**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

**– – – –**

**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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P Mander September 2014

Legendre Transforms in thermodynamics.

In ‘Going around in circles: Legendre transformations’ Peter Mander above suggests that JW Gibbs defined, or at least understood, enthalpy H and free energy (F,G) as Legendre Transforms (short: LT) of the internal energy U. Mander also suggests that this idea is generally accepted in thermodynamics ever since. We will argue against this view.

The assumed usefulness of the LT in thermodynamics was first disclosed by Callen and Tisza at MIT in 1960, almost a century after Gibbs formulated his theory. Before this date there is very little to be found in the thermodynamic literature about any LT, except an article by Lunn (1920), but Lunn transforms temperature T to its reciprocal, a trick not repeated by Callen or Tisza. After 1960 the LT does not become mainstream either: many textbook writers (eg Atkins, Cengel, Devoe, Denbigh, Moran&Shapiro, Smith&VanNess, Zemansky&Dittmar), assumed they can do without the LT. So the opinion that H,F,G are LT’s of internal energy remained a minority position, even after it gained support from the IUPAC (Alberty,2001).

In mathematics, the substitution known as LT is used to solve certain differential equations (short: DE)(Courant,1962,p32). In thermodynamics, not a single DE is solved this way, ie such a (true) LT is not applied. The substitution used in thermodynamics to change the dependent variable at the expense of changing the independent variable has serious limitations. (e.g. U(S,V) is transformed to H(S,p), which can be transformed back to U(S,V).) First, for the needed derivatives to exist, U and H must be continuous functions of V and p respectively. Second, the result of the transform should not be zero, which implies that U should not have linear terms in V and H not in p. Third, Legendre transformations behave very badly if the curvature of U(S,V) changes sign as V changes. (Kennerly,2011). Therefore the functions U and H must be monotonous in V, respectively p, ie they do not have a maximum or minimum with respect to these variables. As the function U(S,V) is unknown for most systems, it is unknown whether the transform can be performed. Note that U is a variable of state for any and every system.

Callen (1987,p138) writes about the application of Legendre Transform in thermodynamics: “the introduction of the transformed representations is purely a matter of convenience”. The question arises whether Gibbs introduced H,F,G as a matter of convenience. The answer is certainly no. From the above one might expect that this application is not without problems and the LT-hypothesis leaves us with many questions. We give a few:

a. Gibbs called enthalpy by the name ‘heat at constant pressure’. At constant pressure the function H(T,p) cannot be transformed back because (dH/dp)T is un-determined. In ‘regular’ thermodynamics H is well-defined at constant p: some even say only-defined at constant p. Similar problems arise for the LT at constant temperature, not to mention constant T and p.

b. It is unclear why U(S,V) is the function to be transformed, or why does the LT for other U-functions U(x,y) gets you nowhere (as well). For mono-atomic ideal gases U(T)=nCvT. The LT in T, becomes LU=T(dU/dT)-U=0: why don’t we get some interesting property of this system ? (Note: U cannot be linear in T.) For any ideal gas the internal energy U is not a function of V: hence the function U cannot be transformed with respect of V, except by a large detour.

d. The conservation principle applies to quantities U and H, but not to F and G: this cannot be explained from the LT point of view, as they are all LT’s from each other.

e. Why would the LT of U with respect to both S and V be equal to the capacity to do technical work ?

or, when does the LT of any physical quantity become a physical quantity ?

It can be concluded that the Legendre Transform in thermodynamics must be classified as a ‘deus ex mathematica’, a kissing cousin of the ‘deus ex machina’ popular in Greek tragedies: at some point in the play the problems can no longer be solved by humans and the Gods have to come to the rescue. This time in vain: the Greeks knew that Zeus sometimes bungles….

References: Alberty RA, (2001) Use of Legendre transforms in chemical thermodynamics. Pure Appl Chem 73:1349-80. / Atkins PW, (1982) Physical Chemistry (2edn), Oxford UP. / Callen HB, (1960) Thermodynamics. J Wiley. / Callen HB, (1987) Thermodynamics and an introduction to thermostatics. J Wiley. / Cengel YA, Boles MA,(2006) Thermodynamics: An Engineering Approach (5edn), McGraw-Hill, NY. / Courant, R, (1962) Methods of mathematical physics. (Vol2) Interscience, NY. / Devoe H, (2012) Thermodynamics and Chemistry (2edn), Prentice-Hall. / Denbigh K, (1971) The Principles of Chemical Equilibrium. Cambridge UP. / Kennerly S, A graphical derivation of the Legendre transform. (2014) / Lunn AC, (1920) A principle of duality in thermodynamics. Phys Rev 15:269-76. / Moran MJ, Shapiro HN, (1998) Fundamentals of Engineering Thermodynamics (3edn), Wiley & Sons, NY. / Smith JM, VanNess HC (1987) Introduction to Chemical Engineering Thermodynamics (4edn) Mcgraw-Hill,NY. / Zemansky MW, Dittmar RH (1981) Heat and Thermodynamics (6edn), McGraw-Hill, NY.

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LT is the kind of mathematical tool which is essential but often understated. Most of the time, because I don’t want to flood students under mathematical concepts I consider as non essentials for the understanding of physics, I just hide it. I obviously use it to switch from constant volume systems to constant pressure ones, i.e. from internal energy to enthalpy, but without talking about its name and its real meaning. It’s always difficult to not mention such important concept, but we often have to do so, because of the global standpoint about thermodynamics (broadly technical vs. physical) and the lack of time in front of students.

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