91Ó°ÊÓ

The classical degree of freedom is described by$E=c\left|q\right|$, where$c$is a constant.

The formula to calculate the partition function is given by

$Z=\underset{q}{∑}{e}^{-\mathrm{β}E\left(q\right)}..................\left(1\right)$

Here, $Z$is the partition function and $\mathrm{β}=\frac{1}{kT}$is the Boltzmann factor, $k$being Boltzmann constant and $T$being the absolute temperature.

Substitute $c\left|q\right|$for $E$into equation (1) and change the summation by integral sign to obtain the partition function.

$Z=\frac{1}{∆q}{∫}_{-∞}^{∞}{e}^{-\mathrm{β}c\left|q\right|}dq.................\left(2\right)$

Simplify equation (2) in step 2 to obtain an expression for the partition function.

$\begin{array}{rcl}Z& =& \frac{1}{∆q}\left[{∫}_{-∞}^0{e}^{\mathrm{β}cq}dq+{∫}_0^{∞}{e}^{-\mathrm{β}cq}dq\right]\\ & =& \frac{1}{∆q}\left[\frac{1}{\mathrm{β}c}{\left[e^{\mathrm{β}cq}\right]}_{-∞}^0-\frac{1}{\mathrm{β}c}{\left[e^{-\mathrm{β}cq}\right]}_0^{∞}\right]\\ & =& \frac{2}{∆q\mathrm{β}c}..................\left(3\right)\end{array}$

The formula to calculate the average energy is given by

$\stackrel{-}{E}=-\frac{1}{Z}\frac{∂Z}{∂\mathrm{β}}.................\left(4\right)$

Here, $\stackrel{-}{E}$is the average energy.

Substitute the value of $Z$from equation (3) into equation (4) and simplify to obtain the required average energy.

role="math" localid="1646905279668" $\begin{array}{rcl}\stackrel{-}{E}& =& -\frac{1}{\frac{2}{∆q\mathrm{β}c}}\frac{∂}{∂\mathrm{β}}\left(\frac{2}{∆q\mathrm{β}c}\right)\\ & =& -\mathrm{β}\frac{∂}{∂\mathrm{β}}\left(\frac{1}{\mathrm{β}}\right)\\ & =& \frac{1}{\mathrm{β}}\\ & =& kT\end{array}$