Cyclic Processes and the Carnot Cycle

Going Round in Circles Usefully

A cyclic process returns the system to its starting state, so $\Delta U = 0$ over the cycle and

$$W_{net} = Q_{in} - Q_{out}$$

An engine cannot simply convert all its heat into work, it must dump some to a cold reservoir. Efficiency is what you get over what you pay:

$$\eta = \frac{W_{net}}{Q_{in}} = 1 - \frac{Q_{out}}{Q_{in}}$$

The Carnot cycle consists of four reversible steps: isothermal expansion at $T_h$ absorbing $Q_h$, adiabatic expansion cooling to $T_c$, isothermal compression at $T_c$ rejecting $Q_c$, and adiabatic compression back. Because every step is reversible, the heats are in the ratio of the temperatures, giving

$$\eta_{Carnot} = 1 - \frac{T_c}{T_h}$$

Carnot's theorem states that no engine working between two reservoirs can beat this, and that all reversible engines between the same pair achieve exactly it, regardless of working substance. That substance-independence is what lets the ratio $Q_c/Q_h$ define the thermodynamic temperature scale, with no reference to any material at all.

The bound is severe. A power station with $T_h = 800$ K and $T_c = 300$ K is capped at $1 - 300/800 = 62.5\%$, and real plants reach perhaps 40%. Raising efficiency means raising $T_h$ or lowering $T_c$, and the cold reservoir is usually the environment.

Run in reverse, the same cycle becomes a refrigerator or heat pump, characterised by a coefficient of performance rather than an efficiency: $\text{COP}_{fridge} = T_c/(T_h - T_c)$, which can exceed 1, moving heat is cheaper than creating it.

Common pitfall: using Celsius in the Carnot formula. The ratio $T_c/T_h$ is meaningful only on an absolute scale; using Celsius for a cycle between 27 °C and 227 °C gives $1 - 27/227 = 88\%$ instead of the correct $1 - 300/500 = 40\%$.