Blundell Thermal Physics · Problem 16.4

Thermodynamic potentials

Physics — Statistical Mechanics · natural-variables-thermodynamic-potentials

Problem

(a) The natural variables for

U

are

S

and

V

. This means that if you know

S

and

V

, you can find

U(S,V)

. Show that this also gives you simple expressions for

T

and

p

. (b) Suppose instead that you know

V

T

and the function

U(T,V)

(i.e., you have expressed

U

in terms of variables that are not all the natural variables of

U

). Show that this leads to a (much more complicated) expression for

p

, namely

\frac{p}{T} = \int \left(\frac{\partial U}{\partial V}\right)_T \frac{\mathrm{d}T}{T^2} + f(V), \tag{16.87}

where

f(V)

is some (unknown) function of

V