Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 7.4-22

FIRST-ORDER PHASE TRANSITIONS

Physics — Statistical Mechanics · permissible-functional-form-equation-of-state

Problem

The constant-volume heat capacity of a particular simple system is

c_v = AT^3 \qquad (A = \text{constant})

In addition the equation of state is known to be of the form

(v - v_0)P = B(T)

where

B(T)

is an unspecified function of

T

. Evaluate the permissible functional form of

B(T)

. In terms of the undetermined constants appearing in your functional representation of

B(T)

, evaluate

\alpha

c_p

, and

\kappa_T

as functions of

T

and

v

. Hint: Examine the derivative

\partial^2 s/\partial T\,\partial v

. Answer:

c_p = AT^3 + (T^3/DT + E)

, where

D

and

E

are constants.