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Physics

Statistical Mechanics

An Overview of Thermodynamics-II Paper No. : Statistical Mechanics

Module : An Overview of Thermodynamics-II

Prof. Vinay Gupta, Department of Physics and Astrophysics , University of Delhi, Delhi

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Principal Investigator

Paper Coordinator

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Prof. P.K. Ahluwalia,Physics Department,Himachal Pradesh University,Shimla-171005

Shimla 171 005

Prof. P.K. Ahluwalia,Physics Department,Himachal Pradesh University,Shimla-171005

Prof. P. N. Kotru, Department of Physics, university of Jammu Physics

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An Overview of Thermodynamics-II

Description of Module

Subject Name Physics

Paper Name Statistical Mechanics

Module Name/Title An Overview of Thermodynamics-II Module Id

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An Overview of Thermodynamics-II Contents of this Unit

1. Learning Outcomes 2. Introduction

3. Thermodynamical Potentials 4. Maxwell relations

5. Thermodynamic Relations

6. Applications of Maxwell’s Relations 7. Extre mum Principles

8. Beyond Thermodynamics 9. Summary

Appendix

A1 Legendre Transformations

A2 Differentiation Tricks for The rmodynamic Relations A3 Jacobians

Learning Outcomes

After studying this module, you shall be able to

 Describe evolution of thermodynamic potentials, which are most important thermo physical entities to know the equilibrium state of the system

 Apply mathematical apparatus of Legendre transformations, partial differentiations and Jacobian determinant to derive various thermodynamic relations.

 Define repeatedly encountered measurable physical quantities like heat capacity at constant volume and pressure, isothermal and adiabatic compressibility, co-efficient of thermal expansion and susceptibility.

 Starting from various thermodynamic potentials derive generic equations of state.

 Derive famous Maxwell’s relations to link non-measurable thermodynamic quantities like entropy with measurable thermodynamic quantities like temperature, pressure and volume.

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 Describe the important extremum principles of entropy, internal energy and thermodynamic potentials to characterize equilibrium state of a system.

 Interpret concavity principle of entropy and convexity principle of internal energy geometrically and see their physical significance.

 Derive stability criterion for a system in equilibrium and use these criterion to arrive at system

independent results that heat capacity at constant volume and constant pressure, adiabatic and isothermal compressibility are always greater than or equal to zero.

 Prove using stability criterion that

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An Overview of Thermodynamics-II 2. Introduction

In this module we shall review the idea of thermodynamic potentials besides which we have encountered in module 5. Thermodynamic potentials have unit of energy and their minima gives the equilibrium state of the system subject to constraints. We confine ourselves only to those thermodynamic descriptions, which shall later be needed in the study of

statistical mechanics.

3. Thermodynamic Potentials

In module V. we saw that out of the given thermodynamic variables to have energy representation, we treated , the internal energy, as a function of i.e. were treated as independent variables and for entropic representation treated as a function of

i.e. were treated as independent variables. We also observed another thing there that in these two representations independent variables were extensive parameters and independent variables turned out to be partial derivatives of an dependent extensive variable with respect to independent extensive variable, keeping other independent extensive

quantities constant. Furthermore, it is always desirable to have independent variables as those quantities which are easily measurable., for example temperature and pressure . Also it is worth noting that temperature is a partial derivative of with respect to ,

, and pressure P is negati ve partial derivative of with respect to ,

, So in other words we are looking forward to functions of independent variables which may be intensive variables like P and T.

For a function of a single variable this amounts to saying find a function which is a function of derivative of i.e. . It is like saying that we wish to generate a function which is a function of the slope of the tangents drawn at each point. This is performed by the famous Legendre transformations (AppendixA1).

Now starting with , such that

(1) We note that there are three pairs of variables occurring leading to different

thermodynamic potentials, each additional pair of variables doubles ths number, Let us look at another encountered in module V,

(2) It contains seven pairs of variables and hence could lead to i.e. 128 thermodynamic potentials.

This is indeed a mind boggling number, however we shall concentrate only on the equation with three pairs of variables and generate eight thermodynamic potentials given in Table 1 below. Out of these eight potentials the first five are well known.

Table 1 Thermodynamic Potentials and Their Total Potentials

` Potential Eq. Total differential of Eq. Independent

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No. potential No. variables Internal Energy

or Simply Energy

(3) (4)

Free energy (5) (6)

Enthalpy (7) (8)

Gibbs Free Energy

(9) (10)

Grand

Thermodynamic potential

(11) (12)

No special name

(13) (14)

No special name

(15) (16)

No special

name (17) (18)

It is a very easy to learn this table, provided we recognize that independent variables in the last column are those variables which appear on the right hand side in the third column as as full differential. This is done through legender transformation:

The trick is, let us start with first pair of variables ( ):

(19) (20) (21) Following these steps gives us a legendre transformed function F, called Helmholtz free energy, which now has T as the independent variable instead of S.

The Grand thermodynamic potential can be obtained by two successive Legendre transformations over two pairs of variables i.e. followed by . Similarly other thermodynamic potentials can be obtained.

Also note that a thermodynamic potential is minimum if , where f is any of the defined thermodynamic potential. For example enthalpy is minimum if , which happens only if i.e. when the entropy is constant, pressure is constant and number of particles remain constant. This means that enthalpy describes a closed, isobaric and

adiabatic collection of particles. Thermodynamic potential describes an open, isothermal and isobaric collection of particles.

With the introduction of these potentials, out of which we focus on , , , and only we can get in each of these representations equations of states as given below in Table 2 Table 2 Thermodynamic Potentials and Equations of State

Potential Total Differential of Potential

Equations of State Eq.

No.

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An Overview of Thermodynamics-II or

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or

(23)

or

(24)

or

(25)

or

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4. Maxwell Relations

Maxwell’s relations are very interesting relations providing a link between measurable

quantities like Pressure , Temperature & Volume which can be measured directly and non-measurable quantities like Entropy , Enthalpy , Helmholtz free energy and Gibbs Free Energy which can not be measured directly. Therefore, it is crucial that we express non-measurable quantities in terms of measurable quantities. Maxwell’s relations provide us exactly this option.

By choosing any two independent variables and taking mixed derivatives of a given

thermodynamic potential with respect to these and remembering that order of derivative does not affect the mixed derivatives leads to what are known as Maxwell’s relations of

thermodynamic potentials. Table 3 below describes this procedure for the thermodynamic potentials given in Table 2.

Table 3 Maxwell’s Relations and Their Derivation Potential

Independent variables chosen

Intermediate steps

Maxwell relation Eq.

No.

Note that

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An Overview of Thermodynamics-II Potential

Intermediate steps

No.

Note that

(28)

Note That

(29)

Note That

(30)

Note That

(31)

Note That

(32)

(33)

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Intermediate steps

No.

Note That

(34)

Note That

(35)

Note That

(36)

Note That

(37)

Note That

(38)

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Intermediate steps

No.

Note That

(39)

Note That

(40)

Note That

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5. Thermodynamic Relations

Before we move further to discuss thermodynamic potentials and ensuing Maxwell’s equations, it will be handy to define some thermodynamic relations which provide

measurables for a given thermodynamic system and also he lp us measure the unmeasurable quantities encountered in Thermodynamics such as entropy and chemical potential

Heat Capacity

Heat capacity of a system is defined as amount of heat required by the system to change the temperature of the system by unity. We can write this mathematically as at constant X, say volume ( )or pressure ( ) keeping mass or number of particles constant.

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The above relations can be related to change in internal energy, provided volume is kept constant or change in enthalpy provided pressure is kept constant.

It is known to us that at constant and number of particles and at constant , we have

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(43) (44) Compressibility:

Compressibility of a system is defined as the ratio of fractional change of volume to applied pressure on the system, we can write it as

(45)

Where, implies that pressure was applied under isothermal conditions and imples that pressure was applied under iso-entropic conditions. Therefore, it becomes respectively isothermal compressibility and adibatic compressibility. Negative sign shows that volume compresses as pressure is applied.

Co-efficient of Thermal Expansion:

Co-efficient of Thermal Expansion is difined as the fractional change in volume per unit change in temperature, under constant pressure and constant number of particles. We can write it as

(46) Susceptibility:

Under the influence of an external field an isotropic magnetic system at constant

temperature (constant entropy) acquires a magnetic moment M, the isothermal and adiabatic magnetic susceptibility can then be defined as

(47) (48) respectively.

6. Application of Maxwell’s Relations

To appreciate the use of Maxwell’s relations, it will be interesting to see that how these relations help us in measuring immeasurable quantities. Following table gives a list of measurable and non-measurable Thermodynamic quantities..

Table 4 List of Measurable and Non-Measurable Thermodynamic Physical Quainties Measurable Quantities Non-Measurable

Quantities

Temperature Entropy

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Pressure Chemical potential

Volume

Number of particles or mass (related by the molecular weight)

Enthalpy (latent heat) of phase change ( )

Constant volume heat capacity Constant pressure heat capacity Isothermal compressibility ( ) Thermal expansivity / Expansion Coefficient

By using Maxwell’s relations and the thermodynamic relations derived earlier, we can measure the non-measurable physical quantities easily.

Variation of Entropy with Pressure in terms of measurable(

Starting from left hand side

Therefore,

(49) Variation of Entropy with Volume in terms of measurable(

Starting with Maxwell’s relationship

and using the trick 4 of differentiation on the left hand side we have

Therefore,

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Ratio of heat capacity at constant volume and heat capacity at constant pressure in terms of ratio of isothermal compressibility and ratio of adiabatic compressibility

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Where in the second step we have used the trick no. 4 (Appendix A2) again.

Therefore,

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Difference between heat capacity at constant pressure to Heat capacity at constant Volume This involves usage of Jacobian determinant and one of the Maxwell’s relation

Starting from the definition of Heat capacity at constant volume,

Using the definition of Jacobian it can be written as

Recalling the definition of Jacobian this can be written as

Using Maxwell’s relation

and as proved earlier We get

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This is a very interesting relationship. Firstly it proves that , secondly it is true for any thermodynamic system, because it has not assumed any thing about the composition of the system revealing the universal sweep of thermodynamics.

6. Extremum Principles

Principle of maximum entropy for equilibrium: This principle characterizes the state of equilibrium as the state of maximum entropy for a given value of internal energy ( volume

and number of particles or any other extensive parameter involved in the system.

Mathematically this means and . Equality here implies t hat system is i n equilibrium.

Principle of energy minimum for equilibrium: According to this principle, for a constant value of entropy, equilibrium state is characterized by a state of minimum energy.

Mathematically this means that and . This principle is equivalent to principle of maximum entropy for equilibrium.

Extremum principle for thermodynamic potentials: Many a times in thermodynamics and statistical mechanics we have to deal with fundamental relations in Helmholtz free energy

representation, Enthalpy representation and Gibbs free energy representation. A natural question to ask is what is the stability criteria for these thermodynamic potentials.

Let us look at the stability condition for Helmholtz free energy F . For a constant value of and for the given system, The enery conservation law of thermodynamics implies that

Since we know that , we can write the above relation as If in a process is held constant,

This equation implies that in a process in which are held constant is minimum. Similarly one can prove that if in a process S,P,N are held constant, then its enthalpy has minimum value. For all processes for which are held constant then gibbs free energy has a minimum value

It is interesting to note that for a system in equilibrium under constancy of respective variables, these functions have a minimum value hence these are called thermodynamic potentials.

Concavity principle of entropy:

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Assume that we have two systems in equilibrium, such entropy of each system is a function

of its extensive variables and Suppose that we bring

these two systems together by removing the barriers or walls between them. The extensive

parameters of the composite system will then be . This

combined system will not be in equilibrium and its various parts shall tend to exchange energy in order to come to equilibrium. In this process of establishing equilibrium, since entropy can not decrease, system must move to increased entropy, so that mathematically this implies

(53) This condition expresses the stability condition for the system, which has an interesting geometrical meaning that it represents a concave function.

To appreciate geometrical meaning let us cause a transformation of extensive variables on the left side such that

and

, where Then

Or using property of extensivity we have

Here left hand side represents value of at a point denoted by

Where as right hand side denotes a point on the straight line joining points

and . Pictorially this is represented in the Figure 1.

Convexity principle of internal energy:

In terms of energy representation the energy minimum principle implies that as a function of extensive variables implies that

, ……..

Figure 1 Geometrical representation of concavity of entropy, a point on a chord always lies below the point on the curve joining the two points

, ……..

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An Overview of Thermodynamics-II Physical significance of stability criterion:

Now we will discuss stability criteria in terms of second derivatives of thermodynamic potentials with respect to their extensive and intensive parameters . In terms of internal energy it implies that if exists then i.e.

(54) Using the values of first derivatives it reduces to

(55)

If this equation can be recast as then it will be positive definite if both A and B are positive.

Therefore we visualize the above equation in the form Where

and recast it as , which can be done by simple mathematical manipulation

Since is always positive definite, it implies that i.e. . This further implies that . This leads to following three conditions:

(56) (57) (58)

These three conditions have great physical significance and lead to following four results, which are independent of the composition of the system

(i) Heat capacity at constant volume is always positive Since

and temperature ,

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An Overview of Thermodynamics-II Therefore, .

Recalling the definition of heat capacity, we find that . (ii) Adiabatic compressibility can not be negative

Since and is always positive

i.e. adiabatic compressibility

(iii) Isothermal compressibility is also always positive Recalling

Using the Maxwell’s relations we have

Which can be further written as

Since is positive the other factor on the left hand side must be negative i.e.

Multiplying this with –V and taking the reciprocal we get (iv)

Recall that , therefore, and hence 8. Beyond Thermodynamics

Thermodynamics offers very powerful tools for calculating macroscopic properties of a system howsoever complex the system may be. However, its basis is purely

phenomenological and requires empirical inputs to set up the fundamental relations. The biggest limitation is that it does not consider the structure of matte r or system at hand while describing the properties of macroscopic systems. This exactly where statistical mechanics

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takes over and starts from the microscopic entities making the matter and through average behavior of these large number of particles in a system arrives at fundamental relations which can be manipulated to get all other thermodynamics properties subject to experimental verification.

9. Summary

In this module we have learnt

 To systematically generate thermodynamic potentials

 To derive equations of state in different thermodynamic representations

 That how by applying Legendre transformations, thermodynamic potentials can be obtained.

 To write thermodynamic potentials in differential form.

 To derive Maxwell’s relations

 That Maxwell’s relations provide a way to link non measurable physical quantities with measurable physical quantities.

 The extremum principles for entropy, internal energy and thermodynamic potentials for a thermodynamic system to be in equilibrium

 Stability criterion and their geometrical interpretation in the form of concavity principle of entropy and convexity principle of internal energy.

 That stability criterion in differential form reveal the global nature of the properties of heat capacities ( and , compressibilities ( and and thermal expansion that they are always positive.

 The need to go beyond thermodynamics to understand properties of a system from microscopic conditions.

Bibliography

1. Pal P.B., “An Introductory Course of Statistical Mechanics”, New Delhi: Narosa Publishing House Pvt. Ltd., 2008.

2. Matveev A.N., “ Statistical Physics,” Moscow: Mir Publishers, 1985.

3. Rao Y.V.C., “ Postitutional and Statistical Thermodynamics,” New Delhi: Allied Publishers, 1994.

4. Fermi E., “ Notes on Thermodynamics and Statistics,” Chicago:

The University of Chicago Press, Phonix edition, 1966.

5. Panat P.V., “Thermodynamics and Statistical Mechanics,” New Delhi: Narosa Publishing House Pvt. Ltd., 2008

6. Greiner W., Neise L., Stocker H., “ Thermodynamics and Statistical Mechanics,” New York, Springer Verlag, 1995.

7. Dittman R.H., Zemansky M.W., “ Heat and Thermodynamics,”

New Delhi, Tata McGrawHill Publishing Company Limited, Seventh Edition, 2007.

Appendix

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An Overview of Thermodynamics-II A1 Legendre Transformations

Legendre transformation provides us a way to convert one thermodynamic potential into another, with a difference in thermodynamics that legendre transform involved is negative Legendre transform compared to when we use it in classical mechanics. First we shall look at it from the point of view of single variable function whose legendre transform we are seeking. Legendre transform produces from a new function involving a change of variable from , where . The new function has the same information as , so that it can be transformed back (not shown here). The steps are

(i) Start with (ii) Find .

(iii) Then , where is the value that maximizes for a given p.

This procedure can be extended to a multivariate function , as is the case with thermodynamics. Hence we can define

, where is the value that maximizes for a given p.

For example, Helmholtz free energy is Legendre transform of internal energy

Furthermore, Gibbs free energy or thermodynamic potential, can be derived as Legendre transform of Helmholtz free energy involving a change of variable from

.

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An Overview of Thermodynamics-II A2 Mathematical Tricks of Partial Differentiation Table 5 Partial Differentiation Tricks

Trick No.

Trick Remarks

1.

2. Chain rule

3. Maxwell’s Relation

4. Take care of the sign

5 Trick 4 follows from this trick

provided we choose

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An Overview of Thermodynamics-II A3 Jacobians or Jacobi Determinants

Table 6 Properties of Jacobi Determinants 1.

= 2.

3.

4.

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