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Sackur-Tetrode problem

  1. Feb 1, 2009 #1
    1. The problem statement, all variables and given/known data

    Use the Sackur-Tetrode formula to verify that the average kinetic energy of an ideall gas is [tex]\frac{3}{2}k_B T[/tex].


    2. Relevant equations

    Sackur-Tetrode:

    [tex]
    S_{tot}(E_A) = k_B[N_A(\frac{3}{2}ln \ E_A + ln \ V_A) + N_B(\frac{3}{2}ln(E_{tot} - E_A) + ln \ V_B)] + const.
    [/tex]

    3. The attempt at a solution

    The average value is the most probable value, because of gaussian distribution. Derivation gives:

    [tex]
    0 \ = \ \frac{dS_{tot}}{dE_A} \ = \ \frac{3}{2}k_B(\frac{N_A}{E_A} - \frac{N_B}{E_B})
    [/tex]

    Am I on the right track? What can I do next? Simply set
    [tex]E_A = E_B = \frac{3}{2}k_BT[/tex]
    and
    [tex]N_A = N_B[/tex]?
     
    Last edited: Feb 1, 2009
  2. jcsd
  3. Jul 8, 2009 #2
    this is probably too late a response for you, but I'll post it for future readers:

    The Sackur-Tetrode equation is:
    [tex]
    S = Nk (ln \left(\frac{4V^{2/3} \pi mU}{3N^{5/3}h^2}\right) +5/2)
    [/tex]
    Just take the derivative with respect to U:
    [tex]
    \frac{\partial S}{\partial U} = \frac{1}{T} = \frac{3Nk}{2U}
    [/tex]
    Rearrange and it gives:
    [tex]
    U = \frac{3}{2} NkT
    [/tex]
     
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