Third law of thermodynamics

The third law of thermodynamics fixes the low-temperature boundary condition that the first two laws leave open. It has three statements, which are equivalent. Nernst's (1906): the entropy change of any isothermal process tends to zero as T → 0, so every substance converges on the same entropy at the bottom. Planck's sharpening (1911): that common value is zero for a perfect crystal, because its ground state is unique — Ω = 1, so S = k ln Ω = 0. And the unattainability statement: no finite sequence of operations can bring a system to T = 0.

The unattainability form is not an extra postulate but a consequence of the first two, proven equivalent by Fowler and Guggenheim in 1939. Any cooling cycle works by exploiting a gap between two entropy curves — for adiabatic demagnetisation, the salt's S(T) in weak and strong fields. Nernst's theorem says all such curves converge to the same S(0), so the gap the cycle exploits closes exactly where the cycle is trying to arrive. Each step remains finite and genuinely cooling; the sequence T_n = T_0·rⁿ is positive for every finite n and reaches zero only as a limit. There are always more steps.

The law's significance is that it cannot be derived from the first two, which constrain only changes and are silent about the low-temperature limit. Classical physics does not merely fail to predict it — it contradicts it, since a classical solid's constant equipartition heat capacity makes ∫C/T dT diverge to −∞ at the bottom, and a continuous classical phase space has no notion of 'one arrangement' at all. Quantum mechanics supplies the floor: energy levels are discrete, so there is a lowest one, and it is usually unique. The third law is the thermodynamic shadow of the fact that the world is quantised — which Nernst measured in 1906, five years before the nucleus and nineteen before matrix mechanics.

Walther Nernst announced his Wärmetheorem to the Göttingen Academy in 1906, seeking a way to compute chemical equilibria from calorimetry alone by eliminating the unknown constant of integration in the entropy. Max Planck sharpened it in 1911 by identifying the constant as zero, using Boltzmann's S = k ln Ω, which made absolute entropy meaningful and every thermodynamic entropy table possible. Nernst received the 1920 Nobel Prize in Chemistry. William Giauque spent his career testing the law calorimetrically at Berkeley, and the discrepancies he found turned out to be the residual entropies of substances such as ice and carbon monoxide, whose ground states are degenerate — denting Planck's sharpening, which carries the qualifier 'perfect crystal', while leaving Nernst's statement about entropy differences exactly true.