A well$-$insulated container consists of two halves of equal volumes V separated by a partition. On one half, there are N moles of a monatomic ideal gas at initial temperature Ti$.$ On the other, there is a vacuum. The partition is suddenly removed, and the gas expands. Recall for a monatomic ideal gas that PV $=$ N k$\_$B T and E $=(3/2)$Nk$\_$B T$.$

Expected answer is shown below, please show your work.

$6\mathrm{.}10$

$($a$)W=0,Q=0,E=0,S=N{k}_{B}ln2,T_f=T_i,P_f=\frac{P_i}{2}.$

Use $dE=TdS-PdV$ for state functions and $E=W+Q.$

$($b$)$ Since this is a non$-$reversible process, $S>\frac{Q}{T}$

$($c$)W=-0\mathrm{.}555N{k}_B{T}_i,Q=0,E=W,S=0,T_f=2^{-\frac{2}{3}}{T}_i,P_f=2^{-\frac{5}{3}}{P}_i$