Question

(Thermodynamics, Maxwell relations. Mathematics) 3 Consider the microcanonical formula for the equilibrium energy E(S, V,N) of some general system.58 One knows that the second deriva- tives of E are symmetric; at fixed N, we get the same answer whichever order we take partial derivatives with respect to S and V (a) Use this to show the Marwell relation (3.70) OS S,N (This should take two lines of calculus or less.) Generate two other similar formulae by taking other second partial derivatives of E. There are many of these relations [52] (b) Statistical mechanics check of the Maxwell relation. Úsing eqn 3.57, derive formulæ for Make sure you have T and P as functions N, y, and S and not ?? Show explicitly that the Maxwell relation eqn 3.70 is satisfied

Answer 1

At equilibrium, dE = T ds - P dv + mu dN rightarrow (1) Rightarrow partial differential E/partial differential s = T, partial differential E/partial differential v = - P Rightarrow partial differential^2 E/partial differential s partial differential v = partial differential T/partial differential v Vertical-bar _S, N and partial differential^2 E/partial differential s partial differential v = - partial differential P/partial differential S Vertical-bar _V, N Therefore partial differential T/partial differential V Vertical-bar _S, N = - partial differential P/partial differential S Vertical-bar _V, N-(2) Again from equation (1) partial differential E/partial differential N = mu and partial differential E/partial differential V = - P Rightarrow partial differential^2 E/partial differential N partial differential V = partial differential M/partial differential V and partial differential^2 E/partial differential v partial differential N = - partial differential P/partial differential N Therefore partial differential mu