Chapter 8Fundamental Equation and the Equationsof State

In Chap. 5, we noted that in combination the first and the second laws lead to animportant relationship: the difference, dU; in the internal energy of two neighboringequilibrium states is linearly related to the corresponding difference, dS; in theirentropy. Of course, also included in this relationship is the heat energy quasi-statically added to the system and the quasi-static work, dWquasi�static D P dV ,performed by the system when it transitions from a state with extensive variables.U; S; V / to one with .U C dU; S C dS; V C dV /. That is

dQquasi�static ! T dS D dU C dWquasi�static D dU C P dV: (8.1)

Mostly, thus far, we have explicitly treated only single phased, closed systems wherethe number of moles, n; is constant. Moreover, in addition to the internal energy1

U; and the entropy S; the only extensive variable treated has been the volume V .More general systems may possess properties such as magnetization, electric

charge and polarization, surface tension, etc.; be composed of more than one varietyof molecules; may separate into different phases; and sometimes even undergochemical decomposition, etc. In order to describe these phenomena, other extensivevariables also need to be included in the expression for dWquasi�static: Thus, evenwhile still considering a single phase system, it is helpful to also include twoadditional terms to the quasi-static work that is performed by the system. To thisend we write

dWquasi�static D P dV �`cX

j D1

�j dnj CX

i

Yi dXi : (8.2)

1And, of course, the enthalpy H – which is an extensive variable – that has also been consideredbefore. Here the inclusion of H , however, is subsumed in that of U because the knowledge of thevolume (and its conjugate variable, the pressure) relates U to H:

R. Tahir-Kheli, General and Statistical Thermodynamics, Graduate Texts in Physics,DOI 10.1007/978-3-642-21481-3 8, © Springer-Verlag Berlin Heidelberg 2012

337

338 8 Fundamental Equation and the Equations of State

The parameters, �j ; are the intensive variables conjugate to the extensive variablesnj ; the latter indicating the number of moles of the j th type of molecules inthe system.2 The variable �j is generally called the molar chemical potentialof the j th chemical constituent. We assume here a single phase constituted of `c

different chemical constituents. Similarly, Xi are the other extensive variables thatmay be needed for a complete description of the system. As mentioned earlier,such variables may represent the magnetization M; the electric polarization P , thesurface area A of a film, etc., etc. The relevant conjugate variables, denoted as Yi ;

would then be the magnetic field H; the electric filed E ; the surface tension T ; etc.,etc. Note the product of a variable and its conjugate, for example, P and V , hasdimensions of energy U:

The Euler equation is introduced in Sect. 8.1. Equations of state are introducedand for a simple perfect gas the three possible equations of state in the energyrepresentation are identified. Two of these equations are well known. The thirdequation of state is introduced and described in Sect. 8.2. Gibbs–Duhem relationsin the energy and the entropy representations are worked out in Sects. 8.3 and 8.4.The fundamental equation for the ideal gas in the entropy representation and thethree equations of state for a simple ideal gas are described in Sects. 8.5 and 8.6.The same is done in the energy representation in Sects. 8.7 and 8.8. The concludingSect. 8.9, is devoted to making a relevant remark.

8.1 The Euler Equation

Combining (8.1) with (8.2) gives:

`cX

j D1

�j dnj � dG D dU � T dS C P dV CX

i

Yi dXi : (8.3)

Here G denotes the Gibbs potential – a thermodynamic potential that is put to muchuse later.

8.1.1 Chemical Potential

We are now more informatively able to identify the role that the chemical potentialplays. First, when all other extensive variables – for example, S; V; and Xi – are

2Although, in the chapter on imperfect gases – see Chap. 6 – we did describe the co-existenceof liquid–vapor phases, it was done without an explicit treatment of the internal energy, U; andthe entropy S; in the co-existent regime. Therefore, we were able to make do without having tointroduce the concept of varying n:

8.1 The Euler Equation 339

held constant, an addition of a single mole of the j th chemical constituent increasesthe internal energy of the given thermodynamic system by an amount �j : Therefore,much like the intensive variable pressure, which provides the motive force forcausing change in the extensive variable the volume, the intensive variable chemicalpotential provides the motive force for changing the extensive variable related tochemical composition. In open systems, the chemical potential is related to the rateat which a given chemical constituent is exchanged with the environment. Equally,in closed systems, the concept can be utilized for considering such phase transitionsas affect changes of physical properties between co-existing phases, etc.3

It is important to note that when more than one phase is present, chemicalpotential of any constituent is not dependent on the magnitude – that is, the size –of the corresponding phase. Rather, it is a function of the intensive variables: thetemperature, the pressure, the relative composition, and the various Yi ’s.

Having made the point that a complete description of a thermodynamic systemmay involve more than one extensive variable of the variety nj andXi ; for simplicityin the following we limit the description to a single chemical component systemthat involves only one n: Also, for further simplicity, only one additional extensivevariable, X ; is included in the analysis. That is, the analysis is limited to a systemwhere (8.3) reduces to the following:

dU D T dS � P dV C �dn � YdX : (8.5)

Thus in the manner of (8.4) we have

�@U

@S

�

V;n;XD T I �

�@U

@V

�

S;n;XD P I

�@U

@n

�

S;V;XD �I �

�@U

@X

�

S;V;n

D Y: (8.6)

Because the internal energy is extensive – meaning its magnitude scales linearlywith the size of the system – its dependence on the extensive variables, such as theentropy, the number of moles n, the volume V; and the property X , must be suchthat it constitutes a first-order homogeneous form. What this means is that if each

3Note: According to (8.3), the rate of change of the extensive function U is completely describedin terms of the rates of change of the extensive variables S; V; nj ’s and Xi ’s. Consequently, U

is a function only of these extensive quantities. Moreover, because dU is an exact differential, itspartial differentials with respect to any of the extensive variables are equal to, what we shall call,their conjugate intensive fields. For instance:

�@U

@S

�

V; nj ;Xi

D T I�

@U

@V

�

S;nj ;Xi

D �P I�

@U

@nl

�

S;nj ¤l ;Xi

D �l I etc: (8.4)

The subscripts in the above equation that include nj need to be summed over all values of j: Fornotational convenience, the sum is not displayed.

340 8 Fundamental Equation and the Equations of State

of these variables is made bigger by a factor4 equal to �; then the resultant internalenergy U must also get enlarged by the same factor �: That is

U0 D U.S

0

; V0

; n0

;X 0

/ D � U.S; V; n;X / ; (8.7)

where

S0 D �S I V

0 D �V I n0 D � nIX 0 D �X : (8.8)

Using the chain rule for partial differentiation of an equation given in parametricform,5 we have

dU

0

d�

!D

@U0

@S0

!

V;0n;

0X 0

dS

0

d�

!C

@U0

@V0

!

S;0n;

0X 0

dV

0

d�

!

C

@U0

@n0

!

S;0V;

0X 0

dn

0

d�

!C

@U0

@X 0

!

S;0V;

0n

0

dX 0

d�

!: (8.9)

If we look only at the display above, this equation looks fierce! In reality, thedifferentials with respect to �; are easy to carry out – see (8.8) – and the resulthas a much tamer look:

U D

@U0

@S0

!

V;0n;

0X 0

S C

@U0

@V0

!

S;0n;

0X 0

V

C

@U0

@n0

!

S;0V;

0X 0

n C

@U0

@X 0

!

S;0V;

0n

0

X : (8.10)

While this relationship is good for all finite values of the variable �; it is particularlytransparent for � D 1: That, according to (8.8), means

U D�

@U

@S

�

V ;n;XS C

�@U

@V

�

S ;n;XV

C�

@U

@n

�

S ;V ;Xn C

�@U

@X

�

S ;V ;n

X : (8.11)

4Usually, � is called the “scaling parameter.”5Note: Here � is the parameter.

8.1 The Euler Equation 341

Introducing the results for�

@U@S

�V;n;X ; etc., from (8.6), (8.11) yields, in the energy

representation, the Euler equation6

U D TS � PV C � n � YX

D TS � PV C G � YX : (8.12)

8.1.2 Multiple-Component Systems

It is clear that if the thermodynamic system under discussion has multiple chemicalconstituents, then much like (8.3), instead of the single n in (8.12), molecularconcentration of additional constituents would also need to be specified. This willresult in the replacement of the Gibbs potential that has a single term, that is, .� n/;

by a Gibbs potential that is a sum over all the `c different chemical constituents, thatis,

G D`cX

j D1

�j nj : (8.13)

Similarly, additional variables could also be included. As a result, the Euler equationwill take the more general form

U D TS � PV C G �X

i

Yi Xi

D TS � PV C`cX

j D1

�j nj �X

i

Yi Xi : (8.14)

This then is the “Complete Euler Equation,” or equivalently, the “CompleteFundamental Equation” – of the given thermodynamic system, in the energy rep-resentation. For a complex thermodynamic system, the knowledge of the completefundamental equation is the holy grail of thermodynamics: for it – according toGibbs – contains all the thermodynamic information (about the given system).

8.1.3 Single-Component Systems

Only a single component system will be treated in what follows in this chapter.

6Occasionally, the Euler equation is also called the “Complete Fundamental Equation.”

342 8 Fundamental Equation and the Equations of State

8.2 Equations of State

8.2.1 Callen’s Remarks

So far the only equation of state we have talked about is

PV D nRT:

In formal terms, however, there are more than one equations of state. In fact, for agiven thermodynamic system the number of equations of state – in the energy orin the entropy representation – is equal to the number of all the intensive variablesrequired for the description of thermodynamic states. In this context, Callen7 refersto the intensive variables, T; P and � as follows:

“The temperature, pressure, and the electrochemical potentials are partial deriva-tives of a function of S; V; N1; : : : ; Nr and consequently are also functions ofS; V; N1; : : : ; Nr : We thus have a set of functional relationships

T D T .S; V; N1; : : : ; Nr/IP D P.S; V; N1; : : : ; Nr/I

�j D �j .S; V; N1; : : : ; Nr/: (8.15)

Such relationships, that express intensive parameters in terms of the independentextensive parameters, are called: “equations of state.”

Callen further writes:“: : : knowledge of all the equations of state of a system is equivalent to

knowledge of the fundamental equation and consequently is thermodynamicallycomplete.”

This statement will be referred to as the Callen rule.The fact that the fundamental equation of a system is homogeneous first-order

in terms of the extensive variables8 has direct implications for the functionalform of the equations of state. It follows immediately that the equations of stateare homogeneous zero-order. That is, multiplication of each of the independentextensive parameters by a scalar � leaves the function unchanged.

T .�S; �V; �Ni/ D T .S; V; Ni/IP.�S; �V; �Ni/ D P.S; V; Ni/I�.�S; �V; �Ni/ D �.S; V; Ni/: (8.16)

7op. cit.8For instance, multiply S; V; nj and Xi each by � and U changes to � U .

8.2 Equations of State 343

It, therefore, follows that the temperature of a composite system, composed of twomacroscopically sized subsystems, is equal to the temperature of either subsystem.

We shall refer to (8.16) as Callen’s scaling principle.

8.2.2 The Energy Representation

Consider a system with internal energy9

U D U.S; V; n;X /:

For such a system, in principle, up to four equations of state may be constructed.A formal display of these equations (compare (8.6)) could be as follows:

T .S; V; n;X / D�

@U

@S

�

V;n;XI �P.S; V; n;X / D

�@U

@V

�

S;n;XI

�.S; V; n;X / D�

@U

@n

�

S;V;XI �Y.S; V; n;X / D

�@U

@X

�

S;V;n

: (8.17)

Clearly, for a general system, involving more than one chemical potential �j andextensive variable Y there would be appropriate additional equations of state.

8.2.3 The Entropy Representation10

Equation (8.12) can readily be re-arranged,

S D�

1

T

�U C

�P

T

�V �

��

T

�n C

�YT

�X : (8.18)

As before, the equations of state are found as derivatives with respect to theextensive variables. That is

�1

T

�D�

1

T .U; V; n;X /

D

�@S

@U

�

V;n;XI

�P

T

�D�

P.U; V; n;X /

T .U; V; n;X /

D

�@S

@V

�

U;n;XI

9Note that the phrase “energy representation” implies that the derivatives being considered hereare those of the internal energy U:10Note that the phrase “entropy representation” implies that the derivatives being considered hereare those of the system entropy S.

344 8 Fundamental Equation and the Equations of State

��

T

�D�

�.U; V; n;X /

T .U; V; n;X /

D �

�@S

@n

�

U;V;XI

�YT

�D�Y.U; V; n;X /

T .U; V; n;X /

D

�@S

@X

�

U;V;n

: (8.19)

8.2.4 Known Equations of State: Two Equations for Ideal Gas

Of all thermodynamic systems, the ideal gases are the easiest to analyze. Indeed, fora simple ideal gas, with f degrees of freedom and no additional extensive variableX ; we already know11 two of the possible three12 equations of state in the entropyrepresentation. That is:

�1

T

�D f N kB

2UD f nNAkB

2 UD f nR

2UI

�P

T

�D N kB

VD nN AkB

VD nR

V: (8.20)

Comparison with (8.19) indicates that these equations of state have been expressedin the entropy representation.

8.2.5 Where is the Third Equation of State?

Clearly, the missing, third equation of state in the entropy representation has to be –see (8.19) – of the form

��

T

�D�

�.U; n; V /

T .U; n; V /

D �

�@S

@n

�

U; V

: (8.21)

Because we have not yet worked out the functional details of the entropy, at thisjuncture it is not entirely clear how the above calculation is to be carried out.Therefore, to pursue the matter further, we need to take a different tack.

11See, e.g., the chapter on the Ideal Gas.12The possible three equations of state in the entropy representation are specified in the first threeequations in (8.19).

8.4 Gibbs–Duhem Relation: Entropy Representation 345

8.3 Gibbs–Duhem Relation: Energy Representation

In order to derive the Gibbs–Duhem relation in the energy representation, itis helpful first to examine the difference between the internal energy of twoneighboring equilibrium states for the simple system being treated above, for whichthe fundamental, that is, the Euler, equation is given in (8.12). To this end, we write

dU D .T dS C SdT / � .P dV C V dP / C .�dn C nd�/ � .YdX CXdY/: (8.22)

Next, we compare this result with what would be the corresponding statement of thefirst-second law (compare, (8.3)), namely

dU D T dS � P dV C �dn � YdX : (8.23)

Subtracting (8.23) from (8.22) yields the so-called Gibbs–Duhem relationship, fora simple one-component system, in the energy representation13

0 D SdT � V dP C n d� � XdY: (8.24)

8.4 Gibbs–Duhem Relation: Entropy Representation

Proceeding in an analogous fashion to that done above, we write first the Eulerequation in a format best suited to the entropy representation. That is

S D U

�1

T

�C V

�P

T

�� n

��

T

�C X

�YT

�: (8.25)

Next, we find its derivative.

dS D Ud

�1

T

�C�

1

T

�dU C Vd

�P

T

�C�

P

T

�dV

�nd��

T

����

T

�dn C Xd

�YT

�C�Y

T

�dX : (8.26)

Now we write the first-second law – compare (8.23) – in the form

13Clearly, for the more general system the Gibbs–Duhem equation in the energy representationwould be

0 D SdT � V dP CX

j

nj d�j �X

i

Xi dYi :

346 8 Fundamental Equation and the Equations of State

dS D�

1

T

�dU C

�P

T

�dV �

��

T

�dn C

�YT

�dX ; (8.27)

and subtract it from (8.26). The resultant relationship is the Gibbs–Duhem equationin the entropy representation14

0 D Ud

�1

T

�C Vd

�P

T

�� nd

��

T

�C Xd

�YT

�: (8.28)

8.5 Fundamental Equation for Ideal Gas

We are now in a position to take a stab at finding the missing third equation of statein the entropy representation.15 It is convenient to begin this effort by determiningthe fundamental equation. Also, for simplicity and convenience, we continue toconsider only the simple ideal gas which does not involve extensive variables ofthe form X .

Start by re-arranging (8.28) as follows:

nd��

T

�D Ud

�1

T

�C Vd

�P

T

�: (8.29)

In order to integrate (8.29), substitute the results of the first and the second equationsof state that were recorded in (8.20). That is

nd��

T

�D Ud

��n fRT

2U

�1

T

C Vd

��RV

RV

��P

T

�

D�

f

2R

Ud

��T

U

�n

T

C RV d

��P V

RT

�1

V

D�

f

2R

Ud

� n

U

�C RVd

� n

V

�

D ��

f

2R n

dU

U� .R n/

dV

VC�

f C 2

2

�Rdn: (8.30)

14For the more general system, in the entropy representation, the Gibbs–Duhem equation would be

0 D Ud

�1

T

�C Vd

�P

T

��X

j

nj d��j

T

�CX

i

Xi d

�Yi

T

�:

15For instance, look at (8.21) and note how it involves �; T; S; n; U , and V:

8.5 Fundamental Equation for Ideal Gas 347

In the top right hand line of this equation, we first multiplied by two different factors

each equal to unity, that is,�

nfRT

2U

�and

�RVRV

�. We then extracted

hf

2Ri

from the

left term and R from the right term. Finally in the second term on the right, weused the fact that P V D nRT: (Note, for monatomic ideal gas there are only threedegrees of freedom for each molecule. That is, f D 3: But for a diatomic ideal gas,at temperatures that are usually available in physics laboratories, f D 5:) Next wedivided both sides of (8.30) by n: Thus, we have found

d��

T

�D �f

2R

�dU

U

�� R

�dV

V

�C f C 2

2R

�dn

n

�: (8.31)

Integration is now easy to do and we get

�

TD �

�f

2

�R ln .U / � R ln .V / C

�f C 2

2

�R ln .n/ � C0 ; (8.32)

where C0 is a constant. Using (8.32)16, (8.25) leads to the following:

S D �n��

T

�C�

U

T

�C V

�P

T

�

D�

fRn

2

�ln .U / C R n ln .V / �

�f C 2

2

�Rn ln .n/ C n C0

C�

U

T

�C V

�P

T

�

D�

fRn

2

�ln .U / C Rn ln .V / �

�f C 2

2

�Rn ln .n/ C n C0

Cn

�f C 2

2

�R: (8.33)

(Note that the relationships�

UT

� D f R n

2and V

�PT

� D R n have again been usedhere.)

While the procedures of statistical mechanics lead to an analytical expression forthe constant C0 – namely

C0 D�

3R

2

�ln

2

4 4�m

3h2N53

A

3

5 (8.34)

16Remember, here we are considering the case where, in (8.25), X�Y

T

� D 0:

348 8 Fundamental Equation and the Equations of State

– the discipline of thermodynamics does not lend itself to doing the same. Rather,the constant C0 can be determined only if the entropy, S0; for some reference stateis known. At a certain reference state “0” let S D S0; U D U0; V D V0 and n D n0:

Then (8.33) gives

n0C0 D S0 ��

fRn0

2

�ln .U0/ � R n0 ln .V0/ C

�f C 2

2

�R n0 ln .n0/

��

f C 2

2

�Rn0: (8.35)

Multiplying the above by�

nn0

�and inserting the result for nC0 in (8.33) yields the

fundamental equation for the ideal gas – however, one that is subject to a boundarycondition – in the entropy representation, in the following convenient form:

S D�

n

n0

�S0 C Rn

��f

2

�ln

�U

U0

�C ln

�V

V0

�

��

f C 2

2

�Rn

�ln

�n

n0

�: (8.36)

8.6 Three Equations of State in Entropy Representation:Ideal Gas

Having derived the fundamental equation for an ideal gas in the entropy represen-tation, the relevant three equations of state, that were defined in (8.21), are readilyfound.

�1

T

�D�

@S

@U

�

V;n

D Rnf

2UI (8.37)

�P

T

�D�

@S

@V

�

U;n

D Rn

VI (8.38)

�

TD �

�@S

@n

�

U; V

D�

f C 2

2

�R

�1 C ln

�n

n0

���

S0

n0

�

�R

��f

2

�ln

�U

U0

�C ln

�V

V0

�: (8.39)

8.8 Three Equations of State in Energy Representation: Ideal Gas 349

Regarding Callen’s scaling principle, note that in all of the three equations of stategiven above, multiplication of extensive parameters .U; U0; V; V0; n; n0; S0/ by �

leaves the equations unchanged.

8.7 Ideal Gas: Energy Representation

8.7.1 Fundamental Equation

The fundamental equation for the ideal gas given in (8.36) is in the entropy repre-sentation. But it can readily be transformed into one in the energy representation.The algebra is straight forward and one gets

U

U0

D�

n

n0

�� f C2f

� �V

V0

��� 2f

�

exp

�2

f R

S

n� S0

n0

�: (8.40)

8.8 Three Equations of State in Energy Representation:Ideal Gas

The relevant equations were noted in (8.17). The first one is

T D�

@U

@S

�

V;n

D�

2U0

fR n

��n

n0

�� f C2f

� �V

V0

��� 2f

�

exp

�2

fR

S

n� S0

n0

�: (8.41)

The second and the third equations in energy representation are

P.S; V; n/ D ��

@U

@V

�

S;n

D�

2U0

V0 f

��n

n0

�� f C2f

� �V

V0

����

2Cff

��

exp

�2

fR

S

n�S0

n0

�I (8.42)

� D�

@U

@n

�

S;V

D�

U0

n0

���f C2

f

���

2 S

f R n

��n V0

n0 V

�� 2f

�

exp

�2

f R

S

n�S0

n0

�: (8.43)

350 8 Fundamental Equation and the Equations of State

It is interesting to check whether Callen’s rule, regarding the knowledge of all theequations of state being equivalent to the knowledge of the complete fundamentalequation itself, is correct. This is especially interesting because an equation ofstate generally involves partial differentials whose integration introduces unknownconstants.

In order to check the accuracy of Callen’s rule (in the energy representation), letus introduce the results of the three equations of state, namely (8.41)–(8.43) into therelevant version of the Euler (8.14). That is,

U D TS � P V C �n

D n

�U0

n0

��nV0

n0V

�� 2f

�

exp

�2

fR

S

n� S0

n0

�

��

2S

fRn

���

2

f

�C��

f C 2

f

���

2S

fRn

��

D n

�U0

n0

��nV0

n0 V

�� 2f

�

exp

�2

fR

S

n� S0

n0

�: (8.44)

Note the last line on the right in (8.44) follows from (8.40).

8.8.1 Exercise I

With the use of the Euler equation in the entropy representation given in (8.25) –remember the terms involving the extensive variable X are not being used here –and the relevant three equations of state given in (8.37), (8.38) and (8.39), show thatCallen’s rule also applies to the entropy representation.

8.8.2 Example I

Given the fundamental equation, S D A�.nV U /13 ; where A is a constant, determine

the three equations of state in the entropy representation.

8.8.2.1 Solution

We have1

TD�

@S

@U

�

V;n

D A

3��

nV

U 2

� 13

I

8.8 Three Equations of State in Energy Representation: Ideal Gas 351

P

TD�

@S

@V

�

U;n

D A

3��

nU

V 2

� 13

I

�

TD�

@S

@n

�

V;U

D A

3��

UV

n2

� 13

:

Callen’s scaling clearly applies because when n ! �n; U ! �U , and V ! �V;

all the equations remain unchanged. Similarly, Callen’s rule also applies because

S D U ��

1

T

�C V �

�P

T

�� n �

��

T

�

D U � A

3

�nV

U 2

� 13

C V � A

3

�nU

V 2

� 13

� n � A

3

�UV

n2

� 13

D A � .nVU/13 : (8.45)

8.8.3 Example II

Re-work Example I, but this time in the energy representation.

8.8.3.1 Solution

In the energy representation, the relevant three equations of state are the following:

T D�

@U

@S

�

V;n

D�

1

A3

��3S2

nV

�I

P D ��

@U

@V

�

S;n

D�

1

A3

��S3

nV 2

�I

� D�

@S

@n

�

V;U

D ��

1

A3

��S3

n2V

�:

Again Callen’s scaling applies because when n ! �n; S ! �S , and V ! �V; allthe equations remain unchanged.

Similarly, Callen’s rule also applies because

U D TS � PV C � n

D�

1

A3

��S3

nV

�.3 � 1 � 1/ D

�1

A3

��S3

nV

�: (8.46)

352 8 Fundamental Equation and the Equations of State

8.9 Remark

Only for special cases can thermodynamic systems be exactly solved and a completesolution of the fundamental equation obtained. Nevertheless, the foregoing studyof an extremely simple system has not been for nought. It has taught us thatthermodynamics of a system in equilibrium can be formulated in two alternativebut equivalent ways: one based on the representation of the internal energy as afunction of the entropy S and extensive variables such as the volume V , the molenumbers nj , etc., and the other, on the basis of the entropy as a function of U andthe relevant extensive variables.