Two Level System - Finding Energy as a function of Temperature

[gysush]

Consider a two level system of N distinguishable particles. We want to find the Energy of the system as a function of the Temperature. The first energy level is E1 and the second is E2.

I computed the entropy. Now if we take a derivative with respect to Energy, we have

1/T = dS/dE where S = f(N,N1) or f(N,N2) depending on how we substitute.

Consider another problem, namely dipoles in a uniform magnetic field, then
E=M*Eo , where M = N+ - N-

Then, 1/T = dS/dE= 1/Eo*dS/dM

We can easily calculate dS/dM and rearrange stuff.

The problem I'm having is how to take dS/dE when E=N1*E1 + N2*E2.

 

[Mute]

One way to do it is to use the chain rule. For example:

$$\frac{dS}{dE} = \frac{dS}{dN_1} \frac{dN_1}{dE}$$

Note that since the total number of particles is constrained to be N, you have that

$$E = N_1 E_1 + (N-N_1)E_2~\mbox{or}~(N-N_2)E_1 + N_2E_2.$$

That is to say: You can choose either N₁ or N₂ to be your only variable. Of course, at the end, you'll have to put everything back in terms of energy at the end of the problem, so don't forget to do that if you want to solve for E(T).