Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 8.2-2

FIRST-ORDER PHASE TRANSITIONS

Physics — Statistical Mechanics · helmholtz-second-derivative-identity

Problem

Prove that

\left(\frac{\partial^2 F}{\partial V^2}\right)_T = \frac{\dfrac{\partial^2 U}{\partial S^2}\dfrac{\partial^2 U}{\partial V^2} - \left(\dfrac{\partial^2 U}{\partial S\,\partial V}\right)^2}{\dfrac{\partial^2 U}{\partial S^2}}

Hint: Note that

(\partial^2 F/\partial V^2)_T = -(\partial P/\partial V)_T

, and consider

P

formally to be a function of

S

and

V