$1.$

$($a$)$ From $1$st Tds equation $(\backslash (\backslash$mathrm$\{$Tds$\}=\backslash$mathrm$\{$du$\}+\backslash$mathrm$\{$Pdv$\}\backslash )),$ derive the change of entropy $(\backslash (\backslash$mathrm$\{$s$\}2-\backslash$mathrm$\{$s$\}1\backslash ))$ in terms of temperature and specific volume for ideal gas.

$($b$)$ If cold ideal gas assumption $($cp and cv are constant$)$ is used, how can the above equation be simplified?

$($c$)$ Express the equation obtained in $($b$)$ in terms of temperature and pressure.

$($d$)$ Obtain the isentropic process relationship between pressure and volume using specific heat ratio.