Partition function, particular energy for microstate

[pieterb]

Homework Statement

The problem is simple: two compartments, allowed to exchange heat with environment (canonical ensemble) are allowed to mix. Show change in U,P and S.

Homework Equations

$$Z_{total} = \frac{1}{N!} Z_{1}^{N}$$
$$Z_{1} = e^{-\beta E_{j}}$$

The Attempt at a Solution

I know how to derive all the required thermodynamic quantities from the partition theorem. I am however stuck at assigning a particular energy to the state my system is in. I am inclined to simply use the equipartition theorem and say $$E_{j} = \frac{3}{2} k T$$. Somehow I feel that is way too simple, since that means my partition function reduces to $$Z_{total} = \frac{1}{N!} e^{-\frac{3}{2}}$$.

Am I right in doing this, or how should I proceed?

 

[mark726]

Your approach is correct. Using the equipartition theorem to assign an energy to each state is a valid approach in this scenario. The resulting partition function may seem simple, but it is a direct consequence of the system having only two energy levels (corresponding to the two compartments) and being allowed to exchange heat with the environment. This simplicity is what makes the problem solvable and allows for the calculation of the changes in U, P, and S.

If you want to verify your solution, you can also use the canonical ensemble formula for the partition function, which is Z = \sum_{i} e^{-\beta E_{i}}. In this case, since there are only two energy levels, the sum can be simplified to Z = e^{-\beta E_{1}} + e^{-\beta E_{2}}. Plugging in the energy values you assigned (E_{1} = 0 and E_{2} = \frac{3}{2} k T), you will get the same result as using the equipartition theorem.

Overall, your approach is valid and you can proceed with deriving the changes in U, P, and S from the partition function you calculated.