A monatomic gas in a 2d Universe - multiplicity

[TFM]

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

 

[TFM]

I have attached the entre workings out. any idewa how to get rid iof that pesky 2...?

TFM

 

[Ambidext]

Sorry for 2 year late reply. You're probably graduated by now. But according to Schroeder's book, for part (a),

you can simply throw away the 2 because the multiplicity is very large compared to the 2, and thus it won't really make a difference. He then apologizes below for sloppy working. But... oh well that's what I just read.