5.5: Partition Functions can be Decomposed into Partition Functions of Each Degree of Freedom
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From the previous sections, the partition function for a system of \(N\) indistinguishable and independent molecules is:
\[ Q(N,V,\beta) = \dfrac{\sum_i{e^{-\beta E_i}}}{N!} \label{ID1} \]
And the average energy of the system is:
\[ \langle E \rangle = kT^2 \left(\dfrac{\partial \ln{Q}}{\partial T}\right) \label{ID2} \]
We can combine these two equations to obtain:
\[ \begin{split} \langle E \rangle &= kT^2 \left(\dfrac{\partial \ln{Q}}{\partial T}\right)_{N,V} \\ &= NkT^2 \left(\dfrac{\partial \ln{q}}{\partial T}\right)_V \\ &= N\sum_i{\epsilon_i \dfrac{e^{-\epsilon_i/kT}}{q(V,T)}} \end{split} \label{ID3} \]
The average energy is equal to:
\[ \langle E \rangle = N \langle \epsilon \rangle \label{aveE} \]
where \(\langle \epsilon \rangle\) is the average energy of a single particle. If we compare Equation \(\ref{ID2}\) with Equation \(\ref{ID2}\), we can see:
\[ \langle \epsilon \rangle = \sum_i{\epsilon_i \dfrac{e^{-\epsilon_i/kT}}{q(V,T)}} \nonumber \]
The probability that a particle is in state \(i\), \(\pi_i\), is given by:
\[ \langle \epsilon \rangle = \dfrac{e^{-\epsilon_i/kT}}{q(V,T)} = \dfrac{e^{-\epsilon_i/kT}}{\sum_i{e^{-\epsilon_i/kT}}} \nonumber \]
The energy of a particle is a sum of the energy of each degree of freedom for that particle. In the case of a molecule, the energy is:
\[ \epsilon = \epsilon_\text{trans} + \epsilon_\text{rot} + \epsilon_\text{vib} + \epsilon_\text{elec} \nonumber \]
The molecular partition function is the product of the degree of freedom partition functions:
\[ q(V,T) = q_\text{trans} q_\text{rot} q_\text{vib} q_\text{elec} \nonumber \]
The partition function for each degree of freedom follows the same is related to the Boltzmann distribution. For example, the vibrational partition function is:
\[ q_\text{vib} = \sum_i{e^{-\epsilon_i/kT}} \nonumber \]
The average energy of each degree of freedom follows the same pattern as before. For example, the average vibrational energy is:
\[ \langle \epsilon_\text{vib} \rangle = kT^2\dfrac{\partial \ln{q_\text{vib}}}{\partial t} = -\dfrac{\partial \ln{q_\text{vib}}}{\partial \beta} \nonumber \]