Maxwell Relations

A pure substance existing in a single phase has only two independent variables. Out of the eight quantities p,V,T,S,U,H,F, and G any one may be expressed as a function of and other two quantities.

For a pure substance undergoing an infinitesimal reversible process

dU = TdS -pdV

1.104

dH = dU +pdV + Vdp = TdS + Vdp

1.105

dF = dU - TdS - SdT = -pdV - SdT

1.106

dG = dH - TdS - SdT = Vdp - SdT

1.107

Since, U, H, F and G are thermodynamic properties and exact differentials of the type dz = Mdx + Ndy (Eq. (1.73)), then from Eq. (1.76),

Applying this to the four equations (Eqs. (1.104-1.107))

1.108

1.109

1.110

1.111

Equations (1.108-1.111) are known as Maxwell relations.

The Maxwell relations need not be remembered. It can be easily found out from the thermodynamic mnemonic diagram. Construct a square with two diagonals as shown in Fig. 1.18.

Fig. 1.18 Thermodynamic mnemonic diagram

Mark the positions at the middle of the sides as well as at the corners. Write down the variables G, P, H, S, U, V, F and T starting from the middle of left hand side and in anti clockwise direction. The relations (Table 1.8) for du, dh, dF and dG can be easily memorized by using the phrase

G reat P hysicists H ave S tudied U nder V ery F amous T eachers

(G,P,H,S,U,V,F,T) by considering this thermodynamic mnemonic diagram. Diagonals are then drawn pointing away from the two bottom corners. Now, if an expression for dG is required (that is located at the middle of the line connecting the points T and p ), we first form the differentials of dT and dp, and then link the two with their conjugates as illustrated below:

dG = -S ( conjugate of T with the minus sign due to the diagonal pointing towards T ) x dT +V (that is conjugate of P with the plus sign due to the diagonal pointing away from P) x dp

Table 1.8 Maxwell relations

	Differential	Conjugate	Maxwell Relation	Remarks
u	du = Tds - pdv	T, -p		u = u(s, v)
h	dh = Tds + pdv	T, v		h = h(s,p)
f	dƒ = -sdT - pdv	-s, -p		ƒ = ƒ(T, v)
g	dg = -sdT + vdp	-s, v		g = g(T, p)