17.8: A Molecular Partition Function Can Be Decomposed into Partition Functions for Each Degree of Freedom

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).