# System in Thermal Equilibrium

1. Feb 25, 2013

### CAF123

1. The problem statement, all variables and given/known data
Consider a system in thermal equilibrium consisting of N particles that have 2 energy states E1 and E2 separated by an energy ΔE.

Given that $$N_1 = \frac{N}{exp(-ΔE/k_BT)},\,\,N_2 = \frac{N exp(-ΔE/k_BT)}{1+exp(-ΔE/k_BT)}$$ show that in the case of the lowest energy state having energy = 0, that the total internal energy of the system is $$E_I = \frac{NΔE}{1 + exp(ΔE/k_BT)}.$$

3. The attempt at a solution

The first part of this question asked to show that N1 and N2 are indeed representations of the number of particles in each energy state. I think I have this, but I don't know how to prove the above. I said that most likely N2 represents the number of particles in the lowest energy state and everywhere I replaced ΔE = E1. (since E2=0)

Many thanks.

2. Feb 26, 2013

### CAF123

Anyone any ideas?

3. Feb 26, 2013

### ehild

Are you sure you copied the expressions for N1 and N2 correctly?

ehild

4. Feb 26, 2013

### CAF123

Yes, what appears wrong?

5. Feb 26, 2013

### TSny

If you add N1 + N2 you should get the total number of particles N. But you can see that your expressions won't produce that. So, you must have copied something incorrectly (easy to do). Hint:The denominators of N1 and N2 should be the same.

6. Feb 26, 2013

### CAF123

So sorry, the expression for N1 should have denominator 1+ exp(..) instead of just exp(..)

7. Feb 26, 2013

### TSny

E1 should represent the lower energy (E = 0) and E2 should represent the higher energy (E = ΔE).

The thought process for finding the total energy is the same as for the following question. If you had 7 boxes that each weighed 10 N and 5 boxes that each weighed 20 N, what would be the total weight of all the boxes? You just need to use your expressions in place of the numbers and then simplify.

8. Feb 26, 2013

### CAF123

Why is this the case? Is it just the case that it is likely that more atoms will have non zero energy?

We have N2 molecules each with energy E => total energy is N2E = NE exp(-..)/(1+ exp(-..). Multiply top/bottom by exp(+..) and I get the result.