$1\mathrm{.}5$ moles of air undergo a Carnot cycle operating between a hot reservoir at unknown temperature and a cold reservoir at $300K.$ The pressure ranges between $0\mathrm{.}1$ bar and $\frac{4}{b}ar.$ The efficiency of the cycle is $0\mathrm{.}50.$ You may use your knowledge of air and $($:$O_2\}$ to solve for the heat capacities and $.$ Note: heat capacities can be taken as constant with temperature for this problem. Do the following:

a$.$ Draw the $Pv$ diagram for this process. Label $T_H,Q_H,T_C,Q_C,$ and show the signs for $W$ and $Q$ for each step. Label each step $($e$.$g$.,$ "adiabatic reversible compression"$)$

b$.$ Determine the temperature of the hot reservoir $T_H$

c$.$ Determine $W,Q,$ and $U$ for each step in the Carnot cycle.

d$.$ Calculate the net work and net heat for the Carnot cycle.

e$.$ Calculate $-\frac{W_{\text{n}\text{e}\text{t}}}{Q_H},\frac{|Q_H|-|Q_C|}{Q_H},$ and $1-\frac{T_C}{T_H}$ and show that they are equal to each other.