PART A $(25\%)$ :

$($a$)$ Develop a general relationship for $(del\frac{T}{d}$elP${)}_S$ in terms of measurable quantities and evaluate the expression for an ideal gas $(15\%)$

$($b$)$ Show that for an ideal gas with constant $C_p,$ an isentropic expansion from state $"1"$ to state $"2",$ yields $P{V}^{(\frac{C_P}{C_v})}=$ constant $(10\%)$

PART B $(25\%)$

A well$-$insulated, rigid vessel is divided into two compartments by a partition. The volume of each compartment is $0\mathrm{.}1m^3.$ One compartment initially contains $400$ moles of gas $A$ at $300K$ and the other is initially evacuated. The partition is then removed and the gas is allowed to equilibrate. Gas $A$ is not ideal under these circumstances but can be described by the following equation of state, $P=\frac{RT}{V-b}-\frac{a}{T{V}^2}$

With $a=42J.K.\frac{m^3}{m}o{l}^2$ and $b=3\mathrm{.}2{10}^{-5}\frac{m^3}{m}ol.$ You may use $C_V=(\frac{3}{2})R$

$($c$.).$ Calculate the final temperature