# The Boltzmann distribution of uniformly spaced energy levels

• corr0105
In summary, the question asks for the fraction of particles in the ground state of a system with uniformly spaced energy levels at 3.2x10^-10 J apart, at a temperature of 300K. The Boltzmann distribution formula is used to calculate this, with an expression for the energy state of each level. A mistake in the original function led to a negative answer, but after correction, the answer remains the same.
corr0105
The question states:
A system has energy levels uniformly spaced at 3.2x10^-10 J apart. Thepopulations of the energy levels are given by the Boltzmann distribution. What fraction of particles is in the ground state at T=300K.

I know that the Boltzmann distribution is:
$$_{}p*$$j=probability that a particle is in state j
=exp(-$$_{}E$$j/KT) / $$\sum$$exp(-$$_{}E$$j/KT)

I created my own expression for the energy state Ej:
$$_{}E$$j=3.2x10^-20(j-1)
My thought process behind this was that to get the energy state of each level you have to multiply by the space between each level times 1 minus the level. In other words, E1=0*3.2x10^-20, E2=1(3.2x10^-10)...etc.

I set this pj* equal to: # particles in ground state (g) / # states (t)
Assuming that the particles in the ground state would have energy E=0 I plugged all of the given values into the equation: T=300, Ej=0, K=boltzmann's constant. Because the problem did not give the number of states my answer is in terms of t. I got an answer of:
-0.25875/(1+t)
Obviously this is wrong because you cannot have a negative number of particles.

ANY IDEA WHAT I'VE DONE WRONG?
thanks in advance for any help

Last edited:
Your Boltzmann distribution function has a couple of minus signs missing.

I'm sorry, you are correct. The post has been edited. It was a typo, but the answer I got still stands.