(7) Maxwell’s Thermodynamical Relations

Maxwell’s Thermodynamical Relations

Discipline Course-I Semester-II Paper No: Thermal Physics : Physics-IIA Lesson: Maxwell’s Thermodynamical Relations Lesson Developer: Dr. Vinita Tuli College/ Department: ARSD College, University of Delhi

Institute of Lifelong Learning, University Of Delhi

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Table of Contents Chapter: Maxwell’s Thermodynamical Relations  Introduction o Thermodynamical variables o Extensive and intensive variables

o Distinction between extensive variables

intensive

and

o Maxwell’s thermodynamical relations  Proof of the above relations  Thermodynamical Potentials

 Value Addition  Summary  Exercise/Practice

 References/Bibliography


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INTRODUCTION

Why Do We Use Maxwell’s Relations? • Certain variables in thermodynamics are hard to measure experimentally such as ‘entropy’. • Some variables like Temperature and Pressure are easily measureable. • Maxwell relations provides a way to exchange variables.

Out of several properties of a thermodynamic system some of these like internal energy and entropy cannot be measured. So thermodynamic relations can relate these properties, with those that can be measured like pressure, temperature, compressibility etc. In thermodynamic relations un-measurable properties can be written as partial derivatives involving both intensive and extensive variables.

Thermodynamics Variables The thermodynamic state of a substance is specified by properties like pressure, volume, temperature, internal energy and entropy. These properties change when the system passes from one state to another. These variables are known as thermodynamic variables. These are called macroscopic co-ordinates. They require a few measurable properties of the system to define them and do not require the knowledge of microscopic co-ordinates.

Extensive and Intensive Variables An extensive variable of a system is a macroscopic co-ordinate, which describes a system in equilibrium and has a value equal to the sum of its values in each part of the system. It depends upon the mass or the size of the substance present in the system. Examples are Mass, volume, internal energy, entropy, length, area, heat capacity, magnetization etc. An intensive variable of a substance is a microscopic co-ordinate which describes the system in equilibrium and has the same value in any part of the system. It is independent of mass or size of the system. Examples are


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Pressure, temperature, viscosity, refractive index, density, specific volume, surface tension, electromotive force etc.

Distinction between extensive and intensive variables In a homogeneous system in equilibrium, Suppose the system is divided into many parts and the macroscopic, variable x of the system has the values x1, x2, x3, ….. in each of these parts of the system respectively, then x is said to be extensive variable, if x = x1 + x2 + x3 + …….., and x is said to be intensive variable, if x =x1 = x2 =x3 = ……… An extensive variable may become an intensive variable by specifying unit amount of substance. Like heat capacity is extensive variable but specific heat is an intensive variable.

Maxwell’s Thermodynamical Relations Measurable quantities


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Maxwell’s Thermodynamical Relations How does one measure the entropy or chemical potential from experiments? These kinds of quantities are not usually directly accessible in the lab. What we can measure, typically, are mechanical quantities like pressure, bulk quantities like volume and density etc. thermal properties like temperature and heat flow (e.g., by slow heat exchange experiments where we can measure temperature changes in a coupled reference body). Of the thermodynamic variables that we have discussed thus far, the following are considered measureable: Measurable thermodynamic variables temperature pressure volume or number of particles or mass (related by the molecular weight) and enthalpy (latent heat) of phase change. There are also several readily-measured material properties that depend on derivatives of thermodynamic variables. These quantities measure the change in a parameter in response to an infinitesimally small perturbation, they are termed as response functions: Measurable thermodynamic response functions constant volume heat capacity constant pressure heat capacity isothermal compressibility thermal expansivity / expansion coefficients. Maxwell relations enable us to express experimentally-inaccessible quantities in terms of the measurable ones just listed. A thermodynamic relation is a rule which is obtained by thermodynamic reasoning and applies to most of the systems. These relations are general and useful as they simplify thermodynamic analysis of the system. From the first and second law of thermodynamics, Maxwell was able to derive six fundamental thermodynamical relations. The state of a system can be specified by any pair of quantities, viz pressure (P), Volume (V), temperature (T), and entropy (S). In solving any thermodynamical problem, the most suitable pair is chosen and the quantities constituting the pair are taken as independent variables. A thermodynamic relation is an exact rule which may be obtained by general thermodynamic reasoning and applies to almost any system. These relations are very general and extremely useful as they enormously simplify thermodynamic analysis.

Proof: Institute of Lifelong Learning, University Of Delhi

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From the first law of thermodynamics,

 Q = dU  W  Q = dU  PdV dU =  Q  PdV

From the second law of thermodynamics,

dS =

Q T

 Q = TdS Substituting the value of  Q , we get

 U = TdS  PdV

….(1)

Considering U, S and V to be functions of two independent variables x and y in general, x and y can be any two variables out of P, V, T and S.

 U   U  dU =   dy  dx    x  y  y  x

 S   S  dS =   dx    dy  x  y  y  x  V   V  dV =   dy  dx    x  y  y  x

and

Substituting these values in equation (1) we get

 S    U   S   U  dx  dy = T dx  dy            x  y  x  y  y  y  y  y   V    V   P   dy   dx    x  y  y  x 

  S   V     P   dy  x  y    x  y

= T 

  S   V    T    P    dy  y  x    y  x Comparing the coefficients of dx and dy, we get

 U   S   V     = T    P  x  y  x  y  x  y

…(2)

 U   S   V    = T    P   y  x  y  x  y  x

…(3)

Differentiating equation (2) with respect to y and equation (3) with respect to x


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 2U yx

 T   S   2 S  P   V   2V  T   P        y.x  y  x  x  y V .x  y  x  x  y

=

2S  2V  T   S  =   T    x.y x.y  x  y  y  x

and

 2V  P   V      P x.y  x  y  y  x The change in internal energy can be brought about by changing V and T, whether V is changed by dV first and T by dT later or vice versa, the result is the same. This is because dU is a perfect differential. 

which implies

 2U  2U = and y.x x.y

 T   S   2 S  P   V   2V        T  P y.x  y  x  x  y y.x  y  x  x  y  2 S  P   V   2V  T   S    T   P        x.y  x  y  y  x x.y  x  y  y  x

…(4)

Since dS and dV are also perfect differentials so order of differentiation will not matter, which implies

2S 2S = x.y y.x V V = x.y y.x

and Equation (4), therefore reduces to

 T   S   P   V   T   S   P   V           =         x  y  y  x  x  y  y  x  y  x  x  y  y  x  x  y

…(5)

Equation (5) is the general expression for Maxwell’s thermodynamical relations. In place of the independent variables x and y, we can substitute any two of the four variables S, T, P and V .So that there may be one mechanical variable (P and V) and one thermal variables (S to T). There may be four sets of possible substitutions (S, V) , (T, V), (S, P) and (T, P),which provide us the four Maxwell’s thermodynamical relations.


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Maxwell’s Thermodynamical Relations First Relation Putting x = S and y = V in equation (5), we get

S = 1, x V = 1 y S = 0, y

and

V = 0 x Substituting in equation (5), we have

 T   P    =    x  y  y  x

y  V (as y = V) and s  S (as x = S). Hence  P   T    =    S V  V  S

….(6)

This is the Maxwell’s first thermodynamical relation. The first relation tells us that for a thermodynamic system, increase in temperature per unit increase in volume at constant entropy is equal to the decrease in pressure per unit increase in entropy at constant volume.

Second Relation We get the second relation by putting x = T and y =V in equation (5),

then

T = 1, x V = 1 y

and

T = 0, y

V = 0 x


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Substituting in equation (5), we have

 S   P    =    V T  T V

…(7)

This is the Maxwell’s second thermodynamical relation. This relation tells us that increase in entropy per unit increase in volume at constant temperature is equal to increase in pressure per unit increase in temperature at constant volume.

Third Relation By putting x =S and y = P, in equation (5) we get

S = 1, x P = 1, y

S = 0 y

P = 0 x Substituting these in equation (5), we have

 T   V    =    P  S  S  P

…(8)

This is the Maxwell’s third thermodynamical relation. The third relation tells us that increase in temperature per unit increase in pressure at constant entropy is equal to the increase in volume per unit increase in entropy at constant pressure.

Fourth Relation By putting x = T and y = P, then equation (5) becomes

T = 1, x P = 1, y


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T = 0 y

P = 0 x

and

Substituting these values in equation (5), we get

 S   V    =    P T  T  P

…(9)

This is the Maxwell’s fourth thermodynamical relation. This relation tells us that increase in entropy per unit increase in pressure at constant temperature is equal to the decrease in volume per unit increase in temperature at constant pressure. These are the four Maxwell’s fundamental thermodynamical relations. Apart from these there are two more relations within the mechanical variables (P, V) and thermal variable (T, S).

Fifth Relation By substituting x =P and y =V

P = 1, x V = 1, y P = 0 y

V = 0 x

and

Substituting these values in equation (5), we get

 T   S   T   S          = 1  P V  V  P  V  P  P V

…(10)

This is the Maxwell’s 5th thermodynamical relation.

Sixth Relation


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Maxwell’s Thermodynamical Relations By putting x = T and y = S

T = 1, x S = 1, y T = 0, y

S = 0 x

and Substituting in equation (5), we get

 P   V   P   V          = 1  T  S  S T  S T  T  S

…(11)

This is the Maxwells 6th thermodynamical relation. Out of the six thermodynamical relations, the one suited for a given problem is used and the problem is solved. Let us see, some of the important applications of these Maxwell’s thermodynamic relations.

Thermodynamic Potentials We can also derive Maxwell’s Relations using Thermodynamical relations . The state of a system can be completely described by any two of the five variables, P, V, T, S, and U. Out of these, U the internal energy is determined by using the remaining four, as shown below : According to first law of thermodynamics,

 Q = dU  PdV And from the second law of thermodynamics,  Q = TdS

dU  PdV = TdS dU = TdS  PdV Thus U can be eliminated and we are left with four state variables, P, V, T and S. This is the reason why only four variable of state are defined. Taking two of the four state variables, P, V, T and S at a time, we get six possible pairs (P, V), (P, T), (P, S), (V, T), (V, S) and (T, S).


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Maxwell’s Thermodynamical Relations The pair (P, V) is connected with an in exact differential quantity  W as  W  PdV and pair (T, S) with  Q as  Q  TdS . Hence after eliminating these two pairs, we are left with four pairs of thermodynamical variables (P, T), (P, S), (V, T) and (V, S) and Corresponding to each pair we can write a thermodynamical relation. These thermodynamical relations are known as Maxwell’s thermodynamical relations. There are four thermodynamic potentials: (i) (ii) (iii) (iv)

Internal energy, U Helmholtz free energy, Enthalpy Gibbs function

F = U – TS H = U + PV G = U + PV – TS

Each of the four Maxwell’s thermodynamical relations can be derived from one of these thermodynamics potentials U, F, H and G. Only four thermodynamic potentials are defined. 1. Internal energy U. The internal energy or the intrinsic energy is the total r of the system For an isochoric adiabatic process : dU = 0 U = a constant the internal energy of system remains constant in an isochoric adiabatic process. 2.

Helmholtz Free Energy F. Helmholtz free energy is defined as F = U – TS

For isothermal isochoric process : dT = 0 dV = 0 dF = 0 F = a constant i.e. the Helmholtz free energy remains constant during isothermal isochoric process.

3.

Enthalpy H. This is known as the total heat and is given by H = U + PV

For an isobaric adiabatic process :


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dH = 0 H = a constant i.e., enthalpy remains constant in a reversible isobaric adiabatic process. 4.

Gibbs Function G or Gibbs Free Energy.

This is defined by the equation G = U – TS + PV (b) For an isobaric and isothermal process : Hence, if the process is isothermal and isobaric then dG = 0 G = a constant Thus, Gibbs function (G) or Gibbs free energy remains constant in an isothermalisobaric process.

Significance of Thermodynamic Potential Since the four function U, F, H and G play in thermodynamics the same role as played by potential energy in mechanics, they are called thermodynamic potentials. Maxwell’s thermodynamical relations can be derived from the thermodynamic potentials. Maxwell’s thermodynamical relations are

 S   P    =    V T  T V

…(i)

 S   V    =    P  T  T  P

…(ii)

 P   T    =    S  P  V  S

…(iii)

 T   V    =    P  S  S  P

…(iv)


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Value Addition James Clerk Maxwell From Wikipedia, the free encyclopedia

James Clerk Maxwell

James Clerk Maxwell (1831–1879)

Born

13

June

1831

Edinburgh, Scotland

Died

5

November

1879(aged 48)

Cambridge, England

Citizenship

British

Nationality

Scottish

Fields

Physics and mathematics


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Institutions

Marischal

College,

King's

College

Aberdeen London

University of Cambridge

Alma mater

University

of

Edinburgh

University of Cambridge

Academic advisors

William Hopkins

Notable students

George Chrystal

Known for

Maxwell's

equations

Maxwell

distribution

Maxwell's

demon

Maxwell's Maxwell

discs speed

distribution

Maxwell's

theorem

Maxwell

material

Generalized

Maxwell

Displacement

model current

Maxwell's Wheel

Notable awards

Smith's Adams Rumford

Prize (1854) Prize (1857) Medal (1860)

Keith Prize (1869–71)

Signature

CONTRIBUTIONS OF THE THERMODYNAMICAL RELATIONS

MAN

WHO

DERIVED

MAXWELL’S

James Clerk Maxwell FRS FRSE (13 June 1831 – 5 November 1879) was a Scottish mathematical physicist. His most prominent achievement was to formulate a set of equations that describe electricity, magnetism, and optics as manifestations of the same phenomenon, namely theelectromagnetic field. Maxwell's contribution was his extension and formulation of Michael Faraday's theory of electricity and magnetic lines of force. Maxwell showed that a few relatively simple mathematical


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Maxwell’s Thermodynamical Relations equations could express the behaviour of electric and magnetic fields and their interrelation. Maxwell's achievements concerning electromagnetism have been called the "second great unification in physics",after the first one realised by Isaac Newton. In 1862, Maxwell calculated that the speed of propagation of an electromagnetic field is approximately that of the speed of light. With the publication of A Dynamical Theory of the Electromagnetic Field in 1865, Maxwell demonstrated that electric and magnetic fields travel through space as waves moving at the speed of light. Maxwell proposed that light is in fact undulations in the same medium that is the cause of electric and magnetic phenomena. The unification of light and electrical phenomena led to the prediction of the existence of radio waves. Maxwell helped develop the Maxwell–Boltzmann distribution, which is a statistical means of describing aspects of the kinetic theory of gases. He is also known for presenting the first durable colour photograph in 1861 and for his foundational work on analysing the rigidity of rod-and-joint frameworks (trusses) like those in many bridges. Maxwell also continued his work on the kinetic theory of gases. By treating gases statistically in 1866 he formulated the Maxwell-Boltzmann kinetic theory of gases. This theory showed that the temperatures and heat involved only movement of molecules. Maxwell's approach did not reject the earlier studies on thermodynamics but it used a better theory to explain the observations. Maxwell has also derived the thermodynamical relations, which relate the un-measurable properties with those properties that can be measured. His discoveries helped usher in the era of modern physics, laying the foundation for such fields as special relativity and quantum mechanics. Many physicists regard Maxwell as the 19th-century scientist having the greatest influence on 20th-century physics, and his contributions to the science are considered by many to be of the same magnitude as those of Isaac Newton and Albert Einstein. In the millennium poll—a survey of the 100 most prominent physicists—Maxwell was voted the third greatest physicist of all time, behind only Newton and Einstein. On the centenary of Maxwell's birthday, Einstein himself described Maxwell's work as the "most profound and the most fruitful that physics has experienced since the time of Newton." It is advised to follow this link to view added information of Maxwell’s relations.


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Summary A thermodynamical relations is a rule which is obtained by a simple thermodynamic reasoning and applies to most of the systems. The usefulness of the above relations lies in the fact that they relate quantities which seem unrelated. They help us to link data obtained in various ways or replace a difficult measurement by another one. They can also be used to obtain values of one variable from the calculations of another variable. These relations are very general and immensely useful as they simplify analysis of thermodynamic systems. The most convenient way to derive these relations is to use partial differentiation. They can also be derived from other methods.

Exercise Fill in the Blanks Q1.

___________________ is the Maxwell’s first thermodynamical relation.

Q2.

___________________ is the Maxwell’s second thermodynamical relation.

Q3.

___________________ is the Maxwell’s third thermodynamical relation.

Q4.

___________________ is the Maxwell’s fourth thermodynamical relation.

Q5.

___________________ is the Maxwell’s fifth thermodynamical relation.

Q6.

___________________ is the Maxwell’s sixth thermodynamical relation.


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Maxwell’s Thermodynamical Relations Q7. ________ & _________ are the Maxwell’s relations apart from the four basic thermodynamical relations.

Q8.

Derive Maxwell’s Thermodynamical relations?

Q9.

List the Maxwell’s fundamental relations.

Q10.

Maxwell’s thermodynamic relation is:

(a) (b) (c) (d) Q11.

 S   P       T T  V T  T   P        V  S  S V  T   V       P  P  P  P none of these Prove the thermodynamic relation

 S   P    =    V T  T V

References: 1. 2. 3. 4. 5. 6.

Thermodynamics, kinetic theory and statistical thermodynamics by F.W. Sears and G.L. Salinger. Fundamentals of Statistical and Thermal Physics by F. Reif. A Treatise on Heat by M.N. Saha and B.N. Srivastava. Thermal physics by C. Kittel/H. Kroemer. Heat and Thermodynamics by M.W. Zemansky and R.H. Dittman. Thermal Physics by S.C. Garg, R.M. Bansal and C.K. Ghosh.


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(7) Maxwell’s Thermodynamical Relations

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