Problems and Solutions on Thermodynamics and Statistical Mechanics

Thermodynamics 89

(2) (a) The convexity principle of free energy says that free energy is a

concave function of T while it is a convex function of M, and if

exists then (:;)T=. -

• (Z>,

(b) Supposing F(M) = X [(f)4- (f)’], we have

2X 6M2

(s)T=&-l) *

(s)T = l/XT < 0 ,

M a2F

When I - I < 8, (s) < 0, i.e., F is not convex.

c1 T

(c) If the convexity principle is untenable, for example if

that is, (g) < 0, then the entropy of the equilibrium state is a mini-

mum and the equilibrium state will be unstable.

T

1090

A certain system is found to have a Gibbs free energy given by

G(p, 7’) = RT In -

[(&I

where a and R are constants. Find the specific heat at constant pressure,

Solution:

The entropy is given by

5

S=-(g) =iR-Rln [ ~ (RG5j2] ’

P

The specific heat a.t constant pressure is