An insulated system contains two compartments $1$ and $2,$ containing N$1$ and N$2$ molecules of an ideal gas, respectively. The compartments are linked by a movable, insulating, impermeable wall. Thus the compartments can exchange volume, while the total volume VT $=$ V$1$ V$2$ is conserved.

$($a$)$ Assume that each box can be subdivided into very small cells of volume v; each cell serves as one particular location where one or more ideal gas particles can be placed. Find an expression for the density of states of the entire system in terms of v$,$ N$1,$ N$2,$ VT$,$ and x$1=$ V$1/$VT$.$ Neglect energies in your microstate counting.

$($b$)$ Find an expression for the value of x$1$ and hence V$1$ that maximizes the density of states.

$($c$)$ What happens if the volume is slightly different than its value at the density$-$of$-$ states maximum? Consider another value of V$1,$ given by V$1=0\mathrm{.}9999$V$*1,$ where V$*1$ is the value found in part $($b$).$ Determine the base$-10$ logarithm of the ratio of the number of microstates at the two volumes, log$[\backslash$Omega $(0\mathrm{.}9999$V$*1)/\backslash$Omega $($V$*1)].$ Take N$1$ and N$2$ to be $10^23.$ Given the principle of equal a priori probabilities, what does this result imply for the frequency with which the volume $0\mathrm{.}9999$V$*1$ will be seen, relative to V$*1?$

$($d$)$ Show thermodynamically that, if P$1>$ P$2,$ the approach to equilibrium involves compartment $1$ gaining and compartment $2$ losing volume. Assume that the temperatures of the two systems are the same.

Expected results, please show complete work:

$4\mathrm{.}3$

$($a$)$~~$(\frac{x_1{V}_T}{N_{1}v}{)}^{N_1}(\frac{V_T(1-x_1)}{N_{2}v}{)}^{N_2}$

Assume that for an ideal gas $\frac{V_1}{v}{N}_1$ and $\frac{V_2}{v}{N}_2$ and derive the number of

microstates.

$($b$)x^{**}=\frac{N_1}{N_1+N_2}$

It is easier to maximize $ln.$

$($c$)log(\frac{(0\mathrm{.}9999V_1^{**})}{(V_1^{**})})$~~$-4\mathrm{.}343*{10}^{14}$

$($d$)$ Use the second law, $d{S}_{T}0,$ and the fundamental equation for the entropy.