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Homework Help: Thermodynamics perfect gas Question

  1. Jul 2, 2009 #1
    1. The problem statement, all variables and given/known data

    The partition function for a perfect gas containing N monatomic particles of mass m at a
    temperature T is
    Z =(1/N!)*VN[2(Pi)mkT/h2]3N/2
    (2a) Use this partition function to find the molar Helmholtz free energy, molar internal
    energy and molar heat capacity at constant volume of the gas.
    (2b) Show that the molar entropy of the gas is given by
    S = R[lnV/NA+(3/2)ln T +5/2+(3/2)ln(2(Pi)mk/h2)]

    and find a simple, general expression for the change in entropy as a function of the temperatures and volumes of the initial and final states.
    (2c) Show that this expression for the entropy of the gas is consistent with the expected
    change in entropy for (i) an isothermal expansion of one mole of gas from volume V1 to
    volume V2 and (ii) heating of one mole of gas from temperature T1 to temperature T2 at
    constant volume.

    2. Relevant equations

    E=-(dlnZ/dβ) with β=kT

    3. The attempt at a solution

    I got the helmholtz free energy using the F=-kTlnZ formula. I tried to get the Cv by getting E and then getting (dE/dT)v but the equation is very messy and long and I'm not sure its right. Is E the molar internal energy or do I need another formula.

    I havn't really attempted the next two parts but help for part 1 would be appreacited thanks
  2. jcsd
  3. Jul 2, 2009 #2


    User Avatar
    Science Advisor
    2017 Award

    Isn't the formula for internal energy (E):
    [tex]E = \beta^2 \frac{d(lnZ)}{d \beta}[/tex]

    otherwise, I don't think the units work out.

    Also, for these types of problems, here's what I like to do. After finding the Helmholtz free energy (F), I like to use the relation:
    [tex]S = - \left(\frac{\partial F}{\partial T}\right)_V[/tex]
    (you should be able to show this easily from the definition of Helmholtz free energy), and then get the internal energy from the relation F = E - TS.
  4. Jul 2, 2009 #3
    I think using F = E - TS and S = -(dF/dT) relation will make things easier...
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