# Gibbs energy

In thermodynamics, Gibbs energy, as compared to "Helmholtz energy", is the isothermal, isobaric. thermodynamic potential, of freely-going reaction systems; synonyms include: "available energy" (Gibbs, 1876), "reaction energy" (Haber, 1905), "free enthalpy" (German, c.1915), "free energy" (Lewis, 1923), or "Gibbs free energy" (Guggenheim, 1933).

## Terminology | Confusion

The exists, historically, certain amount of terminology confusion in respect to what thermodynamic function (and characteristic function[1] symbol) is defined by what has been variously called "available energy" (Gibbs, 1876), "reaction energy" (Haber, 1905), "free enthalpy" (German, c.1915), "free energy" (Lewis, 1923), "Gibbs free energy" (Guggenheim, 1933), or "Gibbs energy", used recently, in reactions where temperature and pressure are held constant.

### Lewis | 1923

In 1923, Gilbert Lewis, in his Thermodynamics (pg. 158), devoted a large footnote to the confusion; commenting on how both Walther Nernst (Theoretical Chemistry, 1909) and Fritz Haber (Thermodynamics of Technical Reactions, 1905) confused what we now called Helmholtz energy ${\displaystyle \Delta H}$ (or ${\displaystyle \Delta A}$ used by Lewis) and Gibbs energy ${\displaystyle \Delta G}$ (or ${\displaystyle \Delta F}$ used by Lewis) in their derivations, therefrom making error filled calculations.

To anchor ourselves, in respect to point of reference, in this confusion, Lewis, in his §14: Criteria of Equilibrium and of Spontaneous Change: the Free Energy, defines the "two thermodynamic functions" as follows:[2]

${\displaystyle A=E-TS}$
${\displaystyle F=H-TS}$

The latter of which he expands as follows:

${\displaystyle F=(E+PV)-TS}$
${\displaystyle F=A+PV}$

Lewis, in respect to these functions, points out that ${\displaystyle A}$ was called by Helmholtz (1882) the "free energy"[3], meant to be synonymous with Gibbs (1876) "available energy", and that Helmholtz called the ${\displaystyle TS}$ quantity the "bound energy", meaning energy that was "bound up" in the system, supposedly in the sense of Clausius' notion of "equivalence values of all uncompensated transformations", and thereby not available for external work. Moreover, Helmholtz, as Lewis points out, referred only to ${\displaystyle A}$, but not to ${\displaystyle F}$, as the "free energy".

Lewis, however, defines ${\displaystyle F}$ explicitly as the "free energy", as follows:

“In any process occurring at constant temperature and pressure, ${\displaystyle F_{A}-F_{B}}$ represents the maximum of work which can be obtained from a given process and applied to useful purposes. It is for this reason that ${\displaystyle F}$ is known as the free energy''.
Gilbert Lewis (1923), Thermodynamics (pg. 158)

#### Historical footnote

Lewis then, to elaborate on history of "free energy" terminology, symbol, and equation usage, gives the following footnote:

“The quantities ${\displaystyle F}$ and ${\displaystyle A}$, or better, the molal or partial molal values of these quantities, are sometimes called thermodynamic potentials, on account of certain rough analogies to mechanics, which we need not stress. While the value of such functions was pointed out by Massieu (1869), employed ${\displaystyle -{\frac {F}{T}}}$ and ${\displaystyle -{\frac {A}{T}}}$, the former of which has more recently been used by Planck (Thermodynamics, 1897), their great utility in the interpretation of the most diverse physico-chemical phenomena was first fully demonstrated in the comprehensive work of Gibbs (1873/1875)[4]. Students of Gibbs will observe that the quantities F and A are his functions ζ [zeta] and ψ [psi]. (We may note also that our H is Gibbs' χ [chi] and our E his ε [eta].)
It was Helmholtz’ who introduced the term free energy, and it is to be remarked that he applied this name to A rather than to F. Later, however, since ΔA and ΔF do not largely differ numerically, unfortunate confusion has arisen in the literature between these two quantities. Several authors who have defined the free energy as A have really used the function F. It has therefore seemed best to retain this useful expression for the latter function, which has in practice by far the greater importance.
It will be seen that F bears the same relation to A that H bears to E. We have previously remarked that the choice of pressure and temperature as the chief variables in thermodynamics, together with the fact that most experiments are carried on at constant temperature or pressure, makes quantities like H, ${\displaystyle C_{v}}$, and F more generally useful than the corresponding quantities E, ${\displaystyle C_{v}}$, and F.”

### Guggenheim | 1933

In 1933, Edward Guggenheim, in his Modern Thermodynamics: by the Methods of Willard Gibbs, building on Lewis, defined the two main thermodynamic equations as follows:[5]

${\displaystyle F=E-TS}$

referring to H as the "heat content" or "total heat". Then comments as follows:

“The function F is due to Helmholtz, and was named by him the ‘free energy’. It is sometimes referred to as the ‘work function’ [A = “Arbeit”, meaning ‘work’ in German]. It was called by Gibbs the ‘force function for constant temperature’. The function G is due to Gibbs, and is often referred to by modern writers as the ‘free energy’. We shall call F the ‘Helmholtz free energy’ and G the ‘Gibbs free energy’. But to avoid any possibility of confusion we shall always refer to the symbols F and G defined according to equations #24 and #25.”
— Edward Guggenheim (1933), Modern Thermodynamics (pgs. 13-14)

#### U vs E

Here, we point out that Guggenheim was using the symbol “E” for the energy or internal energy of a system; whereas Clausius originated the concept of the energy or internal energy of a system with the symbol “U”.

In 1936, Theophile de Donder, in his Thermodynamic Theory of Affinity, employed the symbol “E” for the energy or internal energy of a body. Usage wise, up to this point, among the main 22 thermodynamics authors, from Clausius (1865) to Donder (1936), 36 percent used “E” for internal energy, and 64 percent used “U” for internal energy.[1]

### Zemansky | 1937

In 1937, Mark Zemansky, in his Heat and Thermodynamics, provided a notation table, at the beginning of the book, defining things as follows:[6]

H = U + PV
A = U - TS
G = H - TS

where U is "internal energy", A is the "Helmholtz function", and G is the "Gibbs function". Here, aside from his use of the names “Helmholtz function”, which we now called "Helmholtz energy", and “Gibbs function”, which we now call "Gibbs energy", thermodynamics notation and terminology had become “standardized”, historically speaking.

## Quotes

The following are related quotes:

“A crisis gives a genius an opportunity to manifest and to exercise itself which it does not have in normal times. The unusual amount of free and mobile energy then available and seeking channels of some sort, old or new, readily lends itself to the control of the master mind. Indeed, without this undirected energy in persons and the body politic there is little chance for leadership. But with it everything is propitious to us for achievement and change.”
Newell Sims (1924), Society and Surplus (pgs. 349-5S0) [7]
“Any chemical reaction moves downhill (∆G < 0) means downhill) on a Gibbs energy surface if it can, driven by the chemical potential difference between the products and the reactants. Chemical change continues until reactant and product chemical potentials are balanced, the Gibbs energy change equals zero, and chemical equilibrium is reaches.”
— William Cropper (2004), Great Physicists (pg. 118) [8]

## References

1. Characteristic function notation table – Hmolpedia 2020.
2. Lewis, Gilbert. (1923). Thermodynamics: and the Free Energy of Chemical Substances (amanuensis: Merle Randall) (§14: Criteria of Equilibrium and of Spontaneous Change: the Free Energy, pgs. 155-75). McGraw-Hill.
3. Helmholtz, Hermann. (1882). “On the Thermodynamics of Chemical Processes” ("Die Thermodynamic Chemischer Vorgange"), in: Physical Memoirs Selected and Translated from Foreign Sources, 1:43-97. Physical Society of London, Taylor and Francis, 1888.
4. Gibbs, Trans. Conn. Acad. Arts. Sci., 2, 309, 382 (1873); 3, 108, 343 (1875).
5. Guggenheim, Edward, (1933). Modern Thermodynamics: by the Methods of Willard Gibbs. Methuen.
6. Zemansky, Mark. (1937). Heat and Thermodynamics: an Intermediate Textbook for Students of Physics, Chemistry, and Engineering (2nd edition) (notation table, pgs. xiii-xiv; functions defined, pg. 219). McGraw-Hill, 1943
7. Sims, Newell L. (1924). Society and its Surplus: a Study in Social Evolution. Appleton and Co.
8. Cropper, William H. (2001). Great Physicists: the Life and Times of Leading Physicists from Galileo to Hawking (pg. 118). Oxford University Press.