# Homework Help: Calculating average energy of a system with 3 energy levels

1. Sep 15, 2014

### thercias

1. The problem statement, all variables and given/known data
A system has three energy levels, E1=0, E2 =1 and E3 = 2.
In a certain state of the system, the probability that energy level 1 is occupied is 0.1, that energy level 2 is occupied is 0.8, and that energy level 3 is occupied is 0.1. Is this an equilibrium or a non-equilibrium state of the system? Explain why or why not. What is the average energy of the system in this state?

2. Relevant equations

3. The attempt at a solution
So for the first part, I said that it was not in equilibrium because the system is at equilibrium when the disorder is at its greatest. The disorder is at its maximum when thermal energy is dispersed evenly within the 3 energy levels, and would therefore be equally probable in each three states. Since they are not equally probable, it is not in equilibrium.
For the second part, I'm not sure if I'm doing it the right way.
I put E = E1 + E2 + E3
because of their probability, we get

E = 0.1E1 + 0.8E2 + 0.1E3, and given E = 3/2 KT we get
10E = 3/2KT1 + 8(3/2KT2) + 3/2KT3
10E = 3/2K(T1 + 8T2 + T3)
E= 3/20K(T1 + 8T2 + T3)
Avg energy = 1/3E = 1/20K(T1+8T2+T3)

I'm not sure if I'm doing this right or they want it this way, help is appreciated.

Last edited: Sep 15, 2014
2. Sep 16, 2014

### DEvens

Maybe there's more to the problem statement you didn't mention. But where did the E = 3/2 KT come from? That does not appear in the problem statement. Plus, if I'm remembering correctly, it's 1/2 KT per degree of freedom, so where does the 3 come from? Also, what are T1, T2, and T3?

Equilibrium does not necessarily mean thermal equilibrium. It means the system is quasi stable. That is, it means the probability of being in a state does not change over time.

Even for thermal equilibrium, it does not necessarily mean the probability of being in each possible state is the same. Suppose the total energy of the system is very small, say about 0.01. What does that then mean about the probability of being in state E3?

3. Sep 16, 2014

### BvU

Thercias forgot to mention some relevant equations. If he/she corrects that omission, I'm sure we can guide him/her to a deeper understanding of the matter at hand..

He/she knows more than he/she lets on, because bringing in the kT smells like he/she has been in contact with Boltzmann distributions in the context of this exercise. Right ?