Residual entropy

Residual entropy is the entropy a substance retains as T → 0, in apparent defiance of Planck's statement that S → 0. It arises when the ground state is degenerate: if a substance has Ω equally good lowest-energy configurations per molecule rather than one, then S(0) = R ln Ω per mole instead of zero. Planck's sharpening carries the qualifier 'for a perfect crystal' precisely to exclude these cases, and residual entropy is that qualifier earning its keep.

The canonical case is ice, worked out by Linus Pauling in 1935. Ice's oxygen atoms sit on a perfectly ordered lattice, but its protons do not: the Bernal–Fowler ice rules require two hydrogens close to each oxygen and two far, with one proton per O–O bond, which leaves enormous freedom in which two are close — and cooling never resolves it. Counting: 2^(2N) proton placements on 2N bonds, of which 6 of 16 local arrangements survive the rules at each oxygen, giving Ω ≈ (3/2)^N and S₀ = R ln(3/2) ≈ 3.37 J/(mol·K). Giauque and Stout measured 3.4 ± 0.2 the following year. Carbon monoxide does something similar: CO and OC are nearly the same shape, so the molecule freezes either way round, stranding about R ln 2 ≈ 5.76 J/(mol·K) (measured ~4.6).

A glass strands residual entropy for a different reason entirely: it is not in equilibrium. Cooled through its glass transition, it stops being able to explore configurations at all and freezes a snapshot of wherever it happened to be — so its residual entropy is a record of the cooling rate rather than a property of the substance. Neither case violates the third law in Nernst's form. Residual entropy is a constant offset, and a constant offset cancels out of every isothermal ΔS, which is all Nernst's theorem claims. In practice, residual entropy was how such degeneracies were discovered: it appeared as a stubborn discrepancy between calorimetric and spectroscopic entropies.

The discrepancies turned up in William Giauque's low-temperature calorimetry programme at Berkeley in the late 1920s and early 1930s, which compared entropies integrated from measured heat capacities against those computed spectroscopically. Linus Pauling explained ice's in 1935 with a counting argument he admitted was crude; it matched Giauque and Stout's 1936 measurement within error. The same technique of taking thermodynamic discrepancies seriously led Giauque and Johnston to discover the oxygen isotopes ¹⁷O and ¹⁸O in 1929. Ice's proton disorder remains an active subject: 'spin ice' materials, whose magnetic moments obey the same ice rules, were found in the 1990s to carry the identical Pauling residual entropy, and to host emergent magnetic monopoles.