Overview#

• A heat engine converts thermal energy into mechanical work by operating in a cycle: absorbs heat \(Q_H\) from a hot reservoir, does work \(W\), rejects heat \(Q_C\) to a cold reservoir.

• Thermal efficiency \(\varepsilon = W/Q_H = 1 - Q_C/Q_H\); always \(< 1\) (second law).

• The Carnot engine (reversible) achieves the maximum efficiency between two temperatures: \(\varepsilon_C = 1 - T_C/T_H\).

Heat engine and efficiency#

First law (one cycle; \(\Delta E_{\text{int}} = 0\)):

Thermal efficiency:

• \(\varepsilon = 1\) would require \(Q_C = 0\)—no heat rejected. The second law forbids this: heat cannot be fully converted to work in a cyclic process.

Carnot engine#

A Carnot engine uses only reversible processes. Its cycle: isothermal expansion at \(T_H\) → adiabatic expansion → isothermal compression at \(T_C\) → adiabatic compression.

Carnot relations:

• Efficiency depends only on reservoir temperatures, not on the working substance.

• Higher \(T_H\) and lower \(T_C\) → higher efficiency.

• 100% efficiency would require \(T_C = 0\) K (unattainable).

• Real engines are less efficient than Carnot engines (irreversibilities: friction, heat loss).

Entropy and engines#

For a reversible (Carnot) engine per cycle:

• Hot reservoir: \(\Delta S_H = -Q_H/T_H\) (loses heat).

• Cold reservoir: \(\Delta S_C = +Q_C/T_C\) (gains heat).

• With \(Q_C/Q_H = T_C/T_H\): \(\Delta S_{\text{total}} = -Q_H/T_H + Q_C/T_C = 0\).

For an irreversible engine: \(\Delta S_{\text{total}} > 0\) (second law).

Summary#

• \(W = Q_H - Q_C\); \(\varepsilon = 1 - Q_C/Q_H\).

• Carnot: \(\varepsilon_C = 1 - T_C/T_H\) (maximum efficiency).

• \(Q_C/Q_H = T_C/T_H\) for Carnot.

• Reversible cycle: \(\Delta S_{\text{total}} = 0\); irreversible: \(\Delta S_{\text{total}} > 0\).