Question

Problem 14.7 Callen's horrible example of a partial derivative In his classic book on thermodynamics, Herbert Callen used the following derivative an example. as G,N However, he did not complete the derivation. Your task is to reduce this partial derivative to contain only the standard partial derivatives a, kT, cp, and cy, and, of course, any of the first derivatives, T, S, P, V, /i, This derivation will require all the techniques that you have learned for transform or N ing partial derivatives. Have fun!

Answer 1

Gibbs free energy is given by: G UPV TS Differentiating with respect to P, keeping all other variables constant, we get дG au av + P V- T ӘР as дт ӘР ӘР ӘР ӘР aG V, we get As aP au эт av V T ӘР as S ӘР P = ӘР ӘР Dividing whole equation by V, s OT у дР у дР 1 aU P av T as 1 = 1 + v ӘР v aP Using definition of isothermal compressibility, 1 Ov Кт v дР s aT 1 aU T aS Ркт v дР у дР v aP II

av as Using Maxwell relation: aT aP s aT 0 РКт +у ат у дР т дV 1 aU V v aP дт 0 OP av au PVKTT ат ӘР So, we can write av Әт au S ӘР PVКТ — Т дт This gives: 1 дт av au PVKT TT Sap әт Il