Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physics 333

2145
A classical gas of N point particles occupies volume V at tempera-
ture T. The particles interact pairwise, d(r;j) being the potential between
particles i and j, r,j = Ir, - rjl. Suppose this is a “hard sphere” potential

(a) Compute the constant volume specific heat as a function of tem-
V
perature and specific volume v = -.
N

PV

(b) The virial expansion for the equation of state is an expansion of

• in inverse powers of V:
  RT

L1+-+T+... A2(T).

RT V

Compute the virial coefficient Al.

(Princeton)

Solution

For the canonical distribution, the partition function is

z=-.- ’ ’ /e-”dqdp

N! h3N

=’(-> 27rm 3N/2 .Q,

N! Ph2

where qi represents the coordinates and p; the momentum of the ith parti-

cle. Defining the function f;j = exp[-Pd(r;j)] - 1 with f;, = 0 for r;j > a,

we can write