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5.5: Partition Functions can be Decomposed

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  • Express the partition function of a collection of N molecules \(Q\) in terms of the molecular partition function \(q\). Assuming the N molecules to be independent, the total energy \(E_{tot}\) of molecules is a sum of individual molecular energies

    \[ E_{tot} = \sum_i E_i\]

    and all possible

    \[Q = \sum _{\text{all possible energies}} e^{-E/k_BT} = \sum _i e^{-E_i/k_BT} \sum _j e^{-E_j/k_BT} \sum _k e^{-E_k/k_BT} ... \sum _i e^{-E_i/k_BT} \]

    \[ Q = q \times q \times q \times ... q^N\]

    Here \(\epsilon_i^{(1)}\), \(\epsilon_i^{(2)}\), \(\epsilon_i^{N}\) are energies of individual molecules and a sum of all energies can only come from summing over all \(\epsilon_i\). Gibbs postulated that

    \[Q = \dfrac{q^N}{N!}\]

    where the \(N!\) in the denominator is due to the indistinguishability of the tiny molecules (or other quantum particles in a collection).

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