How Do You Derive Particle Distributions Using the Boltzmann Factor?

[ferret123]

Homework Statement

A system in thermal equilibrium at temperature T consists of N particles that have two
energy states separated by an energy Δε.

If the number of particles in the two states is N1 and N2, show that:

$N{1}$ = N($\frac{1}{1+exp(-Δε/k{B}T}$)) and $N{2}$ = N($\frac{exp(-Δε/k{B}T}{1+exp(-Δε/k{B}T}$))

Homework Equations

$\frac{N{1}}{N{2}}$ = $\frac{exp(-E{1}/k{B}T}{exp(-E{2}/k{B}T}$

Δε=E1 - E2

The Attempt at a Solution

Really struggling to see where to get started with this the lectures and the lecture notes we have are not helping.

 

[TSny]

Homework Equations

$\frac{N{1}}{N{2}}$ = $\frac{exp(-E{1}/k{B}T)}{exp(-E{2}/k{B}T)}$

Δε=E1 - E2

From the way the problem is worded, I think Δε should be Δε = E₂ - E₁

See if you can show $\frac{N{2}}{N{1}}= {exp(-Δε/k_{B}T)}$

Also, what must the sum N₁+N₂ equal?

 

[ferret123]

Well N1 + N2 must equal N?

So now that I have it in terms of Δε I can rearrange for expressions for N1 and N2 then add them for N?

 

[TSny]

You have two equations for N₁ and N₂. So you should be able to solve for N₁ and N₂ separately.