Problem $9$: $(10$ pts$)$ A measurement of the heat capacity of an Ideal Gas is possible by expansion in an adiabatic process. Consider an Ideal Gas which is adiabatically and reversibly expanded from $3\mathrm{.}0$ MPa to $0\mathrm{.}10$ MPa $,$ with its volume going from $0\mathrm{.}25m^3$ to $4\mathrm{.}25m^3.$

$($a$)$ Find $=\frac{c_p}{c_v}$ and $\frac{c_v}{R}.$

$($b$)$ Find the work.