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

[fluidistic]

Homework Statement

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.

Homework Equations

$C^N=h^{3N}N!$ 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: $\mu (T,V,N)=\left ( \frac{\partial A}{\partial N} \right ) _{(T,V)}$

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.

 

[mark726]

Hello there! It looks like you're on the right track with your calculations. However, I believe the issue lies in your use of Stirling's approximation. This approximation is often used when dealing with large values of N, but in this case, N is not necessarily large since it represents the number of particles in the gas. Therefore, using Stirling's approximation may not be appropriate here.

Instead, I would suggest looking at the expression for the chemical potential in terms of the partition function: $\mu (T,V,N)=kT\ln \left ( \frac{Z_N(T,V)}{Z_{N-1}(T,V)} \right )$. You have already calculated the partition function, so you can use that to obtain an expression for the chemical potential in terms of V, T, and N. Then, you can plot this expression as a function of T to see how it varies with temperature.

I hope this helps and good luck with your calculations!