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Homework Help: Finding the equation of state and internal energy of a given gas.

  1. Jan 14, 2009 #1
    Consider a gas contained in volume V at temperature T. The gas is composed of N distinguishable particles at zero rest mass, so that the energy E and momentum p are related by E = pc. The number of single-particle energy states in the range p to p + dp is [tex] 4\pi Vp^2 dp [/tex]. Find the equation of state and the internal energy of the gas and compare with an ordinary (ideal?) gas.

    In how I understand the problem, you must integrate [tex] 4\pi Vp^2 dp [/tex] and [tex] E = pc [/tex] to get the partition function [tex] Z(E) [/tex] then use the entropy [tex] S = k_b ln Z [/tex] to get the equation of state given by

    [tex] \left(\frac{\partial S}{\partial V}\right)_{N, E} = \frac{P}{T} [/tex]

    and the internal energy by,

    [tex] \left(\frac{\partial}{\partial E}\right)_{N, V} = \frac{1}{T}. [/tex]

    Did I understand the problem correctly?
    Last edited: Jan 14, 2009
  2. jcsd
  3. Jan 14, 2009 #2


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    You mean

    [tex]\left(\frac{\partial S}{\partial E}\right)_{N, V} = \frac{1}{T}[/tex]

    for the second equation? Looks good.
  4. Jan 15, 2009 #3
    Yes you're right. Sorry for the mistake. And what I meant was you must integrate [tex] 4\pi Vp^2 dp [/tex] and use E = pc to express the integral in terms of [tex]E[/tex] instead of [tex]p[/tex]. Thank you, I hope I got it correct then. What would be the bounds of the integral?
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