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Statistical mechanics, ideal gas, Helmholtz FE and chem. pot.

  1. Sep 23, 2014 #1

    fluidistic

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    1. The problem statement, all variables and given/known data
    I am trying to solve a problem but I am confused on what's going on with an approximation.
    I have to find the pressure in function of V, T and N of an ideal gas using the partition function, then obtain the chemical potential in function of T, p and N and I must graph it in function of T. I'm having trouble with the chemical potential.


    2. Relevant equations
    [itex]C^N=h^{3N}N![/itex] because the particles that make up the gas are indistinguishable.
    Partition function: ##Z_N(T,V)=\frac{1}{C^N} \int _\Gamma dx_1dy_1dz_1...dx_Ndy_Ndz_Ndp1_xdp1_ydp1_z...dpN_xdpN_ydpN_z \exp \left ( -\beta \sum _{i=1}^N \frac{pi_x^2+pi_y^2+pi_z^2}{2m} \right )##
    Helmholtz free energy: ##A=-kT\ln (Z_N(T,V))##.
    Pressure: ##p=-\left ( \frac{\partial A}{\partial V} \right ) _{T,N}##
    Chem. potential: ##\mu (T,V,N)=\left ( \frac{\partial A}{\partial N} \right ) _{(T,V)}##

    3. The attempt at a solution
    Using the relevant equations I calculated ##A(T,V,N)=-kT\ln \left ( \frac{V}{C^N} \left ( \frac{2\pi m}{\beta} \right ) ^{3/2} \right )## which gave me the famous ##P=\frac{kNT}{V}## so there are chances that I got the Helmholtz free energy right.
    Now the problem begins for the chem. potential.
    In order to calculate mu, I want to express A in terms of N explicitely. I got ##A(T,V,N)=-kNT \{ \ln \left [ V \left ( \frac{2m \pi}{\beta} \right ) ^{3/2} \right ] -\ln (h^3) - \frac{1}{N}\ln (N!) \}##
    I must derivate this with respect to N so I guess it is convenient to use Stirling's approximation. If so, I get ##\mu (T,V,N) \approx -k T - \{ kT \ln \left [ \frac{V}{h^3N} \left ( \frac{2m \pi}{\beta} \right ) ^{3/2} \right ] -\frac{kT}{N} \}##. And here is where I'm desperate. Since N is enormous I get a non sensical result (natural logarithm of 0).
    So I don't know if I did something wrong somewhere... Maybe there was no need to use Stirling's approximation? Hmm.
    Any help is appreciated.
     
    Last edited: Sep 23, 2014
  2. jcsd
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