Statistical Mechanics question: calculate energy difference

trelek2

Homework Statement

In a system of N weakly interacting particles each particle can be in one of M energy states:
E_1 < E_2 < ... < E_M
At T=300K there are 3 times as many particles in E_2 as in E_1.
Calculate E_2 - E_1

Homework Equations

This is not my homework, just a tutorial question, I'm revising and I'm not sure if I'm understanding this stuff. Let me know if this is correct and if not please tell me how it should be done.

The Attempt at a Solution

I consider only particles of Energies E_1 and E_2.

Probability of particle in state E_1=
$$P(E_{1})=Z^{-1}exp(- \frac{E_{1}}{kT})=0.25$$

Probability of particle in state E_2=
$$P(E_{2})=Z^{-1}exp(- \frac{E_{2}}{kT})=0.75$$

$$Z=exp(- \frac{E_{1}+E_{2}}{kT})$$
Hence:
$$E_{1}-E_{2}=kT(ln(1/3))$$

$$\frac{P(E_2)}{P(E_1)} = exp(-\frac{E_2-E_1}{kT}) = 3$$