- #1
TFM
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Homework Statement
Consider a monatomic ideal gas that lives in a two-dimensional universe (“flatland”), occupying an area A instead of a volume V.
(a)
By following the logic of the derivation for the three-dimensional case, show that the multiplicity of this gas can be written:
\Omega = \frac{1}{N!}\frac{A^N}{h^{2N}}\frac{\pi^N}{N!}(2MU)^N
(b)
Find an expression for the entropy of this two-dimensional gas. Express your result in terms of U, A and N.
Homework Equations
N/A
The Attempt at a Solution
I have practically finished part a, with one small exception. for the pi fraction, I have 2pi instead of pi?
Some of my working out:
The space multiplicity:
\Omega_{space} = (\frac{A}{(\Delta x)^2})^N
The multiplicity of momentum:
\Omega_{mom} = (\frac{A_{hypercircle}}{(\Delta p_x)})^{2N}
\Omega_{mom} = (\frac{A_{hypercircle} ^{2N}}{(\Delta p_x^{2N})})
The total multiplicity is the multiple:
\Omega = \Omega_{space}\Omega_{mom}
\Omega = ((\frac{A}{(\Delta x)^2})^N )(\frac{A_{hypercircle} ^{2N}}{(\Delta p_x^{2N})} )
This can be rearranged, and using Heisenberg's Uncertainty princeple, \Delta x \Delta p_x \approx h
this gives:
\Omega = \frac{A^N A_{hypercircle}}{h^{2N}}
For indistinguishable particles:
\Omega = \frac{1}{N!}\frac{A^N}{h^{2N}}A_{hypercircle}
Area of hypercircle:
\frac{2\pi^N}{(N - 1)!}{\sqrt{2mu}^{2N-1}}
and using approximations:
\Omega = \frac{1}{N!}\frac{A^N}{h^{2N}}\frac{2\pi^N}{N!}{2mu^N}
As you can see, close, just that annoying 2...
Any ideas how to get rid of it?
Thanks in advanced,
TFM
