# What is the physical significance of molecular partition function?

What is the physical significance of molecular partition function?

I reason, that the molecular partition function

\begin{align} q &= \sum_i \exp(-\beta\varepsilon_i),& \beta &= \frac{1}{k_\mathrm{B}T} \end{align}

is found in the expression for the populations of each configuration, derived from the Boltzmann distribution:

$$p_i = \frac{\exp(-\beta\varepsilon_i)}{\sum_i \exp(-\beta\varepsilon_i)}$$

The partition function $q=\sum_i\exp(-E_i/k_BT)$ in your question can be regarded as the effective number of levels accessible to the molecule at a given temperature. It also means that in the equilibrium distribution the partition functions tells us how the systems are partitioned or divided up among the different energy levels.

In determining the form of the Boltzmann distribution the total number of 'systems' is N and these are divided up as $n_1$ systems have energy $E_1$, $n_2$ have energy $E_2$ etc. subject to conditions $\sum_i n_i=N$ and $\sum_in_iE_i=E$ where E is the total energy. It is found overwhelmingly that the chance that $n_i$ states have energy $E_i$ is given by $n_i = N\exp(-E_i/k_BT)/q$

In practice the partition function acts as a normalisation term in the calculation all sorts of thermodynamic properties, for example the average (internal) energy of N molecules with quantum energy levels $E_i$ is

$$U=N\frac{\sum E_i e^{-E_i/k_BT}}{\sum e^{-E_i/k_BT} }$$

and the calculating the entropy $S=-Nk_BT\sum p_i \ln(p_i)$ where $p_i=n_i/N =\exp(-E_i/k_BT)/q$ is the probability that the ith state is occupied.

[ If the states are degenerate, with degeneracy $g_i$, then the partition function is $q=\sum_i g_i\exp(-E_i/k_BT)$.]

In the simplest terms and most convenient definitions, it represents the total amount of states that the energy can be in.

The probability expression you wrote emphasizes this. A probability is a part over a whole. So in the case of Boltzmann's distribution, it is one state over the sum of all the states.

• I downvoted this answer because $Q$ doesn't represent the total number of possible states. Jun 24, 2017 at 21:53

We first come across the canonical partition function $Q(\beta) := \sum_i e^{-\beta E_i}$ as a normalization constant for the probability that a given microstate is occupied, as given by the Boltzmann distribution. It is the Boltzmann distribution that allows for all the remarkable properties of the normalization constant $Q$.

We mention the two most important properties. For one, $Q$ is (proportional to) a moment-generating function of the distribution of microstate energies: $$Q(\beta) := \sum_i e^{-\beta E_i} = \sum_{E_i} \omega(E_i)\,e^{-\beta E_i} = \Omega\sum_{E_i}\frac{\omega(E_i)}{\Omega}\,e^{-\beta E_i} = \Omega\,\langle e^{-\beta E}\rangle, \quad \Omega := \sum_{E_i}\omega(E_i).$$ Recall that a moment-generating function is useful because it encodes all possible information about a given distribution; hence $Q$ contains all information about (the energy distribution of) our system. This extends to functions of $Q$, particularly the Helmholtz free energy $A = A(Q)$.

For another, differentiating $\ln Q$ with respect to $\beta$ will generate moments of the Boltzmann distribution. This is unique to the function whose derivative is itself, the exponential function. (We differentiate $\ln Q$ in place of $Q$ in order to get a factor of $1/Q$ to generate normalized probabilities.) This is a very useful and important technique, but I'm not sure that there's an associated physical significance.

Do note that I describe two different distributions in my answer: the distribution of microstate energies and the distribution of microstate occupancy. The canonical partition function is the moment-generating function of the former only. However, it is, in some sense, a generating function for the latter distribution, by the second property above.

It is a dimensionless quantity and provides the most convenient way for linking the microscopic property of individual molecules(such as their discrete energy level moment of inertia and dipole moment) with the macroscopic property (such as molar heat, entropy and polarisation)

From Maxwell Boltzmann distribution law, it can be written as

\begin{align*} N_i/n &= \frac{g_i}{g_o} e^{-E_i/KT} \\ Q &=n/n_o \end{align*}

The value of molecular partition function increases with increase in temperature.