$[9$ points$]$ Processes where entropy remains constant $(dS=0)$ are called isentropic processes.

$($a$)[1$ point$]$ Which of the following is applicable describe to isentropic processes: reversible$/$irre$-$

versible$/$adiabatic$/$isothermal$.$

$($b$)[3$ points$]$ Assuming an ideal gas $(PV=nRT,{}_T=\frac{1}{T}$ and ${}_T=\frac{1}{P}),$ write the equations for

$S(T,V),S(P,T),S(V,P)$ in terms of $C_P,C_V$ and nR $.$

$($c$)[3$ points$]$ For an isentropic process, derive the relationship between P$-$V and T$-$V$.$ Replace $\frac{C_P}{C_V}$ by

$$ wherever it is applicable.

$($d$)[2$ points$]$ Isentropic compressibility ${}_S$ is defined by the following equation :

${}_S=-\frac{1}{V}(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{P}}{)}_S$

For an isentropic ideal gas process, $P{V}^{}=a,$ where $\text{'}$a$\text{'}$ is a constant. We learnt that isothermal

compressibility for ideal gases, ${}_T=\frac{1}{P}.$

Derive the equation for isentropic compressibility, ${}_S,$ for ideal gases, using the $P-V$ relation

for isentropic processes $(P{V}^{}=a).$ Write the ${}_S$ in terms of ${}_T$ and the specific heat ratio $.$