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Thermodynamic Cycles

Heat Engine & Carnot Cycle

Turn a temperature difference into work — and meet the ultimate limit on efficiency.
The piston (top) drives the working gas around the four-stage Carnot cycle traced on the P–V diagram (bottom). The shaded area is the net work per cycle.
Isothermal @ T_H (absorb Q_H) Isothermal @ T_C (reject Q_C) Adiabatic (no heat)
Efficiency
0%
Heat in Q_H
0
Work out W
0
Heat out Q_C
0
i About this experiment — click to learn the physics

What you're looking at

A heat engine takes in heat from a hot reservoir, converts part of it into useful work, and dumps the rest into a cold reservoir. The Carnot cycle is the idealised, reversible version that achieves the highest efficiency any engine can. The gas in the cylinder is taken around a four-stage loop, drawn on the pressure–volume diagram below.

The four strokes

  1. Isothermal expansion at T_H — touching the hot reservoir, the gas expands and absorbs heat Q_H, doing work.
  2. Adiabatic expansion — insulated, the gas keeps expanding and cools from T_H down to T_C.
  3. Isothermal compression at T_C — touching the cold reservoir, the gas is compressed and rejects heat Q_C.
  4. Adiabatic compression — insulated, the gas is compressed and warms from T_C back to T_H, closing the loop.

Work and the first law

Over one full cycle the gas returns to its starting state, so its internal energy is unchanged. By energy conservation, the net work equals the net heat:

W = Q_H − Q_C net work per cycle

On the P–V diagram, that work is exactly the enclosed area of the loop — the shaded region. A bigger temperature gap or expansion ratio encloses more area and produces more work.

Carnot efficiency

Efficiency is the fraction of the absorbed heat turned into work. For the Carnot cycle it depends only on the two reservoir temperatures:

η = W / Q_H = 1 − T_C / T_H the Carnot limit

This is the famous result of the second law: no engine working between two temperatures can beat it. You can only reach 100% efficiency with an unreachable cold reservoir at absolute zero.

Things to try

Widen the gap between T_H and T_C and watch efficiency and the enclosed work-area grow. Bring them together and the loop collapses — no temperature difference, no work. Compare a monatomic and diatomic gas: the adiabatic curves change shape, but the efficiency stays exactly 1 − T_C/T_H.