# How do calculate the prob of being in a state in therm eq?

## Homework Statement

A pore has three configurations with the energy levels shown. The pore is in thermal equilibrium with the surroundings at temperature T . Find the probabilities ##p_1##, ##p_2##, ##p_3## of being in each level. Each level has only one microstate associated with it.

Also, state 1 corresponds to energy of 0, state 2 to energy of ##U_0## and state 3 to energy of ##2U_0##.

## Homework Equations

The boltzmann factor = ##e^{-(U_s - U_r)/kT}##

The probability of being in a state is equal to $$p_i = \frac{e^{\frac{-E_i}{kT}}}{\sum{\frac{-E_i}{kT}}}$$

where the denominator is the sum over all states.

## The Attempt at a Solution

This is for self study and not for a course and so I'm really trying to explain these solutions to myself.

In terms of an attempt, I know that

$$\frac{p_2}{p_1} = \frac{\Omega(2)}{\Omega(1)}$$

But $$\Omega(2) = \Omega(1) = 1$$ so the probability of being in any state is the same. And since the we have 3 energy states, I get that $$p_1 = p_2 = p_3 = \frac{1}{3}$$.

Is that correct?

And if so, how can I use the boltzmann factor to arrive at the same answer.