(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I am trying to solve problem 1.4 from Statistical Mechanics by R.K. Pathria 2nd edition. This is the problem:

In a classical gas of hard spheres (of diameter [tex]\sigma[/tex]), the spatial distribution of the particles is no longer uncorrelated. Rougly speaking, the presence of [tex]n[/tex] particles in the system leaves only a volume [tex](V-nv_0)[/tex] available for the (n+1)th particle; clearly, [tex]v_0[/tex] would be proportional to [tex]\sigma^3[/tex]. Assuming that [tex]Nv_0 \ll V[/tex], determine the dependence of [tex]\Omega(N, V, E)[/tex] on V {For an ideal gas this would be [tex]\Omega \propto V^N[/tex]} and show that, as a result of this, V in the gas law (PV=nRT) gets replaced by (V-b), where b is four times the actual space occupied by the particles.

2. The attempt at a solution

I first tried:

[tex]\Omega(N, E, V) \propto (V-Nv_0)^N[/tex]

and then:

[tex]

\frac{P}{T} = \left( \frac{\partial S}{\partial V} \right)_{N,E} = \left( \frac{\partial}{\partial V} k_B \ln \Omega(N, E, V) \right)_{N,E} = k_B \frac{N}{(V-Nv_0)}

[/tex],

rearranging yields

[tex]

P(V-Nv_0) = k_B N T

[/tex]

which looks a lot like what I need to prove, however I did not prove the factor 4.

A second guess was more like a hand waving argument. Suppose two hard spheres of diameter [tex]\sigma[/tex] in close contact. Together they occupy a space twice the volume of a sphere of diameter [tex]\sigma[/tex]:

[tex]

\frac{4}{3} \pi \left( \frac{\sigma}{2} \right)^3 = \frac{1}{3} \pi \sigma^3

[/tex],

but they exclude a volume of a sphere of diameter [tex]2 \sigma[/tex]:

[tex]

\frac{4}{3} \pi \sigma^3

[/tex]

From here we find the factor 4

Third try:

Probably the best thing to do is to assume the following:

[tex]

\Omega \propto \prod_{i=0}^{N-1} (V-iv_0)

[/tex]

and then continue from thereon, but I don't know how to do this properly.

Any help would be appreciated greatly.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Equation of state of a hard sphere gas

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**