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The Carnot Cycle

The Carnot cycle is the most efficient heat engine allowed by the second law, and it is built from four reversible steps. The gas expands isothermally in contact with a hot reservoir at $T_h$, drawing in heat $Q_h$; it expands further along an adiabat, doing work as it cools to $T_c$; it is compressed isothermally against a cold reservoir, rejecting heat $Q_c$; and it is compressed adiabatically back to the start, warming to $T_h$. Each step is a curve on the pressure-volume plane, and the loop they trace encloses an area equal to the net work $W$ per cycle. Because no heat leaks across a finite temperature gap, the efficiency depends only on the two reservoir temperatures: $\eta = W/Q_h = 1 - T_c/T_h$, the ceiling no real engine can beat. The scene runs the cycle with a working point circling the loop and a piston whose volume and temperature colour track the gas, while a coloured bar shows which reservoir it touches. The lower panel plots that efficiency against $T_c/T_h$ and splits the heat input into the work extracted and the heat dumped to the cold sink: $Q_h = W + Q_c$, the first law made visible.

Figure 1. The Carnot cycle. Top: the P-V diagram with the hot isotherm (red), cold isotherm (blue), and two adiabats (dashed), the enclosed net-work area shaded; a working point circles the loop and the piston-cylinder shows the gas volume and temperature with its current reservoir contact. Bottom: efficiency eta = 1 - T_c/T_h against the temperature ratio, and the heat balance Q_h = W + Q_c. Method: ideal-gas isotherms and adiabats. Source: Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Ch. 4.
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WHAT TO TRY

  • Lower the cold temperature $T_c$: the loop grows taller, the efficiency climbs, and the rejected-heat bar shrinks.
  • Raise $T_c$ toward $T_h$: the loop collapses to a sliver, the efficiency falls toward zero, almost all the heat is dumped again.
  • Watch the piston and reservoir bar: heat flows in only on the hot isotherm and out only on the cold one, never during the insulated adiabats.
  • Switch the gas to diatomic ($\gamma=7/5$): the adiabats become shallower and the loop reshapes, but the efficiency is unchanged because it depends only on the temperatures.