What is the Carnot efficiency formula for a heat engine?

• A. η_C = 1 − T_hot/T_cold
• B. η_C = T_hot − T_cold
• C. η_C = 1 − T_cold/T_hot
• D. η_C = (T_hot − T_cold)/T_cold

Answer: (C)

The maximum efficiency of a heat engine operating between two reservoirs depends only on the temperatures of those reservoirs and is achieved by a reversible (Carnot) cycle. In such a cycle, heat is absorbed at the hot temperature T_hot and rejected at the cold temperature T_cold. Because the process is reversible, the entropy change over the cycle is zero, giving Q_h/T_hot = Q_c/T_cold. The efficiency is η = W/Q_h = (Q_h − Q_c)/Q_h = 1 − Q_c/Q_h = 1 − T_cold/T_hot. Using absolute temperatures (Kelvin) ensures the ratio is correct. Therefore, the Carnot efficiency is η_C = 1 − T_cold/T_hot. This represents the theoretical upper bound for any engine between those two temperatures.