Statistical Mechanics · Problem 1.4

Elements of Ensemble Theory — R. K. Pathria

Physics — Statistical Mechanics · hard-sphere-gas-excluded-volume

Problem

In a classical gas of hard spheres (of diameter

\sigma

), the spatial distribution of the particles is no longer uncorrelated. Roughly speaking, the presence of

n

particles in the system leaves only a volume

(V - nv_0)

available for the

(n+1)

th particle; clearly,

v_0

would be proportional to

\sigma^3

. Assuming that

Nv_0 \ll V

, determine the dependence of

\Omega(N, V, E)

V

[cf. eqn. (1.4.1)] and show that, as a result of this,

V

in the gas law (1.4.3) gets replaced by

(V - b)

, where

b

is four times the actual space occupied by the particles.