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Molecules of an ideal gas have internal energy levels that are equidistant, $E_n = n\epsilon$, where n = 0, 1, … and $\epsilon$ is the level spacing. The degeneracy of *n*th level is *n* + 1. Find the contribution of these internal states to the energy of the gas of *N* molecules at temperature *T*.

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Answer

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**Solution 1**

For a non-interacting ideal gas,

(1)where $\zeta$ is the single-molecule partition function

(2)This partition function can be evaluated as follows $(x\equiv\beta\epsilon)$:

(3)Hence, the sought contribution to the energy is

(4)**Solution 2**

Alternatively, one can reproduce this result as follows. One can imagine that every molecule has two independent internal degrees of freedom of harmonic oscilator type, with energy spacing $\epsilon$ each. It is easy to see that this model gives the same spectrum and degeneracies if the energy is counted from the ground state. With this convention, the average energy of a single harmonic oscillator is $\epsilon n_B(\epsilon)$, where $n_B(\epsilon)$ is the Bose-Einstein occupation number. Therefore, for the entire gas we get $E = 2N\epsilon n_B(\epsilon)$, in agreement with the first derivation.

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