Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 8.2-1

FIRST-ORDER PHASE TRANSITIONS

Physics — Statistical Mechanics · stability-fundamental-equations

Problem

a) Show that in the region

X > 0

the function

Y = X^n

is concave for

0 < n < 1

and convex for

n < 0

n > 1

. The following four equations are asserted to be fundamental equations of physical systems.

\begin{aligned} (b)\quad & F = A\left(\frac{N^5 T}{V^3}\right)^{\frac{1}{2}} \\ (c)\quad & G = BT^{\frac{1}{2}}P^2 N \\ (d)\quad & H = \frac{CS^2 P^{\frac{1}{2}}}{N} \\ (e)\quad & U = D\left(\frac{S^3 V^4}{N^5}\right)^{\frac{1}{2}} \end{aligned}

Which of these equations violate the criteria of stability? Assume

A

B

C

, and

D

to be positive constants. Recall the ``fluting condition'' (equation 8.10).