Fourth Maxwell Relation

4. Fourth Maxwell Relation

By subtracting d(Ts) from both sides of Eq. (16) and using the relation g = h – Ts, we obtain the state equation

dg = v dP – s dT (24) where g represents the Gibbs function (named after Josiah Willard Gibbs, 1839–1903). Equa-

tion (24) is another form of the fundamental equation. The intensive form g (= g(T,P)) is also known as the chemical potential µ.

a. Remarks Since, dg = v(T,P) dP – s(T,P) dT,

( ∂g/∂T) P = –s(T,P) and ( ∂g/∂P) T = v(T,P) (25) so that

–ds = g TT dT + g TP dP and dv = g PT dT + g PP dP. The fourth Maxwell relation is represented by the equality ( ∂v/∂T) P = –( ∂s/∂P) T .

(26) The LHS of this expression is measurable while the RHS is not. Since s = –( ∂g/∂T) P ,

g = h – Ts = h + T ( ∂g/∂T) P or G = H + T ( ∂G/∂T) P . This relation is called the Gibbs–Helmholtz equation.

Furthermore, (

P – h/T –( ∂s/∂T) P = –h/T or( ∂(g/T)/∂(1/T)) P =h (27) Similarly from Eq. (24), ( ∂g/∂P) T = v.

∂(g/T)/∂T) 2 P =(

∂h/∂T) 2

(28) These relations are used to prove the Third law of thermodynamics and are useful in

chemical equilibrium relations. The phase change at a specified pressure (e.g., a piston containing an incompressible fluid with a weight placed on it) occurs at a fixed temperature. In that case, dP = dT =

0 and Eq. (24) implies that dg = 0, i.e., g f (for a saturated liquid) = g g (for saturated vapor at that pressure). Figure 3 illustrates the behavior of the properties h, s, and g when matter is heated from the solid to the liquid, and, finally, to the vapor phase.

Note the discontinuities regarding the entropy and Helmholtz function during the phase change, while g is continuous. At constant temperature, dg T = vdP. Therefore, the area under the v–P curve for an isotherm represents Gibbs function. The Planck function is represented by the relation

r = r(T,P) = –g/T = s – h/T, i.e.,

dr = – dg/T + g/T 2 dT = (1/T 2 )(- vT dP + h dT).

The relations for du, dh, da and dg can be easily memorized by using the phrase “Great Physicists Have Studied Under Very Articulate Teachers” (G, P, H, S, U, V,

A, T) by considering the mnemonic diagram (cf. Figure 4 ). In that figure a square is constructed by representing the four corners by the properties P, S, V, T, and by rep- resenting the property H by the space between the corners represented by P and S, the property U by the space between S and V, etc., as illustrated in Figure 4 . Diagonals are then drawn pointing away from the two bottom corners. Such a diagram is also known as a thermodynamic mnemonic diagram. If an expression for dG is desired (that is located at the middle of the line connecting the points T and P), we first form the differentials dT and dP, and then link the two with their conjugates as illustrated below dG = – S(the conjugate of T with the minus sign due to the diagonal pointing towards T) × dT + V (that is conjugate of P with the plus sign due to the diagonal pointing

away from P) × dP.