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## Homework Statement:

- A particle has three energy levels, ##E = 0##, ##\Delta##, and ##4\Delta##, where ##\Delta## is a positive constant. The lowest energy level is nondegenerate, whereas the other two are both doubly degenerate. Find the internal energy ##U## of a system of consisting of a mole of such particles.

## Relevant Equations:

- The canonical partition function is ##Z=\sum\limits_i g(E_i) e^{-\beta E_i}##, and the internal energy is related by ##U=-\frac{\partial}{\partial\beta} \ln Z##.

Hello, I'm doing some refreshers before going back to school. Stat mech is my shakiest and I'd appreciate some help on this problem.

I know that for a single particle, the partition function will be $$Z = 1 + 2e^{-\beta\Delta} + 2e^{-4\beta\Delta}$$ and so its internal energy is $$\frac{1}{Z} \left( 2\Delta e^{-\beta\Delta} + 8\Delta e^{-4\beta\Delta} \right).$$ My only concern is how does this extend to a mole of particles? Since each particle can be in one of 5 energy states, this means that a mole of particles will have ##5^{N_A}## energy states, with many degeneracies. In this case, is there a more explicit way to express the partition function than the general expression above?

I feel like I'm overthinking things, and it just comes down to something like tacking on an ##N_A## somewhere. If so, I'd appreciate any rough explanation why. Thank you.

I know that for a single particle, the partition function will be $$Z = 1 + 2e^{-\beta\Delta} + 2e^{-4\beta\Delta}$$ and so its internal energy is $$\frac{1}{Z} \left( 2\Delta e^{-\beta\Delta} + 8\Delta e^{-4\beta\Delta} \right).$$ My only concern is how does this extend to a mole of particles? Since each particle can be in one of 5 energy states, this means that a mole of particles will have ##5^{N_A}## energy states, with many degeneracies. In this case, is there a more explicit way to express the partition function than the general expression above?

I feel like I'm overthinking things, and it just comes down to something like tacking on an ##N_A## somewhere. If so, I'd appreciate any rough explanation why. Thank you.