Show the four steps of a Carnot engine in a $P-V$ graph,write the equation for each step,and obtain the work done by the system. Also,derive the efficiency of a Carnot engine.

(N/A) Carnot engine operates through a cycle of four reversible processes:
$(1)$ Isothermal expansion $(A \rightarrow B)$: The gas expands at a constant temperature $T_1$. The heat absorbed is $Q_1 = \mu RT_1 \ln(V_2/V_1)$. The work done is $W_1 = Q_1 = \mu RT_1 \ln(V_2/V_1)$.
$(2)$ Adiabatic expansion $(B \rightarrow C)$: The gas expands adiabatically from $(P_2, V_2, T_1)$ to $(P_3, V_3, T_2)$. The work done is $W_2 = \frac{\mu R(T_1 - T_2)}{\gamma - 1}$.
$(3)$ Isothermal compression $(C \rightarrow D)$: The gas is compressed at a constant temperature $T_2$. The heat rejected is $Q_2 = \mu RT_2 \ln(V_3/V_4)$. The work done on the gas is $W_3 = \mu RT_2 \ln(V_3/V_4)$.
$(4)$ Adiabatic compression $(D \rightarrow A)$: The gas is compressed adiabatically from $(P_4, V_4, T_2)$ to $(P_1, V_1, T_1)$. The work done on the gas is $W_4 = \frac{\mu R(T_1 - T_2)}{\gamma - 1}$.
Total work done by the system: $W = W_1 + W_2 - W_3 - W_4 = W_1 - W_3 = \mu RT_1 \ln(V_2/V_1) - \mu RT_2 \ln(V_3/V_4)$.
Efficiency $\eta = 1 - \frac{Q_2}{Q_1} = 1 - \frac{T_2}{T_1}$.