The Carnot Cycle

The Carnot Cycle

A complete Carnot cycle consists of four reversible steps executed on a working substance (typically an ideal gas):

1. Isothermal expansion at $T_h$: the gas expands slowly while absorbing heat $Q_h$ from the hot reservoir. Temperature is held constant.
2. Adiabatic expansion: the gas continues to expand with no heat exchange. Temperature drops from $T_h$ to $T_c$.
3. Isothermal compression at $T_c$: the gas is compressed slowly while rejecting heat $Q_c$ to the cold reservoir.
4. Adiabatic compression: the gas is compressed with no heat exchange. Temperature rises from $T_c$ back to $T_h$.

Energy Relations

The net work output equals the difference between heat absorbed and heat rejected:

$W_{\text{net}} = Q_h - Q_c$

The efficiency relates these quantities:

$\eta = \frac{W_{\text{net}}}{Q_h} = 1 - \frac{T_c}{T_h}$

A fundamental result of the Carnot cycle is that the heat ratio equals the temperature ratio:

$\frac{Q_c}{Q_h} = \frac{T_c}{T_h}$

This allows us to compute the heat rejected and work output given only $Q_h$ and the two reservoir temperatures:

$W = Q_h \left(1 - \frac{T_c}{T_h}\right), \qquad Q_c = Q_h \cdot \frac{T_c}{T_h}$

Energy Conservation Check

By construction, $W + Q_c = Q_h$, confirming the First Law for the cycle (the working fluid returns to its original state after each complete cycle, so $\Delta U = 0$).