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Homework Help: A monatomic gas in a 2d Universe - multiplicity

  1. Feb 2, 2009 #1

    TFM

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    1. The problem statement, all variables and given/known data

    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:

    [tex] \Omega = \frac{1}{N!}\frac{A^N}{h^{2N}}\frac{\pi^N}{N!}(2MU)^N [/tex]

    (b)

    Find an expression for the entropy of this two-dimensional gas. Express your result in terms of U, A and N.

    2. Relevant equations

    N/A

    3. 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:

    [tex] \Omega_{space} = (\frac{A}{(\Delta x)^2})^N [/tex]

    The multiplicity of momentum:

    [tex] \Omega_{mom} = (\frac{A_{hypercircle}}{(\Delta p_x)})^{2N} [/tex]

    [tex] \Omega_{mom} = (\frac{A_{hypercircle} ^{2N}}{(\Delta p_x^{2N})}) [/tex]

    The total multiplicity is the multiple:

    [tex] \Omega = \Omega_{space}\Omega_{mom} [/tex]

    [tex] \Omega = ((\frac{A}{(\Delta x)^2})^N )(\frac{A_{hypercircle} ^{2N}}{(\Delta p_x^{2N})} ) [/tex]

    This can be rearranged, and using Heisenberg's Uncertainty princeple, [tex] \Delta x \Delta p_x \approx h [/tex]

    this gives:
    [tex] \Omega = \frac{A^N A_{hypercircle}}{h^{2N}} [/tex]

    For indistinguishable particles:

    [tex] \Omega = \frac{1}{N!}\frac{A^N}{h^{2N}}A_{hypercircle} [/tex]

    Area of hypercircle:

    [tex] \frac{2\pi^N}{(N - 1)!}{\sqrt{2mu}^{2N-1}} [/tex]

    and using approximations:

    [tex] \Omega = \frac{1}{N!}\frac{A^N}{h^{2N}}\frac{2\pi^N}{N!}{2mu^N} [/tex]

    As you can see, close, just that annoying 2...

    Any ideas how to get rid of it???

    Thanks in advanced,

    TFM
     
  2. jcsd
  3. Feb 4, 2009 #2

    TFM

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

    TFM
     

    Attached Files:

  4. Oct 20, 2011 #3
    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.
     
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