- #1
jfy4
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Homework Statement
In a classical gas of hard spheres (of diameter D), the spatial distribution of the particles is no
longer uncorrelated. Roughly speaking, the presence of n particles in the system leaves only a volume [itex](V − nv_0 )[/itex] available for the (n + 1)th particle; clearly, [itex]v_0[/itex] would be proportional to [itex]D^3[/itex]. Assuming that [itex]Nv_0 \ll V[/itex], determine the dependence of [itex]\Omega(N,V,E)[/itex] on V (compare to equation (1.4.1)) and show that, as a result of this, V in the ideal-gas law (1.4.3) gets replaced by (V − b), where b is four times the actual volume occupied by the particles.
Homework Equations
(1.4.1) [itex]\Omega (N,V,E) \propto V^N[/itex]
(1.4.3) [itex]PV=NkT=nRT[/itex]
The Attempt at a Solution
Well I took
[tex]
(V-Nv_0)^N \approx V^N - N^2 v_0 V^{N-1}+...
[/tex]
Then I tried to do something similar as to the construction of the ideal gas law by trying
[tex]
\frac{P}{T}=k\frac{\partial \ln (\Omega)}{\partial V}\frac{\partial \Omega}{\partial V}
[/tex]
assuming [itex]\Omega \propto V^N - N^2 v_0 V^{N-1}[/itex] similar to the original derivation.
But to no success. So I'm stumped now. Please help.
Thanks,