4 The Second Law and Entropy

Statements and direction of the second law

The Second law of thermodynamics adds direction to the first law's accounting of energy. The first law accounts for energy conservation as energy moves between a system and its surroundings or changes form; the second law tells which processes can occur spontaneously and why converting energy into useful work has limits.

Two equivalent statements express the second law:

• The Clausius statement says that heat does not flow spontaneously from a colder body to a hotter one.

• The Kelvin–Planck statement says that no cyclic heat engine can take heat from a single reservoir and convert it entirely into work with no other effect.

A refrigerator can move heat from cold to hot, but it requires work from outside. A heat engine can produce work from heat, but it must reject some heat to a colder reservoir.

Entropy and its balance

Entropy, denoted by $S$, is a thermodynamic state property. Its change between two equilibrium states can be calculated along any reversible path connecting them:

Here, $\delta Q_{\mathrm{rev}}$ is the heat transferred along that reversible path, and $T$ is absolute temperature in kelvins. The reversible path is a way to calculate the change; the actual process need not be reversible.

Entropy balance

For a simple system exchanging heat with its surroundings at boundary temperature $T_b$, the entropy balance is

The integral represents entropy transferred with heat, and $S_{\mathrm{gen}}$ is entropy produced by irreversibility. For a reversible process, entropy generation is zero; for an irreversible process, it is positive. A system's entropy can decrease if it transfers enough entropy to its surroundings, but the total entropy of an isolated system—or of a system plus its surroundings—cannot decrease:

Entropy is sometimes informally described as “disorder,” but this is only a rough analogy. Its precise thermodynamic meaning is captured by its definition and balance.

Reversible and irreversible processes

A Reversible process is an ideal limit that can be reversed while leaving both the system and its surroundings unchanged. Real processes are irreversible: reversing them would require changes elsewhere. Common causes of irreversibility include friction, unrestrained expansion, mixing, and heat transfer across a finite temperature difference.

For example, suppose $100\ \mathrm{J}$ of heat flows spontaneously from an object at $400\ \mathrm{K}$ to one at $300\ \mathrm{K}$. The hot object's entropy change is $-\frac{100}{400}\ \mathrm{J/K}$, while the cold object's entropy change is $+\frac{100}{300}\ \mathrm{J/K}$. Their combined change is

The positive total entropy change identifies this heat flow as irreversible. Reversing it—moving heat from the colder object to the hotter one—does not happen spontaneously.

Heat engines and efficiency limits

A heat engine operating in a cycle absorbs heat $Q_H$ from a hot reservoir, produces work $W$, and rejects heat $Q_C$ to a cold reservoir. Its thermal efficiency is

The second law requires some heat to be rejected: $Q_C > 0$ for an engine operating between ordinary finite-temperature reservoirs.

Carnot limit

The Carnot efficiency is the maximum possible efficiency, achieved only by an ideal reversible engine:

where $T_H$ and $T_C$ are the hot- and cold-reservoir temperatures in kelvins. No real engine operating between the same reservoirs can be more efficient.

For reservoirs at $600\ \mathrm{K}$ and $300\ \mathrm{K}$, the maximum efficiency is $1 - \frac{300}{600} = 0.50$, or $50\%$; an actual engine must be less efficient. This limit does not arise because energy is destroyed. It arises because the second law restricts how much of the transferred energy can become work.