3.5 The second law and entropy

The first law says energy is conserved, but it permits much that never happens. It does not forbid a warm room’s heat gathering itself into a cup to boil it, or a gas spontaneously crowding into one corner. The second law rules these out and gives time its direction, through a new state function: entropy.

What the first law allows but nature forbids

Two statements of the second law, equivalent though they look different:

Clausius: no process can have, as its sole result, the transfer of heat from a colder body to a hotter one.
Kelvin: no process can have, as its sole result, the complete conversion of heat from a single reservoir into work.

Both forbid getting something for nothing thermally. The bridge to a quantitative law is the heat engine: a device running in a cycle, absorbing heat $Q_h$ from a hot reservoir at $T_h$ , doing work $W$ , and rejecting heat $Q_c$ to a cold reservoir at $T_c$ .

▶ Carnot efficiency: η = 1 − T_c/T_h Derivation

A reversible engine run in a cycle returns to its starting state, so its entropy change over the cycle is zero. The entropy it gains absorbing heat at the hot reservoir, $Q_h/T_h$ , must be balanced by the entropy it sheds at the cold one, $Q_c/T_c$ :

\frac{Q_h}{T_h} \;=\; \frac{Q_c}{T_c} \quad\Longrightarrow\quad \frac{Q_c}{Q_h} \;=\; \frac{T_c}{T_h}.

By the first law over a cycle ( $\Delta U = 0$ ) the work done is $W = Q_h - Q_c$ , so the efficiency is

\eta \;=\; \frac{W}{Q_h} \;=\; 1 - \frac{Q_c}{Q_h} \;=\; 1 - \frac{T_c}{T_h}.

The efficiency depends only on the two reservoir temperatures, not on the working substance — Carnot’s result. No engine between the same two reservoirs can beat it, because a better one could be run together with a reversed Carnot engine to move heat from cold to hot with no other effect, violating Clausius. ✓

Entropy as a state function

Carnot’s $Q_h/T_h = Q_c/T_c$ says the combination $\delta Q_\text{rev}/T$ integrates to zero around a reversible cycle. A quantity whose cyclic integral vanishes is the differential of a state function — and that function is the entropy $S$ :

dS \;=\; \frac{\delta Q_\text{rev}}{T}.

For a real, irreversible process the entropy generated is more than the heat divided by temperature, and the second law takes its sharpest form: the entropy of an isolated system never decreases,

\boxed{\;\Delta S_\text{isolated} \;\ge\; 0,\;}

with equality only for a reversible process. This is the law’s modern statement. It selects the direction of every spontaneous change, and applied to a system together with its surroundings it is the foundation of the free-energy minimisation in Chapter 4.

The statistical meaning

Boltzmann gave entropy its microscopic interpretation:

S \;=\; k_B \ln W,

where $W$ is the number of microscopic arrangements — microstates — consistent with the macroscopic state. Entropy counts the ways. A gas spread through its container has overwhelmingly more arrangements than one crowded into a corner, so spreading is not a law of force but a matter of counting: the system is found in the macrostate with the most microstates simply because there are more of them. The second law is statistics raised to near-certainty by the size of $10^{23}$ .

The third law

As the temperature falls toward absolute zero, a system settles into its lowest-energy state, and the number of accessible microstates collapses toward one. By $S = k_B\ln W$ the entropy then approaches a constant, taken as zero for a perfect crystal:

S \;\to\; 0 \quad\text{as}\quad T \to 0.

This is the third law. It fixes the otherwise arbitrary additive constant in the entropy — entropies become absolute, not merely differences — and it carries a striking corollary: absolute zero is unattainable, because removing the last increment of entropy from a system would take infinitely many steps.

⏳ The history — Carnot, Clausius, and the invention of entropy

The first law was assembled in the 1840s by Julius Mayer, James Joule, and Hermann von Helmholtz: Mayer argued that heat and work are forms of one thing, Joule made the calorimetric measurements (his water-paddle experiment) that fixed the mechanical equivalent of heat, and Helmholtz gave the systematic statement.

The second law came, oddly, first. Sadi Carnot’s 1824 analysis of heat engines held the key result — that efficiency depends only on the reservoir temperatures — but expressed it in the caloric theory, which wrongly treated heat as a conserved fluid. Rudolf Clausius reconciled Carnot’s insight with the new first law around 1850, and in 1865 introduced the state function defined by $dS = \delta Q_\text{rev}/T$ . He coined entropy from the Greek for “transformation”, deliberately echoing “energy” so the two would stand as partners. Ludwig Boltzmann supplied the molecular meaning $S = k_B\ln W$ in 1877, joining macroscopic thermodynamics to the kinetic theory; the third law was added by Walther Nernst in 1906.