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Internal Energy of a Mole of Particles each with 3 Energy Levels
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[QUOTE="William Crawford, post: 6213862, member: 665076"] In general, $$Z=z^N$$ is the partition function for a system consiting of ##N## distinguishable and noninteracting particles with single particle partition function ##z## ([I]prove this from first principle, good exercise[/I]). Thus, in your case where the single particle partition function is given by $$z = 1 + 2e^{-\beta\Delta} + 2e^{-4\beta\Delta}.$$ The partition function for a mole of distinguishable and noninteracting particles is therefore $$Z = \big(1 + 2e^{-\beta\Delta} + 2e^{-4\beta\Delta}\big)^{N_A}$$ and the internal energy becomes \begin{align*}U &= -\frac{\partial}{\partial\beta}\log Z \\ &= 2N_A\Delta\frac{e^{-\beta\Delta}+4e^{-4\beta\Delta}}{1+2e^{-\beta\Delta}+2e^{-4\beta\Delta}}.\end{align*} [/QUOTE]
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Internal Energy of a Mole of Particles each with 3 Energy Levels
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