Statistical Mechanics: Ideal Gas

[splitringtail]

Homework Statement

Using the microcanonical ensemble, find the entropy for the mixing of two ideal gases, but we need to compute all at once instead of separately for each gas and adding the two.

Homework Equations

[tex]\Omega(E)=\int_{H<E}d\overline{p}d\overline{q}[/tex]

[tex]S(E,V)= k_{B} (Ln[\Omega(E)])[/tex]

The Attempt at a Solution

Now we did an N particle ideal gas in class. I feel this is very similar, but I have the
Hamiltonian

[tex] H = \stackrel{N_{a}}{\sum}} \frac{p^{2}_{i}}{2 m_{a}} + \stackrel{N_{b}}{\sum}} \frac{p^{2}_{j}}{2 m_{b}} [/tex]

Well, since it is independent of the position, then before mixing

[tex]\Omega(E)=V^{N_{a}}_{a}V^{N_{b}}_{b} \int_{H < E} \frac{\stackrel{N_{a}}{\prod}d\overline{p}_{i}}{h^{3N_{a}}}\frac{\stackrel{N_{b}}{\prod}d\overline{p}_{j}}{h^{3N_{b}}}[/tex]

The remaining integral should be a [tex]3(N_{a}+N_{b})[/tex]-dimensional hypersphere w/ a radius of [tex]\sqrt{2 m_{a} m_{b} E}[/tex]

The rest should fall in place, but I wondering if it is that straightforward.

 

[Jhero]

Hello, thank you for your post. It seems like you are on the right track with your solution. The microcanonical ensemble is used to calculate the entropy of a system at a fixed energy, volume, and number of particles. In this case, we are considering the mixing of two ideal gases, which means that the total energy, volume, and number of particles of the system are fixed.

To calculate the entropy, we need to first determine the number of microstates (Ω) that correspond to the given energy of the system. This can be done by integrating over all possible momentum states for both gases, as you have done in your attempt at a solution. However, I would suggest using the product rule for integrals to make the calculation a little easier. This would give you an integral over the momentum states for gas A and an integral over the momentum states for gas B, which can then be multiplied together to get the total number of microstates for the system.

Once you have determined the number of microstates, you can use the equation S = kB ln(Ω) to calculate the entropy. Just be sure to use the correct value of kB for your units.

I hope this helps. Let me know if you have any further questions. Good luck with your calculations!