# Statistical Mechanics: Ideal Gas

1. Apr 11, 2009

### splitringtail

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

$$\Omega(E)=\int_{H<E}d\overline{p}d\overline{q}$$

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

3. 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

$$H = \stackrel{N_{a}}{\sum}} \frac{p^{2}_{i}}{2 m_{a}} + \stackrel{N_{b}}{\sum}} \frac{p^{2}_{j}}{2 m_{b}}$$

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

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

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

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

Last edited: Apr 11, 2009
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