Derivatives of Thermodynamic Quantities

In this section we will establish some generic relations between partial derivatives of thermodynamicquantities, which follow just from the multiple-variable calculus. We will also showby the analysis ofintensive and extensive variablesthat there are three types of special partial derivatives that either(i) reduce to ratios, or (ii) identically equal to zeros, or (iii) identically equal to infinity. Finally, wewill introduce the so-called Jacobian technique, which is a mathematical tool for re-expressing partialderivatives with respect to a given set of variables in terms of some other set of variables.

Maxwell Relations

Consider the derivative (S

V

)

T. (1)

[At the moment we assume that the total number of particles, N , is either an internal observable,like in the systems with non-conserving N (photons, phonons), or kept fixed. The situation withconserving but variable N will be discussed later.] Recalling that

S = (

F

T

)

V, (2)

we have (S

V

)

T=

2F (V, T )V T

. (3)

The function F (V, T ) normally is a regular function of its two variables, and we can swap the partialderivatives:

2F (V, T )V T

=2F (V, T )

T V. (4)

Then, taking into account (F

V

)

T= P , (5)

we see that (S

V

)

T=

(P

T

)

V. (6)

This equality and analogous equalities following just from swapping variables in the mixed second-order partial derivatives are called Maxwell relations. These are general thermodynamic relationsvalid for all systems.

To obtain all the Maxwell relations, consider the following abstract mathematical situation. Somequantity A is a function of two variables, B and C (just for the sake of briefnessthe actual numberof variables can be arbitrarily large). The differential of A is written as

dA = X dB + Y dC , (7)

whereX =

(A

B

)

C, Y =

(A

C

)

B. (8)

Then we have (X

C

)

B=

2A(B,C)C B

=2A(B,C)

B C=

(Y

B

)

C. (9)

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So far, we have obtained just a direct analog of (6). But now we do the following trick. We introducethe quantity

A = AXB , (10)and note that

dA = d(AXB) = B dX + Y dC . (11)Hence, (

B

C

)

X=

2A(X, C)C X

= 2A(X, C)C X

= (

Y

X

)

C. (12)

That is the relation (7) always implies(

B

C

)

X=

(Y

X

)

C. (13)

Eq. (10) is called Legendre transformation.Let us apply Legendre transformation to the function F (V, T ). If instead of the pair (V, T ) we

want to have (S, T ), then the transformed F is nothing else then energy, since

F + TS = E , (14)

anddE = T dS P dV . (15)

And the corresponding Maxwell relation is(

T

V

)

S=

(P

S

)

V. (16)

There are two more Legendre transformations:

H = E + PV , dH = T dS + V dP (Enthalpy) , (17)

G = F + PV , dG = S dT + V dP (Gibbs Free Energy) . (18)

Problem 43. (a) Find two Maxwell relations corresponding to Eqs. (17) and (18).

(b) Derive the relation

CP =(

H

T

)

P

(19)

that establishes an important physical meaning of enthalpy as a function describing the heat absorbed/releasedby a system at fixed pressure and varying temperature: Q = H.

Variable Number of Particles

In the case of variable number of particles, we can produce all Maxwell-type relations starting fromeither of the two previously established differential forms,

d = S dT P dV Nd , (20)

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dE = T dS P dV + dN , (21)and performing all possible Legendre transformations. The starting point is not important, since thetwo forms are related to each other by two Legendre steps.

How many relations are we going to obtain? It is a good idea to count all different Maxwellrelations generated by a differential form

dA = Y1 dX1 + Y2 dX2 + + Ym dXm . (22)Legendre transformation is independently applicable to each conjugated pair (Yj , Xj). We have mdifferent conjugated pairs (Yj , Xj). The total number of differential form which we can produce byall possible Legendre transformation is equal to the total number of m-digit binary numbers, where1/0 in the j-th digit corresponds to transformed/non-transformed j-th conjugated pair. Hence, thetotal number of forms is 2m. With each form we can take any two conjugate pairs, ja and jb, and getthe relation

Yja(X1, X2, . . . , Xm)Xjb

=Yjb(X1, X2, . . . , Xm)

Xja. (23)

That is each form generates C2m m(m 1)/2 relations. The final result is that we are going to haveC2m2

m m(m 1)2m1 different relations, which in the case of m = 3 makes 24.However, among these 24 formal relations there will be a lot of special (and even trivial) ones, as

it becomes clear from the following analysis.

Special Relations

The Legendre transformation = + PV (24)

leads to the differential formd = S dT + V dP Nd . (25)

But we have established earlier that PV , and thus 0, andS dT + V dP Nd = 0 . (26)

This formula is called Gibbs-Duhem relation. If we introduce the entropy density

s = S/V , (27)

we can conveniently re-write Gibbs-Duhem relation in terms of intensive quantities P, T, , s, n as

dP = sdT + nd . (28)

[Reminder: A quantity is called intensive if it does not directly depend on the system size. Aquantity which is proportional to the system volume is called extensive. In other words, if we taketwo identical systems and combine them into one, all the intensive quantities remain the same, whileall the extensive quantities become factor two larger.]

What we see from Eq. (28) is that any three intensive thermodynamic quantities are not indepen-dent. For example, Eq. (28) explicitly states that P is always a function of T and : P = P (T, ),and, more over

P (T, )

= n , (29)

3

P (T, )T

= s . (30)

[The fact that P = P (T, ) and Eq. (29) are already known to us from the Grand canonical formalism.]Similarly, = (T, P ), and

(T, P )P

= 1/n , (31)

(T, P )T

= s/n , (32)as well as T = T (, P ) and

T (, P )

= n/s , (33)

T (, P )P

= 1/s . (34)

Trivial relations. What happens if we formally use Eq. (25) to produce Maxwell relations? We willget relations in terms of pathological expressions of the form

(A

B

)

x,y, (35)

where A is an extensive variable, while B, x, and y are intensive. By fixing x and y we actually fix allthe other intensive quantities, including B, so that the partial derivative is ill defined. Mathematically,it is convenient to treat it as infinity, in view of the fact that its inverse

(B

A

)

x,y, (36)

equals zero since B B(x, y), and thus is independent of A.Example: (

S

P

)

,T= ,

(P

S

)

,T= 0 . (37)

It is also worth noting that if x and y are intensive, while A and B are extensive, then(

A

B

)

x,y=

A

B, (38)

since the only consistent way to represent the function A(x, y,B) is A = Bf(x, y). We have alreadymet an example of such a situation. Namely,

P = (

V

)

T,=

V. (39)

Problem 44. Establish all 24 Maxwell-type relations following from the differential form (20). Among therelations found, identify pathological ones, as well as the ones which reduce to Eqs. (29)-(34).

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Jacobian Technique

Suppose we are looking for the heat capacity at constant volume (and total number of particles) of aquantum gas:

CV = T(

S

T

)

V,N. (40)

To explicitly do the partial derivative (40) we have to know an analytic expression for S as a functionof N , V , and T . [Alternatively, we might do the derivative numerically, which is however rathercumbersome.] Our results for quantum gases are obtained in the grand canonical formalism wherethe natural variables are (T, , V ), rather than (T, N, V ). In Multi-Variable Calculus, there is ageneric technique for solving problems like that based on the properties of Jacobians.

Suppose we have m functions, y1, y2, . . . , ym, of m variables, x1, x2, . . . , xm. If we introduce thematrix M as

Mij =yixj

, (41)

then, by definition, the Jacobian of the given set of functions and arguments is the determinant ofM :

(y1, y2, . . . , ym)(x1, x2, . . . , xm)

= detM . (42)

The left-hand side of (42) is just a convenient symbol for corresponding Jacobian.Jacobians feature some generic properties which we will be utilizing below. The first property

immediately follows from the definition:

(x1, x2, . . . , xm)(x1, x2, . . . , xm)

1 . (43)

The most important property is:

(y1, y2, . . . , ym)(z1, z2, . . . , zm)

(z1, z2, . . . , zm)(x1, x2, . . . , xm)

=(y1, y2, . . . , ym)(x1, x2, . . . , xm)

. (44)

One readily gets Eq. (44) from the chain rule for partial derivatives,

yixj

=m

k=1

yizk

zkxj

, (45)

and the theorem that the determinant of a product of matricesthe right-hand side of Eq. (45) isnothing else than the product of matricesis equal to the product of their determinants.

With (43) and (44), we obtain

(y1, y2, . . . , ym)(x1, x2, . . . , xm)

(x1, x2, . . . , xm)(y1, y2, . . . , ym)

= 1 , (46)

that is(x1, x2, . . . , xm)(y1, y2, . . . , ym)

=[

(y1, y2, . . . , ym)(x1, x2, . . . , xm)

]1. (47)

The above-mentioned properties of Jacobians are reminiscent of corresponding properties of fractions.It is also worth mentioning that if we interchange any two functions in the numerator, then the Ja-cobian changes its sign, the same is true for any two arguments in the denominator, both propertiesbeing obvious, since determinants change signs when any two columns/rows are exchanged.

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Why Jacobians are good for our purposes? Because of the following formula (readily seen fromthe definition):

(u, x2, . . . , xm)(x1, x2, . . . , xm)

=(

u

x1

)

x2,...,xm

. (48)

Example. Consider Eq. (40). We can write it as

CV = T(S,N)(T,N)

. (49)

Note that we do not include V in this Jacobian, since with respect to the quantity and variables ofinterest, the variable V plays a role of a fixed external parameter which does not participate in thetransformation of variables.The situation with CP would be different.

We write

(S, N)(T, N)

=(S,N)(T, )

(T, )(T,N)

=(S,N)(T, )

[(T, N)(T, )

]1=

(S, N)(T, )

/ (N

)

T,V

. (50)

By definition,(S, N)(T, )

=(

S

T

)

V,

(N

)

V,T

(

N

T

)

V,

(S

)

V,T

. (51)

It is also reasonable to take into account the Maxwell-type relation(

N

T

)

V,=

(S

)

V,T

. (52)

Finally, we get

CV = T(

S

T

)

V, T

[(N

T

)

V,

]2 / (N

)

T,V

. (53)

Problem 45. Express the heat capacity at constant pressure (and the total number of particles), CP , interms of partial derivatives of S, P , and N as functions of the three grand-canonical variables (T, , V ). Tryto maximally simplify your final answer by utilizing Maxwell relations.

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