17.7: Partition Functions of Indistinguishable Molecules Must Avoid Over Counting States
In the previous section, the definition of the the partition function involves a sum of state formulism:
\[Q = \sum_i e^{-\beta E_i} . \label{1} \]
However, under most conditions, full knowledge of each member of an ensemble is missing and hence we have to operate with a more reduced knowledge. This is demonstrated via a simple model of two particles in a two-energy level system in Figure \(\PageIndex{1}\). Each particle (red or blue) can occupy either the \(E_1=0\) energy level or the \(E_2=\epsilon\) energy level resulting in four possible states that describe the system. The corresponding partition function for this system is then (via Equation \ref{1}):
\[Q_{\text{distinguishable}}=e^0+ e^{-\beta\epsilon} + e^{-\beta\epsilon} + e^{-2 \beta\epsilon}=q^2 \label{Q1} \]
and is just the molecular partition function (\(q\)) squared.
However, if the two particles are indistinguishable (e.g., both the same color as in Figure \(\PageIndex{2}\)) then while four different combinations can be generated like in Figure \(\PageIndex{1}\), there is no discernible way to separate the two middle states. Hence, there are effectively only three states observable for this system.
The corresponding partition function for this system (again using Equation \ref{1}) can be constructed:
\[ Q(N,V,\beta) =e^0+ e^{-\beta\epsilon} + e^{-2 \beta\epsilon} \neq q^2 \label{Q2} \]
and this is not equal to the square of the molecular partition function. If Equation \ref{Q1} were used to describe the indistinguishable particle case, then it would overestimate the number of observable states. From combinatorics, using \(q^N\) for a large \(N\)-particle system of indistinguishable particles will overestimate the number of states by a factor of \(N!\). Therefore Equation \ref{1} requires a slight modification to account for this over counting.
\[ Q(N,V,\beta) = \dfrac{\sum_i{e^{-\beta E_i}}}{N!} \label{2} \]
If we have \(N\) molecules, we can perform \(N!\) permutations that should not affect the outcome. To avoid over counting (making sure we do not count each state more than once), the partition function becomes:
\[Q(N,V,\beta) = \dfrac{q(V,\beta)^N}{N!} \nonumber \]
As you may have noticed, using Equation \ref{2} to estimate of \(Q\) for the two-indistinguishable particle discussed case above with \(N=2\) is incorrect (i.e., Equation \ref{2} is not equal to Equation \ref{Q2}). That is because the \(N!\) factor is only applicable for large \(N\).