Equation of state of a hard sphere gas

In summary, the conversation is about trying to solve a problem from Statistical Mechanics by R.K. Pathria 2nd edition. The problem involves determining the dependence of \Omega(N, V, E) on V for a classical gas of hard spheres. The solution involves considering the volume (V-nv_0) available for the (n+1)th particle, where v_0 is proportional to \sigma^3. The factor of 4 is obtained by considering two hard spheres in close contact, which occupy a space twice the volume of a sphere of diameter \sigma and exclude a volume of a sphere of diameter 2\sigma.
  • #1
apj
1
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Homework Statement


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.
 
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  • #2
Hey can you please explain more elaborately how you got the factor of 4.
 

1. What is the equation of state for a hard sphere gas?

The equation of state for a hard sphere gas is PV = NkT, where P is the pressure, V is the volume, N is the number of particles, k is the Boltzmann constant, and T is the temperature. This equation relates the macroscopic properties of the gas to the microscopic properties of the particles.

2. How is the equation of state derived for a hard sphere gas?

The equation of state is derived using statistical mechanics and the ideal gas law. It takes into account the volume occupied by the particles and the interactions between them. The hard sphere model assumes that the particles do not have any volume and only interact when they collide.

3. Can the equation of state be applied to all gases?

No, the equation of state for a hard sphere gas is only applicable to gases composed of hard, non-interacting particles. It does not take into account the effects of intermolecular forces, which are present in most real gases.

4. How does temperature affect the equation of state for a hard sphere gas?

As the temperature of the gas increases, the average speed of the particles also increases. This leads to a higher pressure, according to the equation of state. Conversely, as the temperature decreases, the pressure decreases as well.

5. What are the limitations of the equation of state for a hard sphere gas?

The hard sphere model is a simplified representation of real gases and does not take into account the effects of intermolecular forces, particle size, and other factors. It also assumes that the particles do not have any volume, which is not true for most gases. Therefore, the equation of state is not accurate for real gases and is only applicable in certain idealized situations.

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