**Historical background***

It was the American physicist Josiah Willard Gibbs (1839-1903) pictured above who first introduced the thermodynamic potentials ψ*, χ, ζ *which we today call Helmholtz free energy (A), enthalpy (H) and Gibbs free energy (G).

In his milestone treatise *On the Equilibrium of Heterogeneous Substances* (1876-1878), Gibbs springs these functions on the reader with no indication of where he got them from. Using an esoteric lexicon of Greek symbols he simply states:

*Let
ψ = ε – tη
χ = ε + pv
ζ = ε – tη + pv*

As with much of Gibbs’ writings, the clues to his sudden pronouncements need to be sought on other pages or – as in this case – another publication.

In an earlier paper entitled *A method of geometrical representation of the thermodynamic properties of substances by means of surfaces*, Gibbs shows that the state of a body in terms of its volume, entropy and energy can be represented by a surface:

It can be demonstrated from purely geometrical considerations that the tangent plane at any point on this surface represents the U-related function

Now this is none other than Gibbs’ zeta (*ζ *) function. The question is, did he recognize it for what it was – a Legendre transform? A key feature of *On the Equilibrium of Heterogeneous Substances* is the business of finding an extremum for a multivariable function subject to various kinds of constraint, and it is known that Gibbs was familiar with *Lagrange’s method of multipliers* – he mentions the technique by name on page 71, immediately after equation 41. The point here is that the *Legendre transformation* can be phrased in the same terms – for example, the multiplier expression for finding the stationary value of U when T and P are held constant yields the Legendre transform shown above.

But suggestive though this is, it actually gets us no closer to determining whether or not Gibbs was aware that ψ*, χ, ζ * were Legendre transforms. Gibbs gave no indication in his writings either that he knew the transformation trick, or that he had discovered it for himself. We can only estimate likelihoods and have hunches.

*Text revised following input from Bas Mannaerts (see comments below)

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**Thermodynamics and the Legendre transformation**

The fundamental relation of thermodynamics dU = TdS–PdV is an exact differential expression

where the coefficients C_{i} are functions of the independent variables X_{i}. By means of Legendre transformations (ℑ) the above expression generates three new state functions whose natural variables contain one or more C_{i} in place of the conjugate X_{i}

The equation of the tangent plane to the thermodynamic surface generates ℑ_{3}, with ℑ_{1} and ℑ_{2} following procedurally from

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**How the Legendre transformation works**

defines a new Y-related function Z by transforming

into

Proof

dZ = dY – d(C_{1}X_{1})

dZ = dY – C_{1}dX_{1} – X_{1}dC_{1}

Substitute dY with the original differential expression

dZ = C_{1}dX_{1} + C_{2}dX_{2} – C_{1}dX_{1} – X_{1}dC_{1}

The C_{1}dX_{1} terms cancel, leaving

dZ = C_{2}dX_{2} – X_{1}dC_{1}

The independent (natural) variables are transformed from Y(X_{1},X_{2}) to Z(X_{2}, C_{1})

The same procedural principle applies to ℑ_{2} and ℑ_{3}.

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**The Legendre Wheel**

Since exact differential expressions in two independent (natural) variables can be written for the internal energy (U), the enthalpy (H), the Gibbs free energy (G) and the Helmholtz free energy (A), each of these state functions can generate the other three via the Legendre transformations ℑ_{1}, ℑ_{2}, ℑ_{3}. This is neatly demonstrated by the *Legendre Wheel*, which executes the transformation functions

from any of the four starting points:

[click on image to enlarge]

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**Legendre transformations and the Gibbs-Helmholtz equations**

For an exact differential expression

the transforming function

can be written in terms of the natural variables of Y

This Legendre transformation is the means by which we obtain the Gibbs-Helmholtz equations. Taking Y=G(T,P) as an example, ℑ_{1} executes the *clockwise* transformation

while the transforming function

reverses the positions of the natural variables and executes the *counterclockwise* transformation

Setting Y=G(T,P) generates six Gibbs-Helmholtz equations, in each of which one of the two natural variables is held constant. Since there are four state functions – U, H, G and A – the total number of Gibbs-Helmholtz equations generated by this procedure is twenty-four. To this can be added a parallel set of twenty-four equations where U, H, G and A are replaced by ΔU, ΔH, ΔG and ΔA.

These equations are particularly useful since they relate a state function’s dependence on either of its natural variables to an adjacent state function on the Legendre Wheel.

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**Who was Legendre?**

Adrien Legendre (1752-1833) was a French mathematician. He wrote a popular and influential geometry textbook *Éléments de géométrie* (1794) and contributed to the development of calculus and mechanics. The Legendre transformation and Legendre polynomials are named for him.

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