RAMAN RAMA N BHAVA BHAVAN :: VISAKHA VI SAKHAPA PATNAM TNAM Thermodynamic relations Gibbs Fnction and Helmholt! Fnction
Gibbs equation is dU
= Tds − PdV
The enthalpy H enthalpy H can can be differentiated, dH = dU + PdV + VdP Combining the two results in dH
= TdS + PdV
The coefficients T and v are partial derivative of h(s,P), h(s,P),
∂ H = T ∂S÷ P
∂ H = V ÷ ∂P s
Since V > 0, an isentropic increase in pressure will result in an increase in enthalpy Similarly, using the Gibbs function G
= H − TS
dG = VdP − SdT Consequently,
∂G = V ∂ P÷ T
∂G = − S ÷ ∂T P
!ote" # The decrease in $elmholt% function of a system sets an upper limit to the wor& done in any process between two equilibrium states at the same temperature during which the system e'changes heat only with a single reservoir at this temperature Since the decrease in the $elmholt% potential represents the potential to do wor& by the the system, it is also a thermodynamic potential ( The decrease in Gibbs function of a system sets an upper limit to the wor&, e'clusive of )pdv* wor& in any any proc process ess betw betwee een n two two state statess at the the same same temp tempera eratu ture re and and press pressur ure, e, prov provid ided ed the the syst system em e'changes heat only with a single reservoir at this temperature and that the surroundings are at a constant pressure equal to that in the end states of the pressure The ma'imum wor& is done when the process is isothermal isobaric Gibbs function is also called Chemical +otential
Ma"#ell$s relations:
∂T = − ∂P ,-rom equation du = Tds − PdV . ∂V÷ ÷ s ∂S v ∂T = ∂V ,-rom equation dH = Tds + VdP . ∂ P÷ ∂s÷ s P ∂ P = ∂S ,-rom equation dA = − Pdv − SdT . ∂T÷ ∂V ÷ v T ∂V = − ∂S ,-rom equation dG = VdP − SdT . ∂T÷ ∂P ÷ P T The internal energy u = u(T,v) -or a simple compressible substance,
dU
∂U = C dT + ÷ ∂V
dv
v
dS
=
T
C v T
dT
# ∂U + ÷ + P T ∂V
dv
T
Ta&ing entropy as a function of temperature and volume,
∂S = C v /sing thrid 0a'well1s relation, ∂T÷ T v ∂S = ∂P =# ∂U + P -rom this we obtain ∂U = T ∂P − P ∂V÷ ∂T÷ ÷ ÷ ÷ T v T ∂V T ∂V T ∂T v
This important equation e'presses the dependence of the internal energy on the volume at fi'ed temperature solely in terms of measurable T, + and v This is helpful in construction of tables for u in terms of measured T, + and v -or a perfect gas, +v 2 3T
∂ P = R ∂T÷ V
∂U = T R − P = P − P = 0 ÷ ∂V V
v
T
This implies that, for a perfect gas, internal energy is independent of density and depends only on T
dS
C v
=
T
dT
∂P + dv ÷ ∂T v
Similarly it can be shown using -ourth a'well4s relation that
dS
=
C P T
∂V dP ÷ ∂T P
dT +
/sing the above two equations and solving for d+,
dP =
− Cv 5∂P 6 ∂T 7v dT − dv T 5∂V 6 ∂T 7 P ( ∂U 6 ∂T ) P C P
Considering + as a function of T and v, we see that
− C v ∂P = ÷ T 5∂v 6 ∂T 7 P ∂T v C P
Two thermodynamic properties can be defined at this stage, # ∂V # ∂V
β ≡
κ ≡ −
÷
v ∂T P
V
÷ ∂P
T
β is called the isobaric compressibility and κ is called the isothermal compressibility -rom calculus, it can be shown that,Therefore,
C P
− C v = −
T ( ∂V 6 ∂P ) P
(
( ∂V 6 ∂P ) T
Thermodynamic Relations In%ol%in& 'ntro(y
5a7 Entropy as a un!t"on o T and P "8 Consider s2 s 5T,+7 Then the differential in change in the entropy,
∂S = ÷ dT + ∂T p
dS
∂ S dP ∂P ÷ T
Tds = CpdT
− β VTdP
5b7 Entropy as a un!t"on o T and v"8 ds
= Cv dT + β dv T
Tds
κ
= CvdT + T β dv κ
Thermodynamic relations in%ol%in& 'nthal(y and Internal ener&y
dh
=
CpdT + 5v − Tvβ 7 dP
)*+,' TH*MS*NS -*'FFI-I'NT µ #T
=
# # 5T β v − v 7 = v5T β − #7 Cp Cp
ENTROPY CHANGE OF DIFFERENT PROCESS
+39C:SS
=sothermal reversible process
:!T39+; 9- S;ST:
∆S = nC
V
T(
ln
+ nR l n
T#
v( v#
5ve7 nR l n
=sothermal reversible compression
:!39+; 9S/339/!<=!GS
∆S = −nR l n
:!T39; 9/!=V:3S:
v( v#
0
5 8 ve7
v(
−nR l n
v#
v( v#
0
58ve7
5ve7
?diabatic reversible e'pansion
=ncrease in :ntropy due to volume increase will be compensated by decrease in temperature 507
0
0
?diabatic reversible compression
0
0
0
=sothermal irreversible e'pansion
∆S = nR l n
=sothermal irreversible compression
∆S = nR l n
v(
− P( ( v( − v# )
v#
T 58ve7
v(
− P( ( v( − v# )
v#
T 5ve7
ve
0
ve
0
ve
0
ve
0
ve
5ve7
58ve7
?diabatic irreversible compression
∆S = nC
?diabatic irreversible e'pansion
∆S = nC
V
ln
T( T#
+ nR l n
v(
+ nR l n
v(
v#
ve
5ve7 V
ln
T( T#
v#
5ve7
=sothermal free e'pansion
∆S = nR l n
v(
?diabatic free e'pansion
∆S = nR l n
v(
v#
v#
