<aside> <img src="https://em-content.zobj.net/source/microsoft-teams/337/gear_2699-fe0f.png" alt="https://em-content.zobj.net/source/microsoft-teams/337/gear_2699-fe0f.png" width="40px" /> Heat engine: devices that produce work from heat. They operate in cycles such that:

$$ \Delta E=\oint \text dE=0 $$

</aside>

Example:

• The heat absorbed from the hot reservoir at $T_H$ is $Q_H=Q_1+Q_4$
• The heat ejected to the cold reservoir at $T_C$ is $Q_C=Q_2+Q_3$
• From the 1st law we have, for cycle: $\Delta E=0=(-W)+(Q_H-Q_C) \Rightarrow W=Q_H-Q_C$
• The efficiency of the engine is defines as:

$$ \eta=\frac{\text{desired output}}{\text{required input}}=\frac{W}{Q_H}=\frac{Q_H-Q_c}{Q_H}<1 $$

• In the case where heat is extracted the efficiency is

$$ \eta_R=\frac{\text{desired output}}{\text{required input}}=\frac{Q_C}{W}=\frac{Q_C}{Q_H-Q_C} $$

The second law of thermodynamics

Consequences:

• Systems at different temperatures reach equilibrium at an intermediate temperature. The reverse never happens
• Heat naturally flows from hot bodies to cold bodies. For the reverse to occur, we require a refrigerator and a power supply
• It is simple to use frictional work to warm a body, though it is far more difficult to produce work from heat; it requires an engine

Kelvin-Planck statement

It is impossible to construct an engine which, operating in a cycle, produces no effect other than the extraction of heat from a reservoir and the performance of an equivalent amount of work.

Clausius statement

It is impossible to construct a refrigerator which, operating in a cycle, produces no effect other than the transfer of heat from a cooler body to a hotter one

The Carnot Cycle

Carnot’s Theorem

1. A reversible engine is the most efficient
2. All reversible engines, operating between two heat baths, have the same efficiency

$$ \text{Efficiency: } \quad \eta_C=1-\frac{T_C}{T_H} $$