The Third Law and Low Temperatures

The Unreachable Floor

The third law comes in two related statements. Nernst: as $T \to 0$, the entropy change of any isothermal process tends to zero. Planck, stronger: the entropy of a perfect crystal tends to zero itself,

$$\lim_{T\to 0} S = 0$$

The statistical reading makes this natural. With $S = k_B\ln W$, a perfect crystal at zero temperature has exactly one accessible arrangement, so $W = 1$ and $S = 0$. Unlike energy, entropy therefore has a genuine absolute zero, which is why tables list absolute $S^\circ$ rather than differences.

Real systems can retain residual entropy if they freeze into a disordered state. Carbon monoxide, whose CO and OC orientations differ by very little energy, keeps roughly $k_B\ln 2$ per molecule, the crystal is not perfect, so Planck's statement does not apply to it.

A striking consequence: heat capacities must vanish as $T \to 0$. Since $S(T) = \int_0^T (C/T)\,dT'$ must converge, $C$ has to fall faster than $T$. Experiment agrees, metals show $C = \gamma T + AT^3$, the linear term from electrons and the cubic from phonons.

The third law also implies the unattainability of absolute zero. Cooling methods remove entropy in finite steps, and as $S \to 0$ for every accessible state, each step removes less. Reaching exactly zero would take infinitely many steps. Absolute zero is a limit, not a destination.

Practical cooling uses staged techniques: liquid nitrogen to 77 K, liquid helium to 4.2 K, pumped helium-3 to about 0.3 K, dilution refrigeration to a few millikelvin, and adiabatic demagnetisation below that, magnetise a paramagnetic salt isothermally to order the spins, then demagnetise it adiabatically so the spins re-disorder at the expense of the lattice's thermal energy.

Common pitfall: reading unattainability as a practical difficulty to be engineered around. It is a statement about the structure of thermodynamics: each cooling step removes a smaller entropy increment than the last, so no finite sequence of steps reaches zero however good the apparatus.