Carnot Engine

Definition: Carnot Engine

The Carnot engine is a heat engine based on an ideal gas which undergoes the following Carnot cycle:

1. The gas expands isothermally at temperature $T_{\text{H}}$, absorbing heat $Q_{\text{H}}$.
2. The gas expands adiabatically until its temperature drops to $T_{\text{C}}$.
3. The gas is compressed isothermally at temperature $T_{\text{C}}$, releasing heat $|Q_{\text{C}}|$.
4. The gas is compressed adiabatically back to temperature $T_{\text{H}}$.

Theorem: Thermal Efficiency of the Carnot Engine

No heat engine can have a higher thermal efficiency than that of a Carnot engine, which is given by the temperature $T_{\text{H}}$ of the hot reservoir and the temperature $T_{\text{C}}$ of the cold reservoir as follows:

$${\eta }_{\text{Carnot}}=\frac{T_{\text{H}}-T_{\text{C}}}{T_{\text{H}}}$$

PROOF

TODO