Consider a large system of $N$indistinguishable, noninteracting molecules (perhaps an ideal gas or a dilute solution). Find an expression for the Helmholtz free energy of this system, in terms of $Z_1$, the partition function for a single molecule. (Use Stirling's approximation to eliminate the $N!$.) Then use your result to find the chemical potential, again in terms of$Z_1$.

The partition function$\left(Z\right)$ for a system of $N$indistinguishable, noninteracting molecules is given by

role="math" localid="1647058886923" $Z=\frac{1}{N!}{Z}_1^N...........................\left(1\right)$

The formula to calculate the Helmholtz free energy of the system is given by

$F=-kT\ln Z.............\left(2\right)$

Substitute the value for $Z$from equation (1) into equation (2) and simplify to obtain the free energy.

role="math" localid="1647059254929" $\begin{array}{rcl}F& =& -kT\ln \left(\frac{1}{N!}{Z}_1^N\right)\\ & =& -kT\left[N\ln Z_1-\ln \left(Z!\right)\right]\\ & =& -kT\left[N\ln Z_1-N\ln N+N\right]\\ & =& -NkT\left[\ln \frac{Z_1}{N}+1\right]...................\left(3\right)\end{array}$

The formula to calculate the chemical potential of the system is given by

$\mu ={\left(\frac{\partial F}{\partial N}\right)}_{T,V}......................\left(4\right)$

Substitute the value of the free energy from equation (3) into equation (4) and simplify to obtain the chemical potential of the system.

role="math" localid="1647059391759" $\begin{array}{rcl}\mu & =& \left(\frac{\partial \left[-NkT\left[\ln \frac{Z_1}{N}+1\right]\right]}{\partial N}\right)\\ & =& -kT\left(\ln \frac{Z_1}{N}+1\right)-NkT\frac{\partial }{\partial N}\left(-\ln N\right)\\ & =& -kT\ln \frac{Z_1}{N}\end{array}$