Heat
Ens'\"\":
Conversion
of Heat
into
Work
input
Output
Figure 8.2
Entropy
and
irreversibilities ihal generate outflow
entropy inflow
at
ihe
energy
(low
new
in
entropy
lower temperature higher temperature. at the
heat engine containing inside the device. The is larger than the entropy
a teal
t,, and t,. Actual heat engines operates betweenthe temperatures have lowerefficiencies becausethe processes taking place within the device are not perfectly reversible. will be generated inside the device by irreversEntropy 8.2. flow diagram is modified as in Figure irreversibleprocesses. The energy-entropy We now have three inequalities
engine that
ft S:
&(r,/Tfc);
(9)
A0)
actual
The
energy conversion efficiency j;
obeys the Carnal inequality
W/Qk
ijc only in the limit of reversible
i; =
have
can
We
takes in heat at ih and ejectsheal at rt. The Carnot inequality is the basic limitationon any heat in a cyclic The result te!!s us that it is impossible process.
heat tnto work. For a is
efficiency
with
increases
increasing
tJt, -* go. The
waste
usually Th high compared limited
which
are
is
by
materials
an
in
temperatures
constraints.
the
below
all input
conversion
be
ultimately
environmental
input
temperature
are unfortu-
practice
In power plant
steam turbines,
operate continuously for years, the upper temperature willi the strength to about and corrosion 600 K\"by problems 300 K and Th = 600 K, the Carnot efficiency is i;c = $, or
limited
of steel.With = r, 50 percent. Lossescausedby unavoidable irreversibiliiies to 40 about eilicicncies T oobtain typically percent. Higher
reduce is a
this ellkicticy problem
in
high
metallurgy.
temperature
Sources
operates
to
expected
currently
heist engine must
K. Higheilicieney requires
to 300 K. The usable
various
to convert
that
inoperation. The limiting efficiency !00 percent efficiency only when attain
of any
heat
that
engine
the highest
tJt,
so thai r, c;innoi be
about 300
temperature,
we
but
the environment,
into
unfortunately
reversible
xhfxit
low-temperature
ejected
temperature
given
under
obtained
ratio
a device
of
operation
Figure S.3 illustratesseveral
of UreversibHity.
common
sources
of
irrevcrsibility:
(a)
Part of the bypassing
cylinder
the
heat
input
actual
walls
Qh may
energy
duriiig
to the low temperature,byprocess, as in the heat flow into the
flow directly
conversion
the combustion cycle of the
combustion
internal
engine.
(b)
Part of temperature the
r, way not be availableas temthe actual energy conversion process,because resistances in the path of the heat thermal across
the temperature differencex,, in
diflcrencc
tesnperature
drop
-
of
flow.
(c)
part
of ilic
work generated may
be convertedbackto heat
by
mechanical
friction.
(d)
Gas may
expand irreversibly without expansion of an idealgas into n vacuum.
doing
work,
as in Ihe
irreversible
Irreversible wfthout
expansion or
work
heal
bypass
Four sources of trreversihilily in heal engines: the conversion ihermai resislance b> passing energy process, of tile heal flow, frtclional and losses, entropy generation Figure
8.3
irreversible
heal
How
in the
palh
during
expansions.
Refrigerators to move work consume engines in reverse.Refrigerators low temperature r, to a higher temperature the energyConsider rA. no in 8.1. Because entropy flow diagram of a reversibleheat engine Figure its operaiion can be reversed, with an enlropy is generatedinsidethe device, e*act reversal of the energy and Ot>-\\i. F) Equations D) ihrough entropy
Refrigerators i'^ai from a
remam valid
are heat
for
the
reversed
flows.
Chapter S: I hat
The
but
G), to
temperature
of interest
ratio
energy
efficiency
Work
and
the
ratio
tlic
work
value
reversible
in
with
i;
<
i always,
consumed
f\302\260r 'he
W;Qh
y can
be > 1 or
of
tem-
low refrigerator
the Carnot
confuse*/ =
0,/W heat engine;although W = Qh - Qh the work
yc. Do not
efficiency of a
conversion
energy
by
conversion
tlic
at
is called
operation
coefficient of refrigeratorperformance, denoted 52 \302\273;
tile energy
is itot
refrigerator
Q,/W of iho heat extracted This ratio is calledthe coefficient
s
consumed.
its limiting
performance;
y
u
in
< 1.
From
E)
Eq.
and
is
A2)
The Carnot coefficientof refrigerator performance
is
-
Th
This ratio can be larger Actual refrigerators, that generate entropy
or
smaller
than
A3]
T,'
unity.
like actual heat engines, inside the device.In a
always
irreversibilities
contain
as in the ejected at the higher temperature, ettcrgy-entropy that all energy and entropy flows Figure 8.4.With the convention
we now
place
are
positive,
have ah >
in
is of
excess eniropy flow diagram
this
refrigerator
of (8).
e, ,
Further,
& 2 (V'Jfl
W
=
A4)
Qt
- Q, >
\342\200\224
1]Q,
Q,iW
,
=\342\200\242
< yc.
A5)
-
Q,
\302\273
Q,/yc
.
A6)
Air
ami
Conditioners
Heal Pum
Ouipui
Input Figure 8.4
The Carnot coefficient yc
energy conversion Both heat engines
refrigerator.
limit to the actual coefficient ofrefrigerator upper Carnot efficiency t\\c is an upper limit to the actual ;; of
efficiency and
flow in a
energy
is an
as the
y, just
performance
and
Entropy
a heat
refrigerators
are
engine. subject
to restrictions
imposed by the
but the
device designproblemsare totally the design of refrigerators to operateat the temperature In particular, helium or below is a challengingproblemin thermal (Chapter physics law
of
of entropy,
increase
Air Conditioners Air
automobile;
inside during
the
ofliquid
12).
and Heat Pumps or an autothe inside of a the environment.If outside interchange a building connections, an air conditionercan beusedto If Such a device Is calleda heat t| xk a beat
are
conditioners
different.
heat
and outside the winter,
refrigerators
is ejected
that cool
building
lo the
we
heat
pump.
\342\200\224\302\253 r\302\273
Hear and
Chapter Si
heat the
can
pump
Work
on the
limitations
The
much more costly Heat
air
a lower
with
of energy
consurnption
than by direct
I).
(Problem
healing
building
and
install
to
economic
make
pumps
use of heat pumps
is required
conditioning
are
economical,
largely
They
are
than are simpleheatersor furnaces. sense primarily in eiimatic conditions in which to main tain
anyway.
Carnot Cycle
The
of
coefficient
refrigerator
realize process refrigeration is
Carnot
the
of
derivation
a
by which
achieved.
The
energy
conversion made
performance
work is
generated
and best
simplest
efficiency and of the Carnot no statement about how to
from
heat,
or
about
how refrigera-
known such processis the
Carnot
cycle.
Carnot cycle a gas\342\200\224or another substance\342\200\224is and working expanded in four stages, two isothermal and two as in Figure 8.5. isentropic, compressed 1 has and The is At the the the gas expanded point gas entropy aL. temperature xk at r until 2. In the at constant the entropy has increased to the value a,,, point In the
second stage the gasis further now at constant a, until the temperature expanded, to has dropped to the value r,, at point 3. The gas is compressed isothermally 4 and then compressed isentropically to !heoriginals!a!eI. We write aL point contained in the and c,, for the low and high the values of working entropy are the to distinguish these values from and which substance, ch, entropy a, tlic Carnot cycle, at !hc low and high temperatures r, and rv For per cycle flows G\\
~ The
=
\302\260h
&I1
\" \302\260L-
done
work
by the
system in one cycle is
~
W~{Th
of the
rectangle
in
\342\226\240\342\226\240= 0 =<\302\243uta
where jjklV is the work done by the first phase is T = ts during
combine
described substance.
T,)(tfn
- ffj.
A3)
from
follows
SdU
We
area
8.5:
Figure
whicii
[he
(IS) and A9) to
the
system
-SpJV
in one
,
cycle. The
obtain the Carnot efficiency
boat taken up
;/c.
Any
by Figure 8.5 is calleda Carnot cycle,regardlessof the
process working
at
,lCyd.
Figure 8-5
into work, cycle, for the conversion of heat of entropy versus for an arbiirary temperature, working subsiance. The cycle consistsof two expansion phases 4 and 4 -+ 1). (t -* 2 and 2-<3) and wo compression phases
B -> 3 and 4
iscntropic
loop.The
rather than
Even
and outputs such
where
reservoirs.
The
Carnot
healing
Caettot
Example:
cycle-
reservoir
initially
at
is
kind
done is ihc
area surrounded by
area of ihc ilic
the
energy
conversion
need a
cycles
temperature output of heal, bu!
gas
as
exist,
cooling
cycle for an ihc
All
in principle
could
what
(lie
often
are not well-definedreservoirsat constanttemperatures. presenl
and
is done.
fact
a low
and
reservoirs
difference
temperature
in
what
input
temperature
inputs
of each
phase work
cycle is a point of reference!o indicate
The Carnol
be done,
at r^ is the
consulted
heal
one nei
line.
broken
heat
- 2 and 3 - 4), and -* 1}.The
(I
isothermal
high
Carnot
A
as a plot
illustrated
between
steam
the
occupies
We
carry
is invariably
(here
turbines,
working substance and
processes are
idea!gas.
high temperature
in
never truly
an
ideal
;s
reser-
the
reversible.
monaionjic
a volume t', and is in thermal tv The gas is expanded isolherlttally
gas
equilibrium to
a
through
the
with j volume
K, as \\vo[k heai
in
=
Qh
This
work is
furlhcr
:,.
Ihe
In
is delivered by
now
Afler
F.63).
ciioscnlo
=.
the addilional
The volume
gas.
\"
V3
3
the
the
at
from
(SI* and
low
tem-
iseniropic expansionis related
lo
the
=
VJVj.
into contact
.
(tJt,}*2
B2}
a temperature
with
reservoir
satisfy
\342\200\224
f,.
i'j,-
Kj/V'4
the
gas is
initial
has returned
this
from
CH,
ncl work
areas in Figures curves in the p-V
delivered
value
initial
8.6a
cancels
by
the
and 8.6b,
diagram
as
heai:
= ^34-
and
jy4!
on ihe gas; this -* 2 3, by B1). expansion
the work
B5)
until its temperaiseniropicatly of the choice B3}of K4l the gas volume and the cyde is completed,in this last slage
recompressed
i*. Because
temperature
is performed
to
B3)
,
VJV2
compression,
is ejected
work,
to its
-
= (t*A,K'2
To accomplish
disconnected
to ihe
liseii
at this point the work
The
end of ihe
or
ft
temperature has
B0}
disconnect! has dropped to
ihe temperalure
is brought
gas
ust be done on ihe gas.This
Finally,
as the
work
= TftV22'3 ,
point
A'r
the gas is
Next,
\0212
until
iscntropically,
=
Nr^dV/V
labeled
KfVx so thai
=
JpJK area
the
process
the
ii
reservoiris
by
TlV32:3
from
n^cdmmcal
II'l3
indicated
expanded,
peralure
the gas absorbs Die heal Q, from (Rt and delivers sy ^Icm conncclco to Die pislon. i~or an sdcaiQas
Hie process
\302\243.6a. In
Figure
*jj 10 ^n c\\lern>ii absorbed from (he
the
Vlt
|N(rh
work
gas during is the
which
_
\302\253
B6)
T()
W2i done
by
the
the cycle is given enclosed area in
are steeper lhan the
gas
by
during
ihe difference 8.6c.
Figure
isothermalcurves,
the iseniropic
so
thai
in shaded
The isenfropic Ihe area of the
v
y3
Figure 8.6 and
> r,}. \342\226\240t,(xk (a)
the
work
an ideal gas, as a p-V plot. An ideal gas is expanded Two of them are isothermal, at the temperature I* and stages. back. The shaded areas show ofthem arc isentropic,ffomtt to ti.and
The Camot cycle
riicompressed
for
in four
Two done during
compression stages,
and
two expansion stages, the net work done during (c) the
(b) the the
work
cycle.
done
during
the two
2nd
is nnitc
loop
and Work
8: Heat
Chapter
i^
to trie
ctintii
-
relation
Camot
the
of
We
of heai from
a
reversible
All
have
peratures
so, We
ihe
~-
is jusi
which
r,)/Tj,,
and on the performance
law of increaseof entropy.
of
Q{\\n)
devices
ihe lower
I with
d\302\243\\ ice
different
with
(hat mows not only ihe
2 back io
the
The
li^otit), uiihout
ihe annihilationof entropy
would
and
any
qt <
efficiencies,
ij2,
efficiency is operaied in
entire wasteheat Ql2 th,
iemperaiure
higher
The overall
as well.
heai
Q{in) to work
heal
- (ij
into work
of ihe
reversible
(Figure 8.7)ihat
refrigeralor efficient device
amouni
of heat
consequences
as a
more
W/Qj,
of Thermodynamics
Law
conversion
two
combine
could
in such a way reverse
so that
B0),
B7)
usually is formuiaiedwiihom memion of Kelvin-Planck formulation in Chapter 2: for any cyclic process lo occurwhose effcci is ihe exiraction sole reservoir and the performance ofan equivalent amount of work.\" conversion devices thai operaie beiweenihe same ienienergy = W/Qh- Were ihis not the same energy conversionefficiency ;j
is impossible
\"Ii
.
of ihermodynamics stated ihe classical
law
eniropy,
in
Second
the
direct
are
refrigerators
second
given
Imvc
iVc
G).
limiis on the
The Carnoi
m F^Rurc 8,3.
jtic rccmn^lc
TjloaflV^).
Cftk was
Conversion and
Energy
-
W(rfc
The heat absorbed from
of 3r\302\243*i
but
ihe
from
an additional
result would be ihe conversion net
violate
waste
ihe law
heai.
This
of
would require
of increase of entropy.
jhai all reversibledevicesIhat operatebeiween ii is sufficient same temperatures energyconversionefficiency, to calculate this cfiiciency for any particular device to find the common value. ~ = The Carnot cycledeviceleadsloi;c (xh ri)/rft for the common value. eslablished
we have
thai
Now
the
have the
same
of Path Dependence We have processes,
and
Meat
Work
carefully used the words heat and work to characterizeenergy iiself. and not to characterize properties of the system
We look plane,
returned
It
is not
to spe;ik of the heat content or of the work system. in the Around a closed the Cinnot once more: p-V loop cycle a net amount of work is generatedby the system, and a net amount of
meaningful
heat
transfer
is
content
of a
around
Ihe
at
But
consumed.
to
precisely
changed.This
means
the
system\342\200\224on
the
initial
that
there
being
once
loop\342\200\224is
no property of the system has exist two functions Q{a,V) and W{a,V)
condition; cannot
taken
Path
and
of I hat
Dependence
Work
different reversible energy conversiondevicesoperating between the same temperatures tj, and r, coutd have diiTercni efiiciencies f'jj > >ji conversion energy h would he possible to combine them 100 pd cfikictio into a single device wilti '\342\226\240>\342 the lessdiicienl deuce ! us a refrigerator that moves not only ihc eniire w;i>:c using ficas ofstie more efficient device 2 backto the higher bui an addiriotiii Qti semporaiure, would shcii be completely converted amount Q[in) of licaE us wt;il. Tlifs additionat huat Figure
8.7
If uvo
to work.
sudi th;n the heal Q^ and siSilc(aa,VB)toa slate {ab,Yb)
If such
loop
the are
fimcitons extsicd, the
necessarily
would
be
nut
zero,
work given
Wub
by the
transfers
;md we
to
required
differences
of heat
the system
curry
and of
h:ivc shown that
in
Q
from a
W:
and
work aroumla closed transfers
the
;tre
not
zero.
The transfers of hc:itand work between state (a) ;ittd slate (b) depend on the the two states. This path-dependence is expressedwhen taken between path we say (hat hwt and work' are noi siaic functions. Unlike entropy, temperature, energy, heat and work are not increments dO and (fIS'that we introduced and free
inlrtnsic
atlributes
in (l)and
of
B) cannot
(he
system.
The
bedillerentials
8:
Chapter
and Work
that
Twotrr
Figure 8.8
sibie proceisesin
or electrical
mechanical
which
potemial
For this reason we designated of mathematical functions Q(a,V)and \\V(o,Vy the increments by dQ and d\\V, rather than by itQ and dW. Without the path not exist that of heat and work there would dependence cyclical processes
permit the generationof work Work
Irreversible
We consider the is
heal.
from
energy transfer
or electrical
mechanical
a purely
of
processes
8.8. In
Figure
system that
delivers
or
friction
same as if the energy of &2 is increased by
Processes
no
way
to
If newly irreversible
the
reverse
to heat, either by
that
dU
reversible this
change
work in
in order
process
the
at led
and
in energy
entropy
tfWtev
the
S2
The
^ J^e
entropy
created entropy.
are irreversible because there is
to deslroy the newly conversion
place.
is newly
entropy
is created
of
state
of work
created
to heat,
entropy.
we say that
been performed.
has
work
If we look only
This
ilU2fc-
entropy arises by
created
the process
\302\253
do2
new
which
in
transferred to &2 is converted The finai electrical resistance. by had been added as heat in the first
with zero
work
pure
entropy change. The eneray mechanical
each process &i
to i\\a
net change this change
in
entropy,
as
the
a reversible
irreversible, the actual
work
in
a system,
there
is no
way to tell whether
was reversible or irreversible.For a we can define a reversible heat dQICV
change and
a
of heat and work that would accomplish If process. part of the work done on the system is a given change is larger to accomplish required amount
Work
Ineveniblt
the
than
reversible
work, tn\\'il
of energy
conservation
By
dU =
+
if\\Viim
that
so
actual heal reversible heat.
The
Example:Sudden
ideal gas. As an example of an irreversible processwe expansion of an ideal gas info a vacuum. Neither heal nor *= 0 and dx = 0- The stale is identical with the stale final
is transferred, so thai ttU that refills from a reversibleisothermai with a reservoir. The work IV,,, done
work
i\\
:o
Vj is,
equai
to
the
gas is negative;the gas transfer into the system:
heal
change
is equal
-
W[cy
lo Q,cJt,
a1-al'=
agreement
in
Ihermai
expansion
equilibrium
from volume
with
B8) and
B9)-
C0)
posittve work
does
>
Wtet <
0;
on the piston
in
an
amount
0.
C1)
or ~Wlcy/x
In the ineversible process of expansion into because neiiher heat nor work Rows entropy = 0- From C1) we obtain Qi\302\253c.
in
reversible
the
-Nilog(^/K,).
on the
Q[cy= The entropy
in
from F.57).
done
work
gas
the gas
with
expansion
on the
\\V,n= The
the
ihan
less
be
of an
expansion
more llic sudden
once
consider
transferred hi the irreversibleprocessmust
=
the into
C2)
NlogfiyV,).
this enlropy from the system
vacuum the
is newly outside:
created lVitltt
*=
Chapter 8: Heat and
Work
Systems between
8.9
Figure
heat is transferred
need not
which
be
at the
but no
work
on!)'
same temperature
for the process to be reversible.
In our discussionof irreversible is created insidethe This
systems.
heat transfer,
work
ai
In this
2. lower
the
process
heat t2.
temperature
transferred
fs
= (I/I, is from high to
heal flow so
but
actual reversible
They
entropy
energy
other
Pure
transfer.
place between out an example in worked a system a! rt to a system if it takes
=\342\226\240
that
chitT >
low
0.
C4)
remain
processes
constitute
generation.
a
natural
i!Qt
temperature;
is negative;
r3
- ri
is
.
transfer
energy
C5)
-ii.re..
1/I2)rfe,
0.
The energy transfer between two not be irreversibleif only work but AH
system by
created entropy is
The newly
negative,
from
in
entropy
We have.'.
i!U2
The
of work Jo the
We
temperatures.
the new
that
assumed
is nol she only source of irreverstbility nol involvingany work, is irreversible
iwo systems having different Chapter
the delivery
during
system
we
systems
no heat
with
different
need
is transferred (Figure8.9).
are invariably somewhat irreversible, processes the backbone of the theory of thermalphysics. which is the equilibrium limit assume hereafterthat the words
limit,
We shall
temperatures
without a further qualifier, referto reversible processes.
of
heat
vanishing
and work,
Pressure
TEMPERATURE
AT CONSTANT
WORK
AND
HEAT
Temperature or Constant
at Constant
Work
and
Heal
OR CONSTANT PRESSURE isoiherma!
F ~
energy
the ioia! work performed on a system in is io Ihe in Heimlioltz increase the free equai process For a reversible process tfQ = ida ~ d{ia), ihe system. show
We
work.
Isothermal
a reversible
\342\200\224 za of
V
because dx =
0, so
that
=
dW
in
additional
system to
terms
of
work
that
the
the
work
(he
done
dU
Hclrnholiz
is required
reservoir.
the
ideal
for
~
appropriate than
ftiuciion, more
process
dV
- d(za) \302\253dF.
processes the Helmhoitz free energy
in such
Thus
ihai
Often
gas the energy V is equal lo the
Isobaricheat
and
work.
Many
the
the
energy
When
V.
free energy,
to make up heat
does
noj
is
C6!
ihe
natural
we treat
encrgc:;';
an isotEicrltui
we automatically include for
transfer is
the
heat
transfer
ihe
from the
ihe major part of the work; isothermal process, and
in an
change
heat transfer. energy
transfer
processes\342\200\224isothermal
or
not\342\200\224
take place pressure, particularly those processes that is said to be the A to at constant pressure open atmosphere. process an isobaricprocess. A is the boiling of a liquid as in Figure 8.10, example simple
lake
place
at constant
in systems
n
F
Figure
displacing
8.10
=/<\342\200\236<<
a liquid boils under aimospheru: pressure,the vapor the aimosplierc does work againsi ihe atmospheric pressure. When
Heal and Work
ChapterS:
where the
pressure on the its
changes
system
environment
If positive,
system.
sense \"free.\" if
this
in
is
and
and is not extractablefrom
thjs reason it
is
thus obtain the
appropriate
eflccthe work
ilW =
itW
+
=
the
the
energy
enthalpy
which
V plays isi
itU +
d{pV)
for other
~d(pV) the system, =
itQ
the
from
of the is part environthe to
delivered
is
work
system
d{pV) -
\342\200\224
the
purposes.
For
work.
We
total
defined as ~ HQ ,
till
C7)
function
new
H
called
the
on
performed
d{PV)
where we have defineda
=
is provided by
this work
io subtract
pressure. If (he
atmospheric
negative, the
environment
often
external
by dV, the work -pdV
volume
total work doneon the
is the
piston
plays
= V
+ pV ,
the role in
processes at constant
C8)
processes at constant pressurethai
volume. The term/it7in
is the
C8)
to displace the surrounding atmospherehi order to to be occupiedby the system. is in these definitions the idea Implicit there that are other kinds of work besides ihut due to volume changes. Two classes of the constant pressure processesare particularly important: in whichiiodTcctive work is done. The hcattransfcr ist7Q ~
work
vacate
required
the space
d\\V
=> dF
where we have defined another new G
the at
Gibbs constant
free
energy.
temperature
~ F
+ pV
+ d{pV)
used
energy
\302\253 U
+
pV
10.
- xa ,
D0)
effective work performed in a reversibleprocess and pressure is equal to the change in the Gibbs
The
of the
in Chapter
C9)
function
system. This is particularly where the volume changes as the reaction The Gibbs free energy ts used extensively free
=> dG ,
in
useful
proceeds
in
at
chemical
a constant
reactions
pressure.
Chapter 9, and Ihe enthalpy
is
Heat and
isobaric
an
Consider
electrodes
ant) SO4~
H+
~
ciccHoiylc
noi re net
ilia t do
w&in
of dilulc sutfurie lnc acid [r igtirc
where
s. 11J.
\302\253 2H+
+
positive electrodes gas and clocirons: _. +
the
net
H2O-\302\273
Whch carried out
An
sulfurtc and
in
slowly
a vessci
temperature.
prcsburcaiKicoitMiiiil
hydrogen
move
electrolysis
acid. oxygen.
cell
The overall
A
to lhe
negaihe clecirode
g;is:
D2) water
decompose
they
with the
- H3SO4+'iOj + 2c!^.
H;O
above liircc stepsis
of the
Figure 8.11
in i o
H2.
where
the
SO4~
as dilute
^titfiir
D!)
ions through the ceil lhe hydrogen up electrons and form molecular hydrogen
lake
The sulfatc ions move to release of molecular oxygen
sum
Tnc
arc immersed philiiuiin tc sci^i dissocjaics
SO.,\"\"
-+
The
which
iti
is passed
current
llicy
acid
ions:
H2SOa When a
Constant Pressure
Temperature or
at Constant
Work
rcucikm H2
+
ctttuiluin
in lite
D3)
cell; {44}
1O3.
open to lhc almosplicrt!, ihc prticessis at power negligible pjft of the olcciricul injml
An electrical result
The process is an
temperature and constant pressure.
is the
current
passes
decomposition
example of work
being
.
through
goes
an electrolyte,
ofwaterinto done
conslain
gaseous
at constant
.
iulo
such
resistance
of the electrolyte. The effective work required to to the molar Gibbs free energies of Use reactants:
heating
is related
water
W
=
AG
G(Hj) -
-
G(H2O)
\\G{02).
Gibbs free energy differenceAG
list the
tables
Chemical
=>
decompose1moleof
- 237 kJ
as
D5)
per mole at
room
temperature.
In electrolysis tins work is performed by a one Vo. If I is the time required to decompose
(not
the
of water.
male
flows under an external I x I is lhe total Q \302\273
voltage charge
the cell, and we have
tlirough
flowing
heat!)
/ that
current
W ~QV0. to
According
are
there
D3),
one water molecule,
in decomposing
involved
electrons
two
D6)
llcnce
=
Q We
D5) 10obiain
D6) to
equate
minimum
because
VQ
A
merely
If V
<
reaction
the
Vo,
the
simplesetupof Figure al all.
happen
Ev
8.11
is possible,
and oxygen
hydrogen
f'o between
voltage
arrangement on board the
sources
for
water The
focced
cell are shown
to J.
under
through
and,
tf
Apollo
O. M.
1969.
(V
-
VQ)
x
lhc
/will
(Figure 8-12). Such adevice producesa arc connected, eucnwl current will fuel cell. Fuel cells were used as power
pcessure the
electrodes
hydrogeci-osygen
Gemini and Apoilo* ipacccta.fiand
technological limitation of cell the current density Apollo
the
\342\200\242
excess power
on
dtinking
produced
incidentally
astronauts.
the
of an
characterise
Va, the
flow,
the systems
gaseous
principal In
V >
current
finite
between
from
proceed
is called a
elecirode areas are required
The
will
the electrodes
This
obtain a
poierciuit
is right to Icfl provided gaseoushydrogen at lhc negative electrode. In lhe oxygen will the gases are permitted to escape,and for V < Vo nothing however, lo construct the electrodes as poroussponges,wiih
D4)
positive electrodeand
at the
available
D8}
be applied to barrier
must
reaction equationD4).When. dissipaicd as heat in trie electrolyte.
area.
place.This requiresa
,
~AC/2NAe
larger lhan Vo reduces to zero
voltage
alone
=
Vo
of the
sides
{law.
lo take
for electrolysis
D7)
voltage
or 1.229volts.
be
condition
the
.
two
x 10scoulomb.
= -1.93
~2/v>
lo
ceil
clearochcmi'cal in
K.I
Figure
End criU used
Bockrisand
produce
Ni
in
was
only
a
few
current,
hundred
currents. The
reasonable
Us two
low
is their
cells
fuel
operating
ranges as
fuel
per
unit
electrode
hence
mA/cmJ;
large
current-voltage characcell
and
as electrolytic
3.
and
NiO
S. Sriili^asan,
uniict ibn fwf cells:
Pt
Their
as decltoiici.
electrochemistry,
ami K.OH racier McGraw-Hill,
sVian
H .SO\302\253
New York,
Porous electrodes ecl! is an electrolysis cell operated in and oxygen supplied as fuels. The reverse, hydrogen fuels arc forced under prow tin: through dmroOes porous an The and oxygen rea electrolyte hydrogen sqwratcJ by Tree is delivered to form the excess Gibbs waicr; energy Water forms at the positive outside as electrical energy. Figure
A fuel
8.12
with
electrode and
is removed
there.
Elcnrolysis
Figure
8.13
celt or
fuel
Tliecui
ic of
indie
anges.
cell,
\342\200\224
aa electroiyiic
Heal and Work
ChaptcrS;
Chemical Work
Work performed by it is
because
work,
independent variableson which a reversible
for
then
chemical
the
derivatives by their lerm
zdo
xda -
familiar
represents
The -pdV.termis
mechanical
the
=
transfer
reservoir,
both
from
systems.
system
&x to
dN2
\342\200\224 \342\204\242
dN =i
dN,
*=
(dV1
= 0),
The result summarize
{a} The particle
(b)
=
and
~pdV
of heat,
fidN
terms
be reversible:
to
E0}
pJN.
pdN
term is
the chemical work: .
ftdN.
0. All
+
d\\Vcl
and
if
all
chemical
into
fadNt
processes
E2) gives art
the
=
be supplied to the
that must
dVi
our definition
E1}
the Work
system .&;. The chemicalpotentialsare /it and p2. If is the number of particles transferred, the total chemical
= OWcl
dWc
work
the
and
the partial
replaced
is
performed
The
have
is chemical, arc usually two systems involved, both in contact there is the sum of the contributions and the total chemicalwork In the arrangement of Figure 8.14a pump transfers particles
In particle a heat
= U(atV,N),
If U
D9}
understood
-pJV+
work;
volume change, dV
is no
is one
system
,
5.1). By
of heal
dWc
work
fidN
(Table
equivalents
Ihe transfer
from
+
pdV
represent the performance of work, ali
If there
the
U depends.
energy
thermodynamic identity of Chapter 5. Herewe
by tlie
with
potential.
process
dU =
the
the
with
particles are transferred, the number of particlesin
When
of the
associated
system is called chemical
to a
of particles
transfer
the
additional
of ihe
propcriics
potential
the
system,
The difference in
+ pldN2
pump
= (p2 -
is a\\Vr if
{52}
nt)dN.
there is
no volume work
arc reversible meaning
of the
chemical potential.
We
chemical potential:
system is the from a reservoir at
of a
work required to transfer one chemical
zero
chemical potential between two
net work requiredto movea particle from
one
systems
system
potential. is equal
to the
other.
to the
8.t4
Figure
chemical
chemical
{c} If ihe
work is the work performed when particles arc moved one system to another, with the two systems having different ITilie two volumes do not change, ihe work is pure potentials. Chemical
from
rcs'ersibly
work
two
the
;
amount
potential;no
in diffusive
are
systems
is
work
per pariicie is
required
the
in chemical
difference
equilibrium ihey have the move a particle from
io
potentials.
same chemical one
syslem
10
the other,
{d} The difference in internal chemicalpotential to the potential systems is equal but opposite
systems in diffusive
5}
{Chapter
barrier thai
two
between
maintains the
equilibrium.
gas. We considerlhc work per particle required to monatomic idea! gas from -Sj wuh concentration t\\^. to -S. with concentritiion n2 > ii,, both sysiems being al the same leinpcrntnre tFtgure 8.151 If \302\253 the work dV cati be calculated from ihe contains only a chemical work term, which 0, difference in chemical no matter how is actually performed. The the process potential, chemical with concentrations is difference between two ideal gassystems different potential
Example
i Chemical
move revcrsibly ins
02 This
rcs\\ih
isothciniiiily
compressN W
work
-
/fl
the
of a
=
t[10gOl2/\302\273iQ)
-
logOlj/Hfl)] =
E3)
Tlog0l2/Jl,}.
io lite n\\cch:tnit;tl work per particle required io compress the gas from the concentration \302\253j to the concern mi ion ux. Tltc work required io of ideal from an initial voluitw i\\ to a littal volume ''. is an particles g;ts
is cqtial
-
= -\302\247pdV
Hence
an ideal
for
atoms
mechanical
-Nx^tiVjV^
work per
NTlog{IV^}
particle is iSogtoj/iij),identical
= WtlogfHj/ii,}. to
the
result
{5A}
E3). The
ChapterSt Heat and
Work
Reservoir'
-
ilnergy
exchanges-
8.15 Isothermal diemiciit work. The amount of chemical does not change if the process is performed isoihermally with thermal equilibrium wiih a common targe reservoir.
work
Figure
work
chemical
the
ideniiiy
of
lence or
convcriibiiiiyof
Magnetic Work An
and
form
important
the
wiih
kinds
dilferem
Superconductors
of work
isoihcrmal
,-
is magnetic work.Themostimportant
is to
a
conductivity
to
in
illustrates the equi-
compress
that
electrical
per panicle
sysiems
of work.
superconductors, and Below some critical temperature Tc is electrical conductorsundergo transition from work
of magnetic
both
a superconducting
this
application
their
than
less
usually
application
here.
is treated
state wiih a
normal
slate wiih an
many
20K,
finite
infinite
apparently
conductivity. superexpel magnetic fields from their interior. If the into a cooled below the critical temperatureand ihen inserted would shield the the infinite magnetic field, we might expect that conductivity
Superconductors
is first
superconductor
interior from even
8.16).
ir i|ie
This
duciivity
shielding
the
penetration
by a
occurs magnetic field. However, ihe expulsion
is cooled below Tc while in a magneticfield (Figure superconductor that active expulsion, called ilic Metssncreffect,shows supefcon-
ilian an infinite conductivity.The Mcissncreffect the currents lhat are spontaneouslygeneratednear surface,
is more
about lG~5cm iliick. The magnetic
field
expulsion field.
by
in a
layer
is not
expulsion
Superconductors are said tobeof type It if the in a range of fields above some low nonzero,
is caused
We
always complete. is incomplete, but stilt
shall
restrict
ourselves
ark
and
Superconductors
(
)
8-16
Hgure in
a constant
transition the
Mcissner
applied temperature
supercon6tiding sphere cooled ihe magnetic field; on passingbelow tlic iiiics of induciion B arc ejectedfrom effect in a
sphere.
Thresholdcurves
Figure 8,17 fidd
versus
lempcratui;:
conductors. A below
In
TiX
Temperature,
in
K
specimen
ihe curve and
for scv \342\226\240i ;jpe
noiii..;! iib
,-->I[-lOUl
fill
111
and Superconductors
I Curt
Magnetic
Figure 8.19 arcu
A in
produces
transition that
it
electronic
is the
The superconductingstaleis a by differences
The of
iicai
in
the
with the
field,
iower
capacity 8.1 S) -di
superconducifvhy
magnetic
heat (Figure
capactiy
superconductor
sol
superconducting
a magnetic Ikld
crystai structureof the
B.
metai
x =
energy.
distinct
.
phase,
as confirmed
of the normai and the supcrconduciingstales. cxiiibhs a pronounced discontinuity at ihe onset xt; when superconductivity is desiroyed by a
ihe discontinuiiy free
rather than the
A
transition.
a phase
undergoes
system
n
Beiow
disappears. The stable phasewill t =
of liie superconductingphaseis lower
tc than
in
zero
magnetic
that
of the
fieid
normal
be
the
phase
the free
energy phase. The free field, as we show
energy of the superconduciingphaseincreasesin the magnetic beiow. The free energy the normal of phase is approximately independent of the field. Eventually, as the the free energy of the superfieid is increased, is will exceed that of the normal phase. The normaiphase superconducting phase ihen the stable phase, and superconductivity is destroyed. of is The increase the free energy of a superconductor in a magneticfield in the interior calculated as the work required to reduce the magnetic field to zero the Meissner the is account for of to required superconductor; the zero value in the effect. Considera superconductor form of a long rod of uniform crosssectioninsidea long solenoid that produces a uniform field B, as in Figure 8.19. is The work required to reduce the field to zeroinsidethe superconductor a counteracting the equai to the work required to create within superconductor field We know from electromagnetic B thai exactlycancelsthe solenoid fteld. B is given the work by theory that per unit volume required to create a field (SI)
(CGS)
B2/2{t0\\
E5a)
E5b)
Figure 8.20
Tlic free
noi\\ti\\^i
t\\otii\\3.t meiat
syndic
melai
energy
Al a $\342\200\236.
fkUi
magnetic
and Work
dcnsiiy
F* of
is
in
x <
zero
SI
unite
land by BBJ/8n
lhai Fs(r,/iJ larger
=\342\226\240
deiisiiy is lower
ihe
in
siiperconduciing si ale is ide siabic
vertical se;ilc in ftgute
equally
so
uniis),
Ba3/2j@. If Bu is BM ihe free energy siaic than in normal field
st,iie. The
ilic
Bj/l/i,,,
Fs by
CGS
siaic, and
the
An
F^i.O).
+
F5(r,0)
criikjil
liian ihe
in
the
xt
magnate
dial Fsti,O)is lower than field increases applied magnetic
field, so in
applied
temperature
is a superconductor
a
appcox.iiv\\u.lcty
intensity of the
of [he
independent
Heal
Si
Chapter
of i tic
f Jr-O)-The
is ai
drawiiij;
10 U5
applies
no* ihc nornmt
origin
und
t
ii\\
Us
= 0.
Tiiis is the amount by is raised by constant
magnelic field Ba
Applied
which
free energy
the
density
application of an externalmagnetic
the
in
bulk
in an
field,
superconductor
experiment
at con-
temperature.
There
because
is no
free energy increase for
comparable
there is no
screeningof
the
normal
conductor,
Thus
field.
applied
the
(SI)
E6a)
{CGS)
E6b)
the field magnetic energy density of both phasesversus tne free will rise of the energy superconductingphase ultimately {Figure 8.20), above that of the normal phase,sothat in high fields the specimen will be in the normal phase, and the superconducting is no longer the stable phase phase This is the explanationof the destruction of superconductivity by a critical En
a
plot
of
the free
magnetic field Wiih
Bf.
increasing
superconducting
decreases. Everything
the free
temperature
phase decreases as else
being
equal,
superconductor will lead to boili a high field. The highestcriticalfields are found highest
crilical
temperatures,
t
energy differencebeI ween --\302\273
rr,
a
and
high
critical
amongst
and \\ice versa.
the
stabilization
temperature
critical
normal
magnetic
energy
and
and
field
type I a high critical
the superconductors
in a
with
the
SUMMARY
1. Heat is
contact
by thermal
energy
process dQ = xda,
reversible
2.
of
transfer
the
only
by a change in theexEerna! parameters the system. The entropy transfer in a reversibleprocessiszero is performed and no heat is transferred work
The
Carnot
the
is
Work
of energy
transfer
that
describe
3.
t!ie ratio WjQh
limit to
4.
limit to
upper
of
5. Thetotal
work
the
efficiency, j;c = (zH to the work generated
the ratio
on
performed
the
of
QijW
&
roof
6.
the
the
Hclmholtz
is the
upper
system.
effective work performed on a system pressure in a reversible process is equal
The
=
C
energy
7. The
tJ/tj,
in a reversible temperature free energy F se U - ro-
at constant
system
process is equal to the changein
\342\200\224
when
heat added. ~ - r,), is the performance, -,'c t,/(ta heat extracted to the work consumed
of refrigerator
coefficient
Carnot
The
conversion
energy
a
In
reservoir.
a
with
U
to
particles
the
8. Tlic changein by an
external
the
to
change
- w + pK.
work
chemical
constant
at
system the
on
performed
free
a system
in
the
reversible
and
temperature
in the
Gibbs
transfer
of
free
t.lN
is pdN. energy
density
of a
superconductor {oftype
in magnetic field B is B2/2{i0
SI
and
S2,8;i
{) caused
in CGS.
PROBLEMS
/. Heat pump,
per
unit
of
heat
(a)
Show
delivered
that
inside
fora reversible heal pump tile energy required the building is given by the Carnoi F): efficiency
Chapter S: Heat and Work
2.
In absorption refrigerator. not as work, but as heal
Absorption
process
is
fuel,
supplied
home and
xh. Mobile
>
i(,h
(a) Give an
refrigerators the energy driving the a gas from flame at a tempenisure
cabin refrigeratorsmay
energy-entropy
flow
be
diagram
this type,
of
similar
to Figures
with
propane
8.2 and
involving no work at all, but with energy and entropy > ta > ij. (b) Calculate the ratio QJQhh, three temperatures rfch at r \302\253r,, where QM is the heat input at r = xhh. Assume heat extracted a refrigerator,
such
Hows
the
at
8.4 for for
ihe
reversible
operation.
Camot engine. Considera Carnotengine uses as the working that substance a photon gas. (a) Given V2, delerwne i,, and r, as well as Vl and and is heat the work done What the taken and Vi VA. (b) Qh up by the gas during the first isothermal Are they equal to each other,as for the ideal gas? expansion? Do the two (c) isentropic stagescanceleach other, as for the ideal gas? (d) Calcuit with the total work done by the gas during one cycle.Compare the heat Calculate is the Carnot taken up at rh and show that the energy conversion efficiency 3, Photon
efficiency.
4. Heat
The
cascade.
engine\342\200\224refrigerator
of a
efficiency
heat engine is to be
tow-temperature reservoirto a temperature r,, by means of a refrigerator. The consumes of the work produced part by the heat engine.Assume that both tlie heat engine and the refrigerator operatereversibly.Calculate the ratio of the net (available) work to the heat Qh supplied to the heat engine at temperature ift. Is it possible to obtain a higher net energy conversionefficiency
valuer,., refrigerator below
in
this
the
lowering
by
improved
the
of its
temperature
environmental
way?
5. Thermal
pollution.
a water
with
river
A
temperature
T} =
2CTC
is
to
be
the low temperature reservoirof a large power with a steam plant, = of 500JC. If the amount of temperature Th ecological considerations limit heat that can bedumped the rtver to \\ 500 MW, into what is the largest electrical
used as
output that the plant can deliver?If improvements would
permit raising
Th
by
what
lOQ'Q
in
effect would
hot-steam
this have
technology
on the plant
capacity? air conditioner operatesas a Carnot cycle between an outside temperature Th and a room at a lower temperarefrigerator \342\200\224 this heat room gains heat from the outdoors at a rate A[Th temperature7\"j. The T,); is P. T he to the unit the air is removed by condilioner. cooling power supplied (a) Showthat the steady state temperature of the room is
6.
Room
ah
A room
conditioner.
T,
\302\253
(Th
+
PjlA)
[
4-
P/2AJ
-
T,,2]1'2.
(b) ir the outdoorsis at 3VC and the room is maintained at 17\302\260Cby a cooling A of the room in W K\021.A good power of 2kW, find the heat losscoefficient
Amer. J. Physics 2S2 K
air conditioners
of room
discussion
and the
7. Light bulb
Carnoi
In a
19 A978).
46,
is given by
A
a refrigerator,
in
that draws
S.
Gcoihermal
to
generate
to drive
steam
A
energy.
electricity
rockdrops,
to
according
a
dQh
W. D.
Teeters,
bulb is
100 W light Can
tOOW.
be
at
the
left
inside
burning
a
below room
cool
refrigerator
mass M of porous hot rock is to be hot by injecting water and utilizing the resulting the temperature of ihe result of heat extraction, = -~MCdTh, where C is the specificheat of the
very
a turbine. As
LefT and
K.
378
tempera!ure? uiilized
S.
realistic unit the cooling coils may
outdoor heat exchanger at
refrigerator
H.
large
assumed to be temperature independent.If the plain operatesat the Carnoi ihe tola! amount W of electrical estractable from the limit, calculate energy of the rock was initially Th = T,, and if the plant is to rock, if the temperature be shut down when the temperature has dropped to Th = Tf. Assume that the rock,
lower reservoirtemperature T,stays end of
At the
the
calculation,
give
-.
constant.
a numerical
(about 30km3). C=Hg\"!K\"\"', T,-= 600 77 the units and explain all steps! For comparison: The Watch in the world in 1976was between1 and 2 times 10!4 produced 9. Cooling of nonmetatlic capacity
to Ti,
T
\302\253 0
as
by
means
solid
to
We saw in
T~Q.
solids
spin
\\
fermions
C.
electricity
kWh.
Chapter 4 that the heat
at sufficiently
(varying.) low-temperaturereservoir,and for reservoir has a fixed temperature 1\\equal to the solid. Find an expressionfor the electrical energy 10, ltrerersibte
7\",
total
10u kg \302\273 20
low temperatures is proportional to cool a piece of such a solid to Assume it were possible the solid as its uses of a reversible refrigerator that specimen
nonmetallic ~ aT3. C
of
-
value, in kWh. for M - 110 C, C,
which initial
tile
required.
a gas of expansionof a Fermtgas. Consider at a volume in A/, initially temperature V;
of mass
gas expandirreversibly
into
a vacuum,
wiihout
high-temperature Tj of the
temperature
N noninteractiilg, if
doing work, to a
~ 0. final
Lei the volume
large temperature of the gas after expansionif Vf is sufficiently for the classicallimit to apply? Estimate the factor by which the gas should be value. Give numerical for its temperature to settle to a constant final expanded in kelvin for two cases: (a) a particle massequal values for ihe final temperature = mass to the electron as in metals; and (b) a particle 10\"cm\023, mass, NjV = stars. white dwarf equal to a nucleon,and N/V 10JO,as in V}. What
is the
Chapter
9
Gibbs
Free
Energy
and Chemical Reactions
FREE
GIBBS
Example:
262
ENERGY
Comparison
of G whh
265
F
266
IN REACTIONS
EQUILIBRIUM
Equilibriumfor IdealGases Example:
of Atomie
Equilibrium
Example;pH
and
the
Ionszatton
Example: Kinetic Model of Mass
267
and Molecular of Water
Hydrogen
269 265 270
Action
SUMMARY
272
PROBLEMS
272
1. Thermal ExpansionNearAbsolute 2. Thermal lonization of Hydrogen
3. lonizationofDonor
Impurities
4. Hiopolynicr
5.
Patticlu-Antiparticle
Growth Equilibrium
in
Zero
272
273 Semiconductors
2 73
2/3 274
Free Energy
9: Gibbs
Chapter
The Helmholtzfree energy
constant
and
volume
chemical
many
F
introduced
are performed
system at many experiments, and in particular at constant pressure, often one aimo
to introduce another function at eonslant pressure ;tnd temperature.
define the Gibbs free energy
G
thermodynamie
- xa +
U
the
treat As
in
equilibrium Chupicr 8, we
A)
pV.
energy, and
this the free
call
to
as
G a
often
3 describes a
in Chapier
useful
configuration
Chemists
Reactioi
But
temperature.
reactions, is
h
spherc,
Chemical
ENERGY
FREE
GIBBS
and
often
physicists
call
potential.
The most importantproperty of the Gibbs free energy is thai it is a for a system S in equilibriumat constant pressurewhen in thermal a (H. G with reservoir The differential is of dU ~ ida
dG =
a system {Figure9.1)in
Consider at
temperature
so
that
t and
the
differential
The thermodyiiamicidentity zdds
B) becomes
cdx
+
pdV
+
contact
Vdp. with
contact
minimum
a heat
reservoir
contact wilh a pressure reservoir
that
= 0 and dt p, but cannot exchangeheat. Now dp = 0, becomes dG oflhe system in the equilibrium configuration
dGi
so that
-
thermal
in mechanical
the pressure
maintains
the
tt
=
-
dUs
B)
+
C)
is
E.39)
= dUi
dG^ \302\253ndNx.
xdax + pdV*
-
But
dGj
ttdNi
dNs
\302\273 0
= 0,
,
pdVs,
wh
ence
.
i eser
System
rese rvoir
Hea
\342\226\240oir
\342\226\240\342\226\240\342\226\240v\"'Jif
m,
a heat
v.hh
NPSu
qualiz
Pressure
whkh
reservoir
ger lo
on Hie
e pressure
and
reservoir
in
mechanical
barysiai or pressure maintains a constant pic
with a
equilibrium
system.
The
is
barysial
insuSalcd.
cserv
-
is tlic
which
condition for
at
constant
therefore, the
natural
variations
irreversible
change
G3
sign associated place entirely wiihin
laking
respect
w'Hh
&
to system
varia-
number. Theseare,thereraiher
a minimum,
be
musi
wilh
and particle
minus
the
from
direciiy
extrcmum
temperature, for G(N,z,p).
pressure, variables
That ihe extremumof follows
Gj to be ;m
the
will
than a maximum, eniropy
in (i);
increase
a and
Any
ihus
decrease Gs. With
B),
-
The
difierentiai
E)
may
Comparison of{5)and
be writicn
F)
gives
adx +
E)
as
the relations
0)
(8)
thermally
Chemical Reactions
Free Energy and
Gibbs
9:
Chapter
, = V.
Three see
Maxwell
relations
from these
be obtained
may
E) by cross-differentiation;
1.
Problem
In the
Gibbs
free energy G
=
V
\342\200\224 za
+
pKthevariabfesr
and pare
intensive
are two identical systems pui together. But U, a, V, and G are linear in the number of particlesA':their value doubles We when effects,. two identical interface systems are put togeiher,apart from one particle say that V, a, V, A7 and G are extensive quantities. Assume that only species is present. If G is directlyproportionalto N, we must be able to write
quantities: they
not
do
value when
change
G= ep is
where
quantitiespand r. If wjih
temperature,c;idi
does
not
of N because it
independent
identical
molecules,
j.V
in
change
two
the
A0)
,
Nip(p,x)
is
a
function
only
of the
intensive
and temperaiire |>ui together, the Gibbs free energy volumes
process.
of gas at
It follows
equal pressure
from this argument that (II)
We
in G)
saw
that
A2)
so that
must
be
identical
with ;j,
G(,V,p,t)
and (iO) becomes
A3)
is equal to 'h for s single-componentsystem I or G for an ideal gas, sec B1) below. Gibbs free energy per particle,G'/A'. If more than one chemical species is present, A3) is replacedby a sum ov
Thus
the
chemical
poiemiaf
all species:
A4)
Gibbs FreeEnergy
The
becomes
identity
ihermodynamic
xda = dV
+
-
pdV
A5)
^e/INy,
and E) becomes
shall
We
the
develop
G ss: YJ^jfif
that
ts
reacting
\302\243njdNj
(!6)
+ Vdp.
adz
of chemical
the property equilibria by exploiting with respect to changes in the distributionof in a t, p. No new atoms comeinto the system
theory
a minimum at
molecules
~
\302\253
dG
constant
themselves reaction; the atoms that are presentredistribute snucics to another molecular species.
ti \302\273f
Omipurhtw
iiiutupfri
Let
F.
with
sttf
its
is Jitiacut
wluit
molecular
one
from
:ihuuJ
\\\\vi
rd.ttions
iwo
= p(iVrT,K)
{cFfdN)tty
A7)
and
*=
$G/dN)tiP
We found
lM
in F.18)
for an ideal
(IS)
fi{T,p).
gas
MN.r.F) -Tlog(WKnQ) , so
that
ti(N,z,l')
is
not
and therefore
of N
independent
She iaiegra!of{S7). Thai is, f is not direciiy proportional lo N number of particles is increased. Instead,from F(t,V,N)^
But
she
Gibbs
free energy for
C(r,/\302\273,N)
t-he
\302\273 F
if ilie
system
we cannot is kept at
+
PK
gas
potential
in
the form
cotislanl volume
ihc
as
B0J
is
=
N/V
- l] +
Arr[!og(p/rny)
==
p/r.
-Vr
PO
,
NrIog(;j/THQ)
of ihe ideal gas Saw
JV;i(r,f) .is
- I].
\302\273.
by use
F =
write
F.24),
Ni[\\og(NJVnQ) ideaJ
A9)
We readily
identify
in B1)
the
chemical
us
/((t,p)
= i
lo\302\243(pJxnQ)
,
B2)
9; Gibbs Free Energy
Chapter
Reactions
Chemical
and
G \302\253 in sit.V) in We see [hat N appears by reference to the result iVjih.pV unavoidably but not in /j(t,p) in B2). The chemical potential is the Gibbs free energy per particic, A9), but il isnol ihcHctnihoHz free energy Of course, we are free lo wrile p its cilher pet panicle.
A9) or
122},as is convenient.
IN
EQUILIBRIUM
We may write the
REACTIONS
equation of a chemicalreactionas v,A,
+
v2A2
+
= 0
\342\226\240 - \342\226\240
+
v,A,
B3)
,
B4)
species Hj
in
the
reaction
equation.
species, and the Vj are the coefficients Here v is the Greek lettermi. For the
of the reaction
Clj = 2HC1 we lave
+ =
Ai
the chemical
Aj denote
the
where
- Clj;
A,
H,;
Aj
=
fj=l;
Vi=l;
HC1;
\302\273j=-2.
B5)
of chemical
discussion
The
conditions
of constant pressureand
energy isa minimumwith
The
differential
of
respect
constant
presented
In
temperature.
in the
to changes
for
equilibrium
proportions
reactions
under
the Gibbs free of the reaclants.
G is
dG
Here//j
equilibria is usually
-
- Z fj^i
+
adz
Vi!p.
B6)
is the chemical potentialof species j, asdefined by \\is = {BGJdNj)s-p. At = 0;then B6) reduces pressure dp = 0 and at constant temperaturefa
to
07)
rfC-1/,/W,.
j
The changein
the
potentials of the zero.
Gibbs
free
re\302\243tctbnts\302\273 Xti
energy comlit^rtufii
in
a
reaction
G
depends
is &i cxtrcmum
on the chemical tjo uu musl miu
for
Equilibrium
The change coefficient
dN} in
the
the chemical
v,m
of species/
of molecules
number
equation
]>>/.; - 0.
We
may
Ideal
Gases
is proportionalto the dN} in
Write
the form
\"
JNj dfif indicates
where in
how many
\302\253
,
rjd$
B8)
timesthe reactionB4)
takes
place.
The
change
dG
becomes
B7)
dG
In equilibriumdG
=
0, so
\302\273
dft-
vjfi
I
B9)
that
C0)
This is the conditionfor
pressure
and
for Ideal
0 when
utilize
We
F.48)
nj
is the
concentration
-
which
is
the
depends internal
Uul ihcmull
when
p and
genera!
the constituents
equilibrium condition acts as an ideal gas.
potential of speciesj as
of species j
-
logc;) ,
C1)
and
\302\253QJZ/int)
C2)
,
on ttie temperalure but not on ttie concentration.HereZ/int) function, F.44). Then C0) can be rearrangedas partition 5\302\273gnj
*
of
rflogMj
cj a
the
of
form
useful
we assume that each lo write the chemical fij
where
constant
Gases
simple and
We obtain a Y^Vjf-j
of matter at
temperature.*
Equilibrium
=
transformation
in a
equilibrium
is moregencrat; onceequilibrium
t arc specified..
=
\342\226\240
$>,!(*<:,
is reached,
,
ihe rcaclion
C3a)
does nol proceedfurihi:r,
Chapter 9: Gibbs
Free
and Chemical
Energy
Rcactia
C3b) The left-hand sidecan berewritten
as
C3c) side can
the right-hand
and
be expressedas C3d)
Here
the equilibrium
called
K(t},
constant, is a function only of the
temperature.
Wiih{32)\\vehave
s
K{t)
free energy
internal
the
because
C4)
nil\302\253/'
is Ffim) =
-TiogZj{int).
From
and
C3c,d)
we have
C4)
C5)
Fk-Vj
law of mass action.The result
as the
known
concentrations of the reactanlsis in
of any one
the conceniraison
concentration of one or moreofthe the equilibrium
To calculate
a consistent on our
choice of the
without
happen l\\2
^
a conscious
2H, ilie simplest
is
It
effort
not on
the
of the dissociatedpanicles(here
2H)
of
the
ground
state
of the
A
change
equilibrium
it is
essential
to choose in
partition
function
The
depends
Zji'mi)
different
need
zeros
properly the energy or free
for the energy
to arrange ibis, but it docs not part. For a dissociation reaction such clioosethe zero of the internal energy
diilicuh our
procedure is to
of each compositeparticle(here
C4),
eigenstales.
to give
be related
reaction.
in ihe
difference
as
must
reactants
different
in the
energy of each reactant.We
internal
zero of the energy
a change
reaclants.
other
the value of each
because
consistency
alone.
temperature
force
constant K(x}in
of the
zero
the
way
here
of the
a function
reaciam will
product of the
the indicated
that
says
H2 a!
molecule} res!.
to coincide
we piacc
Accordingly,
composiie particle at
~\302\243fl,
with the
where
is \302\2433
energy the energy the
energy
Equilibrium for
required
in
to be
laken
is
and
positive.
Examplei Equilibrium
of
atomic
rcacuon Hj
for the
aciioil
mass
hydrogen imo
particle into its constituents
the composite
to dissociate
reaction
the
Ideal Gases
atomic hydrogen
and molecular hydrogen. = 2H ttz ~ 2H = 0
or
The siatcmcnl of ihe for
the
dissociaiion
law
of molecul
is
C6)
denotes ihc concenmuion of nlolecutar
Here [llj] atomic
of
li
h>drogen.
hydrogen,
and
fjt]
i
foltov-sihai
l37) [h^tpo17^'
lhai is, the relume concentration proportional lo the square rooi of K is
equilibriumcon:.lani
given
logK
in
of the
terms
of
atomic
ihe
concent
at a given is inversely temperature hydrogen ration of molecular hydrogen. The equilib-
by
=
log^tHj)
internal free energy of
-
2IognQ(H) -
H,. per
molecule.
Spin
F(U2)/i, factors are
C8)
absorbed
in
F(H,}.
is H,, the of energy is laken for an H atom at rest. The more lightly bound is Kt leading of Hj in ihe 10 a higher more negative is F{Ht), and ihe higher proportion al absolute zero. eV per molecule, mixture. The energy to dissociate Ht is 4-476 be said lhai ihe dissociationof molecular into atomic hydrogen is an It may hydrogen of dissociaiion: The En associated wiih ihe decomposition example entropy gain entropy [he It is believed of Hj into two independent in loss particles compensates binding energy. reaction in not The that most the is H and of Ht: intergalactic hydrogen space present as in the direction of H by the low values of ilie concentration of Hj. equilibrium is thrown Here
[he zero
Hjdrogcn
is very dilute
Example:
pi! and ihe limitation of water.
in
intergalactic
space.
In
liquid
waicr
ihe ioni^lion
process
09}
H2O*~>H+ + OH\" eeds
to
o.ximalcly
txicnt. Al room tcinpcraluic she coiiccmraikm product
a slight by
the reaciion
cquilibfium
i
D0}
where ihc jonac ^nefcasco
aGGin&
oy
lo
as required
can
ions
decrease
of
z\\t\\
iictu
ionizaiion
be increased
\342\200\224
fOH
^=
\021
of H* ions is
wai^r
\302\273^ntJ ihe concentration of C3i 1 tons wtll decrease constant. Similarly, ihe concentration [H*][O!!'] * a hase lo ihe water, and ihe H concentration will adding
by
siaie oi wafer
The physical process
prolon
product
is more
ions ate
H*
suggests\342\200\224ihc
of H2O molecules.This
with groups*
per liicr. inpurcualcr\302\243H*l donor. The concentration
moles
in
lo aa as a
to llic
the
accordingly.
ihe
is said
mainiain
\"
of OH
arc given
concentrations
An acid
!0\021molr''.
Chemical Reactions
Free finersy and
Gibbs
9;
Chapter
protons,
affect ihe
not sigiltficantry
does
than ihe equation
complicated
not bare
associated
are
but
of
ihe reaction
letmsof
ihe
vulidily
equation. it
convenient
is ofien
lo express theacidily
oraikalinii) otasolulionin
pti,
defined as
pH s -log10[H+].
The
pH
concentration in The
base ten of iht; hydrogen ion concentrais ihe negative of the togarilhm = liier The of water 7 because !CT7mair'. oCsolution. is per pH pure [H*] acidic sotuiions have pH near 0 oi even have an negative; apple may pH - 3. lo is basic. has a of 73 it 7-5; slightly plasma pH
solution
ota moles
slrongesi blood
Human
Kttictic
Kxati\\ple2
nB,
action.
modelof/nms
AB. We suppose that t!ie concentrations
moieculc
nAB denote
is
AB
of A,
where the me constant C describes rale constant D describesthe reverse atoms A and B. in thermal equilibrium = Oand so that du.a:di
the
B combine to form a collision of A and B. Let ha, formed in a biaiomic of nAB is B, and AB respectively. The rale of change Suppose
formation
a function
of
temperature
derived earlier by Suppose
\"
The
by
o[AB
some
ptolon.
505A958).
catalytic
in
a collision
of A
result
is consistent
=
law
ihe
its component ace constant,
D3)
,
D/C
with ihe
B, and
with
process, the ihemiat decay of AB into the concermaiions of all consiiiutents
only. This
action
of mass
that we
thermodynamics.
AB is noi formed
dominant
one surrounding
standard
A and
atoms
that
nAB
formed
D!)
principally
by
the bimolecular
collision of
A
and
B, but
is
process such as
species present A [cvie* is given
is
most by
molecules surroundlikely it*1 4HjO, acomplc*of4 water P(oc_ Roy. Soc tLondon) A147. M. Eigcn and L. Dc Macycr,
,
\342\226\240 .
Here E is ihe
So long as ttie
catalyst which intermediate
to its original slate AE ts so short lived that
is returned
product
up as AE, ihc ratio iiaiiu/iUb in equilibrium direct above. process A + B<--AB treated
is lied the
actually The
proceeds, ihc equilibrium must in equilibrium of the direct equality
of detailed
be
the
and
ai ihe
end
IdealCases
for
Equilibrium
seeond step. quantity of A
of the
no significant
us if AB were formed what route Hie reaction by same. The rates, however, may differ. inverse reaction rate's is culled the principle must
No
be
the same
in
rnaiier
balance.
Comment: Reactionrates. The law of mass action expresses ihc condition satisfied by ihe concentrations once a reaction aboui how fasi lias gone to equilibrium, li iclis us nothing AH as it proceeds, ihc reaction proceeds. A reaciion A + B = C may bat evolve energy before the reaction can occur A and B may have 10 negotiate a potential as in barrier, is called the activation energy. Only moleculeson the high Figure 9.2. The barrier heigh! will not be able to get end of their will be able to read; others distribution energy energy over the potential hill. A catalyst speeds up a reaction by offeritig an alternate reaction path a lower energy of activation, but it does not change the equilibrium with concentrations.
Schematic
AH measures ihe energy evolved in the reaction the equilibrium concentration ratio [A][B]/[C]. The barrier to be negotiated is the height of the potential energy the reaction it determines the rate at which the can proceed, and takes place.
Figure 9.2
The quantity
nnd determines activation before
reaction
coordinate
Chapter 9: Gibbt FreeEnergy
Reactions
Chemical
and
SUMMARY
1. The
Gibbs free energy
G3 is a minimum
thermal
in
2. (cG/3r)H,= -a;
=
= W,,(r,ri
4
ofmass
law
for a
action
of
the
pV temperature and pressure.
V;
= p.
(SG/SN),.P
chemical reaction is that IK'
a function
iff +
at constant
equilibrium
tfG/cph.,
3. C(r,p,W) The
-
U
- -^w.
alone.
temperature
PROBLEMS
/. Thermal expansion near absolutezero,
(a)
the
Prove
three
Maxwell
rela-
relations
,
(dV/di)P
=
-{dts/dp),
,
(aVldN)p = +(ap/ap)^ , (Qj/cr)jV
two
omit
(lie help Coefficient
of
subscripts
of D5a) tjicrniiti
approaches
D5c)
appear similarly in D5b) and D5c).It is commonto that occur on both sides of theseequalities, Show wilh (b) the third law of thermodynamics tjiat the volume Coeffi-
should
subscripts
those
~Ea/dN)x.
D5b)
D5a) should be written
Strictly speaking,
and
=
D5a)
and
expansion
zero as t
-* 0.
2.
Thermal
ionization
in
hydrogen
c +
wheree isad
H*ftH,
electron on a proton H*. (a) of the reactants satisfy the relation of an
the
that
Show
of atomic
the; formation
Consider
of hydrogen.
the reaction
the
as
eicutron,
concentrations
equilibrium I
D7)
[e][H+J/[HJs\302\253acxp(-//T),
/ is
where
(he energyrequited to tontze
refers to the electron.Negiecithe
not aSTed
and
electrons
concentration
atomic
of
spins
and
the
~
{im/2zh2K11 this assumption does
hydrogen,
particles;
tiQ
is known as the Saha equation.Ifall the arise from the ionization of hydrogen the then protons atoms, of protons is equal to that and the of the electron Electrons,
concentration is
The result
result.
final
the
by
given
]
D8)
0]~[H]\"V'3\302\253p(-//iT).
problem arises in
A similar
thermal ionization
semiconductor
of impurity
The
is
electronic
excited
[H(exc)] andT -
with
[e]
/ is
a simple
the ionization
proportional
energy, the square root of the
to !
electrons to the system, then (He
state, conditions
for
ts not
this
that
shows
which
concentration.
atom
(b) Let [H(exc)]denotethe first
/,
Here
problem.
concentration
elccjron
If we add excess will decrease.
C)
^1 and not
involves
\"Boltzmannfactor\" hydrogen
the
I
exponent
B) The
u-jth
connection
of electrons.
are donors
that
atoms
in
physics
No lice lhat: A)
hy-
adsorption
which atthe
of
concentration
equilibrium
the is \\l above ground surface of the Sun,with
5000K.
of protons
concentration
H atoms in the state. Compare
[H]
=
1023 cm~
3
| I
3,
of donor
hnization
impurities in semiconductors.
\\ A
pentavalent
impurity
atom in crystalline silicon introduced in place ofa tetravalent the role of in free space, but with silicon acts likea hydrogen atom e2/e playing e2 and an effective mass m* playing the role of the electron mass m in the state of the of *hc kmizalion energy and radius o( the ground description the diclctltic consta.t\\t free silicon electron. For impurity atom, and alsofor the e ~ 11.7 and, 0pproxima!e!y, m* =0.3 m. U there are 10\" donors p^r cmJ, K. 100 electronsat cent ration of conduction estimate the con
(called
4.
a donor)
monomer
polymers
4-
the
Consider
Siopotymergrowth.
of linear
A'mer
made
= (A'
up
of
chemical
identical
-f l)mer.
of a sotution basic reaction step is
equilibrium
units. The
Let K^ denote liieequilibrium
constant
for
Chapter
9: Gibbs free
this reaction,
Reactions
Chemical
and
Energy
(a) Show from the law
mass
of
\342\200\242 [\342\226\240 \342\226\240]
lhat the concentrations
action
satisfy
+
[N
from the
(b) Show
1]
=
[If \"/K.KjKj
Iheory of reactionsthai
\342\200\242 \342\200\242 \342\226\240
K.v
ideal
for
gas
D9)
conditions
(an ideal
solution):
where iVmer
MN is
,
^Bnti2/M,^rm
wQ(W)
the mass of the Nmer molecule,and
molecule,
(c)
Assume
ratio [N
concentration
4-
N t]/[N]
\302\273 j,
at
so
thai
room
FN
E1) is the
nQ{N) =
Hq{N
free energy +
if there
temperature
1),
of one
Find
is zero
the free
basic reaction siep: that is, if AF = FKi.l ~ Fs ~ fj =-0. = in a bacteria! Assume as for ammo acid molecules ceil. The 10aocm~3, [I] molecular weight of the monomer is 200.(d) Show that for the reaction to go in the direction of long molecules we need AF < This ~0.4cV, approximately. condition is not satisfied in Nature, but an ingenious is followed that pathway simulates the condition. An elementary is given by C. KiUel, Am. discussion J. energy
Phys.
in the
change
40,60A972).
for the expression equilibrium, (a) Find a quantitative = = n+ n~ in the particle-antiparticle concentration n reactionA+ 4- A\" = 0. The reactants may be electronsand positrons; protons Let the mass of and antiprotons; or electrons and hoies in a semiconductor. the either particle be M; neglect of the particles. The minimum energy spins A\" is A-Take the zero of the energy scaleas the release when A* combines with with no particles energy present, (b) Estimate n in cm\023 for an electron (or a 300 that K. with a A such A/t = 20. The hole is hoie) in a semiconductor T \302\253* viewedas the antiparticic to the electron. Assume that the electronconcentration
5. Pavtkte-antipartkk thermal equilibrium
is equai to
the
hoie
concentration;
assume
aiso
titat
the
particles
are in
of (a) to let each particle have a spin of classical regime, (c) CorrectIheresult Particles that have amiparticfes are usually fermions with spins of \\.
the 3.
10
Chapter
Transformations
Phase
PRESSURE
VAPOR
276
EQUATION
the Coexistence Curve,p Versus
Derivation of Point Triple
278
t
284
Latent Heat and Enthalpy Model
Example:
System
WAALS
DER
VAN
284
for Gas-Solid
285
Equilibrium
287
OF STATE-
EQUATION
Mean Field Method
CriticalPoints
288
the
for
Gibbs Free Energy
van
der
the
van
of
2S9
Gas der Waals Gas Waals
291
Nucieation
Fe
from
agnet
295
298
OF PHASE TRANSITIONS
THEORY
LANDAU
302
Ferromagnets
Example:
First
294 ism
Order
302
Transitions
305
PROBLEMS
1.
2. Calculationof 3. Heat 4.
and
Energy,
Entropy,
for
dTjdp
Gas-Soltd
7. Simplified
Note: tn
305 305
Water
Order
the
305
305
Equilibrium
305
Equilibrium
of 6. Thermodynamics First
der Waals Gas
of Vaporization of Ice
5. Gas-Solid
8.
of van
Enthalpy
the
Superconducting
306
Transition
Model of the SuperconductingTransition Crystal
first section
307
307
Transformation
s denotes c/iV,
the
entropy
per atom.
In
the
section
on fcr
Chapter 10: Phaie Transformations
PRESSURE
VAPOR
of pressure versus
The curve
-
EQUATION
by
together
appropriate conditions can of a system is a portion phase
and under
or solid
a liquid
in
energy
another
one
with
interact
of matter at constant substance. The curve is a real gas in which the atoms
of the
isoihcrmsof
the
We
or molecules associate
free
the
a quantity
for
volume
temperature is determined calledan isotherm. consider
phase,
A
that is uniform in composition.
Two phases real
ofa in
atoms
the
in
and
solid
we say
As
There
are isotherms at
in
isothermsfor
and gas
soiid
which
holds also
equilibrium
liquid-gas
low icmperaturcsfor for
the
which
coexist. Everything equilibrium
solid-gas
solid-liquid equilibrium. and
on a
coexist
may
vapor*
section of an isotherm only
the the
phase\342\200\224exists, only a single phase\342\200\224the the pressure. There is no more reason to cali this phase a gas than we avoid the issueand callit a fluid. Values of the critical temperature
great so
no
fluid
temperaiure
a liquid,
if
isotherm lies below a critical temperaturerc. Above
of the
temperature
how
the
other.
coexist and
the
Liquid critica!
phase.
gas
liquid for
and the
each
with
boundary between them. An isotherm p-V plane in which liquid and gas coexist 10.1, part of the volume contains Figure
a definite
show a region in
gas may
equilibrium
with
coexist,
may
for severalgasesare Liquid and gas
iO.t.
in Table
given
matter
extent of an isotherm a to from zero pressure infinite along they coexist at most only pressure; of atoms, fixed number section of the isotherm. For a fixed and temperature the there will be a volume above which all atoms present are in gas phase. at room bell an sealed A small drop ofwater placedin evacuated temperature jar will A
water
of
entirely. the
evaporate the
from
atoms
relations
are
by
suggested
for
the
the entire
along
the bell jar
filled
with
gas at
H2O
some pressure.
may already saturated with moisture There is a concentration of water, however,above which into a liquid drop. The volume vapor will bind themselves
exposed
The thermodynamic
conditions
coexist
never
leaving
entirely,
evaporate drop
will
to air not
Figure
JO.i.
conditions
equilibrium
for
of two
the
coexistence
of two
systems that are
in
phases are the
thermal,
diffusive,
Vapor Pressure
IO.I
Figure
is constant, but Efiei^
is only
and its
the s^nclc
vapor are in
of a
isotherm
Pressure-volume
such that liquid temperature is, t < tc. in the two-phase
and
Gas
+ gas
Liquid
Liquid
gas phases
region of
Hquid
reyj
gas
at a
may coexist,that 4- gas the pressure
may change. At a given lempcrature a Jjtjuiu v*iluc of tltc ptcssurc for Vvh^cri at we move the Jf this pressure equilibrium.
volume
some of the gas is condensed to liquid, but remains unchanged as Jong as any gas remains.
down,
piston
pressure
Table 10.1 Critical temperatures T,.
of
gases Tt, in
K
in
He
5.2
H2
Nc
414
N.
Ar
151
210
Oj
Kr
Xc
289
CO,
K
33.2
126.0
1543 647.1
7
the
.W.2
Equatio
10: Phase
Chapter
Transformations
and mechanical contact. Thesecondilions are for
or,
=
18\\
Ml
=
the
phases
pressure
temperature
=
the general point in the p-x plane alone is stable, and if liquid phase
At a
Metastablephases
may
may
phase
have
a lower
by
occur,
be the pressure for
which that
+
dp;iQ
the
divides
It is a
pl
=
and
liquid
.
A) phases.
gas
in the
species
Note
that
the
two phases must be
B)
/ijl
two ns
coexist: If /i, < ng the gas phase alone is stable.
phases <
Pi
do not
or superheating.
supercooling
A
metastable a
chemical potential.
temperaturez0.Suppose Pa
p2;
a transient existence, sometimesbrief, sometimes long,at another and ntore stable phaseof the same subslance
Derivationof the CoexistenceCurve, Let pQ
=
for which
temperature
has
fit
potentials are evaluatedat the common of the liquid and gas, so that
/l,(p,T)
the
t2:
The chemical
coexist.
common
and
Pi = Pg
/Jj,;
where the subscripts / and g denote the chemical potentials of the same chemical if
=
tj
and gas,
liquid
*j
equal
that
+ rfi.Thecurve
p, t plane into a
the
p
two two
in thep.T
Versus
phases, phases
r
liquid and also coexist
gas, coexist at the at the nearby point
plane along which the two
phase diagram, as
given
in
Figure
phasescocxist
10.2
for H2O.
condition of coexistencethat C)
D)
dt).
We
relationship betweendp a series expansion of each sideof D) to
C) and
Equations
make
D) give a
and
dx.
obtain
-.
E)
ion
Cot
ojthc
Figure
10.2
relationships
the
jiB in
and
Phase diagram of H.O. The chemical poienmls /t,. ;i,. solid, liquid, and gas phases ate of the
shown.The phase
here
boundary
bciwcen
ice
and water is not cxacily vertical; the slope is actually negative,although large. After very Iniemalionat Critical Tobies.Vol. 3. and P. \\V Proc. Am. Acad. Sci.47, 4-i| A912) Qridgman, forms of ice. see Zemamky, for the several p.
-100
100
0
200
Temperature,in In
the
limit
as dp
300
375.
400
\302\260C
and dz approach zero, F)
by
C)
which
and
E). This
result may
is the differential
be rearrangedto give
curve equation of the coexistence
or
vapor
pressure
curve.
The derivatives of the chemicalpotentialwhich
in terms ofquantities
accessible
to
measurement.
occur
In the
may be
expressed treatment of the Gibbs
in G)
Chapter 10: Phase Transformations
free
With
9 we
in Chapter
energy
the definitions s
v
the volume
for
relations
the
found
per moleculein
and entropy
1 (cG\\
V
5 =
V/N,
o/W
(9)
each
we have
phase,
(dp\\
)JJ Then
for
G)
becomes
dp/dt
01) ~
Here
sa
st is
moleculefrom
of entropy
the increase the
gas, and
the
to
liquid
the
of
ihc
is
t\\
vg
we transfer one increase of volume
when
system \342\200\224
from the liquid to the gas. It is essential to understand thai the derivative dp/dz in (I!) ts not simply taken from the of state of the gas. The derivative refers to the very equation of i in which the and and special interdependent change p gas liquid continue
of the system
we
when
to coexist.The number varied, subject only The be
of to
of molecules
numbers
quantity
added
~
sa
to the
one molecule
transfer
the
s, is
when
decrease
added
in the
the
system
as
vary
Here
constant.
a
moiecule
is
volume
the
Ns and
outside
in the
is transferred
reYersibly
the
of
temperature
from
molecule
one
transfer
(Va
the
are
system
from
constant.
process, the
must
that
the liquid
(If heat is
temperature
to the gas.)The quantity
of
will
heat
transfer is l1Q ~
by virtue
will
gas phases, respectively. liquid related directly to She quantity of heat
sysfern to
the
A\\
phase
and
to the gas, while keepingthe
not addedto
=
+ Na
N, in
in each
molecules
\" TE*
of the connectionbetween
5'}'
and
heat
A2) the
change
of entropy in a
reversible process.The quantity L
=
tEs
-
A3)
of the
Dcritaiion
defines She latent heat of
Coexistence
vaporization, and is easily
measured
Cur
by
elementary
calonmetry.
We let A*\302\253
the change
denote
to
she
We
gas.
of volumewhen
combine
A1), U3),
A4}
i-g-v,
is transferred
molecule
one
from the
liquid
and A4) to obtain
A5)
Clausius-Clapcyron equationor the vapor pressure equation of this equation was a remarkableearly The derivation of accomplishment Bo!li sides of and are determined thermodynamics. (!5) easily experimentally, the equation has been verified to high precision. We obtain a particularly useful form of A5} if we make two approximations: that volume the (a) We assume by an atom in the gas I'j: occupied vg \302\273 is known
This
phaseis replace
as the
much
very
larger
in the
than
liquid (or solid)phase,so that
pressure
atmospheric
(b)
so
that
may
Av by vg:
A6)
may
the ideal written as
that
assume
We
i'9/i;( ^
be
!03, and the
approximation is very
gas law pVg
=
Av
With these
{16)
vt/yr
Avsva= At
we
S
Ngz
applies
to the
good.
gas phase, A7)
zip.
approximations the vapor pressureequationbecomes
of temperature, molecule. Given L as a function curve. this equation may be integrated to find the coexistence the the heat L is independent of temperature over latent if, in addition, Thus the of interest, we may take L = l.,Q outside integral. range temperature when we integrate (IS) we obtain where
L
is the
latent
heat per
r /dp
(lv
A9)
10: Phase
Chapter
=
logp
Traasfortnat
\342\200\224Z-0/i+
where p0 is a constant.We
one
where
LQ as
defined
to one
instead
refers
If Lo
nioiecuie.
) =
constant;
,
p0CXp(-L0/T)
the latent
heat of vaporization of
mole, then
A'cta, where No is t!ie Avogadro constant. For water the latent heat at the iiqufd-gas transition is 2485J g~' at O'Cand 2260 J g~' at lOO'C.a substantial variation with tcmpcraiure. as !ogp The vapor pressure of water and of ice is plottedtn Figure !0.3 R
is the
versus 1/T.
gas constant,
R =
The curve is linear over
substantial
-Crit
ca! p
with
consistent
regions,
the
Jin.
\\ id
wa
er
X 103 \342\200\224 1 atm
pressure
Vapor
ofwaier
and of
The vertical scale is dashed line b a straiglif
\\
\342\226\240-
102
\"
is 1/T. The
ice
iine.
S
I
\\
10
Si*\"
vc
\\ s \342\200\242Ice
V
1
\\
1.5
2.0
2.5
3.0 3.5 4.0 lO'/T,
in
K~\302\273
4.5
5.0
2
Tcmperamre, Figure
4
3 in
K
Vapor pressure versus iemperaiurcfor 4He. After H. van Dijk Research offheNauonal Bureau of Standards 63A, \\2
10.4
eial.. Journal of A959)-
10-4, vapor pressure of 4He, plottedin Figure of temperatures between I and 5 K. The phasediagram of 4He at low temperatures was shown in Figure 7.14. Notice tUat the liquid-soHd eoexistence curve is closely horizontal below 1.4K. We infer from this and (I!) [hat the entropy of the liquid is very nearly equal to the entropy of the solidin this region. It is remarkable that the entropies a normal should be so similar, because liquid is much more disordered than a of a normal liquid is considerably higherthan that solid, so that the entropy
result
approxmiate
is widely
used
in
the
B0). The
measurement
10: Phase
Chapter
Transformations
of a normal solid. But
3He,
the
slope
is a
4He
the
of
liquid. For another quantum liquid, curve is negative at low temperatures
quanhim
liquid-solid
entropy of the liquid is lessthan the entropy solid has more accessible statesthan the liquid! Liquid 3He a Fermi gas, has a relatively low entropy for a liquid it approximates because has a low enSropy when t \302\253 which a large proportion generally zF because of the atoms have Sheir momenta ordered the Fermi into sphere of Chapter 7.
(Figure
region the
in this
and
7.15),
of the
solid. The
Triple
point.
The
poin!
triple
of
t
is that
a substance
point p,, t,
in
She
p~z
all three phases, vapor, liquid,and solid, are in equilibrium. plane = = solid Here ng /i( /js. Consider an equilibrium mixture ofliquidand phases in a volume enclosed that somewhat larger than occupied by the mixture alone. The remaining volume will contain in the vapor, equilibrium wish only bo!h condensed phases,and at a pressure equal to the common equilibrium at which
The
of
of
pressure
vapor
both
temperature
point
iriple
at
substance
She
This pressure is the triplepointpressure. is not identical wish the melting temperature
phases.
somewhaton pressure; triple under common equilibrium the
the
For
the
water
defined
that
such
Latent
the
pressure of the two condensedphases. is 0.01 K above the atmospheric temperature = 273.16K. The Kelvin scale is O.Oi\302\260C = T, vapor
point, of
triple
water is exactly273,16K; seeAppendix
ofa
Tile latent heat
and enthalpy.
heat
phase to she gas phase,is equal
the liquid
of the
temperature;
melting
pressure
point
triple
two
phases
difference
of
enthalpy.
The
H
s
at
constant
V
-f
pV
= dU -f
=
On the coexistencecurve jig
L Values
of//
at coiisnint
are
\302\253 tAa
tabulated;
dV
-f- pdV
~
//,. Thus
\302\273 At/
+
-
phases, where H is
=
Hr-
1
difference
When
called
the
we cross the
applies:
-
ihey are found
\342\200\224-
Vtlp.
\302\273 Ml
=
B2)
,
jt,)dN
at constant
pressure:
T
+
pdV
(}tg
pAV
from
latent heat is also equal to the
coexistence curve,the thermodynamic identity Tito
the entropy
t times
B.
as
transformation,
phase
to
The pressure. the two between is dH
differential
is the
lemperaSure
point
temperatures depend meiSing temperature
Melting
pressure.
atmospheric
pressure Ha
- llt.
by integrationof the
B3} heat
capacity
of the
Derivation
Coexistence
Curve, p
Ver
B5)
jc,,
Example: Model system for a solid in equilibrium
gas-sotidequilibrium.
We
construct
a simple model can derive ihe easily a to apply liquid.
io de-
as in Figure 10-5. vapor Roughly the same model would of N atoms, each bound as a harmonic oscillator of freImagine the solid to consist u to 3 fixed ol force. The binding center oleach atom in ihe ground siaicis frequency energy that is, ihe energy of an atom in its ground to a free atom at rest. \302\243fl; state is \342\200\224 co referred \342\200\224where The energy states of a single oscillator arc ntioi h a or zero is r.o, positive integer in o:.e we suppose itial each ntom can oscillaiconly (Figure 10.6). For the sake of simplicity dimension. The result for oscillators in [hrce dimensions is left as a problem. ol a single oscillaior in llie solid is The Junction pariiiion
describe
pressurecurve
Z, =
for
ihis
with
a gas,
We
model.
\302\243\302\253p[-(n/10>
= expOWt)
eo)/r]
= \302\243\302\253p(--,,Aa./T)
B6) The Ziee energy
F, is
F* = u* ~ The Gibbs free
energy
in the
Gs
Tff,
-
B7)
-tlogZ,.
solid is, per atom,
\302\253
Vt
-
to-, +
pt',
=>
Fs
+
pvx =
//,.
Figure
10.S
wiih
aionis
pressureis a energy ol the in
than
aioms
in a
Aloms
in the
in
cqmSibn
gas phase. The equilibri ol temperature.
function
The
solid phase is
in [he
atoms
ihe
solid
phase, bm the cmiopy of
gas lo be lends
in
higher
the
at may
temperature
high
be
in ihe
iIk
atoms
all ot
gas.
\\
gas phase. . The
equilibrium configmaiion is dfiietiiiiacd of the iwo ciTccls.A! low cotmtcrpkiy
lanpemSurc most of
lo
;src in ll
most of
ihe a
y
ihe
10: Phase Transformation
Chapter
(u. The lor of frequency ssumed 10 be % below that
in!he
o(a
gas phase.
Ground Male < aloui
bound
The pressure
volume
i>i
per
in
in ihe
gas phase: c, \302\253 vr
that
o[
solid phase is much
[he gas with than
smaller
but Uie which il is in contact, atom in [he the volume vt per
\342\226\240
neglect the term absolute activity is
If we
the
is equal to
solid
the
atom
pv, we have
(or
the chemical
potential of
,/t) =
[hesolidp, S
We
to be
' with
equilibrium
p
K
we
insert
nQ from
V
=
gas
phase,
and we
take Ihe spin of
C0) \302\253q
The gas is in
B9)
gas approximation to describethe zero. Then, [rom Chapter 6,
the ideal
make
atom
whence \342\200\236
exp(-logZJ
the
f
the solid
t\\
'\"a
when
inQexp(-E0/t)[l
).f
=
or ).\342\200\236
- exp(-/1<0/t)].
C1)
C.63J:
C2)
=
(j^j
The
mode!
simplest
interactions
the
he was
below,
liquid-gas phase transition is that ofvander Waals,\\vho gas equation pV = Nx to take into accounlapproximately between we atoms or molecules. By the argument that of a
ideal
the
modified
led to a modified equation of stateof the (p
van
the
as
known
- Nb) = Nx ,
+ N2a/V2)(V
der Waals
equation of state. Tliis is written
molecules, and the constant /' is a measureof their of (Figure 10.7). We shall derive C3) \\vitli..tlie help We shall then trcal ~-(SF/DV)liN. in order to exhibitthe liquid-gas For an ideal gas we have, from
The hard core the
b is
had
gas
at
repulsion
To
we
now
add a
volume
-NT{log[)fQ(V
correction
a is
between
two
repulsion
range
= properlics of the model relation p
general
V,
be treated
can
distances but
the
the volume per molecule. We replace ~ of instead Nb). Thus, C4),we N/{V
this
in
constant
for
the
C4|
+ i].
-NT[log(na/n)
in C4) by
=
atoms
F.24),
free
approximately as
volume
V
~
the concentration
therefore
F
short
tlic
the ihcrmodynamic
short
not the
available
N
for
transition.
=
F(idealgas)
if
C33
the a, b are interactionconstantsto be defined; of the long range attractivepart the of mieraeiion
a measure
give
form
V. The
volume
of State
Equation
OF STATE
DER WAALS EQUATION
VAN
Waats
Der
Van
when
Sb, n
=
N/V
have
- Nb)/N] + intermolecular
C5}
1}. attractive
forces.
Figure 10.7
The
iiucraaion
energy
between
The repulsion plus a long range aiiracHou. short range repulsion can be described that each molecule by saying approximately
has a hard, impeneirable
coie.
10: Phase Transformation*
Chapter
Mean Field
Method
There exists
a
taking
the effect
of weak long range
particles system.The most gases and to ferromagnets.Let of two atoms separated a distance
of a
known
widely tp(r)
the average value of at r = Ois
is n,
gas
the
atom
interactions
potential
for
the
among
method are to energy of interaction
the concentration
r. When
method,
of the
applications the
denote
by
the
the mean field
called
method,
approximate
simple
account
inio
of atoms in
ihe total interactionof all other
on
atoms
C6)
where -2a denotes
value
the
of
ihe
We exclude the
convention.
useful
integral \\dVip{r). The factor of two hard core sphereof voiume b from
is
of integraiion. In writing C6) we assume that the concentration ihe volume accessible to the moleculesof the gas. That ihroughout we use the mean value of n. Tins assumption essence is the of the mean
volume constant
approximation. of concentration
in
we
language
energy
say
the
\"
molecules.
interacting From
regions
thaS
it follows
C6)
of a
that
The
factor
exact
s MJ =
is
Heimholu
free
of a
energy
F(vilW) =
The
pressure
*=
obiain
the
the
free
C7)
-N2a/V.
it arranges
problems;
only once in
|-N(N - I), which
We add C7) to C5) to
change the energy and
-\\BNna)
is counted
molecules
of bonds
number
is,
V by
volume
j is common to self-energy
\"bond\"between two
is
\342\226\240
the interactions
gas of N moleculesin AF
n
field we ignore the increase concentration uniform of strong attractive potential energy. In modern mean field method neglectscorrelationsbetween
assuming
By
a
the
we
der
v;m
approximate
Waais
that an
interaction
the total energy.The as |N2.
approximation
for the
yas:
-.Vi{log[\302\273u(l'r
Sh)/N~\\
+ 1}
- Nza/V.
0$)
is
C9) '\342\226\240-^\302\253'\027^5-F
Critical Pawls for
the
Figure
suggests
pressure be
used
of intermolecular near ihe boundary Y. The van dci Waats argument lhal ihcse forces contribute art internal Nxa!Vl which Is lo be addedlo ihe
10.8
forces that of a volume
Direciions
on molecules
ad
as ihe
pressure
in
gas taw.
the
1
Q
o o
o
of volume Khas N not b. The volume molcculcSi V S'b. Intuition by molecules is occupied suggests that iHis fece volume should be used in V. llic gas law in place of the coiiiaincr Volume
V
Figure
each
O 0
O o
Q
(p
Waals
10.8
and
coniainer
of volume
o
\302\251
der
Hie
10.9
+ N2a/V2){V
equatum of
~ Nb)
\302\253 A'l
staSe. The terms in
a
D0)
.
and
b arc
interpreted
in
10.9.
Points
for She van
ne the
quantities
der
Gas
Waals
pc ^ a!21b2\\
Vc
s
3Nb;
xc
^ 8a/27b.
D1)
Chapter
10:
Phase
Trans/or,
. -
0.95tt
P/Pc
0
Figure
In termsof
these
der Waals
Tile van
10.10
critical
the
van der
the
quantities
equation of
3
Waals equation
\\(V
f7HJAH
This equation is plotted in temperature tc. The equation
stal
Courtesy of R. Cahn.
temperature.
Figure may
10-10 be
l\\
8t
3/
3t/
for
written
becomes D2)
near the temperatures in terms of the dimensionless
several
variables ps
9sVfV.\\
pjPc\\
t
e t/tc ,
D3)
D4)
This si!
gases
result
!ook
ts
known alike\342\200\224if
as the they
correspondingstates. In termsof p, V, t, the van der Waals equation. Valuesof a
law of obey
Free
Gibbs
usually obtained
and b are
Realgasesdo one
At
same p,
at the
substances
not
the
obey
P~V
curve
phases.
and
coincide,
At a
horizontal
corresponding
local
the
Here
and
tc the critical
Above
tt
of the
van
der
G
gives
N.
of pressure
function
as
a function
we
cannot
by
gas
of
Nt{log[\302\273u(K
V,
t,
N;
It is
want
Nb)/N]
the natural
put G
conveniently
(9.13).
the characteristics With G = F + pV,
of
exhibits
pressure.
instead ofvolume.We
obtain ;i(t,p)as G(z,p,N)/N
relation
Waals
-
-^
-\342\200\224^
This equation p, t, Unfortunately
Vet
Gas
constant
-
=
pc,
\342\226\240-
Waals
der
the !iquici-gasphase transition at have from C8) and C9) the result
G(x,VtN)
=
respectively.
exists.
free energy of the van
The Gibbs
liquid
9 = 1;i 1. We call critical volume,and criticaltemperature, p = 1;
by D4) if
pressure, separation
Energy
of the
@i-
no phase
Free
Gibbs
and minimum
maximum
a
f has
constant
at
V
is no separation between the vapor and of inflection point
satisfied
are
conditions
ihe subsiances.
there
aThese
states of
point, the curve of p versus
horizontal point of inflection.
tc. States of two
pc and
observed
Gas
M'aah
(let
van
to high accuracy.
equation
the critical
point,
called
Vy x are
the
to
fitting
by
of the
Energy
variables
into an
G(z,p,N)
+ 1],
analytic
because
we
D6)
for G are form
we can
as
/i that determines the phase coexistence
temperature
results of numerical calculationsof G versus plotted pare 10.! 1 for temperatures below and at ihe critical temperature. At any the lowest branch represents the stable phase; the otherbranches
represent
unstable
=
ft,
in Figure
a
then
fig. The
phases.
The pressure
at which the branchescrossdetermines
gas and liquid; this pressure is calledthe equilibrium !0.12. for G versus t are plottedin Figure vapor pressure. V < Vs In which only ihe 10.13 shows, on a p-V diagram, the Figure region and the V > V2 in which liquid phase exists region only the gas phase exists. between and The value or The phasescoexist of V2 is determined Vx V^. Kj by the condition that /i](r,p)== }ia{x,p) along the horizontal line between Vx and the
transition
between
Results
V2- This
will occur if the
shaded area belowthe Jine
is equal
to the
shaded area
10;
Chapter
Phase
Ttansfo
-0.40
t\342\200\224
--
rasure
,-Vapor[
/ \\
Figure 10.11
der
Waals
(a) Gibbs free of stale:
equaiion
(b) Gibbs
free energy
versus
stale;i = %..
above
the
To see
line.
have
difference
dG
=
versus
pressure
i = 0.95tc. Courtesy
pressure
for van der
of
for van R.
Cahn.
Waals equation of
ihis, consider dG
We
energy
Vdp at
\302\253 ~adx
+
Vdp +
constant i and constanttola!number
of G between V\\
and
D7)
pdN. of
particles.
The
V1 is
G,-G,=
fWp,
D8)
j
1 ~p
-
0.95
nqb =
!
LiqutaS^
\302\273bic
-
G/Nrc
k
i
t
'
i
0.990
0.988
0.986
0.984
\342\226\240
't/t, (a)
Figure
versus temperature of A. Manoliu.
Gibbs free energy
10.12a
of slale a! p
=
0.95 pe. Courlesy
for
van
der Waals
= p
nqb
equalio
1.0
=
1
-
\\
\\
-
\\oas
-2.70
0.90
I.OO
1.10
1.05
t/t,
(b) Figure
of slate
10.12b
a! the
Gibbs crilical
free energy pressure
pc.
versus temperature for van
der
Waais
cquatiO!
Chapter
Id: Phase
Transformations
T
~
conslan
- ^Liquid -Cocx
\\
c
fir
Stetice
X
<\302\243
i
a
as
|\\ s.
v.t
10.13 Isotherm of van lemperafure belowliie critical Figun e less
than
above
lliegas phase exisls. Between in stable equilibrium lies along the
line
and is
liquid
and
liquid
and
volumes
bul ihe When
equals
the volume
/^{r,p)along
Hie
we require Nuclcatiou.
of the
magnitudes
j.tg
Let
V
that
and
V2
coexistence
is available.
shaded
areas,
areas are
equal, line
coexistence
horizontal
=
Vx
an inhomogencousmixture of two phases. The gas phases coexist. The proportion of liic be such lhai ihe sum of fheir gas phases must
integral isjust ihesum ofthe Ihe
volumes
only
Kj
Ihe system
volumes
For
phase exisls; for
!hc liquid
Vt only
a
der Waalsgasat lempcralure.
one
-
G?(t,/>)
drawn
in
and one positive. Gj(t,p) and [is{x,p}=
negalive
the
figure.
In equilibrium
/*,. Aji
\342\200\224 \342\200\224be
;i,
ftg
ihe
chemical
potential
difTerence between
the vaporsurroundinga smallliquid and the liquid in bulk (an infinitely droplet is if A/i iarye drop), positive, Ihe buik liquid will have a lower free energy than the gas and thus the will be more stable than the gas. However,the liquid surface the free free of a liquid drop is positiveand tends lo increase energy Al the of'he small radii the surface can be dominantand energy liquid. drop can be unstable the change with respect to the gas.We calculate in Gibbs drop of molecules is the concentration R forms. freecnergy when a drop of radius If\302\273, in the liquid, AG
=
G,
-
= G9
-{
+ 4nR2y
D9)
Ferromagtutism
the surface
y is
where
drop
will
grow
free energy per unit Gt <
when
An
Gr
0 =
=
ti&G/dR
This
is the
fend
to
critical radius
R tlie
larger
drop
will
to
tend
E1)
that
because that,
drop
will
energy.
too, will
At
lower
free energy.
the
must that energy barrier (Figure10.14) fluctuation in order fora nucleusto grow beyond in E1) D9):
The free
(&G)C
to
free
the
Jower
will
spontaneously
grow
R the
At smaller
a drop-
of
because
spontaneously
evaporaie
E0)
\302\2532y/H,A/i.
nuclcation
for
when
+ SnRy ,
-4nR2iii&n
Rc
The liquid
is attained
of AG
maximum
unstable
tension.
surface
or
area,
If we
assume [hat
express
Aji as
the
vapor
=
for
-r
the vapor pressure tn of the bulk liquid (R ~*
Kcrro
300 K
at
water
and p
E2)
like an
behaves
p is
pressure
by a ihcrni.i! substitution of by
overcome
R,. is found
(lenPW/nMl*I}.
Aft
where
be
ideal gas, we can use Chapter5
,
tlog{p/peq)
the
gas
co).
phase
and
use
~
We
y
pe
72erg
the
equilibrium
cm\022 to
vapor
estimate Rt
x !0~6cm.
= l.J^tobel
ism
magnet
means a magnetic has a spontaneous magneticmoment,which field approxithe mean moment even in zero applied magneticfield.We develop defined as the of ihe magnetization, approximation SO the temperature dependence each moment niagnelic magnetic per unit volume. Tlie centra! assumptionis that alom experiences an effective fieid BE proportional to she magnetization:
A ferromagnet
BE
where
I is
a conslans. We
take
the
\302\253 AM
external
E3)
,
applied
field as
zero.
Chapter
10:
Phase
Tram/or
barn
Critical *\"
or
t for growH,
3n nuclei
cortdcnsut
fx
'
1
1
L
ondensation
\\
,
nuclei
|
__
evaporate
\342\226\240'\342\226\240\342\226\240-
1
Figure 10.14 of
function
Excessfree
drop
radius
energy
of drop
relative lo
gas, as
R, both m reduced units. The
gas
is
because the iiquid has the lower free energy for but the surface energy ofsmaii dropscreates an energy of nuclei of the liquid barrier ihat inhibits the growth fluctuations eventually may carry nuclei over ihe phase. Thermal
supersaturated this
curve
as drawn,
barrier.
a system with
Consider
a concentration h of magneticatoms,
magnetic moment fi. In Chapter3 we magnetization in a field B:
and of
M
In the mean field
.
.
approximation
an
found
each
exact
of spin
E3)
Me . \342\226\240\"\342\226\240_\342\226\240_\342\226\240
this
.
for a
becomes,
n/nanh(/dMA)
for the
result
= HjitanhOiB/t).
,
.
j
E4)
ferromagnet, '..
'.
E5)
Fcrromu^cth,
1.0
0.8
0.6
0.4
0.2
0
soiuiion of \302\243q. for the Graphical E6) reduced magnetization ,,i as a function of temperature. The reduced magnetization is defined as m = M/n/t. The left-hand side of Hq. E6) is piolicd as a straight line in 10.15
Figure
The right-hand side is tanh(m/0 and is three difierenl of for values the reduced plotted t = r/n;i'^ *= i/tt. The three curves temperature to the temperatures 2x,,zt, and Q.5rt. The correspond with unit slope.
versus
curve for
=
i
in
the
2 intersects
as appropriate for ihe is no externai applied magneiic
m s= 0, t =
(or this
it) is
murks
ictnpetalure for
Curve
lo
tangent
i =
0.5 is in
tmersccts ihc straight 1
-\342\200\242 0 the
magneiic
we write it temperature
iiie
tn f
s
line
moments
are
Figure
and
10.15. The
M
versus
shall
We
M.
=
left
of
sides
and n/i. As
0.94
l.so
that all
absolute zero.
see that
E6)
ianh{Hi/r). this
equation
separately
intercept of the two curvesgives
temperature of interest.The critical
curvesof
up at
iined
-
in
region s= \302\273i
whence
z/n[i2X,
We plot the right in
ferromagnciic in at about
moves up to
hi
as
the suatght Sine m at ihc origin; ilic onset of fcrroiiiagnetisiu.The
solutions of this equation M exist in the temperature range between0 and v To solve E5) terms of the reduced magnetization ??i \302\253 and the reduced M/h/i
nonzero
with
region (iherc curve for / = i
paramagnetic The fieid).
intercept
a transcendental equation for
al
in only
iine
straight
x obtained
in ihis
temperature
ts
way reproduce
f
the
=
I,
;is functions of \302\273t, value of m at Ihe = or zt h/i^L The
roughly the featuresofthe
10;
Chapter
Phase
Transformation
of Figure 10.16 Saiuration magnetization as a nickel function of temperature,together with ihc theoretical curve fci spin \\ on ihe
mean fieid theory.
resuiis, as
experimental
in
shown
magnetizationdecreases
smoothly
OF
THEORY
LANDAU
Landau gave a systematic transitions
consider
free
to
applicable
at
systems
F =
energy
with
minimum
\302\243,
system,
electrons bonds
have
can
a certain
be indepaideally
mean
field theory
the
It isnot helpful
system
can
to
all possible
consider
by a single order
be described
might be the magnetization in a ferromagneti in a ferroelectric system, the fraction of polarization in a superconductor, or the fraction of neighborA-B
in ait
value
alloy c,
=
specified,
equilibrium the order parameter the Landau iheory we imagine Ihat \302
AB. In thermal cc{c).
In
and we consider the LandauTree
FL(\302\243.x)
(\"unction
energy
=
the energy and entropy are taken when the value \302\243 no! necessarily c0. The equilibrium specified where
of phase transi-
xi, which
Greek
bondstotola! will
to what variables?
dielectric
superconducting
of the
variety of systems exhibitingsuch transitions.We their Heimholtz constant volume and temperature, so that \342\200\224 in The ta is a minimum big questionis,a equilibrium.
suppose here that
parameter the
TRANSITIONS
a large
respect
variables. We the
U
at z
zero
PHASER
formulation
for nickel As t increases the = rc, called the Curie temperature
10.16
Figure to
E7)
order value
has
parameter
\302\2430{t)is
the
value
the
of
Landau
Transitions
of Phase
Theory
makes FL a minimum, at a given the actual Hdmholtz t, and f(r) of the system at i is equal lo thai minimum:
c, that
F(t) Plotted as a function than one minimum.
\302\273
S
Fl($0,t)
of
for \302\243
r, ilic
constant
The lowest of these transition another phase i is increased. of
function ferroelectric
are
systems
exam
well-behaved function something
this.
We
that \302\243
it
can
not be
lowest
In a
state.
equilibrium the
becomes
more
as
minimum
Landau (\"unction is an even applied fields. Most ferromagneticand ferro-
pies of of
should
that
of
absence
the
in \302\243
the
have
for which Ihe
to systems
ourselves
E8)
Co-
Landau free energy may
minimum
restrict
* \302\243
determines
first order We
if
FL(\302\243r)
free energy
taken
also
that
assume
be expanded
for
For
granted.
F 1(^,1) is a
in a
sufficiently
power series in
\302\243\342\200\224
function of
an even
as \302\243,
assumed,
entire
The
dependence of
temperature
coefficients
g6- These
g2,gx,
ga\\
FL(\302\243,x)
is
are matters
coefficients
in the
contained for
expansion or theory.
experiment
changes sign at ^(x) example of a phase transitionoccurswhen a temperature i0, with y4 positive and the higher terms negligible.For simplicity
The simplest take
we
linear
g2(t)
in i: ~
<72(r)
the
over
these
Witli
The
temperature
The
which
of interest,
F0)
and we take g4
as
in that
constant
range.
idealizations,
form F0)
certainly
range
temperature
- to)* .
(r
fails is not
and cannot be accurateover a very wide temperature range, on tembecause such a linear dependence at low temperatures consistent
equilibrium
has the
with
value of
the is \302\243
law.
third
found
it
at the
minimum of
FL{$
;t)
with
respect
roots
f~Q
and
?
= (to
- r)(a/g4).
F3)
Phase
Chapter
iff;
With a
and ga positive, the root c,
energy
function
Transformations
=
above i0;
at temperatures
F1)
0 corresponds
J=Xt) =
The other root, c,2 energy
of
FdZ'J-)
as
of F(r) with
in Figure
ihe
for \302\243:
temperature
minimum of the free
here the HelinhoHzfreeenergy
is
F4)
0g(l).
t) corresponds to the minimum of the free below t0; here the Helmhoitz is free energy
temperature
of
a function
Figure 10.18,and shown
-
(a/gj(to
at temperatures
function
The variation
~
to the
10.17. The variation ihree is shown in temperatures representative is of the equilibrium value of \302\243 dependence is shown
in Figure
10.19.
Our model describesa phasetransition
elergoes
to
continuously
Figure for an
zero
as the
in
which
temperature
the value of the order paratnis increased to t0. The entropy
10.17 Temperature dependence of (he free energy of the second order. ideatized phase t ransition
Figure
10.18
free energy
Landau
function
As the reprcscniaLivc temperatures. the equilibrium value of \302\243 gradually posiiion
of the minimum
Figure
10.19
temperature, curve is not realisticat use Of Eq. F0): the third thatdf/rfr
increases,
polarization
a second-order
->0asi-*0.
at \302\2432
;is tic fined
by
t0 the
of the tree energy.
Spontaneous for
versus
temperature drops below
low
law
versus
phase transition. The because of the temperatures of thermodynamics
requires
Chapter
10: Phase
\342\200\224
Transformations at t
is continuous
dFfdz
temperature f0.
a
Such
Transitions
nonzero
a
with
latent
The real
second order transitions;
is no
there
latent
heat at the
transition
definition a second order transition. order heat are called first we transitions; a remarkable world contains of diversity arc ferromagnels and superexamples is by
transition
them presently.
discuss
= rQ,so that
best
the
superconductors.
Example:
Landau which
In the mean field approximation, ferromagnets the satisfy To show moment a field consider a n atom of theory. this, 3, pin magnetic magnetic we shall set equal to ihe tijean field >M as in E3). The interaction energy density is Ferromagnets.
V(M) =
j iscommon to self-energy
ihe factor
where
F6)
-\302\261
g{M) = constant in Ihe
regime
in
FL{M) ~
At
transition
Ihe
M
which
\302\253
n/j.
Thus
constant -
temperature
\\M2(/.'~
the coefficient of M*
with
First Order
Transitions
A
latent
lion order
at
heat
constant transitions
in metals transformations
the
discussion
density is given approxi-
F7)
,
fursciion
\342\200\224A+
-
entropy
M2J2nn2
the free energy
i0 in agreement
The
problems.
per
lermsof
vanishes,
unit
volume
higher
is
order.
so ihat
F9}
\302\273}i2>.,
following
F8)
E6).
transiphase transition. The liquid-gas In ihe physics of solids first transition. is a first order pressure and in phase transformaare common in ferroelectric crystals a first order iransition describes and alloys. The Landau function
characterizes
a first order
when the expansion coefficientg*
is
negative
and
gb Is
positive.
We
consider
first
l(K20
Figure first
Landau
free energy
at representative function has equal minima For t below rf the as shown. \302\243
order
versus
function
transition,
i1
temperatures.
Order
Transitions
m a At
xc
at a is minimum finite absolute is a there iatger values of ^; as r passes through tc in the position of the absolute disconimuouschcinge minimum. The artows mark the minima. the Landau
The extrema of
this
function
are
given
by
at
=
\302\243
0 and
o
the roots
FigureJ0.20: G1) -
Either
\302\243
0 or
G2) At with
transition
the c,
~
0 and
temperature
with the
rc the
root c^O.
free energies will
The
value
of
be
xc will
equal
for
not be
the phases
equal to r0,
Chapter
Phas
10;
T
and the
order parameter\302\243 (Figure
weui earlier, where \302\243 transformation
show
may
those
differ from
results
These
xc.
at continuously as in supercooling
zero
to
hysteresis,
hysteresis exists in
t0 -
coexistencecurve in the p-x planebetween Clausius-Clapeyron equation: dp
L is
the two 2.
The
the latent heat and
van
heat L
order
transfor-
but
supcrsaturation,
no
is the
An
two
phases
must
satisfy tiic
L
_
der
Waais
volume
difference per
~
pV is the
the energy
'
is a minimum
of state
(P +
atom between
necessarily
V +
enthalpy.
is -
N2a!V2){V
Nb) =
Nx.
function
energy
and entropy
not value \302\243,,
H
where
equation
free
Landau
- Hlt
~ H,
.
4. In the
first
A
phases.
latent
3. The
tf.
or
1. The
where
continuously to zero at phase transition treated
transition.
order
a second
rrt in K
order
second
the
in
~
not go
does
10.21}
-20
-40
_60
are
the
taken
when
thermal
with respect to
when \302\243
parameter has the specified The function Fl equilibrium value \302\2430. the order
the
system
is
in
thermal
equilibrium.
A first
5,
phase transition is characterizedby
order
a
Intent
heal
and
that
the
by
hysteresis.
PROBLEMS
/,
dcr
van
ofthe
entropy
and
energy,
Entropy^
(c)
the
-
enthalpy//
//(r./>) arc
results
given
2. Calculationof equation
U =
INi.-
= U
+ pV
=
|Nt
-
-jNr +
to first order
near p
-
~
the
for
G6)
terms o, h.
\\V;ials correction
the vapor
from
Calculate
G5)
2N2ajV\\
2NuPfx.
der
van
~ 1 atm
G4)
i%
NhP
the
in
heat of vaporizational
wmcr. The
G3)
N2.ii!V.
+ N2bx/V
water.
for
(IT{dp
of elT/dp
value
the
en-
+ 1}.
Nb)JN]
N[log[nQ(V
H{i,V)
All
Show
is
energy
the
thai
Show
(a)
gas is
Waats
o(b) Showthat
IVaah gas.
of van der
enthalpy
Hquid-vypor
100Xis 2260
Express
Jg\021.
pressure equaof in
equilibrium the result
keivin/atm.
3. Heat of vaporization Hgal of vaporization
ice
4.
Gas-solid
we
let the
at
ice.
of
the
latent
soiid move in \302\273 haS)
heat
the
range
the heat
Jmol\"'
of vaporiza-
of interest.
of the exampleB6}-C2) in dimensions,
three
vapor
per atom
pressure
is tQ
which
(a) Show that in
the
is
\342\200\224
\\i.
the gas-solid equilibrium under the exof the solidmay be neglected over the temthe ofthe be cohesive Let -e0 solid,per atom. energy
Consider equilibrium. that the entropy assumption
GaS'SoU'd
temperature
a version
Consider
equilibrium.
osciSiators in the
(b) Explain why extreme
vapor over iceis3,SSmm
of water
pressure
~l\302\260C.
high temperatureregime(t
5.
The
mniHgai OX. Estimatein
and 4.58
~2=C
10: Phase
Chapter
Transformatio
Treat the gas as idea!and
of ihe
energy
system
minimum
of
ihe
+ F,
number
total
the
the
independent
container,
thai the
(a) Show
volume
of the much
total HeimhoHzfree
is
F = F, where
that the
the approximation
Make
monatomic.
accessible to the gas is the volume V of Smaller Volume occupied by the Solid,
free energy
\"
+
that in ihe
G8)
(b) Find
constant,
is
Ng
to N^show
respect
- I] ,
N,r[!og(N,/l'il0)
N = N,
of atoms, with
+
-Ufa
the mini-
equilibrium condition 09)
(c) Find the equilibrium
vapor
6.
of the
Thermodynamics
pressure.
superconducting transition, (a) Show
th
2/i0
SI units for Be. Because Bc decreases with side is negative. The superconducting phase
temperature,
increasing has
(SO)
[i0 ih
.
in
that
the
lower
entropy:
it
the
right
is the
more
ordered phase. As t ~+ 0, the entropy'in both phases will go to zero, consistent r? with the third law. What for the sliapc of the curve of Bt versus docs this imply = = this ihe SIiow llutt result hits 0 and hence (b) At r = xtt we have Bt a^ aN. following consequences: A) The two free energy curves do not cross ;if tt but are the same: as shown in Figure 10.22.B) The two energies merge, Usfr,.) = heat with the transition at r \342\200\224 associated tt. U.vW- C) There is no Intent What is the latent heat of the transition when out carried in a magnetic field,
at r < i{7 (c) related
that
Show
Cs and
CN, the
heat capacitiesper unit
volume,
are
by
(81)
Figure S.iS is a than
linearly
dominated
plot of Cj'T
with
by Cs.
T1
vs
decreasing
Show that
and
r, while Hiis
implies
shows
that
Cs decreases
Cs decreasesas yz.
For
t
much \302\253
tc>
faster
AC
is
_-
-0.2
X. Normal
.1
Superconductor
*STC=1.180K
0.5
lure.
Temper;
K
of Experimental values of the Tree energy as a function in the superconducting state and in the normal stale. Below the transition is lower temperature T, = 1.180 K (he free energy in !he transition slate. The (wo curves at the supcrcondtiding merge heat is second order {(hereis no laient tempcra(ure, so thai the phase transition a! Tc). The curve and of transiiion in zero is measured FN is magnetic field, Fs normal slaie. measured in a magnetic to in (he field suftkien! pu! the specimen
Figure 10.22
temperature
Tor aluminum
Courtesy of
N.
E. Phillips.
model of the superconducting transition. TheBc(i}curves that iutve shapes close to simple paraboias.Suppose
7, Simplified superconductors
=
Bt(i)
that
Cs
linear in r,
as for
Assume
and plot
calculate
heat
8.
crystal
of
order
two.structures,
siable form
low
of the
than linearly as (Chapter 7}. Draw
gas
the i dependencesof
and ihe latent First
Fermi
of
the transition.
temperalure
substance.
separated atomsat
form
by a and
he
then
(83}
also t -* 0. on the resultsof
the two
heat capacities,
Cs(rc)/Cv(r(}= 3. crystal that
can existin
'hat and /?. We suppose the /J structure is the stable high
the
energy
6 to
Problem
the
energy
Cs is
that
Assume
entropies,
Consider.'!
If the zero of the infinity,
two
Show that
transformation.
denoted
I
most
d/r(}2].
faster
vanishes a
-
Bt0[l
of
scale
density
is
taken
eilher
is the
a slrucmre semperalure
as the
1/@) at r
==
0
stale of uill
be
Chapter 10:
Pha:
negative. The phasestable
velocity of sound ve U?@). to lower values phase, corresponding
thermal excitationsin
phase.The !he energy.
the
larger
Soft
free
by the
is
$
thermal
excitation,
tend
systems
the
of
the
-~rc2r'i/3Oi;3/i3,
by
given
velocity of
all phonons.(b)
Show
value of U{0);thus
amplitudes than in the a the entropy and the lower
stabie at high tempera!ures,hard
to be
low.
in the
the lower
/J phase is lower than vx in the clastic moduli for /?, then the
have larger the larger
will
phase
in
the (a) Show from Chapter 4 that phonons in a solid at a temperature much at
systems
the
have
\342\200\224 0 will
If the
<
Ux@)
a
t
at
free
energy
density
icssthan the Debye temperature
Debye approximation that at the transformation
v
with
will
a finite
actual
is
transformation
defined
real solution if t'p phase
<
transformations
as the
trans-
thermal energy
(85), U
refers to unii volume.
that
must
the
that
L = 4[U,@)and
(84)
This example is a simplified model in solids, (c) The latent heat of
Show
(84)
as the
iv
system through the transformation.
In
taken
temperature
>-v.-').
There be ofa classof
contributed
t/,@
be
latent
to carry the heal for this model is supplied
(85)
11
Chapter
Mixtures
Binary
310
SOLUBILITY GAPS OF
ENTROPY
AND
ENERGY
MIXING
wrth Interactions Nearest-Neighbor Example: Binary Alloy of Structures Mixture Two Solids with Different Crystal Example: Low 3Hc-\"He Mixturesat Temperatures Example: Liquid for Simple Solubility Gaps Phase Diagrams PHASE
EQUILIBRIA
AND
SOLID
BETWEEN
31-1 318 319 320
321
LIQUID
322
MIXTURES
Advanced Treatment: Eutectics
325
SUMMARY
330
PROBLEMS
330
Potentials in Two-Phase Equilibrium Energy in 3He-4He and Pb-Sn Mixtures
1. Chemical 2.
3.
Mixing Segregation
Coefficient
of
Impurities
4. Solidification Range of Binary 5. Alloying of Gold into Silicon
Alloy
330
330 331 331
33i
11:
Chapter
Mixtures
Binary
of materials science,and large parts of applications chemistry are concerned of with the biophysics, properties multicomponentsystems
and
Many
phases in coexistence.Beautiful, unexpected,and important occur in such systems. We treat the fundamentals of the subject
or more
two
have
effects
physical
in this chapter, with
Mixturesarc
of
systems
Mixtures
ternary and quaternary mixtures.If molecules, A
to form
three
and
mixtures
are called
four constituents
\"oii
expression
phases, such as oil and their and water do not niix\" means [hat
or
more
mole-
and not
arc atoms,
constituents
when its constituents are intermixedon an A mixture is heterogeneous phase, as in a solution.
a single
two
contains
with
the
species.Birjury
is homogeneous
mixture
scale
situations.
an ailoy.
is called
mixture
the
simple
different chemical
or more
two
two constituents.
only
from
drawn
examples
GAPS
SOLUBILITY
have
that
distinct
The
water.
atomic
when
it
everyday
does not form
mixture
a single homogeneousphase.
Thepropertiesof
differ
mixtures
solidification
and
melting Heterogeneous
Consider
mixtures
may
a gold-silicon
from
the properties
ofmixturesareofspeciai at lower temperatures than
properties melt
alioy: pure Au
melisat
lO63cCand
substances. The
of pure
interest. Heterotheir
pure
constituents.
Si at
I404X,
at 37OCC. This is not but an ailoy of 69pet Au and 31 pet Si melts (and solidifies} ihc result of ihe formationof any low-melting Au-Si microscopic compound: a two mixture of almost phase investigation of the solidified mixtureshows pure
Au
side
by
side
with
almost
pure Si (Figure I I.I}.Mixtures
with
such
because and they are of practicalimportance precisely of their lowered melting points. What determines whether two substances form a homogeneous or a heteroare in equilibrium mixture? What is ihe composition of ihe phaseslhat heterogeneous can be of mixtures with each osher in a heterogeneous mixture?The properties at a fixed semperaiure will underslood from the principle that any system evolve to the of minimum free energy. Two subsiances wiil configuration is the configuradissoive in each oilier and form a homogeneous mixtureif that will free energy accessible to the components. The subsiances of iowest configuration
properties
are common,
Gaps
Solubility
SO/tm When a mixture of 69 pci Au and 31 pet Figure II.I Heterogeneousgold-silicon alloy. Si is melted and then solidified, the mixture Au a into segregaies pure phase of almost \302\253iih a almost coexistent of Si aboui phase (Sight phase) pure {darkphase).Magnified is that of the lowest-melting Au-Si mixture, the 800 times. The composinon given eutectic a later so-called mixture, concept explained in she text. Photograph courtesy
ofStephan Justi. a
form
side by side is
phases
then we say I hat A
if [he
tnixlure
heterogeneous
the
mixture miMure
hclerogeneous
free energy of
combined
tower ihaii the free energy exhibits will
of
the
Uvo
the
separate
mixture:
homogeneous
a solubility
melt at
gap. a lower [cmperalurethan
the
separate
free energy of the homogeneousmeltis lower than the two combined free of solid energies separate phases. ihis we assume for simplicity that the external Throughout chapter be and we sel pV = 0. Then volume changesdo not neglected, may
substances
work,
and
if I he
the appropriate
Ilian the Gibbs free We
discuss
compounds
the
energy
free energy is G.
We
tire
will usually
HelmhoUz
free
the
pressure involve
F rather
energy
simply speak of
com-
free
energy.
well-defined binary mixtures of constituents Ihat do no! form with each other. Our principal interestis in binary Consider alloys.
Chapter H: Binary
Altitun
or more
two
have
concerned
are
biophysics,
science, and large parts of chemistry with the properties of multicomponent systems
materials
of
applications
Many
phases
coexistence.
in
Beauiiftil,
that
and important
unexpected,
Ihe fundamentals physical systems. in this chapter, with examples drawn from simplesituations. We treat
in such
occur
effects
and
of
the
subject
SOLUBILITYGAPS are
Mixtures
or more
of two
systems
molecules, A
is homogeneous
mixture
Binary species. four constituents
and
three
are
constituents
an alloy. when its constituents
is called
mixture
the
chemical
different
constituents. Mixtureswith ternary and quaternary mixtures.If the
have only two
atoms,
arc intermixedon an
a Single phase, as in a solution. A mixture two water. contains or more distinct phases, such as oil and \"oil and water do no! mix\" that means their mixture expression a single
melting
solidification
and
mixtures
may
a
an
of 69
alloy
from
the
properties
side
by
alloy:
pet Au
side
are common,
properties
of their loweredmelfing What heterogeneous
determines mixture?
and
evolve dissolve configuration of
with
they
a
two
compound: phase
mixture
pure Si (Figure 11.1).Mixtures
almost
and
Au-Si
are
of practical
importance
microscopic
of almost with
such
precisely because
points.
whether two substances form a homogeneousor 3 heteroWhat is the composition of the phasesthat are in equilibrium
be other in a heterogeneousmixture?Theproperties ofmixtures can wiil from the principle that at a fixed temperature any system to the configuration of minimum free energy. Two substanceswill in each other and form a homogeneousmixture if that is the configurawill lowest to the free energy accessible components. The substances
with each understood
substances. The
of pure
of mixtures are of special interest.Heterolower temperatures than their constituents. Si at 1404\302\260C, pure Au melts at 1063\302\260C and pure 31 pet Si melts (and solidifies} at 370\302\260C This is not
of the formationof any iow-me!ting investigation of the solidified mixture shows Au
form
at
melt
the result
pure
everyday
does iiot
properiies
Consider gold-silicon but
The
it
phase.
homogeneous
The properties of mixtures differ Heterogeneous
atomic
is heterogeneous when
to form
scale
mixtures
are called and not mole-
Solubility
10/mi
H.I Figure jure 11.1 Siisn is melted
Heterogeneous gold-siliconalloy. When a mixture of 69 pet Au and 31 pet and ihen solidified,!hemixiure into a phase of almost pure Au segregates a codxisieni of almost Si aboui wiih (dark (lighi phase) phase pure phase). Magnified 800 limes. The composition given is !hal of ihe towesi-melting Aii-Si mixiure, ihc eulectk: mixture, a concept explainedtater so-called in ihe texi. Photograph courtesy
ofSiephanJusii. mixture if the combined free energy of the two separate is free of the lower than the by side phases homogeneous mixture: energy we say that the mixture exhibiis a solubility gap. then A hcierogeneous mixture will melt at a tower temperature than the separate comif the free energy of the homogeneous melt is lowerthan ilie substances free of the two separate solid phases. combined energies Ihis we assume for simplicity that the external pressure Throughout chapter and we set pV ~ 0. Then volume changesdo not involve be neglected, may a heterogeneous
form
side
and
work,
than
the
appropriate
We
compounds
discuss
free eneryy
free energy G. We
the Gibbs
binary wiih
each
mixtures
will
is the Hetmlioltzfree energy usually
of constituents
other. Our
simply
speak
rather
F
of the free
energy.
that do not form weil-defined
principal interest is in
binary
alloys.
Consider
Chapter Hi
a mixture
Binary
Mixtures
of
aloms
JVA
the composition of the system
We express
x the sysiem
Suppose
per atom
in
\302\273
1
A'b/N;
-
Suppose
further
curves shape.
B}
jVa/N.
an
average
free energy
with
C)
two
homogeneous
Any
respect
to
two
separate
x
<
at
points,
of this shape are common,and mixture
in
derivative
second
ihe
x, < is unstable
\302\273
form shown
the functional
lias
that/(.\\)
we can draw a line tangent to the curve at
cause this
x
x of B aioms;
= F/N.
contains a range in which
energy
the fraciion
iermsof
forms a homogeneous solution,wiih
/
Free
B.Thetotal
by
given
this curve
NB atoms of substance
A and
ofsubsiance
is
atoms
of
number
in the
we
wilt
Figure
11.2.
Because
d2f/dx2 is negative, x = xx and x \342\204\242 x^. see
later what
composition range D)
xp
phases of
may
composition x, and x^. We
is that ihe average free energy per atomof the mixture segregated a the i\" line and in ihe on the Thus the straight given by connecting points [}. point a lower has free energy than entire composition range D) the segregatedsystem shall
show
the homogeneoussystem.
Proof
i
The
free
energy
of a F
where ,V,
Nfi are
and
=>
the total
segregated mixture of the two phasesa and NJix,)
+
NfJ\\xfi)
numbers of atoms in
j$ is
E)
. phasesa
and
ji, respectively.
These numbers satisfy the relations
which may
be
solved
for .V,
and Ny.
0)
Gaps
Solubility
Free energy per alom as a function for a of composition, a aiom of system gap. tf the free energy per has a shape such that a tangent can be drawn homogeneous mixture touches the two x and that curve at diftereiit /?, (lie composition points the two points is unstable. Any mixture with a range between in this two phases with the composition range will decompose into composition _v, arid ,\\f. The free energy of the two phase mixture is It. given by the point / on the straight line, below the point tl.2
Figure
with
E) we
From
a solubility
obtain
fjix)
(S) JV
for
the
free energy
straight line through
the
the point i
in
thef-x
points
of the
two
plane. a and
system.
phase
If we
/?. Thus/
set in
.v.
(he
=
is linear in x and is a v,c see (hat the line docsgo A'3 or.v^, result
This
between
interval
on the straight lineconnectinga and
p.
.y4 and
xfi
is given
by
Binary Mixtures
II:
Chapter
We have not yet
made useof
(he
that
assumption
(he straight
line is tangentto
3 and /J, and therefore our result holds for any straight line points two points 2 and/Jin commonwith/(.v). Bui fora given vaiue of x, (he lowest free energy is obtainedby drawing (he lowest possible straight line that has (wo points in common on opposite sides of a-. The lowest wiih/(-v), possible line the is shown. The and straight (wo-pointtangent x3 compositions x? are
f{x)
the
at
that has
the limits of the solubility of (he system. gap Once (he system has reachedits lowest free (he (wo phases must be in energy, to diffusive with both atomic species,so thai their chemical equilibrium respect satisfy
potentials
/*a> = We show
point tangent with as in Figure 11.2.
the
two
/jB
are
(9)
Pb*-
given
by (he
of the two-
intercepts
plot at x
edges of thc/(x)
vertical
*=
0
and
a ~
1,
MIXING
OF
ENTROPY
AND
ENERGY
jja and
i that
Problem
in
=
Pb\302\273
/*\302\253;
The Heimhoitz free energy F ~ U ~ to has contributions from Ihe energy and from the A and B on We treat the effect of mixing two components entropy. both terms. Let uA and \302\273a be the energy per atom of the pure substancesA and B, referred
atoms
to separated
at
Tlie
infinity.
energy
average
per atom of the
constituents is u
=
(uANA
+ vsyn)/N
which defines a straight line in
the
per
mixture
atom
separate homogeneous
of
the
homogeneous
constituents. is
niixture
u~x
-
nA
4-
-
(\302\253B
A0)
uA)x ,
Figure 11.3. The average energy be may larger or smallerthan for the
plane.
In (he example of Figure11.3,(he energy than the energy of the separate larger
of
the homoge-
constituents.The
of mixing. If (he re term in the free energy is negligible,asat 1 = 0,a positive mixing mixture not will that a is stable. mixture means such energy homogeneous Any in the the \342\200\224ia then separate into two phases. But at a finite temperature term free energy of the homogeneous the tends to lower mixturealways free energy. a contribution, the ofa called mixturecontains entropy of mixing, Theentropy of the separate components. The mixing that is not present in (he entropies the different arises when of atoms species are interchangedin position; entropy of such interthis a different state of (he system.Because operation generates
energy
excess
is called
\342\200\224
the energy
and Entropy
Energy
11,3
Figure
in a
sysicm
Energy with
per atom
a positive
as a function
of composition
energy.
mixing
of Mixing
A
simple
a solubiliiy gap may occur is thai of a in which the system energy per atom of the homogeneous mixilire Him of ihe separate phases, so that is greater than 1 s. The 0 for att c mposi ing e rgy i e bci differ een the u[x) curve and the straight line. example
for which
states a mixture has more accessible and hence the mixturehas the higher entropy. In C.80) we calculated the mixing entropy A, ^B,. to find
changes
two
tlie
than
erM
of
separate
substances,
a homogeneous
alloy
(ID
as
plotted
in Figure
11.4. The
that the slope at the ends of the
N dx which
goes
to
+ co
as x
curve of aM
range
composition
-
X)
the important property is vertical. We have
x has
versus
-
bgx
-+ 0 and to \342\200\224 co as
- ioj
x -*
I.
A2)
Chapter II:
Binary
Mixtures
=
da^/elx
Ffgure
11.4
Mixing entropy.
interchange of two
atoms
Tn
species leads to a new
system. The logarithm of liie number of mixing
Consider
is the
slate
in this
related
slates
way
of the is the
entropy.
now the
quantity
u[x)
which
of two constii uenis an
mixture
any
of different
\342\200\224 X
free energy
per atom without
-
(a
the
-
A3)
ffjtf
mixing
entropy
contribution.
The
is usually nearly the same for (he an, non-mixing part of the entropy,
entropy
curves
dependence
to fo(x),
~-zaM/N in
Figure of/0(.v)
we obtain at
11.5. In drawing itself,
because
the for
Energy
Free energy temperatures.The curve Figure 11.5
per
atom
fQ is the
versus composition, free energy per atom
und
Entropy
of Mixing
at three without
the
a parabolic mixing entropy contribution. For ilHistraiion composition dependence is assumed, and the temperaturedependence of/0 is The tliree solid curves represcnl the free energy neglected. including the mixing for the temperatures 0.8 rM, 1.0 tM, and 1.2 rM, entropy, where there is a solubility rw is the maximum temperature for which gap.
our
The
this is
argument
construction
separation
phase
at 0.8 rw
is apparent.
follow irrelevant. Three importantdeductions
of the/(x)
from
the
curves:
(b)
f{x) turns up at both ends of the composition contribution. infinite range, entropy slope of the mixing Below a certain temperature rM there which is a com position range within than is ihe the second derivative of the fo[x) curve stronger negative second it derivative of the positive -taM contribution,thereby making values of x. to draw a common tangent to f(x) at two different possible
(c)
Above Ty
(a)
At
all
finite
temperatures
because
the
of the
curve
has
a positive
second derivative at al! composilions.
Binary Mixtures
11:
Chapter
We conclude that the A-B system with below the solubility gap temperature tM.
widens with composition
only
range
solubility of A
in
B and
as t -* of B
in
The
0.
At
A,
a result
finite
any
energy
can reach the
a
gap
edgesof
there
temperature
earlier
obtained
exhibii
of the
will
range
composition
the gap
but
temperature,
decreasing
mixing
positive
in
the
finite
is a
3. The
Chapter
below Positive is that the mutual solubility is limited only tw. We now discuss three examples. mixing energies arise in different ways.
new result
Consider an alloy A^jB, with nearest-neighbor interact'ionSi in than the attractive interinteraction between unlike atoms is weaker interaction between as bonds. There are like atoms. For simplicity we speakof the interactions be ihe potential three different bonds: energies of A-A, A-B, and B-B. Let uAA) uAB and uBB each bond.These binding energies will usually be negativewith respect to separated aloms. We assume the atoms are randomly distributed among the lattice sites.The average of ihe bonds surrounding an A alom is energy Binary
Example:
alloy
(he attractive
which
=
uA
where (t - x)is ihe propoition mean field approximationof
-
A
of A and x is the
=
A
The total energy is obtained by summing the average energy per atom neighbors,
The
factor^
can be written
ip[(l
Figure
mixing
II.5.
proportion of B.This for B atoms,
-
+
X)HAD
result
is wiiucn
in ihe
over
both
A5)
N\"UD.
atom
types. Ifeach
atom has
p nearest
is
+
xJUAA
2jcA
aiises because eachbond is sharedby
- *Kb + the
two
atoms
A6)
*3\302\253db]-
it
The
connects.
result A6)
as
u =
is the
-
A4)
tO. Similarly,
Chapter
\"a
,
xuAa
+
x)uAA
energy.
On this
ip[(l
model
-
the
x)uAA
mixing
+
xum]
energy
as a
+ uM.
function
A7)
ofx
is a
parabola, as
in
Energy
A solubility gap
occurs whenever (/'//dx1
< 0, that
= -2P[fAB
^r
-
and
Entropy
of
Mixing
is, when
i(\302\253AA
+
B0)
O3-
From A2),
x{l -
N dx2 The
sign holds
equal
for
,x =
$. Wilh
T*i
-
x)
these results{19)yields
M\302\253ab
-
iO'AA
+ ^a)]
B2)
as die lower
of the temperature for a solubility limit gap. are many reasons why mixed bonds may be weaker than ihe bonds of the sepafaic constituents.If the constituent in radius, the difference introduces atom* of an alloy differ clasticstrains that water molecules raise the energy. Water and oil \"do not mix\"' because There
carry a large water
strong
electric
molecules.
as
the
moment
dipole
This attraction
weaker
oil-oil
Example: Mixture of two
that leads to a strong electrostatic attraction which are only is absent in water-oil bonds,
between about
as
bonds.
solids
with different
crystal structures.
Consider a homoge-
of gold is the facesilicon. The stable crystal centeredcubic structure in which nearest equidistant every atom is surrounded by twelve of silicon is the diamond structure in which structure every neighbors. The stable crystal aiom is surrounded by only four equidistant nearest neighbors, if in pure Au we replace a wiih the small fraction xof the atoms by Si, we obtain a homogeneousmixture Au^.Si., 1 fee crystal structure fraction x of the of Au. Similarly, if in pure Si we replace a small aioms by Au, we obtain a homogeneous mixture Au, -,5^, but with the diamond crystal siructure of Si.There are two different free energies, one for each crystal structure (Figure range, or else pure Au and 11.6).The two curves must cross somewhere in the composition curve consists of the lower Si would not crystallize in different The structures. equilibrium a sotubility with a kink at the crossover point. Such a system exhibits of the two curves, in the on side of the curves shown either crossover The figure are schegap composition. to the in the actual extends so close Au~Si system the unstable range schematic; edges of the from x = 0 to x = 1. that it cannot be representedon a fult-scale extending plot diagram homogeneouscrystalline
mixture
of gold and
structure
Chapter
11;
Binary
Mixture
\\/
Figure
11.6
Tree
energy versus
homogeneousmisiuresfor mixture
crystallize
free energy curves
which
in ^ilfcrem
composition for cryslallinc [he [wo constituents of the
crystal
are involved,
one
Two
structures.
for each
differe
crystal structure
Different crystal structures for the pureconstiiucmsarean cause or solubility important in crystalline solid mixiures. Our a/gument to mixtures of ihis kind, provided gaps applies the two structures do not transform coniiriuously into each other wilh changing composicomposition. This when is a tacit assumption in our discussion, an assumption not always satisfied the two crystal structures are closely similar. The other we make throughout assumption this is that no stable compound formation occur, should in the presence of comchapter the behavior of the mixture be more formation compound complex, may
vs^~7cz:\":\"s'.\".r:'~ Exampk: mixture
with
Liquid
SIU-*
a solubility
-..,.-,...
~
-\342\226\240\342\226\240\342\226\240 ---\342\200\224.-.\342\200\224 --...-.. ... \342\226\240\342\200\242\342\226\240-\342\226\240->\342\226\240\342\226\240-\342\226\240\342\226\240\342\226\240-\342\226\240\342\200\224
He m*.Uuivi
gap is the
at W
miMuniof
liquid mix-
tanpcraiures.
The
ilie two iiefium
isotopes JHeand JHe, atoms
moat
interesting
;-.-,,
of
ocmii fcimjon^ unti of the 'aHer bosons, 1 lie re js a soluoiltty u\302\273io sn the mtx turc oclow 0.S7rCj ii1/ in i igure 11.\027. 1 Ins property ss utilised m the Iicliuitt cJj'tiiion refftccr^tor have a must be positi\\e to The origin of the solubility gap. {Chapter 12). Tito mi.\\ing energy low temperais tht! folio\" ing: 4Hc aloiiisarc bosons.At suliieiently positive mixing encray temperatures almost jli \342\226\240*! le afoins have state orbii;tl of the sysicm, vvherc occurs) the ground they
tiie Toruicr
30
20
10
40
Phaie
Diagrams for
50
60
.70
SimpleSolubility
80
Gaps
100
90
\342\226\240\" Atomic
Figure
11.7
Liquid
percent
mixtures of
JHe and
He
pure
\342\226\240'He
4Hc.
kinetic energy. Almost trie entire energy of the mixture is contribuicd by t!ie which are fermions.The of a degenerate Fermi atoms, energy per atom gas increases v,ilh concentration 7. This energy has a negative secondderivative as n1'*,as in Chapter
kinetic
zero
3He
Pltase
Diagrams
Tor Simple
Solubility Gaps
dependence of solubiiily gaps,as in the 11.8. The two compositions xx and xf arc plaited horizontally, Figure The .v^ and xf branches merge at the vertically. corresponding temperature maximum temperaturet,m for which a solubility gap exists. At a given temperatemperature, overall composition falls within the raoge enclosed mixture whose by any of actual curve is unstable the as a homogeneous mixture.The phasediagrams
A
phase
mixtures
form of
diagram
represents
with solubility
the temperature
according gups may be more complex,
(he free energy relation/(.v),
but
the
underlying
principles
to
the
aclual
are ihesamc.
II;
Chapter
Binary Mlxtur
Slabk
Decomposilion
/
\\
1
1 1 1
Uns
Figure 11.8 Phase diagram gap. A homogeneous mixture i if
temperature
the point
curve. The mixture
will
curve
boundary
system
PHASE
(*,i)
a solubility
be unstable stability boundary
tlic
ai
form two
separate phasesof the the intersections of the stability boundary line
for
with
temperature
r. The
stability
calculated quantitatively
a parabolic
BETWEEN
EQUIUBRIA
with
x will
below
Tails
shown here was
of Figure lt.5,
system
oCcomposition
then
given by curve with the horteontal compositions
a binary
for
the
for
fo{x).
LIQUID
AND SOLID MIXTURES a small
When
of
the
fraction
phenomenonis mixtures.
of a
solid that forms We
the homogeneous liquid mixturefreezes,
is almost always
readily
understood
consider
a simple
different
from
that
composition
of the liquid.
The
liquid and solid model, under two assumptions; (a) Neitherthe from
the free
energies for
Phase
solid nor she liquid has a solubility
constituenl
is
A
The free energies
melting
temperature
ta or pure
Semperature tb of pure constituentB. and ta ra. Tor the solid and fL(x) Tor the liquid, are
between fs{x)
atom,
per
(b) The
gap.
Solid Mixtures
Between Liquid and
the melsing
Jhan
lower
a SemperaSure
consider
We
Equilibria
11.9a. The two curves intersect at some comLeSus draw a commonlo boSh aS .\\ ~ xs posision. jangenS curves, touching/j = and fL a! x xL.We can define three composision ranges, each with differcnS shown
in Figure
qualitatively
internal
equilibria:
x < xL, the
(a) When
in
system
is a
equilibrium
homogeneous
liquid.
a system in equilibrium consistsof two phases, solid phase of composition xs and a liquid phaseof composition xL. x > xs the system in equilibriumjs a homogeneous When solid. (c) a arc The and of and a so!id in compositions xs xL liquid equilibrium phase of the temperature dependent. As ihc temperature decreasesthe free energy solid decreases more rapidly tlKll of the liquid. The Ungctitiai points in than Figure 11.9amove to the Icfi, Tliis behavior is rcprcscnScdby a phase diagram stinihlr to the earlier representationof the equilibrium curves for composition
(b)
When
mixtures
xL<
with
xs, the
x <
11.9b the curve
In Figure
separation.
phase
for
xL
is called
the
curve. Hquidus curve; the curve for xs is the soltdus have been determined experimentallyfor vast numbers The phase diagrams of binary mixtures. Those for most of the possible binary alloys are known.*
For
Figure phase diagrams are more complicatedthan for a 11.9b, simple system, germanium-silicon. When is lowered in a binary liquid mixturewith the the temperature phase of Figure lj.9b, solidification takes placeover a finite diagram temperature a liquid with the range, not just at a fixed temperature.To see this, consider is lowered, initial composition xiL shown in Figure 11.10.As the temperature \342\200\224 of the solid formed is given solidification begins at t composition x,. The is changed. 'hat of the In the so the remaining liquid by xtsi composition the example xiS > xiL, so that the liquid moves towards lowervaluesof x, where if solidification is lower. The temperature has to be lowered temperature the of the moves solidification is to continue. The composition along liquid = at t The solid formed the curve until solidification is compleied tA. liquidus metal
most
homogenizeafterward
in
for
a long
slandard
Constitution
iabutatlons
of
binary
arc
by
solids
many
M.
in
atomic
solid
may homoge-
temperature remains is too slow, and the
if lhe diffusion
\"
indcnniieiy.
Hansen,
of binary alloys, firsi second supplement, alloys,
Constitution
Edioti,
for
''frozen
The
equilibrium.
particularly
diffusion,
time. But remains
\342\200\242
The
atomic
by
homogeneity
R. P.
and is not in
in composition
is nonuniform
high
the
alloys
was drawn
which
Coitsilxatlon of binary supplement,
allays,
McGraw-Hill,
McGraw-Hill, 1969. .
McGraw-Hill.
1965; R
A.
1958; Shunfc,
11.9
Figure
ihis example
Phase
equilibrium
neither phase
cxhibhs
btiwccn
liquid
a solubiliSy
and
solid
mixtures.
In
gap. We assume
the free energies for i|ie two plxiscs; The upper figure (a) shows The curves xL ihc lower figure (b) shows ihc corresponding diagram. phase and xs in She phase diagram are called ihe liquid us and She solid us curves. = 940cC and The phasediagram is She Gc-Si phase diagram, wish TCt
tA
x < xlt.
<
-
I412\"C
7\342\204\2425i
324
Phase
Figure
Mosi
11.10
Scmperalurc,
but
liquid over
higher-nwiliiig
consiiluenl
lower-mching
consliSucnt
solidification
lemperalure
Advanced Treatment;
mixiuresdo
a finite
not
temperature
liquid
at a
range
from
and Solid
Mixtm
sharp t,
'o ta.
The
first, thereby enriching the liquid phase and thus lowering She of ific liquid. precipilaSes
in ihc
down
tower meltingtemperatureof the system: a mixture of 69pet An and solidification
Liquid
solidify
Eutecltcs. There are many
liquid phase remainsa
compositions
Between
Equilibria
to
temperatures
constituents.
31 pet
starts at a
binary
The
Si starts to
in which
systems
the
below the alloy is such a
significantly go!d-si!icon
solidify at
higher temperature.When
370\302\260C.
we
At
plot
other
the
of alloy oflhe onset of solidificationas a Function composition, obtain Mixtures with two we the two-branch liqutdus curve in Figure 11.11.
temperature
solidification minimum temperature liquidus branchesare calledcutectics.The is She eutectic is the eutectic SemperaSure,where She composition composition. is a two phase solid, wiih The solidifiedsolid at the eutecticcomposition nearly pure gold sideby side wiih nearly pure silicon, as in Figure 11.1.In the solidAu-Si mixture shcre h a very wide solubility gap. The low mching point oFthe for the free occurs eutectic composision becausethe homogeneous energy melt is lower than the free energy of the two for at solid, temperatures phase or above She cutectie temperature. Such behavior is common among systems thai exhibita solubilitygap ill the solid but not in the liquid. The behavior of eutccjj'cs can be understood from the free energy plotsin Figure11.12a. We for the solid as in Figure 11.6, assuine_/^(.\\\")
11:
Chapter
Mixtures
Binary
1600 404\302\260
1400
1200
U
1063\302\260
1000
E
/
/ \\
/ \\ 0\302\260
1/
-31
Pure
11.11
Figure
Au
Euieciic
and
range
Figure
but
II.12a
below
10
iempctaiure indicaie
37OX
complcic
80
90
100
Pure Si
silicon
percent
go!d-siljconalloys.The
iogeihcr ai ihe euieciic daia poinls ai ihe mixiurc docs noi
to difTcrcnt
corresponding
temperature
diagram of
ihe experimcnial
composiiion
constituents.
phase
60
50
40 Atomic
branches ihai come line
30
20
10
0
Hquidus
iwo
consisisof
T, = 37O;C.The horizonial ihai ihroughoul the eniirc
iis solidificalion
unlit
ihe
euieciic
crystal structures a and ^ for the two pure confor a temperature above the cutectic
is constructed
the melting
iemperature of cither consihuent,
so
that
to energy of the liquid reachesbelow the common tangent phase curves. We can draw two new two-point tangents tltat give even lower free energies. We now distinguishfive different ranges: composition is a (a) and (e). For x < xaS or x > x^, the equilibrium state of the system solid. In the first range the solid will have the crystal structure a; Homogeneous in the second the structure is range ($. (c). For xlL < x < XpL, the equilibrium state is a homogeneousliquid. and For is in a liquid (b) (d). x^ < x < xaL or x^L < x < Xp$, phase equilibrium with a solid phase. the As is lowered, faS and fa decreasemorerapidly than/L, temperature and of the the range H.12b homogeneous liquid becomesnarrower. Figure shows the corresponding the two curves. solidus phase diagram, including tUe free
the solid
Figure 11.12 Free energies(a)and sysiem.
At
to
the
theeulecltc
common
t, Jhe
temperature
tangent
to f^
energy fL
a homogeneous
A mixture
meits at a
above
(he
and fps, as in
tangent,
of Jhe liquid phase is tangential
free energy
which fL touches the tangent is the iies
(bj in a
diagram
phase
Figure
eutcctic
although
11.13.
The
composition At x < xt, the
composition.
fL may be beiowthe free energy
at
free of
solid.
of composition equal to
single temperature,just like
the
a
eutectic
pure
composition
substance.
solidifies
The solidification
and
of
Chapter 11: Binary
Mixtures
FiCe energies
Figurel!.!3
compositions
away
From
and
ends
at
the eutecttc
and
ends
at
a higher
in
a euseclic
system at t
, andati
< xr.
starls at a
higher temperature at the starts eulectic temperature. Melting temperature the
euieciic
composition
temperature.
The minimum properly of the utilized. The Au-Si eutectic plays
melting
a large
temperature
role
in
of eutectics
semiconductor
is widely device
tech-
welding of electrical contact wires madeofgoldtosilicon devices. Lead-tin exhibit a euieciic (Figure alloys at a i83\302\260C solder below that of pure tin, to 11.14) give melting temperature 232;C. is to whether a sharp melting temperature or a melting range According comor a different citlicr the exact cutectie compositionB6pet lead) desired, Salt the because of the low is on ice melts ice composition employed. sprinkled
technology:
eutectic
the
cutectie
temperature
permits
-2L2\"Cof
low temperature
the H2O~NaC!
eutecttcat 8.17moipet
NaO.
The in character, solfdus curves of eutectic systems vary for the greatly die ioclt Pb-Sti system (Figure Il.M) die solid phases in equilibrium with contain :tn appreciable fraction of tltc minority const [merit, and this fraction in other increases with decreasing systems this fraction may be temperature, small or may decrease with or both. The Au-Si system decreasing temperature, with is an example: The relative concentration of Au in solid Si ill equilibrium of only an Au-Si melt reaches a ma\\imum value 2 x !G~6s.-ound i 300\" C, and it drops off rapidly at lower temperature. In our discussionof the free energy curves of Figures 31.12 and 11.13we assumed the lite composition tltat at which the liquid phasefree energy touches
10
\020
20
pure Sn
30
40 Atomic
50
60
percent
lead
70
90
100
pure Pb
10/tm Figure
of [he Jackson.
11.14 Pb-Sn
diagram of the Pb-Sn s> stem, after Hatlicn.{b) Microphotngrapf; of J. D. Hunt and K. A. magnified about S0Otimes.Courtesy
{a} Ptiasc eutmic,
S29
11:
Chapter
Mixtures
Binary
curves lies between the
solid phase
tangent
to the
In some
systems this point lies outsidethe
and
were
fL
i
iti Figure
interchanged
i
interval,
Such
.\\2a.
and
compositions
xlS
as
and/t
if either/aS
arc caiied
systems
xfiS.
or/flS
peritectic
systems.
SUMMARY
1.
side by side is lowerthan
phases
separate
the combinedfreeenergy
gap when
a solubility
exhibits
mixture
A
free
the
of the
energy
two
of
homogeneous
mixture.
2, The in
the alloy
For
position.
3. The mixing
for
energy
uM
for
p nearest
4. The
5. Mixtures
minimum
we have
A} _IBI,
px(l
-
is
interactions
nearest-neighbor =
species are interchanged
of different
atoms
-
j(uAA +
uBB)]
.xL versus
t
x)[uAB
,
neighbors. is
Hquidus
equilibrium for a solid
when
arises
entropy
mixing
the
curve
composition
with a solid.
a
for
The solidusis the compositioncurve
phase in equilibriumwith
a
in
phase
liquid Xs
i
versus
liquid.
two branches to the liquidus curve solidification temperature is called the
with
The
eutectics.
called
are
eutectic
temperature.
PROBLEMS
L Chemicalpotentialsm two-phase
[eniials ;iA and /jB of phase mixtureare given with 1. liquid
the vertical Mixing 3He-4He
energy
die two by
the
intercepts
B of
and
oFthe two-point
edges oFthe diagramat x
=0
x
and
the chemical an equilibrium
tangent
mixtures
in
in
the
Similarly,
solubility
oFPb
11-2
Figure
andPb-Sn mixtures. The phasediagram of 3He Figure 11.8 shows that the solubility
-> 0.
potwo
\342\200\224 1.
in 3He~*He
finite (about 6 pet) as r residual Figure.11.14 shows a finite remains
that
Show
equilibrium.
atomic species A
Pb-Sn
in solid
phase
oF
liq-
in 4He
diagram
of
Sn with decreasing
t.
do
What
such
residual
finite
solubilities
about
inipty
the
Form
of the
Function
u(.x)?
3.
Segregation
In
this limit the
1to A, wish A' \302\253 Let B be an impurity the oF can be as linear Free non-mixing parts expressed energy both solid phases. of x, as fQ(x) = /0@) + x/0'@),for and functions liquid Assume thai the liquidmixtureis in equilibrium with the solid mixture. Calculate concentration the coefficient. ratio k ~ xs/xL, called the segregation equilibrium For k \302\253 then a and substance be many systems may I, purified by melting and partial resolidificatioti, discarding a small FractionoFthe meit.Thisprinciple used in the purification of materials,as m the zone is widely of semirefining \342\200\224 = = \302\243 ! T 1000K. Give a numerical value for eV and semiconductors. for/os' /Dt'
4.
of impurities.
coefficient
of a
range
Solidification
binary
alloy.
Consider the solidificationofa
binary
of the that, regardless diagram of Figure.' 11.10.Show B in component initial the melt will always become fully composition, depleted ion the id i Seas the time remnant That sol the last of the meit solidifies. is, by will not be complete until the has dropped to TA. temperature
alloy
5.
wish
Alloying
the phase
of gold
hto
silicon,
(a) Suppose a
and onto a Si crystal, subsequently diagram, Figure 11.11,estimate how
silicon crystal. the estimate
The
for
densities
800\302\260C.
of Au
heated deep
and Si
1000A
to she
400\302\260C-
gold
layer From
will
of Au the
is evaporated Au-Si phase
penetrate
are 19.3and 2.33gcm\"\023.
into the (b)
Redo
12
Chapter
Cryogenics
COOLING BY EXTERNAL IN
AN
Gas Liquefactionby
WORK
334
ENGINE
EXPANSION
for
Effect
337
Effect
Joule-Thomson
the
Example: Joule-Thomson
van
der
Waals
Helium,
Pumped
Cooling:
to 0.3
K
HeliumDilution Refrigerator;
341 342
Miilidegrees
DEMAGNETIZATION:
ISENTROPIC
QUEST
333
339
Liride Cycle Evaporation
Gas
346
ZERO
FOR ABSOLUTE
NuclearDemagnetization
348
SUMMARY
350
PROBLEMS
350
1. Helium
2. Ideal
as a
van
der
Waals
359
Gas
35i
Carnot Liquefier
3. Claude
Cycle
4. Evaporation
Helium
35!
Liquefier
352
Cooling Limit
for 5. Initial Temperature
Demagnetization
Cooling
352
physics and techiioiogyofthe productionoftow temperatures. the physical principles of the most important cooling methods,
is the
Cryogenics discuss
We
lowest
the
to
down
cooling
a gas
of
temperatures.
principle oflow temperature generation
The dominant
by kiting it do work be a conventional may
against
is the
lOmK
to
down
during an expansion. The
a force
gas; the free electron gas in a semiconatoms dissolvedin liquid 4He. The force semiconductor; or internal to the gas. Below be external against which work is done may 10mK the dominant cooling principle is the iscntropic demagnetizationof a
gas
employed
or
the
ihe cooling methods chain lhat starts cooling
discuss
We laboratory
to the
Household
cooling
evaporation
liquid helium
below its
COOLING
BY
In the
method boiling
EXTERNAL
EXPANSION
AN
in
the
they occur
in which
order
by liquefying helium and
a
in
from
proceeds
1 ;iK. sometimes lowest laboratory temperatures,usually lOmK, and automobile air conditioners utilize the cooling appliances
same
IN
gas of 3Hc
substance.
paramagnetic
there
virtual
that
temperature,
is used
in
the
to about
for
laboratory 1
cooling
K.
WORK
ENGINE
isentropic expansion
of a monatomicidealgas
lower pressurep2, the temperaturedropsaccording
from
to a
pi
pressure
to
(i)
by F.64). temperature working
process
and Ti = 300K; then the temSuppose p, = 32atm; p2 = iatm; will drop to T; \342\200\224 75 K. We are chiefly interested in helium as the and for helium A) is an excellentapproximation if the cooling gas, is reversible.
The problems in implementingexpansioncoolingarise of actual expansion processes.The problems irreversibility of good low temperature lubricants. by the nonexistence and cooling cycles follow Figure 12.1.The compression
the
from are
partial
compounded
Actual expansion
expansion parts
of
itisng
by
Work in on
External
Heal
{Expansion
ejection
Expansion
ngine
Working\342\200\224
volume
gas ts Simple expansion refrtgeraior.A working is the heat of into the compressed; compression ejected environment. The compressed room temperature gas is heat counleriiow further in the exchanger. It then precooled to a does work in an expansion engine, where it cools volume. Afkr extracting temperature below that of the working hea{ from the working the gas returns to the compressor volume,
FEgure
via
{he
I2.I
heat
exchanger.
Ens
Chapter 12:
Cryogenics
separated.The compressionis
the cycle arc
exchanger by contact
flow heat
temperatureof the
in
cooling
the
on
the
expansion
of the
design
The
work
done the The total
by
plus the work
She
=
W
{Ul
on the
performed
important
fij and
the expansion
work p{Vl gas.
the displacement
plus
gas
into
flowing
difference
enthalpy
Vx
a given
to
refer
mass of
is the energy U2 the the gas against pressurep2-
with
engine
the gas
of
the
gas
work
The
is the difference
For a monatomic ideal work
the
of
boih
to move
V% required
p2
by ihe engine
extracted
exchangeris as
is the
engine
expansion
where
leaving
energy
im-
requirements
cooling
gas: The iota!energy
and output
compressor,
by
the
iruernat energy U^
engine is the
heat
the
via
compressor
expansion engine.
extracted
the input
between
the
the
reduces
The design of Sheheat
engine.
counter-
in a
precooled
return gas stream at the low cooled to itslowesttemperature a low friction turbine.The cold gas extracts
The heat exchanger greatly
exchanger. imposed
usually
temperature
gas is then
load and then returns to
the cooling
from
room
near
room
above
cold
the
with
The
load.
engine,
expansion
heat
as
the
or
at
performed
The hot compressedgas is cooled to temperature. by ejecting heat into the environment. The gasis further
- {Uz +
4- PlVt)
gas
U
j=
pV =
and
|Nr
=
PlV1)
Hs
~~
B)
H2.
Nr, hence H =
The
\\Nx.
engine by the gas is -
\302\273
W
|N{t,
C)
r2).
The countefHowheat exchangeris an enthalpy device: it is an exchange expansion engine which extractsno externalwork. Most use expansion engines to prccool the gas closeto its Hquefiers gas It is impractical to carry She expansion cooling to temperature. liquefaction of of a liquid the point liquefaction: the formation phase inside expansion
enginescauses
mechanical
is
a
usually
Joule-Thomson
liquefiers
usually
temperatures,
with
multiple
The principle of
to
the
electron
potential potentials.
cooling by
gas
in
or
more
stage
expansion expansion
of an
below.
exchangers. isentropic
semiconductors.
When
ideal gas
electrons How
is applicable
from
a
semi-
high electron concentration into a semiconductor she the electron gas expands and does work concentration, against barrier between the two substances that equalizesthe two chemical is used electronic cooling, called the Peltiereffect, The resulting
semiconductor wuh electron
heat
final liquefaction
Helium and hydrogen engines at successive tem-
discussed
stage, two
eontain
The
difficulties.
operating
with
a lower
Gas Liquefaction by
down
to about
So 135K Gas
195 K quite routinely;in by the
Liquefaction
units
multistage
a
12.2.The work
~plt(Vj doneon
Here
dVt
the
is
in
gas
pushing
it through
is negative
The overall processis at constant expansion valve acts as an expansion = 0 in If B), we have H\\ ~ li2 in the
11
\342\204\242
ideal
\\Nr,
so
that
down
\342\200\224in
ts
r2
of
al! gases.
causes
significant
will condense.
gas
lower
pressure
p2,
the displacement and {he expansion valve
work
a
engine
sec this,
that extracts effect.
Joule-Thomson
There is
the expansion.
the
gas on the downstreamside. To
enthalpy.
At are
interactions
the
between
difference
{he
+p2(\"/2 recovered from the and dV2 is positive.
work
displacement
temperatures
pressure p, is forcedthrough
valve into space with
an expansion
called
constriction
as in Figure
Gas at
is simple.
implementation
practical
Effect
Joule-Thomson Effect
Intcrinolccular attractive interactionscausethe condensation icmpcratures slightly above the condensation temperature that work strong enough against them during expansion of the cooling of the gas. If the coolingis sufficient, part This process is Joule-Thomsonliquefaction.
The
Joule-Thomson
achieved.
been
hnvc
the
notice that the
zero work. With For an ideal gas
zero coolingeffect
for
an
gas.
gases a small temperaturechange work done by the molecules duriiig expansion. in real
|2,2
The Joule-Thomson
through an expansion value. If be a temperature change during
done against initially
will cool
the
below
The sign
the
eflccl.A
the the
gas
is pushed
notlflieal. ihere will because of work expansion
gas is
forces.
If the
on Joule-Thomson expansion-
internal
of the temperature
temperature is inversion temperature, riB,, the gas
intermolecuhtr
a certain
of
because
valve
Expansion
Figure
occurs
12:
Chapter
Cry
Liquefaclion dala
U.I
Table
n.
K
CO,
195
cm
112
902 77.3 20.4
o,
N,
H, \342\200\242He
4.SS
JHe
3.20
the
The las
liquid. Jrti
T( ano\" oot
T,,.,
Tt,
K
(jas
lo
for
U/mol
304
B050)
25.2
191
A290)
155 126 33.3 5.25 3.35
umn,
measured
em'/mol
223
66
6.82
67
621
5.57
205
0.90
28.1 34.6 28.6
51 B3)
0-082
320
0.025
50.8
n walls
pressure.
mosphcric
der Waais
effect for
have
ran
\302\253=
JWt
gas, where a and
corrections caused by corrections
opposite
the
short
is
0.14
for
because its lrjple us LNG give daia
quamilics for shipping we of air. For helium,
signs. The
the critical
far
li'aab
gas.
+
{S2fV){bx
b are
positive
range repulsion and
initial such
poinl
fuel.
boih
c
Liquid for ihe
temperature. an expansion
for common
== lab
in
found
gases
A0.75)
that
D)
constants. The the
= 2/rt,
tempera!urc, defined by A0.46). lemperature. iln, is the inversion
We
- 2a)
tola! correciion changes tinv
where xc
0.7!
12.1.
Table
Joule-Thomson
a van
8.7
'
H for
45
be la'ken up
can
thai
on the depends change during a Joule-Thomsonexpansion All gases have an inversiontemperature below which TIn, above which heals (he it cools, gas. inversion temperatures
Example:
314
893
of natural gas, which is liquefied in huge and niirogen are separaied iu lhe liquefaction ei isotope 4Hc and for 3He.
in
Mite
34.4
Carbon dioxide solidifies
are listed
wall
8.18
tndkai
Atl/V,
V,.
AH,
K
long sign
range at
last
two
atiraction.
arc the The correc-
terms
the temperature
E)
at fixed For t < iin, the enthalpy here in expansion the work done against the increases; temperature increasesas the volume In a process at consiant enihafpy attraciive interactions between molecules is dominani. this increase is compensated by a decrease of the \\Nt ierm, that is, by cooling the gas. For The
temperature
Gas Liquefactionby i a
ioIccuIcspenetrate
farther
lhe repulsive
Joutc-Thot
s because now the anl: ai lhc higher
fixed lemperamre inio
the
work
done by the
lempcraluie
the
regio
wilh litjueficrs the Joule-Thomsonexpansioniscombined heat exchanger, as shown in Figure J2.3.The combination is a cycle in 1895 to called a Lindc cycle, aficr Carl von Linde who used such air starting from room temperature.In our discussion we assume that liquefy is ihe same the expanded the heat at from exchanger gas returning temperature as the compressed it. We neglect any pressure differencebetween gas entering the output of the heat exchanger and the pressure above the liquid.
Linde
cycle.
In gas
a counlerflow
To
and
from
comprcs
Figure by
12.3
The Lindecycle.Gas
combining
a countcrflow
JT expansionvaJv.
Liquefied
gas
Joule-Thomson
expa
Iieatexchanger.
Figure
as a
Performance of helium inpui pressure,
12.4
fund
ion ofihe
for various values of
ihc
inierna! refrigeration load available
ii^uciicrand
ihc
ihe
heal
through
See Problem Plenum,
still
exchanger A. J.
1971.
p. 1S7.
K
if
gas boiledoff
rather
Croft
cunes give at 4.2
coid helium
3. Afier
The solid
in
than
boiled
QiM (he
= toad
curvesgive tfDUl
A&mWL-dcryo&mcs
-
ihe
Hia, ihe
is placed
by lhc load
oiT into
Liude cycle,
of 1 aim and
pressure
ouiput
temperature.
The broken
coefficient
liquefaciion
input
an
by the
operating
litjueficrs for
inside ihe
is relumed
liie atmosphere. (C. A. Baiiey, ed.),
Evaporation Coaling;
The
comhimtiion
hc;\302\273
fraction
is
X
Constant
liquefied.
=
Hla
lhe input
of
heat
the
and
output
under lhe pressurepoal.
peraturc
-
+
enthalpy ihat the
suppose
J)H9Ui'
\302\243!
lhe enthalpy
tfHl) is
exchanger-
consimu
K
\302\2436}
mo!c of llou, = H(Tin.pBJare the enlhalpies per both at lhe common pressures, upper temperature
H(TiMp-a) and
gas at
combination;
to 0.3
requires lhat
cniluilpy =
Here
Hie
enter
//;\342\200\236 -IWii,
is ;i
valve
exchanger-expansion
arrangement. Let one moleof gas
PumpedHelium,
From
F)
per moleof liquid
its
at
boiling
lcm-
the fraction
we obtain
17)
called
the
coefficient.
liquefaction
when
lakes place
Liquefaction
>
//\342\200\236\342\200\236, Hia;
> H{Tia,p-J.
HiT^J the
Only
Joule-Thomson
will
take place.
with
rapidly
If
Figure 12.4 shows lhe experimentally. from them Tor helium. The liquefaction
known
are
G)
calculated
coefficient
coefficientdrops
numerator
lemperalure of lhe heat exchangermaHer. at this temperature cools the gas, liquefaction
expansion
The three enthalpiesin liquefaction
(8)
at the input
enthalpies
the
is, when
thai
Tiat
increasing
decrease of the
of the
because
denominator. To obtain useful liqueinversion ;. > 0.!, input temperatures below one-thirdof the liquefaction, say For are this usually required. temperature many gases requires precedingof and the engine. The combination of an expansionengine gas by an expansion is invariably a Linde engine cycle is called a Claude cycle. The expansion 12.1preceded by another heat excitauger, as in Figure in
Coofing:
Evaporatfon from
Starting
evaporation latent
and
G)
heat
liquid
the
Pumped Helium, lo 0.3 K helium, the of !hc liquid
simplest route lo lower temperaturesis
helium, by cooling of the of vaporization liquid Iteltum
The heat extractioncauses
ihc
:itonu'c
ihc
forces
of the
increase
that
cnuscJ
Tltomson cooling tlte initial initial state is a liquid.
further
cooling:
helium staie
is
pumping is extracted
work
io liquefy in a gas, while in
lieiium
away
along
is done the
first
evaporation
vapor,
with the
f
j
y '
vap,-..
against the interpi;>ce.
hi JouL-
cooling
the
17:
Chapter
Table
Cryogenics
12.2
3Hc reach
Tempera turds,
in
which
kctvifi.at
1 he
vapor
of 4He and
pressures
values
specified
p(lorr)
The
lowest in
helium
0.79
0.28
0.36
vacuum
1.27
1.74
2.64
0.47
0.66
1.03
1-79
cooling of
by evaporation
technology
pressure
vapor
gas and
0.98
liquid helium 14). As the ternperalure drops,the (Chapter (Table 12.2} and so docsthe raie ai which can be extracted from the liquid helium
accessible
tempcralure
isa problem equilibrium
0.66
drops
its heat ofvaporization
bath. cooling
Evaporation
Helium
in
such a5
Dilution
classical
K io
of
pressure
refrigeration
principles
0.0! K. is
dominated
evaporation refrigeratorin a We saw in Chapter 7 that
cooling
everyday ait
conditioners.
Militdegrees
Refrigerator:
Once the equilibriumvapor 0.6
cooling principle
household refrigerators and freezersant! in is in the workingsubstance. difference only
devices The
dominant
ts the
lose their by
very
clever
dropped to
I0\023
torr,
range utility. The temperaiure
dilution
helium
the
3He has
liquid
from
which is an
refrigerator,
disguise.* quantum are bosons, while
3He atoms are fermions. is not important at temperatures appreciably higher distinction This of \"fie, 2.17 K. However,the two than transition the temperature superfluid Below as altogether different substances at lowertemperatures. behave isotopes like 0.87 K. liquid 3He and 4He are immiscibleovera wide composition range, in Chapter 11 and is shown in the phase oil and water. This was discussed *He
of 3He-4He mixtures in diagram the range labeled unstable wil!
atoms
11.7.
Figure
A mixture
decomposeinto
two
with
in
composition
whose
phases
separate
area. that are given by the two brandies of the curve enclosing compositions 3He phase. 3He phase floats on top of the dilute The concentrated 3He to about 3He in As T -\302\273 the concentration of the dilute drops 0, phase 6 pet, and the phaserich in 3He becomes essentially pure 3He. Consider a liquid
*
For good reviews, sec D. S. Belts.Contemporary 36, 181A968);for a general review or cooling
Physics techniques
9.97 {1968): IC. 1 K see W.
below
Lounasmaa, Repts. Prog, Phys. 36, 423 A973); O. V. Lounasmaa, below t K, AcademicPress,Hew York, 1974. A very elementary
methods Scientific
American
221,26
(t%9).
.
\342\226\240
Wheatley.
Am-1
J. Huiskamp
Experimental accoun!
and principles
Phys. O. V. and
Is O. V. Lounasmaa, .
12.5
Figure
dilution Cooling principle of ilm helium wiih a JHc-4He nmiure. When from ihc pure ]He fluid and
Hlc is in equilibrium mixiure, sHe evaporaics
3He-4He
mixture wiih more than 6 pet 3Hea*
range, near the bottomof Figure11.7.At
atoms have condensedinto
these
refrigerator. Liquid is added io the absorbs heat in ihc
4He
a temperature
temperatures
in the almost
millidcgree all the 4He
Their entropy is negligible which then behave as if they were of the mixture. If the 3He concenalone, as a gas occupying the volutne present the excess condenses into concentrated liquid 3He and exceeds concentration 6pct, heat If concentrated liquid 3He is evaporatedimo the 4He latent is liberated. the latent heat is consumed.The principle rich of evaporation phase, cooling can again be applied: this is the basts of the heliumdilution refrigerator. To to obtain see how the solution of 3He can be employed refrigeration,
comparedto
that
of
the
the
remaining
ground
state
orbital.
3He atoms,
the equilibrium between the concentrated3Heliquidphase the and tile lliai JHc:4lic nilio of dilute gas-like plliise (Figure 12.5).Suppose with the dilute phase is decreased,as by dilution pure *He. In order to restore 3He aiomswil! the equilibrium from the concentrated concentration, evaporate
consider
iHc
3He liquid.Coolingwill
result.
be a cyclic process the 3He-4He mixturemust again. separated is tile different common method Tile most distillation, equilibrium using by 12.6 shows a schematic 3He and *Hc of 12.2). (Table Figure vapor pressures on these principles. The diagram is highly built diagram of a refrigerator In in actual refrigerators titehcat exchangerbetween oversimplified. particular, An alternate chamber and the still has an elaborate multistagedesign. the mixing of 4He method* to separate the. 3He--4He mixture utilizes tile superfluidity of reasons it is less commonly used, below 2.17 K. For a variety, practical
To
obtain
although
Us
performance
is excellent.
.
\342\226\240 .
Chapter
12:
Cryogenics
3He pump
loop
Key: Liquid \342\226\240 I 'lie
Dilution Refrigerator:
Helium
AtitliJegre
Hdium dliulion refrigerator. Prccooledliquid a mixing 3He enters chamber a( (he tower cud of the assembly, wlicrc cooling takes place by ihe quasiof the 3He atoms into the denser JHc-Jf1cmixed cvaporaiion underneath. phase 12.6
Figure
The quast-gas of JH atoms dissolvedin liquid *He then diffuses through heat exchanger into 3 still. There the JHe is disiilledfrom the 3Hc-4Hc
a countcrfiow
mixture
a useful 3He evaporation and circulation selectively, and is pumped olf.To obtain heat must be added to the still, 10 raiseUs temperature to about 0.7 K, at which
rate,
vapor pressure is ssiH much smaller. Thus, the 4He does not a nearly stationary appreciable extent; ihe *Hc moves riirough JHe is returned to ihe system and is condensed background of 4Hc.The pumped-off in a condenser that is cooled to about I K by contact with a pumped 4He bath. The constriction below the condenser takes up the excess pressuregenerated by the the *He
temperature
lo any
circulate
circulation in rhe
first
pump over ihe pressure in siill. ihcn in the counter/low
still. The liquified JHe is cooled further, heat exchanger, beforere-entering tlic miung
the
chamber.
In the convendilution refrigerator has a low temperaturelitnft. conventional evaporaiioii this limit arose because of the disappearance of refrigerator the but the quasi-gas phase of 3Hepersists down to t = 0, However, phase, gas ihe heat of quasi-vaporizationof JHe vanishes to x2, and as a proportionally Ihe rate heat removal from the mixing chamber vanishes as i1. TI'S result, low device;* limit is about 10 mK, In one representative temperature practical was capable of a temperature of 8.3 mK has been achieved:ihe same device
The helium
removing 40/AVat 80mK. ihe
design
there is
SmK
below
Temperatures
of Figure
I
2.6,
no needto cooi the
mixingchamberdrops removed from The
dilution
the
off the
3He
incoming its
below
single shot operation. If, 3He supplyafter some time of opeiation,
be ncltievcu by
can
we shut
sleady
itself, state
and
value,
of the
ihe temperature until
has
3He
all
in
been
chamber.
refrigerator
is not the
oniy cooiingmethodin
the
inillikelvin
known the peculiar propertiesof JHe.An alternate method, in Figas Pomcranchsik cooling, utilizes the phasediagramof 3He,as shown and between Figure 7.15, with its negative liquid slope of the phase boundary ant! solid 3He, The interested reader is referredto the reviews by Huiskamp Lounasmaa, and by Lounastnaa, citedearlier.
that utilizes
range
'
N.
H.
Pcnnings,
84, 102A976}.
R. de
Bruyn
Ouboicr,
K. \\V. Tacoois.
Phjiica 8
SI. !0! A976).
and
Physiea
B
DEMAGNETIZATION:
ISENTROPIC
QUEST FOR ABSOLUTEZERO 0.01
Below
doniimim
K the
cooling process
is the isciitropic(adiabaiic)
dcm;ig-
paramagnetic substance. By this process, temperaturesof I niK have been attained with electronic paramagnetic systems and j /(K with nuclear systems. The method dependson tlie fact that at a fixed temperaparamagnetic temperaturethe of a system of magnetic momentsisloweredby application of a entropy slates are to because accessible the system fewer magnetic field\342\200\224essentially small. when ilic level splitting is large than when the level is splitting Examples of the dependence of the entropy 2 on tlie magnetic field were given in Chapters of a
iictizatioii
3.
and
We first apply a will
magnetic field
a value
attain
without
will
then flow
remain
t2 into
to the
appropriate
reducedto B2
the
changing
the
tj-
the spin
specimen
system only
of
value
entropy
means
which
unchanged, <\342\226\240< When
at constant
Bt
temperature ij. The spin excess If the magnetic field is then Bj/tj. the of spin system,the spin excess
that B2/z2
will
is demagnetized
from
the
system
#i/ri-
equa'
12.7-
can
vibrations, as in
of interest the
the temperatures
At
Bi,
isentropically,entropy
of lattice
entropy of the will be usually negligible; thus the entropy ofthe spin system during isentropie demagnetization of the specimen. Figure
\302\253
HBz
lattice
essentially
constant
\\Latttce
1
Total
w
Spin
Lattice
Time\342\200\224-
Before Time
Figure
12.7
cooling
of
Time\342\200\224\342\200\242
Before Time
New
equilibr:
at which
magnetic field js removed
field
removed
demagnetization the total entropy of the S in of the lattice should be small entropy with the entropy of the spin system in order to obtain significant the lattice. During
isentropie
specimen is constant. The comparison
equilibrium
at which
magnetic is
New
Fig-
vibrations
initial
.
is
Quest for
Demagnetization:
lsentropic
Zer
Absolute
as a function of Icmpcralure.assum Entropy fora spin \\ sysiem of field 100 Bx gauss. The specimenis magnetiz magnetic Thu cxlcrna! ntagnctit insulated isothermaiiy ihcrmaMy. along ah, and is then field is 1 timed on a reasonable off along/>c. Ill order to keep the figure sculc llic initial temperature tlie field are lower woi and external than magnetic Tj
12.8
Figure
inicrna!
an
used
in
The
random
practice.
steps
out in the
carried
cooling processarc shown
field is
applied at temperaturetx
the
the
with
giving
surroundings, {At? ~ 0)
insulated
and the field
with
the
specimen
isothermal
in
thermal
in good
removed; the specimenfollows
6c, ending up at temperature t2. The thermal contact is broken helium provided by gas, and the thermal
gas with The
the contact
constant ai
is
t,
by removing the
a pump.
population
of a
magnetic subievelis a function
is the magnetic momentof a ofthe
contact
path ab. The specimenis ihen
path
entropy
12.8. The
Figure
population
spin.
distribution;
The
spin-system
hence the
only
entropy
of
ntB/x,
is a
spin entropy is a function
function
only
m
where only
oimBjx.
localinteractions is theeffeclive field that corresponds to thediverse among temperature r2 reached in an spins or ofthe spins with the lattice, the final
If SA the
isenlropic demagnetization experimentis
(9} rt the initial temperature.Results which as CMN, Figure IZ9 for the paramagnetic salt known . . magnesium nitrate, where
B
is the
initial
field and
are
denotes
shown
in
cerous
Final
Figure 12.9
removed
as
cniirciy, and
fields
iiiiiial
field
magaclic
bul
0.6
0.5
in K
temperature,
Bf versus
experiments ihe magnetic
In fhese
nil rale.
After
Final
0.4
0.3
0.2
0.1
0
final
field
was
not
indicated values. The were idcnliait in all inns. S. Still and J. H. Milncr.
m tfic
only
icmpcraiurcs
unpublished results 61\" J. 6, by N, Kuril, Kuovo Cimcnio (Supplement)
cilcd
1109A957).
The processdescribed so into a cyclicprocess thermally
is
far
a single
shot
disconnecting,
by
demagnetized working substance from at t,, and repeating the process.*
converted
process. It is easily in one
way
or
it to the load, reconnecting
the
another, the
reservoir
Nuclear
Dcmagnelizalion
nuclear
Because much
are
weaker
paramagnet.
\342\200\242
C.
similar
lhan 100 limes lower with a nuclear paramagnet in The initial temperature of the nuclearstage
temperature
C. B.
V. Hctr.
Rosenblum,
lhan
arc weak, nuclear magneticinteractions a to reach electronic interactions. We expect
moments
magnetic
W.
Barnes,and
A. Slcyerl.
and i.
Daunt.
J.
G.
A.
Barclay.
Rev. Scj, insi. 25. IGS8 j|954); 17, 3S! A977).
OHgcnics
electron
with
an
a
nuclear
spin-
W. p. PraH,
S. S.
t\\'uc!ear Demagnetization
Iniiial magnetic
0.6
in
Held
KG
1
Initial
B/T'm
\\QS
G/K
Nuclear demagnctizaflons of copper skirting from 0.012 K and various fields. After M. V. Hobdcn imd N. Kuril. Phil. Mag.
Figure
I2J0
in
nuclei
the
metal,
-1.1902!1959).
must be lower than in an cooling = = lfwestartatB SOkGandT, 0.01 10 percent decrease on magnetization is experimen!
over
This is
to
sufficient
ihe
overwhelm
T2 ss 10\027K.The
temperature
electron spin-cooling experiment. * 0.5, and the entropy K,then/fiB/*87\\ of the maximum spin entropy. a final lattice and from (9) we estimate nuclear
first
oui by Kurli and coworkers on Cu nuclei at about 0.02 K as attained by electron stage
the
B\302\261
moments
of
=
3.1
the
and
I0~6K. reached in this experimen!was 1.2 so line of Ihe form of (9): 7\\ = T|C.1/B) Bin wilh ^auss, of the magnetic mointeraction field gauss. This is the ciTcctive
Cu nueiei.
nuclei
Temperatures
cooling
first
The
cooling.
demagneltzallon
load was
The motivation conduction
at
below
the
electrons
temperature
1//K
for
have been
using
help
of the
first
nuclei
ensure
in a
metal rather than
rapid thermal
contact
stage.
achieved in experiments in
the system of nuclearspinsitself,
results
The
x
in an insulatoris that
of lattice
from a
starting
metal,
lowest temperature in Figure 12-10fil a llial
was carried
experiment
cooling in
particulatly
in
which
experiments
the
combinations of cooling experimentsand
that were
nuclear
resonance
magnetic
experiments.*
SUMMARY
1. The
dominant
two
a gas
of
cooling
of the
principles
by letting
work
do
it
of low temperaturesarc ihe a force during an expansion
production
and the iscntropicdcmagncii/atioii of a
against
substance.
paramagnetic
is done work cooling is an irreversible process in which interatomic forces in a It is used as last the attractive against gas. cooling
Joule-Thomson
2.
stage
in
gases.
low-boiling
liquefying
3. In evaporation cooling the work is also doneagainst but starting from the liquid phaserather than the
the 5.
dilution
helium
of magnetic
system
moments,
an
when
external
moments may magnetic nuclear moments, temperatures
using
devices
cooling device in
which
4He.
lowering of the
utilizes the
The
strength. By
evaporation
household
of cooling
laboratory
gas of 3He atomsdissolvedin
demagnetization
Isentropic
is an
refrigerator
gas is the virtual
basis
the
forces,
Using different
phase.
gas
working substances, evaporation coolingforms and cooling devices, automobile air conditioners, (in the range 4 K down to SGmK).
4. The
interatomic
the
magnetic
temperature
of
field is
reduced
be electronic or nuclearmoments. in the microkelvin range may
a in
be
achieved.
PROBLEMS
L Helium for
as a
helium
Use
of the
and
liquid
data
in
6
it as such
in
helium Table
for \342\200\242See,
example,
.
a
der
van
Waals
tlte liquefaction coefficient X gas. Select the van der Waals
a way that Tor one and that 2a/b is the
12.!. Approximate the Hout
849A970).
(a) Estimate
H'aafcgas.
by treating
coefficientsa volume
der
van
-
M. Chapcllier,
Hi(q
M. Goldman,
AH
4- f(rin
V.
mole 2Nb is the actual molar
actual inversion temperature. denominator in G) by setting -
H. Chau
xliH)
A0)
,
and A.
-.-._...\342\226\240
Abragara,
Appl.
Phys-41,
where All is the latent heat of vaporizationof liquid how this helium. (Explain arises if one treats the as an ideal gas). The approximation expanded gas /. as a function of the molar volumes Yin and resulting expression gives Vatll. Convert to pressures by approximating the l\"s via the ideal gas law. (b) Inaert numerical values T = 15 K and compare with for 12.4. Figure Carnot liquefier. (a) Calculatethe work mole of a monstomic ideal gas if the liquefy Assume that the gas is suppliedat roomtemperature
2. Ideal
one
pressure p0 at whidi the liquefied 7\\ be the boiling temperature of the
of vaporization.Show
that
under
gas gas
these
\\VL thai
iiqticfier To.
be required to operated rcversibiy. the same and under
would
is removed,
1 atmosphere. typically and A// the latent at this pressure, conditions
Let
heat
A1)
To derive A1) assumethat the gas is first cooled at fixed pressure p6 from To the fixed between to Tfc, by means of a reversible refrigeratorthat operates upper temperature Tb ~ To attd a variable lower temperatureequalto the gas ~ 1\\.After reaching Tb the Initially temperature. T, = To, and at the end T, at the lower the latent heat of fixed extracts temperavaporization refrigerator of temperature Tb. (b) Insert To = 300 K and values for Tb and AH characteristic liter of liquid helium. helium. Re-express the result as kilowatt-hours per Actua! helium liquefiersconsume5 to lOkWh. liter. 1 mols\021 in which cycle helium Hqucfier. Considera heliumliquefier enters the Lrnde stage at T(o = 15 K and at a pressure pla = 30 aim. all the in liter hr\"'. Suppose that liquefied (a) Calculate the rate of liquefaction, helium is withdrawn to cool an externalexperimental the releasing apparatus, load in boiled-off helium vapor into the atmosphere. Calculate the cooling the the it is this watts sufficient to evaporate heliumat rate liquefied. Compare if the liquefier is operated as a closed-cycle with the load obtainable cooling the apparatus into the liquid collectionvesselof the refrigerator by placing the heat so lhat the still cold boiled-off helium gas is returnedthrough liquefier, and ex(b) Assume that the heat exchangerbetween exchangers, compressor ideal that return is the (Figure 12.1) sufficiently expanded gas expansion engine the same temperature Tc as the that leaves it with pressure is at essentially pout compressedgasenteringit with pressure pc. Show that under ordinary liquefier must extract the work operationthe expansion engine
3.
Claude
of gas
Te -
TJ ,
A2)
Chapter
12:
per mole
Cryogenics
of compressed gas. HereTin,
pin,
pBUt, and
X
have
the
same
meaning
Undo cycle sectionof this chapier. Assume the expansion engine between ihe and operates isemropically pressure-temperature pairs {pc,Tc) Estimate (Pia>Ti(l). From A2) and the given values of (pia,Tia),calculate(pc,Te). (c) the minimum compressor power required to operate the iiquefier,by assuming the compression is isothermal from to pc at temperature that Tc ~ 50\"C. poai Combinethe result with {hecooling loads calculated under (a) into a coefficient of refrigerator for both the modes of operation. Compare with performance,
as in ihe
Carnot
iimit.
cooling limit. Estimate the lowest temperatureTmia that can if the cooling load is 0.1 W 4He evaporation cooling of liquid = and the vacuum I02filers\021. Assume (hat the pump has a pump speed S helium vapor pressure above the boiling is equal to the equilibrium helium lo TBliJ1, and assume that ilic helium gas warms vapor pressurecorresponding the to roorn and expands accordingly before it enters tip temperature ptunp. Nota: Tlte molar volume of an ideul and atmospheric at room gas icmpcramrc pressure G60torr) is about 24 liters. Repeat the calculation for a mtjch smaller heat load (I0~3 W) and a faster puinpA0J is defined liter s\"\021). Puntp in speed 4. Evaporation be achieved by
Chapter
14.
temperature far demagnetizationcoaling, Considera paramagnetic field ofiOOkG with a Dcbyc temperature {Chapter 4) of 100K. A magnetic or lOtcsia is available in the laboratory. the temperature to which the Estiniate that salt must be prccoolcd other means order cooling by in significant magnetic process. Take may subsequently be obtained by !he isentropic demagnetization 5. Initial salt
to be I Bohr ion the magnetic momentofa paramagnetic By signifimagneton. 0.1 of the initial we to understand temperature. cooling may cooling
significant
13
Chapter
Statistics
Semiconductor
ENERGY
FERMI
BANDS;
LEVEL;
355
AND HOLES
ELECTRONS
358
ClassicalRegime of
Law
362
Action
Mass
362
intrinsic Fermi Level
363
/r-TVPE AND Donors
Fermi
SEMICONDUCTORS
P'TYVE
363
and Acceptors
364
Semiconductors
Extrinsic
in
Level
365
Semiconductors
Degenerate
368
Impunly Levels
Occupationof Donor
369
Levels
Example:
Gallium
Semi-Insulating
Arsenide
373
p-n JUNCTIONS
Reverse-Biased
p-n
NONEQUIUBRIUM
SEMICONDUCrORS
Abrupt
377
Junction
Quasi-Fermi
Drift and
Flow:
379
Diffusion
Example:Injection
381
Laser
Example:
Carrier
379 379
Levels
Current
372
Through an Impurity Level
Recombination
383
SUMMARY
385
PROBLEMS
387
1. Weakly
2.
Intrinsic
387
Doped .Semiconductor and
Conductivity
Minimum
Conductivity
387
3. Resistivity and Impurity Concemraiiou 4. Mass Action Law for High Electron Concentrations
387
5. Electron
387
and
Hole
Concentrations
in InSb
6. Incomplete lonizationof Deep Impurities 7. Built-in Field for Exponential Doping Profile
8.
Einstein
Relation
for
High
Electron
Concentrations
387 3S7
388 388
13:
Chapter
Ha
ht n,
Carrier
^
conduct v\302\273'ilti
conccnlration
= cffcciivc =
effective
38S
Lifetime
388
Pair Generation
Electron-Hole
iiiiiiui^zny
388
Laser
9. Injection
10. Minority 11.
Statistics
Semiconductor
[tin xind valence b jfiu\302\243 \\ elects oeis snd
of holes
hoics^ donors
*inu
3cccplovs.
I he
^
c^uantiiEn
conccnlnU/on
for condudion
quamum
coiKcntralion
for
ctccirons;
holes.
densities of states for the conduction In the semieonductoi tileralurc n,. and % ate called ihe effective and valence bands. Notice iKal we use ;i fo( tin; chemical potential or Fcimi level, and we use Ji foi
cai-iicr mobilities.
ENERGY
The is
FERMI LEVEL;
BANDS;
ELECTRONS
HOLES
AND
of the Fermi-Dirac disiributionto eiecirons in semiconductors application central to the design and operation of all semiconductor and devices,
to much of modern electronics.We
of semiconductorsand
treat
devices
semiconductor
of the physics thermal physics. of the physics of
those
below
thus
aspects are parts of
that
is familiar wjlh the basic ideas as 'n texts on solidslatephysics the crysialline soiids, treated in and on semiconductor the We assume the devices cited references. general of bands and of conduction electrons and hotcs. Our principal by concept energy aim is to understand the dependence of the alt-important concentrations of conduction electronsand of holesupon the concentration and the impurity that
assume
We
eleclrons
the
reader
in
temperature. A
semiconductor
band t
=
and
0 al!
is a
system
with
electron
orbitats
grouped
into two
energy
energy gap (Figure 13.1). The lower band is the valence at the upper band is the conductionband.* In a pure semiconducior valence band orbitats are occupied and alt conduction band orbitals
bands separatedby
an
are empty. A full band cannot carry any current, so that a pure in a semiconductor at r = 0 is an insulator.Finite conductivity
semiconductor
follows either
in the conduction from the presence of electrons,catled conduction electrons, orbitats in the valence band, called notes. band or from unoccupied Two different electrons and holes: mechanisms rise to conduction give of electrons from the vatencc band to the conduction band, Thermalexcitation of or the presence that change the balance between the number impurities of electrons available to fill them. of orbitats in the valence band and the number and the energy the band We denote valence the energy of the top of by \302\243,., rence of the bottom of the conductionband by e{. The differed
is the energygap of the semiconductor.Fortypical eV. 0.1 and 2.5 electronvolts.In silicon,e, ^ 1.1 \342\200\242 We tieai both bands of bands wtih additional
as single
for out (imposes it bands; gaps wiihtn each gioup.
does
semiconductors Because
t ^
noi mailer thai
Eg
is between
1/40 eV at boih
may
room
be groups
Chapter 13: Semiconductor
Statistics
Empiy
band
Conduction
Energy
atr =0
gap
Filled
air =0
t3.1
Figure
Energy
Air = conduct the
structure of
band
orbilajs occur in
a puic semiconductor
bands
\\vhjch
we usually
havec,, \302\273t. Substances 2.5 eV are usually insulators. Table 13.1 gives semiconductors, together with other properties Let
nt
ihe
denote
concentration of
insulator.
exlerid
gap.
energy
temperature,
or
through the crystal. Gallorbitaisuplothe top of the valence band are filled,and ihe the bands is called ion bantl is empty. The energy interval between
The electron
the
pure semiconductorthe ii, =
crystal is electrically neutral. Most semiconductorsas usedn\\ devices impurities that may become thermally
gap of more than later.
electrons and two
about
for selected
gaps
energy
needed
of conduction
concentration
holes. In a
with a
wiH
nh
the
con-
be equal:
B)
\302\253*.
if the
\342\226\240temperature.
positively
tiiat
Impurities
charged
in the
give
an
have
ionized
electron
been
In the
inteniionaiiy
doped with
semiconductor
to the
crystal
process) are called donors.Impurities
(and that
at room become accept
Bands; Fermi
Energy
data
structure
Band
13.1
Table
Lml; Elec
Energy
Q
ions and
c Idea
at
gaps
300 K
itions
effective
|
olcs
li
free
1300K
\",.
eV
'
2.7 x 10
Cc
0.67
1.0 x !O 9
GaAs
1.43
Let
4.6 X
!he
\302\253j+be
'
10
10\"
0.58
11.7
0.35 0.71 0.42
0.07
0.073
15.8
13.13
12.37
0.39
0.015
band (and become negatively of positively
concentration
charged acceptors. An
ts called
1.06
0.56
17.S8
the
in
charged
acceptors.
of negatively
concentration
to
vacuum
ilia
10\"
6.9 x 10'* 6.2x 10\"
the valence
are called
process)
relative
cicctr on mass
x to'a
1.5 x
4.9 x 10 '
from
electron
an
1.1 x 5-2
4.6 x 10 '
0.18
of the
c
t.[4
1.35
masses,
constants
in/ftn
c
Si
,,p InSb
units
in
Dielectric
f-statcs
Dcnslty-
i concentr
antun
=
-
/[/
charged donors and
The difference C)
na~
donor concentration.The electrical neutrality
the net ionized
the
na~
condition
becomes
=
which
specifies The electron
distribution
the difference concentration
[i is
the chemical potentialof the
electrons. In semiconductor theory the Fermilevel. called Further, almost
level
always
reserved
is designated
of a meia!
which
in
ihe
limil
r -*
by ef we
for ihe
or
by
at
The
electrons.
To \302\243.
theory
temperature.
confusion
avoid
as cf our
and
with
the
the Fermi
Fermi
energy
which stands for tile Fermi
previous
usage
to
is always potential is the character fi
chemical
semiconductor
e refers
subscript
electron and hole mobilities,and
designated
any
E) /
electron
the in
0, we shall maintain
chemicalpotential
D)
no~~t
betweenelectronand hole concentrations. may be calculated from the Fermi-Diracdis-
exp[(e where
\342\200\224
Hj*
6:
of Chapter
function
=
An
of the
level
letter /j for the
;i and
Given
the distribution
The
t, the number of conductionelectronsis obtained function /,(e) over all conduction band orbitals:
of
number
[l \302\243
- /.(*)]
= I
ftU).
V)
VI!
VB
is overall valenceband
the summation
summing
is
holes
*\\ =
where
by
Here
orbitals.
we have
introduced
the quantity
at energy e is unoccupied.We say a hole\"; that is the distribution [hen/h(e) function for holes just as f\302\243t) is the distribution function for electrons. Comof with shows that the involves hole occupation probability Comparison (8) E) \342\200\224 e where the electron y. y. occupation probability involves c p. = = The concentrations and nh nt NJV NJV depend on the Ferm't level. But what is the value of the Fermi level? It ts determined by the electrical \342\200\224 = as An. This is an neutrality requirement D), now written nh{y) nt{y) the must for to solve we determine the functional implicit equation y.; equation
which
that an orbital probability the unoccupied orbital is \"occupiedby the
is
ne{y) and
dependences
nh{y}.
Classical Regime We
assume
that
by the
defined
regime
concentrations are
and hole
electron
both
requirements
that
\302\253 1
fr
and
fh
in
as
\302\253 I,
the
classical
in Chapter
This will be true if, as in Figure 13.2, the Fermi level lies insidethe energy and ts separated from both band edgesby energies that large enough -
exp[-(\302\243c
To few
satisfy times
inequalities
are
satisfied
(9) both larger (9)
\342\200\224
place
n)
(gc
than
in many
/i)/t]
\302\253 1;
and
(/t
-
exp[-0*
-
eu)
to
have
ej/t]
\302\253 1.
be positive
6. gap
(9)
and al
least a
The a semiconductor is callednondegenerate. and limits on the electron and hole concentrations
t. Such upper
applications.
/J.E.)and /h(g)reduceto classical
With
distributions:
(9)
the
two
occupation
probabilities
Classical
':.-\"-\\~'\\
I
Regi
Conduciion I
band
13.2 Occupancy oforbiials as a finite temperaiure, to the Fermiaccording The conduclion and valence bands may be represented Dirac disSribution function. numbers in terms of temperature-dependenteffedive Nc, Nc of degenerate orbiials The located aS the iwo band edges e,, \302\243\342\200\236. n(, n( arc ihe corresponding quantum
Figure
=
We
use
F)
and
-
exp[-(e
A0) to
write
=>
the
total
number
cxp[-(,,
-
of conduction
A0) electrons in the
form
N,
A1)
Statistics
13: Semiconductor
Chapter
where we define
N, \342\200\224 \302\243 is \302\243c
Here
band
the
A2)
ej/r].
conduction electron referred to
of a
energy
\302\243exp[~(s Cfl
conduction
the
ec as origin.
edge
The expression for
lias the maihematical fomt of a partition function a similar conduction band. In Chapter3 we evaluated sum denoted there by Zlt and we can adapt that rcsuil lo the prescuJ problem with an for modificalion band siructure effects. Because of the approximate rapid decrease of cxp[-{\302\243 - e()/i] as c increases above its minimum value for
one
at ee,only
Nc
in the
electron
ihe
of orbitals
distribution
a
within
range
of a few
above
x
cc really
evaluation of the sum in A2). The orbitalshigh in the band make a negligible contribution. The important is that near the band edge point the electronsbehave very much like free particles. Not only arc the electrons of the semiconductor, but the energy mobile,which causes the conductivity distributionofihe orbitalsnear the band edge usually differs from that of free in the particles only by a proportionality factor in the energy and eventually matters
sum
in ihe
for Z%.
We can arrange for a suitable proponionaliiyfactor by use of a device we calculated the called the densify-of-sfates effective For free particles mass. in C.62), but for zero spin. For particlesof spin | the partition function Z\\ result is larger by a factor of 2, so that A2) becomes
Nc = Zj
=
this gives
Numerically,
NJV =- 2.509 x where
Tis
1019 x
dependenceas this
Nt A3),
formally
for but
ihe same temperature a by proportionality facior. We exhibits
semiconductors
actual differs
by wriiing,
in magnitude in
io
analogy
A3),
Nc *= 2(mSz/2nti2K'2V
where Experimental
is more
(H)
.
G/300KK'2cm\023
in kelvin.
The quantity
express
A3)
2(mx/2nh2)i!iV.
2nQV=
me*
is
values
than
called
the
arc
given
a formality.
effective mass for denshy-of-btates in Table 13.1. The introduction
In the theory
of
electrons
A5)
, electrons.
Experi-
of effective
masses
in crystals
it
is
shown
that
Classical Regime
Ihe dynamical behavior of electrons and
forces such as electricfields, free electron
the
from
Ihe density-of-staJes We
the
define
that
is
holes,
the influence of external effective wilh masses different
under
of particles
dynamical massesusually masses, however. mass. The
quantum
coticetilralion
for ??\302\243
eleclrons
conduction
from
different
are
as
NJV -
nc =
A6)
By A0 the conductionelectronconcentration
=
ne
NJV
becomes
A7)
Jo the assumption The earlier assumption (9) is equivaleni that n,. \302\253 nc, so that the conduction electrons act as an ideal gas. As an aid to memory, we may level at ;i. IVanriiuj: think as wiili the Fermi of Ne arising from N( orbilalsat \302\243\302\243, In is invariably called the effective density of the semiconduclor literature nt
statesof
conduction
the
Similar
reasoning
band.
gives the
number of holesin
valence
ihe
band:
- e)/t]
wiih
the
definition
A9)
We define
the quantum concentrationnv
e
where
wk*
concentration/^
is
llns
NJV
dcnsity-of-sJaJcs
s
for
holes
as
2{in^z/2jihi)m.
effective
mass for
holes.
By
(IS)
ihe
hole
s NJV is
nh
\342\200\224 aBexp[\342\200\224(/(
\342\200\224
e,-)
B1)
gives the carrier concentrationin
A7), this
Like
positionof
ceniraiion and the
edge.
In
of the
valeuce
is
of the
independent
'V'k = 'Wexpf-fE,.where the and
energy gap
the common
concentration
t^
Fermi level so longas the
concentrations
Then
classical regime.
in the
valeuce band effective density of states
Action
The productn^nh are
the
called
con-
quantum
to the
band.
of Mass
Law
is
nu
(he
of
terms
relative
level
Fermi
literature
semiconductor
(he
the
s=
\302\2439
ec
-
eff.
In
O/r]
s=
ncnuexp(~
we have
semiconductor
a pure
B2a)
,
eJz)
value of the two concentrations is called the of the semiconductor. By B2a),
\342\200\224
>ibt
ut.
carrier
intrinsic
B2b)
The Fermi level independence of the retains its value even when ne <\302\243 nh, as atoms,
impurity We
then
may
both
provided
product in the
concentrations
means
n^iij,
this
that
product
presence of electrically charged remain in the classical regime.
B2a) as
write
B2c)
The mass
of
value
action
the
depends
product
law of
only on
the temperature.This result
semiconductors, similar to
chemical
the
mass
action
is
the
law
(Chapter 9).
Intrinsic FermiLevel For
an
intrinsic
sidesof A7)
and
semiconductor
eB
= e^
nh
and
we
may
equate
the right-hand
B2b):
neexp[-(ec Insert
ne =
\342\200\224 and
eB
divide
- ^)/r] = by n,.exp(
(vO\022exp(-\302\243/>*).
\342\200\224\302\243(/r):
ej/2r].
B3)
and Acceptors
Donors
logarithms to obtain
We lake
=
/i
{U,
+
+ \302\243,.)
= l(cc
|t Iog(u,/u,)
of A6) and B0). The Fermilevel (he middle of lhe forbidden gap, but by use
is usually
that
amount
Pure
an
semiconductor lies near the exact middle by an
intrinsic from
displaced
B4)
3rlog(\302\273ifc7\302\273i,*),
small.
AND p-TYPE SEMICONDUCTORS
w-TYPE and
Donors
for
+ e,) +
Acceptors
are an
semiconductors
idealization of Httle
Semicon-
interest.
practical
to usually have impurities intentionally added in order semiconthe concentration of either conductionelectrons A more with conduction electrons than holes is called \302\273i-type; a semiin devices
used
Semiconductors
or holes.
increase semiconductor
n and electrons is called p-type.The letters p and in carriers. Consider a silicon positive signify negative majority crystal atoms. which some of the Si atoms have been substituted by phosphorus hence each P has Phosphorus is just to the right of Si in the periodic table,
semiconductor with
fit
into
the
Si it replaces. valence band; hence a Si crysial
filled
than the
more
electron
one
exactly
more holes than
do not electrons P atoms wiil contain
extra
These some
with
more conduction electronsand, by the law of muss action, fewer holes than \302\253 Si crystal Next consider aluminum atoms. Aluminum is just to the left pure of Si in the periodic fewer than the Si table, hence Al lias exactly one electron it replaces. As a result, Al atoms increase of holes and decreasethe the number of
number
electrons.
conduction
Most impurities in the same columnsof the periodic behave in St just as P and Al behave. What matters
electrons
relative
from
Impurities
Similar reasoning For the presentwe may
enter
assume
that
is the
to Si not the total number of other columns of the periodic table will
and
can be appliedto other lhat
assume
band or
the conduction
each acceptor
fill
one
and Al will
number of valence
electronson behave
not
for
semiconductors,
donor
each
as P
table
the
atom.
so simply.
example
GaAs.
atom contributes one electronwhich hole also in the valence band. We
atom removes one electron,either
the
from
valence
are called the approxifrom the conduction band.Theseassumptions all impurities when ionized are either approximation of posiimpurities: fully ionized A\". donors D+ or negatively charged acceptors positively charged electrical The condition D) told us that neutrality
band or
An
=
nt
\342\200\224~ nfc
nj+
\342\200\224
na~.
.
.
.
B5)
J3; Semiconductor
Chapter
Becausenh
=
Statistics aciioti law,
mass
the
m,V\302\273*from
equation for
we secthat
B5)
to a
leads
quadratic
nc\\
B6)
V-\".^!-^1.
root is
The positive
=
i{[{A'O2
\302\273*
~
because
and
nh
-
^
\302\253 \302\273* l([(AnJ
Most
the
often
+ V]1''2
- An],
nr or
either
is much
ft*
[(AnJ +
an n-type
=
An*]\"*
=x
A/i
+ nffAn
p-typc semicondiiclor
Ji. ^ The
majority
n?/\\An\\
proportionalto Level
Fcrmf
By use
An
-
the
B9)
ln*l\\bt\\.
B7) becomes
nh
An;
** n^/An
and B7)
is negative
\302\253 n,;
while
extrinsic
the
carrier
mmoriiy
C0)
becomes
{A/i| +
nk ^
\302\253 n,.
h;V1A*i1
limit
^
B8)
C1)
jAffj.
is nearly
concentration
equal to
is inversely
jAnj.
in Extrinsic
of the massaction
having to calculate
the
solving
m +
carrier concent ration in
the magnitude of An,
/ij,by
B7)
+ (hj&nJ]1'2
|^[i
&n is positive and
semiconductor nt
In a
B8)
nt.
In
than n,:
larger
be expanded:
then
can
B7b)
extrinsic semiconductor. The squarerootsin
defines an
Condition
B7a)
\302\273
\\An\\
This
,
An}
compared to the intrinsic con-
is Urge
concentration
doping that
so
concentration,
+
have
we
An
+ V]1/J
A7) or
Semiconductor law Fermi
concentrations without Tlic Fermi level is obtained from n, or
we calculated level
B1) for/c
first.
the carrier
Scum
Degenerate
Figure various
the
13.3
The Fermi icvet in
doping
band edges.
A
sniaii
silicon
as a function
The Fernifleveis
concentrations. decrease
of liic energy
of lempcraturc,
are expressed relative
gap
wiih
icmperaiure
for
Ic
has
been negieaed.
now use B7)
We may
to
find
ft as
a function of
temperature and doping level
13,3 gives numerical results for Si. With decreasing Figure eiiher Fermi level in an extrinsic semiconductorapproaches She valence band edge.
ths
temperature
An.
the
cr
conduction
DegenerateSemiconductors one
When quantum
of
concentration,
carrier.
the carrier we
The calculation
is increasedand approachesthe use the classicaldistribution no longer A0)
concentrations may
of the carriorconcenir.iiion now
Fermi gas in Chapter 7. The sum is written equal to [he number of electrons, states times the distributionfunction: of the
N
over as
follows
all
occupied
an
iiiteura!
the
quan-
for
iliiit
treatment
orbitais, which n over the density yf
IS:
Chapter
Statistics
Semiconductor
where for free
llie panicles of mass \302\273j
is
of stales
densily
C4)
Thai is, \302\251(\302\243)(& is ihe make ihe by
in
n,V;
Lei x
of orbitals
number
in
the
transition to conductionelectronsin \342\200\224 oblain by m,*; and \302\243 by e ec. We
s (e -
and
er)/r
ij
=
~
(fi
et)/t. We
interval
energy
+ (\302\243,e
we
semiconductors
use the definition A6)
of
To
ck).
iV
replace
obtain
J(c to
C6)
The integral /(;;)in When
ee
-
p.
as the
is known
C6)
\302\273 x
we
have
\302\273 1, \342\200\224ij
Ferml-Dirac integral. so
that
-
cxp(x
jj)
\302\273 1.
In
this
limit
C7)
the
result
familiar
for
the ideal
gas.
several limes electron concentrationrarely exceeds the quantum concentration nc.The deviation between the value of/i from C5) and the approximation then can be expanded into a rapidly C7) converging = series the r of ratio power njn^ calledthe Joyce-Dixon approximation:*
In semiconductors the
\342\226\240 \",/\"\342\200\236
C8)
\302\253-\342\200\224i-S-f) t3.4
Figure
compares
the exact
relation C6) witll
the
approximations
C7)
and
C8). \342\200\242
W.
B. Joyce
1.483S6
and B. W.
x lO\"*:^,
Dixon.
App!.
- -4A2561x
Phys. Lclt.
1Q-6.'
31,354 A977).
If the
right side
of C8) is
wrine
Slmicomlitct
Dtgcncnitt
7
-6
-5
-4
-3 -2
-I i)
above
conduciion
band
edge Er. The
Joycc-Dixon approximation
C8).
=
0
(|.
1
2
3
4
5
6
- i,)h
dashed curve reprcsenisihc firsi
icrm
of ihc
When
Sta
Semiconductor
13:
Chapter
longer small comparedto n(,
neisno
law must be modified.In Problem4
ask
we
Ihe
ihe
of the
expression
io show
reader
mass action
ihat
C9) If
the
Itself
gap
here
will
on
depend
Impurity
carrier concentrations, the value
on the
depends
localized
P
atom
be used
n, to
concentration.
Levels
The addition of impuritiesto a conduction or valenceband into
as
of
Ihe
where the
gap,
energy
from
orbitals
some
moves
semiconductor
the
orbiials now appear
in a silicon crystal.If the electron to the Si conductionband, the atom as a positively charged ion. The positiveion attractsthe electrons in appears the conduction an electron band, and the ton can bind just as a proton can bind an electronin a hydrogen atom. the However, binding energy in the phosphorous
iis extra
released
has
consider
We
states.
bound
semiconductoris severalordersof the energy is to be by square paniy
column
V donors
donor
corresponds
in
the binding
of the static dielectric constant, and for 13-2 gives the ionizationenergies Si and a Ge. The lowest orbital of an electron bound to \342\200\224 = to an energy level Asd below the edge of the st st
effects. Table
mass
of
because
because
mostly
lower,
magnitude
divided
conduction band (Figure13.5). There
is
one
set
orbitals
of bound
for
every
donor.
argument applies to 'he valence band, as in Figure
A parallel from
set ofbotmdorbitals as
Aej. Ionization
ionization
an
wiih
6rrteV.
For
zinc,
T:.Mc 13.2 column
energy
Asa ~
\342\200\224
il!
the
all
for
most
acceptor,
!uiiu..iio.i encr^e*ofcuhmm acceptors
in Si
arui
Ge,
are
listed
VI donors
column
important
\302\243,,,ofthesameorder
Ea
in Si energies for column III acctipcors
In GaAs the ionizationenergies
closeto
holes and acceptors.Orbiialsare split off For each acceptor atom there is one 13.5. in Table
except oxygen are =
&\302\243a
24meV.
V
in mcV
Ace
11
AI
13.2.
cp OIS
Ga
In
16
49
45
57
65
12.7
10.4
10?
iOS
11.2
Some
of
Occupation
'
\"'
!'
At,
Letch
Donor
\302\261
Donor
13.5
Figure
and acceptor
impurities generate orbitals deep inside the forbidden ionization multiple orbitals corresponding to different of Donor
Occupation A
level
donor
can
Hence there are [wo
different
occupationsof these level is occupied one
two
by
spin.
opposite
gap,
energy
with
sometimes
states.
Levels
be occupied
As a
in Uic
levels
impurity
orbitals
electron,
result, the
by an electronwith with
orbitals are
not
the donor
either
spin
nt
spin down.
energy. However, the
the same
independe
up or
of
each
other;
Once
the
cannot bind a second electron with
occupation probability
for
a
donor
level
is not
but function, by a function given by the simple Fcrmi-Diracdistribution is vacant, the the orbital treated in Chapter 5. We write that donor probability so that the donor is ionized, in a form slightly different from E.73):
the origin to singly occupied donor orbital relative that of the energy. The probability the donor orbital is occupied by an electron, the is is so thitt donor neutral, given by E.74):
Here
Ed
is
the
energy
of a
Statistics
Semiconductor
13;
Chapter
In the ionized conditionA\"
Acceptors require extra thought. each of the chemical bonds between
ihe
atom
acceptor
of
and the
the
acceptor,
surrounding
semiconductor atomscontainsa pair of electrons with There antiparaliel spins. is only one such state, hencethe ionized contributes one condition term, only exp[(/i - cJ/t], to the Gibbs sum for the acceptor, lit the neutral condition A the
of
one electron
acceptor,
the missing electron may
is missing
the
from
up or
haveeithcrspin
bonds.
surrounding
spin down, the
is representedtwice in the Gtbbs sum for the acceptor, by Hence the thermalaverageoccupancy is '
A
condition
neutral
The
exp[{^ ~
2 +
A,
the
with
\"\021+2
Efl)/i]
exp[(Efl
orbityl
acceptor
a
-
Because
neutral condition term 2 x J ~ 2.
\"}
{~
$x]\"
unoccupied,
occurs
with
probability
______
The
of
value
From
An
\342\200\224
== nd+
or D2)
D0)
_ __________
is the
na~
difference
D3)
of concentrations of D*
arid
A\342\204\242.
we have
D4)
D5) The
condition
neutrality
This expressionmay of
functions
the
position
represent the positive
four
visualized
be
in
and all negative
posicivechargesequal
by a
Fermi
of the terms
be rewritten
may
D)
D6);
logarithmic
level
(Figure
the two
charges.The actual total
the
negative
as
plot of n\" and n* 13.6}.
The
solid lines representthe Fermi
level
as
four dashed
occurs
sum
func-
lines of
aii
where the total
charges.
holes can be neglected;for electrons can be neglected. If one of the two impurity nu~ \024* \"i 'he be the can neglected, species majority carrier concentrationcan be calculated in The closed formConsider an it-type semiconductor with no acceptors. For
~
nd+
-
-^
jio~
\302\273
nh
as
in
Figure
13.6, the
Occupation of
Figure
13.6
intersection approximation
with
will A7)
the
be
13.6 is now given by the intersection point of the n* the interdonor concentration is not too high, the the straight portion of the incurve, alongwhich approxiWe rewrite this as
in Figure
point
neutrality
curve
of Fermi teve!and eteciroi coniaining both donors and acceptors.
determinaiion
Graphical
sciniconducior
an n-1ypc
DonorLetch
?!e curve,
on
holds.
U ihe
cxp(/
\302\243i)/t]
=
\302\2537>
(\302\273A)\302\253PfcA);
(\302\273>Jexp[(\302\243,
tj/t]
= njn*
,
D8)
s
nc*
Is
electron
the
-
neexp[~(et
the
with
be present
would
that
concentration
the Fermi level coincided
Eli)/i] = ;i,exp(-A^/t)
donor
level.
~
nd* to
the
in
~
Here
if
conduction
band
~
the donor
ec
Aed
. {49} Ej is
iontzacton energy. insert
We
set
D4) and
into
D8)
nt
obtain
E0)
+
nf3
is a
llus
=
shallow
weak that
donor
8\302\273j\302\253 \302\273/, the
A
for
-v
\302\253 i.
x ~
With
large and
=? 1 +
xI1*
we
8\302\273d/He*
-
ionization.For Table 13.2, so
the
for
that il.4 pet of the donorsremain
the subjectof
Problem
+
-ix2
;tj(l
gives P
=s 0M5nc
ne*
expanded by use of
\302\253
2it//ne*
parentheses
example,
that
\\x
is sufficiently
doping
\342\226\240\342\226\240\342\226\240
E3)
,
obtain
^ \302\253\342\200\236 \302\273\342\200\236
The secondtermin
E2)
I}-
close to nc.ffthe
may be
root
square
+
is
solution
+ (S\",,//)/)]1'3-
V([l
nt* is
levels,
E1)
i\302\273rfn,*.
quadratic equation in n,; ihe positive \302\273\342\200\236
For
=
\\n,nt*
in
the first
-
order departure from complete
300 K, we = from D9). If \302\253j Si at
un-ionized.
E4)
2 |
The
have
Ae,j
0.0!nf,
limic
Eq.
of weak
^
t.74r E4)
from
predicts
ionization is
6.
gallium anaiM*.: Could pure GaAs be prepared, it would have catikt concciUfdtioita! room temperature 10'cm\023. Wjih such a tow of\302\273,< !O art conceuiraiion of carriers, bs; closer @~\" less than a nicial, the conductivity would an as would be useful than to a sciniconductor. G;iAs insulator conveniiona! [
an intrinsic
StuM-itiiHtuting
p-n
However, ic with
is possible
concentra
high
near tnerinste carrier Cm-3) of oxygen impuriiy levels near the middle to achieve
lions (tOi5-IO17
in GaAs by
concentrations and
Junctions
chromium
doping
two impu-
together,
have iticir of ihe energy gap. Oxygen enters an and is a donor in GaAs, as expecicd from the posiiion of O in the periodic Cable relative to As; the energy is an acceptor v> i'h an level* is about 0.7 eV belowts. Chromium level about Q.S4eV below energy et. a GaAs Consider boih and chromium. The ratio of liiS two crysiai doped with oxygen conceniraiions is not critical; anything with an O:Cr raiio between abouc 1:10and 10:1 will do. If the conccitiraiions of all olher are small compared with those of O and impurities Cr, (he position of ihc Fermi level will be governed by ihe equilibrium betweenelm -ons on O and holes on Cr. The of Figure construction t3.6 applied io tliis system shows that over the indicated concentration raiio rangedie Fermi is pinned io a range between level t.5i above the O fevel and l.5r below (he Cr level. With ihe Fermi level pinned near the imrinsic middle of the energy gap, the crysiai must act as nearly Gallium arsenide doped in this way is called semi-insulating GaAs and is used extensively to to10 il cm\\ substrate as a tiijjh-resisiivity for GuAs devices. A similar [!0a prodoping procedure is possible in inP, with iron ihe taking place of chromium. impurities
ttiat
As site
p-n JUNCTIONS
Semiconductors
are almost never uniformly doped. An underan semidoped requires understanding of nonuniformiy of structures called p-n junctionsin which the doping semiconductors, particularly to n-type within 'he samecrystal.We consider ton from /vtype changes with posi! a semiconductor at .v = 0 which the doping changesabruptly crystal inside from a uniform donor concentration nd to a uniform acceptor concentration as in Figure i3.7a. This is an exampleof a p~i\\ junction. More complicated na, a device structures are made up from simple bipolar transistor h.-is junctions: used
m
devices
understanding of devices
iwo closely spacedp-n junctions, ofthe sequence or n~p-n. p-i\\~p in the built-in electrostaticpotential Vbi, even step p~n junctions contain a With no externally absence of an externally applied voltage 13.7b). (Figure the are in of diffusive junction applied voltage, the electrons on the two sides of tlic two which means that the chemical potentials(Fermilevels) equilibrium, within Fenni level ihe band sides are the same. Because ihe posiiionof the level forces :i of the Fermi siructurc depends on the localdoping,constancy shift
in
the
shift
is
eVN.
electron The
energy
potential
bands stop of
in
ihe
crossing
height eVbii%
required to equalize the total chemical intrinsic chemicalpotentialsare unequal,
an
junction example
potential
as
discussed
of
two
(Figure 13.7c). of the potential systems
in Chapter
5.
The step
when ihc
A p-n junclion. (a) Dopingdislribulion.!l is assumed 13.7 the that from doping changes abruptly n-type to p-type. The two levels arc usually different, doping (b) Electrosiaiic poteniiat. The Figure
buitl-in two
voltage sides
wilh
concentrations, shifted
relaiivc
generate the
diffusive between ihe equilibrium electron concenlralions as wet! as hole level must be (c) Energy bands. Becauseihe Fermi Vbl
ealabtishes
differenl
to each buitl-in
other, (d) Spacecharge
voltage
and
to
shift
the
dipolc
energy
required
bands.
to
We assume that
the two doping concentrationsnd,
nondegenerate range, as
defined
Hf
are
If the
donors
the p
side, then the
\302\253 nd
\302\253 \302\273c;
\302\253 \302\273( na
the n
side and
on
the
from \302\261
side
n
Jij,
\302\253 \302\273t.
E5}
the acceptorsfully
on
ionized
electron and ho!econcentrations satisfy =z vd\\
ne
one on
extrinsic but
by
ionized
fully
in the
lie
tta
and
the other
on thep band
The conduction \302\273\342\200\236.)
s:
nh
E6)
na,
side. (We have droppedthe on the
energies
i\\
p sides
and
superscripts
follow
from
A7): E7)
\302\253\302\253-/*-xlbgl^/nj;
H~ xlo&{nJnc) =
ccp = by
eVtf
-
For doping concentrationsnd A
is 0.91
eV in silicon
in electrostatic
step
-
=
E8)
~
must
satisfy
the
at room
temperature. to shift
is required
potential Poisson
to
other.
each
find eHi band
the
The
= es -~9.2t,
edge
electrostatic
energies
on
potential
equation
~
(SI)
Hfl=s0.0inL.,we
0.0inrand
E9)
.
Tlog(W>,2)
\302\243CJ. e\302\253
the two sides of the junction relative
tlogK1/\"^).
Hence
B2c).
which
;1-
= ~~ ,
F1)
of the semiconductor. space charge density and e the permittivity varies. must be whenever In the vicinity of the junction
where
p
is the
(Figure 13.7d}. concentration is less Positive
space than
the
charge donor
on
the
n
concentration,
side
means
indeed,
that
as the
the electron
conduction
Statistics
Semiconductor
13:
Chapter
raised relative to the fixed Fermi decreaseof the electronconcentration ie. band edge ts the
Take Then
ofthe
origin =
ec(x)
electrostatic \342\200\224
and
e
\302\243c(~-a))
ne{x) =
ThePoisson
potential at
x = -co.soihai
d.x i/x^
(/.x \\il\\-J
t
dx =
f
{
x =
r
/
i6d\\
, ,1
e
J
0:
F5)
]
\\dxj interface
F3)
^
J
^
AM2
oc) Integrate uiih the initial condition
the
F2)
to obtain
_
At
ntcxp[e
] Jdtpfdx
by
an exponential
becomes
A7)
^ Multiply
predicts
A7)
is
F1)
equation
level,
0 we assume that'
F6) Vn is
where
the n side.
part of
that
the built-in electrostaticpotentialdropthat
The exponential
on the right-handsideofF5)can
be
on
occurs
and
neglected,
we obtain
E for the
-v
ofthe
component
[peii,/c)(K,-
electric
field E
F7)
*/e)Vn
= -dipjdx. at the interface.Similarly, FS)
\302\243=[{2eMB/\342\202\254)(^-t/e)]\022,
where
ihepstdcTlie we
potential drop part of ilie built-inelectrostatic be the same; from this and from two \302\243 fields must
I-; is that
that Vn
on
occurs -V
Vp
=
Vbi
find
(\302\243^')\022
F9)
rse-BiasedAbrupt is the Sameas if On
field E
The
lite
from
to a
junction
the
Junction
been depleted
electrons had
side all
ii-type
p~n
distance
, G0)
-(Vw-2r/f)
no
with
at
depletion
>
\\x\\
theory as a measureofthe
layer into
on the
p side,
The totaldepletion
width
1f
=
we assume
- 4.25 \302\243
\302\273o
wK
nd =
i0licm\023;e
and w
x 104VcmwI
Reverse-Biased Let
of penetration
depth
a voltage
V
be
4.70
the
to a
applied
p side n
to the
p side
in bulk contains
side
law. As
action
holes,
and
increased
\302\273 side,
contains a very a very
by the
are
2z/e
== 1 volt,
we lind
iO^5cm.
p-n junction,
holes
and low
which
of such sign that that
means
voltage
V
raises
p side the
cond uction
drive
will
the
of
concentration
little
current
approximately
applied voltage,Figure
ihe same 13.S.
The
as
if
built-in
the
field
at
a
electrons
side to the p side.But conduction electrons, and
The distributions
flows.
is at
potential
from the ;i
low concentration of holes,consistentwith
a result, very
potential
x
-
Vbi
Junction
p-n
Abrupt
=
~ l(ko;aiid
negative vohagerelativeto the n side, energy of the electronson the p side.This from
used in semiconductor device of the space charge transition
n side.
the
Similarly,
wa is
distance
The
\\vK.
the
the ifie
mass
of electrons, voltage
were
the interface is now
given by
G3)
Chapter
13: Se
..-\342\226\240
