Second law, heat engines and entropy

1. Thermal reservoir

A thermal reservoir (or heat source) is a system with a heat capacity so large that the heat exchanged with other bodies does not change its temperature appreciably. A reservoir is therefore described by a single quantity, its temperature $T$, which remains constant. The atmosphere, a lake, the boiler of a power plant kept at a fixed temperature are good approximations of reservoirs. Heat engines are described by stating which reservoirs they exchange heat with.

2. Reversible process

A process of a system is reversible if it can be traversed in the opposite direction, bringing back to the initial state both the system and the surroundings with which it has interacted. A reversible process is quasi-static, that is, made of equilibrium states, and free of dissipative effects. Moreover, heat is exchanged in it only between bodies whose temperatures differ by an infinitesimal amount.

Real processes are irreversible. The typical causes are friction, heat exchanges between bodies at different temperatures, the free expansion of a gas into vacuum and the mixing of different substances. A reversible process is therefore an ideal model; it serves as a benchmark because, as Carnot’s theorem shows, a reversible engine has the maximum efficiency between two given reservoirs.

1. Heat engine

A heat engine is a system that undergoes a cyclic process, absorbs heat from one or more reservoirs, releases part of it to other reservoirs and produces positive work. We denote by $Q_{\text{in}}>0$ the sum of the heats absorbed in a cycle and by $Q_{\text{out}}>0$ the sum of the absolute values of the heats released. Since $L=Q$ in a cycle, the work produced in a cycle is

In the simplest case the engine operates between two reservoirs at temperatures $T_1>T_2$; then $Q_1$ and $Q_2$ denote the heats exchanged with the hot reservoir and with the cold one, taken with the sign of the convention of the first law. For an engine $Q_1>0$, $Q_2<0$ and $L=Q_1+Q_2>0$. If the working fluid of the engine goes through a quasi-static cycle, the work is the area enclosed by the cycle in the $(V,p)$ plane, positive when the cycle is traversed clockwise.

2. Efficiency

The efficiency of a heat engine is the ratio of the work produced in a cycle to the heat absorbed in the same cycle:

For an engine between two reservoirs the formula becomes $\eta =1+\frac{Q_2}{Q_1}=1-\frac{|Q_2|}{Q_1}$. The efficiency is a pure number and measures what fraction of the heat absorbed becomes work. From the formula it follows that $\eta <1$ whenever the engine releases heat; the second law states that this release is unavoidable. For example, an engine that in each cycle absorbs $2000\mathrm{J}$ and releases $1500\mathrm{J}$ produces $L=500\mathrm{J}$ and has efficiency $\eta =0.25$.