# Atomic Partition Functions¶

## Boltzmann Probability¶

Boltzmann came up with two big equations, where $$W=$$ number of microstates and $$R=$$ gas constant

$\begin{gather} S=k_B\ln W\\ R=k_BN_A \end{gather}$

## Partition Function¶

Fundamental postulate of statistical thermodynamics

The Expectation value $$\big\langle x\big\rangle$$ is an average over possible values

Is an effective measure of the “accessible number of energy states”

Where:

• $$N=$$ Number of molecules
• $$V=$$ Volume
• $$T=$$ Temperature
• $$E_j=$$ Energy of the microstate
$\begin{matrix} Q(N,V,T)=\sum\limits_j^{\text{states}}e^{-\beta E_j(N,V)}:& \hskip{1cm}\beta=\frac{1}{k_BT}\\ or\\ Q(N,V,T)=\sum\limits_j^{\text{states}}\exp\bigg(\frac{-E_j(N,V)}{k_BT}\bigg) \end{matrix}$

Note

This is the theoretical backing of the FPD

So we can calculate the expectation value of energy fro the partition function

Where:

• $$\big\langle E\big\rangle=$$ Expectation energy of the system
$\begin{matrix} \big\langle E\big\rangle=\sum\limits_j\frac{E_j(N,V)e^{-\beta E_j(N,V)}}{Q(N,V,\beta)}:& \hskip{1cm}\beta=\frac{1}{k_BT}\\ \hskip{1.7cm}or\\ \big\langle E\big\rangle=\sum\limits_j\frac{E_j(N,V)\exp\bigg(\frac{-E_j}{k_BT}\bigg)}{Q(N,V,T)} \end{matrix}$

### Ensemble Partition Function¶

The canonical partition function is based on the sum of molecular partition functions $$q(V,T)$$

\begin{align} Q(N,V,T)&=\sum\limits_i\exp\bigg(\frac{-[\sum_j\epsilon_j(V)]_i}{k_BT}\bigg)\\ &=\frac{[q(V,T)]^N}{N!} \end{align}

### Molecular Partition Function¶

Info

\begin{align} q(V,T)&=\sum\limits_i\exp\bigg(\frac{-(\epsilon_{trans}+\epsilon_{rot}+\epsilon_{vib}+\epsilon_{elec})_i}{k_BT}\bigg)\\ &=\sum\limits_i\exp\bigg(\frac{-\epsilon_{trans,i}}{k_BT}\bigg) \sum\limits_j\exp\bigg(\frac{-\epsilon_{rot,j}}{k_BT}\bigg) \sum\limits_k\exp\bigg(\frac{-\epsilon_{vib,k}}{k_BT}\bigg) \sum\limits_l\exp\bigg(\frac{-\epsilon_{elec,l}}{k_BT}\bigg) \end{align}
To avoid counting degenerate states, we can use the energy levels, instead of the energy states and multiply by the number of degenerate states. Given the degeneracy of rotational states $$\mathrm{g}_J=2J+1$$
\begin{align} q_{rot}(V,T)&=\sum\limits_{j,states}e^{-\beta\epsilon_j}=e^{-E_{j=0}/k_BT}+e^{-E_{j=1}/k_BT}+e^{-E_{j=1}/k_BT}+e^{-E_{j=1}/k_BT}+...\\ q_{rot}(V,T)&=\sum\limits_{j,levels}\mathrm{g}_je^{-\beta\epsilon_j}=1e^{-E_{j=0}/k_BT}+3e^{-E_{j=1}/k_BT}+... \end{align}