The Carnot cycle

The ideal reversible engine

The Carnot cycle is an ideal reversible heat engine operating between a hot reservoir at temperature $T_H$ and a cold reservoir at temperature $T_C$. It is important not because real engines exactly follow it, but because it gives the maximum possible efficiency for any engine between those two temperatures.

Temperatures must be in kelvins.

Four steps of the Carnot cycle

For an ideal gas, the Carnot cycle consists of four reversible processes.

First, reversible isothermal expansion at $T_H$. The gas absorbs heat $Q_H$ from the hot reservoir and does work.

Second, reversible adiabatic expansion. No heat is exchanged, and the gas cools from $T_H$ to $T_C$ while doing work.

Third, reversible isothermal compression at $T_C$. The gas rejects heat $Q_C$ to the cold reservoir.

Fourth, reversible adiabatic compression. No heat is exchanged, and the gas warms from $T_C$ back to $T_H$.

The cycle then repeats.

Carnot efficiency

For a reversible Carnot engine,

rac{Q_C}{Q_H}=rac{T_C}{T_H}

The efficiency is

e_C=1-rac{Q_C}{Q_H}

so

e_C=1-rac{T_C}{T_H}

This is the maximum possible efficiency for any heat engine operating between the same two reservoirs.

Why temperature difference matters

Carnot efficiency increases when $T_H$ is higher or $T_C$ is lower. This is why power plants often use very hot steam and try to reject heat to the coolest practical environment.

However, $T_C$ cannot be zero kelvin, and materials limit how high $T_H$ can be. Real engines also have irreversibilities that lower efficiency below the Carnot value.

Reversibility and entropy

During the isothermal heat absorption,

Delta S_H=rac{Q_H}{T_H}

During isothermal heat rejection,

Delta S_C=-rac{Q_C}{T_C}

For a reversible cycle, total entropy change of the universe is zero:

rac{Q_H}{T_H}-rac{Q_C}{T_C}=0

This gives

rac{Q_C}{Q_H}=rac{T_C}{T_H}

Thus Carnot efficiency is directly tied to entropy conservation in a reversible cycle.

Carnot theorem

Carnot's theorem states that no engine operating between two heat reservoirs can be more efficient than a reversible Carnot engine operating between those same reservoirs. Also, all reversible engines between the same two temperatures have the same efficiency.

This result is independent of the working substance. It applies to ideal gas, steam, or any other reversible engine.

Practical meaning

The Carnot cycle is an upper bound, not a practical blueprint. Real processes must happen in finite time, while reversible heat transfer requires infinitesimal temperature differences and therefore idealized slowness. Real engines need power output, so they must operate irreversibly to some degree.

The big idea

The Carnot cycle sets the theoretical maximum efficiency for heat engines. Its efficiency depends only on the hot and cold reservoir temperatures: $e_C=1-T_C/T_H$. It reveals that the second law imposes unavoidable limits on converting heat into work.