The Carnot Cycle — Four Steps in Order

Step 1 — Isothermal Expansion (A → B)

• Temperature: $T_{1}$ (hot reservoir), held constant
• Volume increases: V_A → V_B
• Heat absorbed from hot reservoir: $Q_{1}$ = nR$T_{1}$ ln(V_B/V_A) > 0
• Work done by gas: $W_{1}$ = $Q_{1}$ (since $\Delta U$ = 0 isothermal)
• PV curve: rectangular hyperbola at $T_{1}$

Step 2 — Adiabatic Expansion (B → C)

• No heat exchange: Q = 0
• Temperature drops from $T_{1}$ to $T_{2}$
• Volume increases: V_B → V_C
• Work done by gas at expense of internal energy: $W_{2}$ = nCᵥ($T_{1}$ − $T_{2}$)
• PV curve: steeper hyperbola

Step 3 — Isothermal Compression (C → D)

• Temperature: $T_{2}$ (cold reservoir), held constant
• Volume decreases: V_C → V_D
• Heat rejected to cold reservoir: $Q_{2}$ = nR$T_{2}$ ln(V_C/V_D) > 0 (positive, heat leaves system)
• Work done on gas: $W_{3}$ = −$Q_{2}$
• PV curve: rectangular hyperbola at $T_{2}$

Step 4 — Adiabatic Compression (D → A)

• No heat exchange: Q = 0
• Temperature rises from $T_{2}$ back to $T_{1}$
• Volume decreases: V_D → V_A
• Work done on gas restores internal energy: $W_{4}$ = nCᵥ($T_{2}$ − $T_{1}$) = −nCᵥ($T_{1}$ − $T_{2}$)
• Returns gas to initial state A

Net Result per Cycle:

• Net heat absorbed: Q_net = $Q_{1}$ − $Q_{2}$
• Net work done: W_net = Q_net = $Q_{1}$ − $Q_{2}$ = area enclosed by cycle on PV diagram
• Efficiency: η = W_net/$Q_{1}$ = 1 − $Q_{2}$/$Q_{1}$ = 1 − $T_{2}$/$T_{1}$