Problems and Solutions on Thermodynamics and Statistical Mechanics

Stdiaticd Physic3 335

Notice the scaling property 4(7r) = 7-n4J(r) for any 7. From this, and
from scaling arguments (e.g. applied to the partition function) show that

U = apV + bNkT, k = Boltzmann's const.,

where the constants a and b depend on the exponent n in the pairwise

potential. Express a and b in terms of n.

(Princeton)

Solution:

The partition function of the system is

z(T, V) = - / e-PEdpdr

1

h3N

3N/2 c

Replacing T with AT, and noticing that

This can be rewritten as

The free energy

F(XT, X-3/"V) = -kTXln z(XT, X-3/nV)

kTX In X + XF(T, V).

We differentiate it with respect to A, take X = 1, and get

T (g)v - :V (g)T = -3N ($ - :) kT+ F.