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Chemistry LibreTexts

17.3: The Average Ensemble Energy

We will be restricting ourselves to the canonical ensemble (constant temperature and constant pressure). Consider a collection of \(N\) molecules. The probability of finding a molecule with energy \(E_i\) is equal to the fraction of the molecules with energy \(E_i\). In the collection of \(N\) molecules, how many molecules (\(n_i\)) have the energy \(E_i\)?. This is the directly obtained from the Boltzmann distribution

\[P_i = \dfrac{n_i}{N} = N \dfrac{e^{-E_i/k_BT}}{Q}\]

This is because the fraction of molecules \(n_i /N\) having the energy \(E_i\) is

\[\dfrac{e^{-E_i/ k_BT}}{Q} \label{BD1}\]

which is the same as the probability of finding a molecule with energy \(E_i\) in the collection. The average energy is obtaining by multiplying \(E_i\) with its probability and summing over all \(i\):

\[ \langle E \rangle = \sum_i E_i P_i \label{Mean1} \]

Equation \(\ref{Mean1}\) is the standard average over a distribution commonly found in quantum mechanics as expectation values.  The quantum mechanical version of this Equation is 

\[ \langle \psi | \hat{H} | \psi \rangle \]

where \(\Psi^2\) is the distribution function that the Hamiltonian operator (e.g., energy) is averaged over; this equation is also the starting point in the Varational method approximation.

Equation \(\ref{Mean1}\) can be explicitly solved by introducting the Boltzmann distribution (Equation \(\ref{BD1}\))

\[ \langle E \rangle =  \sum_i \dfrac{1}{Q} e^{-E_i/ k_BT} \]

The relations between Q and pressure and entropy are given by Eqns. (2.168) and (2.171) respectively.

The pressure \(p\) can also be obtained as the ensemble average of the following partial derivative

\[ \left(\dfrac{-∂E}{∂V}\right)_T \]

giving

\[ p =\sum_i - \dfrac{\partial E_i}{\partial V} P_i = \sum_i \dfrac{\partial E_i}{\partial V}  \dfrac{e^{-E_i/ k_BT}}{Q} = \dfrac{k_BT}{Q} \dfrac{\partial Q}{\partial V} \]

\[= k_BT \dfrac{\partial \ln Q}{\partial V} \]

The entropy is given by

\[S = k_B \ln (Q +  \dfrac{\langle E \rangle}{k_BT} \label{3.41}\]

It is not uncommon to find the notation changes: \(Z\) instead of \(Q\) and \(\bar{E}\) instead of \( \langle E \rangle \) .

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