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).

Contributors and Attributions

• www.chem.iitb.ac.in/~bltembe/pdfs/ch_3.pdf