Calculating Total Internal Energy of a System in Thermal Equilibrium

[CAF123]

Homework Statement

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 $$E_I = \frac{NΔE}{1 + exp(ΔE/k_BT)}.$$

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.

 

[CAF123]

Anyone any ideas?

 

[ehild]

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

 

[CAF123]

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


ehild

Yes, what appears wrong?

 

[TSny]

Yes, what appears wrong?

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.

 

[CAF123]

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.

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

 

[TSny]

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)

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.

 

[CAF123]

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

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

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.

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.