How to Calculate the Total Number of Particles in Different Energy States?

[NewtonApple]

Homework Statement

A system of N particles has three possible energy levels namely; 0, E and 4E. How many particles does one expect in the second state at temperature T?

Homework Equations

It's a sample problem for our finals. Our Text book is Statistical Mechanics by Roger Bowley and Mariana Sanchez.

The Attempt at a Solution

Three Energy levels

$E_{1}=0$, $E_{2}=E$, $E_{3}=4E$

Let us first fill the $E_{1}$ state with 3 particle.

N distinguishable ways of selecting the first particle
N-1 different ways to select second particle
N-2 different ways to select third particle

So the total number of ways to place first three particles in state $E_{1}$ is

$N(N-1)(N-2)=\frac{N!}{(N-3)!}$​

Generally for $n_{1}$ particles placed in $E_{1}$ is,

$\frac{N!}{n_{1}!(N-1)!}$
​

for $E_{2}$ state,

$\frac{(N-n_{1})!}{n_{2}!(N-n_{1}n_{2})!}$​

for $E_{3}$ state,

$\frac{(N-n_{1}n_{2})!}{n_{3}!(N-n_{1}n_{2}n_{3})!}$​

Total number of particles in all three state will be

$P=\frac{N!}{n_{1}!n_{2}!n_{3}!}$​

Substituting values

$P=\frac{N!}{0!1!4!}$​

Am I on right track?

 

[DrClaude]

Am I on right track?

I don't think so. You haven't even invoked temperature in any way.

If you had only one particle, what would be the probability of finding in in state 2 when the temperature is T?

 

[NewtonApple]

ok, I try to re attempt it after going through chapter 6 of the book.