Clapeyron, Maxwell, Gibbs

A quadrilateral that becomes a rectangle

Clapeyron was born in Paris in 1799. In 1834 the engineer gave Sadi Carnot's 1824 argument — ten years unread, written for an audience that was not yet looking for it — its first analytic picture. In his Mémoire sur la puissance motrice de la chaleur, Clapeyron drew the reversible cycle Carnot had described in words as a closed curve on a pressure–volume indicator diagram: two isothermal legs, two adiabatic legs, a quadrilateral that becomes a parallelogram once the legs are shrunk to infinitesimal size. At the time heat was still widely treated as a conserved substance; the diagram outlived that belief by decades.

The quadrilateral is easier to see if the axes are changed. Plot the same cycle not in pressure and volume but in entropy and temperature. An isothermal leg holds temperature fixed, so it is horizontal; an adiabatic leg holds entropy fixed (that is what reversible-and-adiabatic means), so it is vertical. A four-sided reversible cycle between two temperatures and two entropies is therefore exactly a rectangle in the (S,T) plane, for any working substance whatever — not just in the infinitesimal limit. What is not automatic is that the rectangle's area equals the actual mechanical work delivered around the curved path back in the pressure–volume plane. That equality is Carnot's theorem, and ch-clapeyron-gibbs-verify.py §1 checks it directly: it builds an ideal-gas Carnot cycle between 300 K and 400 K, integrates the actual P–V work along all four legs, and compares the total against (T₂−T₁)(S₂−S₁). They agree to one part in a million, limited only by the numerical integration.

Maxwell's 1871 construction, and the entropy he did not yet have

James Clerk Maxwell used Clapeyron's parallelogram again in 1871, in his textbook Theory of Heat, to reach the four relations that carry his name today — without differential calculus applied to state functions that did not yet exist under that name, and almost entirely by plane geometry. He extended the isothermal and adiabatic sides of the original parallelogram outward, built four further parallelograms sharing its base or its height, and showed each also had unit area once the scale is fixed by setting a unit rise in temperature equal to a unit conversion of heat to work. Reading the resulting eight line-segments in pairs as reciprocal slopes gives exactly the four Maxwell relations:

Two things this modern layout hides. First, the thermodynamic potentials H, G and A on the left did not exist as named objects in 1871; Maxwell's geometry produced the relations without them. Second, the variable playing the role of S above was not entropy in Maxwell's book at all. Clausius had stated the first form of the second law in 1850, fifteen years before he had a name for the quantity it turned on. Maxwell instead used Rankine's thermodynamic function Φ, introduced in 1854 and identical in substance to the entropy Clausius eventually named in 1865 — and he used it because he still misunderstood what entropy was. Maxwell believed, in 1871, that entropy meant the available energy of a system, which does not have the dimensions temperature needs for a conjugate variable; Φ did. The mistake was not a matter of taste, and it was not harmless bookkeeping — it was Maxwell holding the wrong concept while building an exactly right construction on top of it.

The tangent plane that fixes Maxwell's word

Josiah Willard Gibbs was born in New Haven in 1839 and died there in 1903. In 1873, in two papers for the Transactions of the Connecticut Academy, he went further than a plane figure. Paper I compared indicator diagrams; Paper II built a surface in the three coordinates volume, entropy and internal energy — (V, S, U) — whose shape encodes everything a single-phase substance's thermodynamics can say. A vertical slice of that surface at constant volume gives energy against entropy, whose slope is temperature; a slice at constant entropy gives energy against volume, whose slope is minus the pressure. The tangent plane at any point therefore carries both T and P at once, and Gibbs used exactly that fact: if two points on the surface share a tangent plane, the two states have the same temperature and the same pressure, and can coexist. He showed algebraically that the quantity U − TS + PV — a Legendre transform of U, later named the Gibbs free energy, G — is equal at any two such points, without publishing the geometric argument that gets there.

Gibbs also corrected Maxwell's dimensional error, in print, politely. Maxwell absorbed the correction into a later edition of Theory of Heat and apologised to his readers for the earlier mistake. The two men never fully overlapped in person on the subject, but they overlapped in matter: in 1874 Maxwell built a clay model of Gibbs' (V,S,U) surface, had it cast in plaster, and sent the cast to Gibbs at Yale — a three-dimensional readers' report on a paper Maxwell had understood quickly and almost no one else had understood at all.

A contact structure, not a coincidence

Put U, S, V, T and P in one five-dimensional space and write the 1-form α = dU − TdS + PdV. A thermodynamic potential U = U(S,V), with T and P read off as its partial derivatives, is exactly a surface on which α vanishes when pulled back — the modern definition of a Legendrian submanifold of the contact manifold (ℝ⁵, α). Gibbs' tangent-plane condition is the geometric picture of that vanishing. The equality of mixed partials that makes each Maxwell relation true is exactly the closedness that keeps such a surface from being an arbitrary, inconsistent graph. None of this was named in 1873; the explicit contact-geometric formulation of equilibrium thermodynamics is a twentieth-century reading — put forward by Robert Hermann in 1973, developed by Ryszard Mrugała in 1978, and given its best-known statement by Vladimir Arnold, who argued at a Gibbs symposium in 1990 that Gibbs' geometric method just is the contact-geometric method, encountered a century early and without the name.

The line about Gibbs' papers

A remark is often repeated about Gibbs' 1876–1878 memoir On the Equilibrium of Heterogeneous Substances — the paper that Papers I and II were building toward — to the effect that only one other scientist alive could follow it, and that when Maxwell died in 1879 the number of people who fully understood Gibbs' work in his own lifetime dropped to zero. The line is genuinely old and genuinely in circulation; it is not, so far as this chapter's sourcing reaches, traceable to a dated, first-hand attribution. It is carried here as folklore with real historiographic standing, not as a documented quotation, and the claims table below records it that way rather than silently upgrading it.

Claims and their status

References