$2.(15$ points$)$ A nozzle is designed to expand air from a chamber in which the pressure and temperature are $800$ KPa and $\backslash (40^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ respectively to a Mach number of $2\mathrm{.}5.$ The throat area of the nozzle is $\backslash (0\mathrm{.}05\backslash$mathrm$\{$~m$\}^\{2\}\backslash ).$ Find

a$.$ The exit area of the nozzle

b$.$ The mass flow rate through the nozzle when operating at design conditions

c$.$ The back pressure for which it will be a normal shock wave on the exit plane of the nozzle

d$.$ The range of back pressures where there will be a normal shock wave in the nozzle

e$.$ The range of back pressures over which there will be oblique shocks at the exit $($overexpanded nozzle$)$

f$.$ The range of pressures over which there will be expansion waves outside the nozzle $($underexpanded nozzle$)$