Boltzmann Factor Type Question

[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.

 

[Nickie]

As a scientist, it is important to have a strong understanding of the concepts and equations that are relevant to your research. In this case, the Boltzmann factor is an important concept in thermodynamics and statistical mechanics that describes the probability of a system being in a particular energy state at a given temperature.

To start, let's look at the equation for the Boltzmann factor:

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

This equation relates the number of particles in two different energy states (N1 and N2) to the energy difference between those states (Δε). We can rewrite this equation as:

N{1} = N{2} * \frac{exp(-Δε/k{B}T}{exp(0)}

where we have used the fact that E1 = 0 and E2 = Δε.

Now, we can substitute in the given information about the system in thermal equilibrium:

N{1} = N{2} * \frac{exp(-Δε/k{B}T}{exp(0)} = N * \frac{exp(-Δε/k{B}T}{1}

where we have used the fact that N1 + N2 = N.

Finally, we can use the fact that exp(-x) = 1/exp(x) to rewrite the equation as:

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)}

This shows the relationship between the number of particles in the two energy states and the temperature, as well as the energy difference between the states. We can see that as the temperature increases (T gets larger), the value of the Boltzmann factor decreases, meaning that there is a higher probability of particles being in the lower energy state. This is a fundamental concept in thermodynamics and statistical mechanics, and understanding it is crucial for understanding many physical systems. I hope this explanation helps with your understanding of this problem.