# Statistical Mechanics question: calculate energy difference

## 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:
$$E_{1}-E_{2}=kT(ln(1/3))$$

How do you know the true probabilities are 0.25 and 0.75 without knowing the partition function. The particle also has a probability of being in a higher state.

The only thing you know is the ratio of the probabilities:

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

Which would make E_2 < E_1 for the single particle (opposite of what you said initially).

Edit: Dealing with N particles can be tricky since you have a lot of combinations. You can have 1 particle in E_1, 3 particles in E_2, and the N-4 particles can occupy any of the higher states. So you might want to limit yourself to a single particle, or a set number of particles.

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