A segment on Gibbs free energy, from a ten page science cartoon on fugacity, by Lucas Landherr. [11]
In chemical thermodynamics, Gibbs free energy, or Gibbs energy, is the free energy of an isothermal-isobaric system. Gibbs free energy, in more detail, is a state function of a closed isothermal-isobaric system in chemical equilibrium. The status of Gibbs free energy is summarized as follows: [8]

“The single most important concept in chemical thermodynamics is that of the Gibbs free energy G a function of state which provides the criterion for deciding whether or not a change of any kind will occur.”

Etymology
In the 1910s and 1920s, the isothermal isobaric potential went by the name of free enthalpy, among German writers. [5] The symbol of G was assigned to Gibbs, in great predominance, in 1933 by English thermodynamicist Edward Guggenheim who stated: [4]

“The function G is due to Gibbs, and is often referred to by modern writers as ‘free energy’. We shall call G the ‘Gibbs free energy’.”

There has been some effort, by the IUPAC, to expunge the middle clarifying term 'free' so to call G simply 'Gibbs energy', but this, for most purposes, has been a failed effort, and it is puzzling why the effort was even made in the first place?

Overview
Changes in the value of free energy can be used to determine if a reaction is thermodynamically favorable. In an equilibrium system at constant temperature and pressure, the the Gibbs free energy is quantified by what is called the Gibbs function:

$G = H - T S\,$

where H is the enthalpy, T the temperature, and S the entropy of the body; an expression representing the part of the energy content of a system that is available to do external work, also known as the free energy G. [6] In an equilibrium system at constant temperature and pressure, G = H–TS, where H is the enthalpy (heat content), T the temperature, and S the entropy (decrease in energy availability). [6] The function was named after American engineer Willard Gibbs. In expanded form, showing enthalpy as a function of internal energy U and pressure volume work pV, the Gibbs function becomes:

 Top: The relationships between the equilibrium constant K, Gibbs free energy change ΔG, and the direction of a chemical reaction. Bottom: the spontaneity criterion rules for reaction direction. [7]

$G = U + p V - T S\,$

The formula was introduced by American mathematical physicist Willard Gibbs in 1873 and called by him "available energy". [1] In chemistry, for simple reactions, the change in Gibbs free energy, between the initial stage and final stage of a reaction, for a system of reacting species, takes the form:

$\Delta G = \Delta H - T \Delta S \,$

where ∆H is the change in enthalpy, T the temperature, and ∆S the change in entropy for the reaction.

Affinity | Spontaneity criterion
A spontaneous or energetically favored reaction will satisfy the following condition (spontaneity criterion):

To determine the feasibility or reaction possibility for a given reaction, the value of ∆G must be determined. To do this, precalculated values of enthalpy change and entropy (per species) are generally listed in standard thermodynamic tables, as shown below (left).

Prior to 1882, wherein German physicist Hermann von Helmholtz showed that free energy was a measure of the chemical "affinity" between the reactants, these reaction calculation tables took the form of affinity tables, the first of which was constructed in 1718 by French physician and chemist Étienne Geoffroy, as shown below (right).

 The historical transformation of affinity tables (1718) into free energy tables (1905), linked via the equation A = –ΔG, meaning that the affinity of an closed isothermal-isobaric reacting system is equal the negative of the Gibbs free energy change, as was proved in German physicist Hermann Helmholtz’s 1882 paper “The Thermodynamics of Chemical Processes”, captures a great density of the underlying corpus of modern science.

Chemical equilibrium
In 1884, Dutch physical chemist Jacobus van't Hoff showed that for a generic reversible reaction of the form:

$x A + y B \rightleftharpoons z C + w D$

where x, y, z, and w are the stoichiometric coefficients, and in which equilibrium constant is defined by follow equation:

$K = \frac{[C]^z [D]^w} {[A]^x [B]^y} \,$

where [A], [B], [C], and [D] are the concentrations of the various reactants and products at the equilibrium, that the following relation will hold:

$\ \Delta G^\circ = -RT \ln K$

The form of this expression, as indicated here, to note, was not derived by van't Hoff exactly, but rather he refers to the previous work of German chemist August Horstmann who suggested that thermodynamics could be applied to chemical equilibrium issues. [3]
 Left: an annotated version (by Ronald Kriz) of Willard Gibbs original 1873 graphical thermodynamics depiction of "available energy" (section AB), as he called it; latter to be named "free energy", by Hermann Helmholtz (1882); which eventually diverged into two types: Helmholtz free energy (isochoric-isobaric processes) and Gibbs free energy (isothermal-isobaric processes) names assigned by Edward Guggenheim (1933). [9]

Goethe's elective affinities
See main: Goethe's human chemistry
In 1809, German polymath Johann von Goethe used Swedish chemist Torbern Bergman's 1775 chemistry textbook A Dissertation on Elective Attractions and specifically its fifty-row, fifty-column affinity table, showing thousands of possible chemical reactions between the known chemical species, to write the famed novella Elective Affinities, a chemical treatise on the origin of love, in which the characters react according to their natural affinity preferences, producing or absorbing work, and forming or breaking bonds along the way. This was the start of the science of human chemistry. [2]

In 1995, American chemical engineer Libb Thims, unaware of Goethe's previous work, began to try to figure out the same type of logic for reactions between people using thermodynamic tables. It took seven years before the technical and conceptual issues behind the problem began to make any sort of sense.

Graphical representation
See main: Graphical thermodynamics
The following diagram shows the first ever two-dimensional (middle, left), three-dimensional (middle, right), and physical-scale (right) representation of Gibbs free energy, which is represented by section AB and originally called "available energy" (the free energy namesake was introduced in German physicist Hermann Helmholtz's 1882 article "On the Thermodynamics of Chemical Processes"):

 American engineer Willard Gibbs’ 1873 figures two and three (above left and middle) used by Scottish physicist James Maxwell in 1874 to create a three-dimensional entropy (x), volume (y), energy (z) thermodynamic surface diagram for a fictitious water-like substance, transposed the two figures of Gibbs (above right) onto the volume-entropy coordinates (transposed to bottom of cube) and energy-entropy coordinates (flipped upside down and transposed to back of cube), respectively, of a three-dimensional Cartesian coordinates; the region AB being the first-ever three-dimensional representation of Gibbs free energy, or what Gibbs called "available energy"; the region AC being its capacity for entropy, what Gibbs defined as “the amount by which the entropy of the body can be increased without changing the energy of the body or increasing its volume.”

In 1873, Gibbs published his first thermodynamics paper, “Graphical Methods in the Thermodynamics of Fluids”, in which Gibbs used the two coordinates of the entropy and volume to represent the state of the body. In his second follow-up paper, “A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces”, published later that year, Gibbs added in the third coordinate of the energy of the body, defined on three figures. In 1874, Scottish physicist James Maxwell used Gibbs' figures to make a 3D energy-entropy-volume thermodynamic surface of a fictitious water-like substance. Thus, in order to understand the very difficult concept of Gibbs free energy one must be able to understand its interpretation as Gibbs defined originally by section AB on his figure 3 and as Maxwell sculpted that section on his 3D surface figure.

Modern chemical thermodynamics
See main: Chemical thermodynamics, History of chemical thermodynamics
The distillation of the rather complicated formulation of available energy or free energy, in somewhat laymanized standard terminology, as applied to standard temperature and pressure volume-changing earth-bound reaction, in the context of what has come to be known as "modern chemical thermodynamics" came about in through the following to publications:

 Gilbert Lewis (1875-1946)American physical chemist 1923 Together with his graduate student Merle Randall, published the 1923 textbook Thermodynamics and the Free Energy of Chemical Substances, which resulted to replace the notion of "affinity" with the notion of "free energy" in the corpus of modern science, in his chapter sub-section "The Driving Force of a Chemical Reaction", he famously situated the "driving force" thermodynamic view of chemical process and introduced what he defined as a "universal rule" as follows: “It is a universal rule that if any isothermal process is to occur with finite velocity, it is necessary that: [This applies to] a chemical process which is in some way harnessed for the production of useful work. In the far more common case of a reaction which runs freely, like the combustion of a fuel, or the action of an acid upon a metal; in other words, systems which are subject to no external forces except a constant pressure [exerted by the atmosphere]. In such cases w’ = 0, and it follows that no actual isothermal processes is possible unless:Therefore if we know the value of ΔF for any isothermal reaction, and if this value is positive, then we know that the reaction, in the direction indicated, is thermodynamically impossible.”The quantity w’ above is what Lewis defines as "net work" namely work done by a chemical reaction, less the pressure volume work (done by the reaction expanding against the atmosphere), that can be connected to a motor or other electrical system for a use (purpose). He continues: “We may think of:as the driving force of a chemical reaction.”Lewis and Randall's book would go onto become the most-cited thermodynamics textbook of all time. Edward Guggenheim (1901-1970)English chemical thermodynamicist 1933 In his Modern Thermodynamics by the Methods of Willard Gibbs, building on the previous work of Lewis and Randall, he stated the conditions for what is "natural" and "unnatural" in isothermal-isobaric surface-attached reaction conditions (earth-bound freely-running processes), namely the Lewis inequality for a natural process (dG < 0) and the Lewis inequality for an unnatural process (dG > 0).

Human-Social-Economic Gibbs free energy
See also: Social free energy, Social Gibbs free energy, Economic free energy, Social Gibbs energy, Human Gibbs free energy, etc.
In human thermodynamics, human free energy or "human Gibbs free energy" is the measure of the Gibbs free energy, Gibbs free energy change, or differential of Gibbs free energy of human chemical reactions (see: human chemical reaction theory). [2]
 Left: a rendition of relationship between Gibbs free energy G and the “creation” (synthesis) of a human or rather "human molecule"; from American physicist Daniel Schroeder’s 2000 Thermal Physics textbook. [10] Right: video of free energy applied to society. Right: an so-called Gibbs spontaneity chart (V) , showing the basic criterion for spontaneous and nonspontaneous reactions. (Ѻ)

References
1. Gibbs, Willard. (1873). "Graphical Methods in the Thermodynamics of Fluids", Transactions of the Connecticut Academy, II., pg. 309-342. April-May.
2. Thims, Libb. (2007). Human Chemistry (Volume Two), (preview) (human free energy, pg. 465; ch. 10: "Goethe's Affinities", pgs. 371-422). Morrisville, NC: LuLu.
3. (a) Vant’ Hoff, Jacobus H. (1884). Études de Dynamique Chimique (Studies in Chemical Dynamics) Amsterdam: F. Muller & Co.
(b) Van’t Hoff, J.H. (1896). Studies in Chemical Dynamics (Chemical Equilibrium, pgs. 143-54; Affinity, pgs. 229-36; Determination of the Work Done by Affinity, pgs. 237-41; Applications, 241-50; Electrical Work Performed by a Chemical Change, pgs. 251-73; Principle of maximum work, pgs. 224-28, 241-42); Revised and Enlarged by Ernst Cohen (trans. Thomas Ewan). 286-pgs. London: Williams & Norgate.
4. Guggenheim, Edward, A. (1933). Modern Thermodynamics by the Methods of Willard Gibbs (pg. 11). London: Methuen & Co.
5. Kubo, Ryogo. (1976). Thermodynamics (Divertissement 8: On the Names of Thermodynamic Functions). North Holland.
6. Clark, John O.E. (2004). The Essential Dictionary of Science. Barnes & Noble.
7. Allen, James P. (2008). Biophysical Chemistry (pg. 56). Wiley.
8. Warn, J.R.W. and Peters, A.P.H. (1996). Concise Chemical Thermodynamics (pg. 76). CRC Press.
9. Kriz, Ronald D. (2009). “Triple Point Thermodynamic Processes Associated with Gibbs free energy”, SV.VT.edu.
10. Schroeder, Daniel V. (2000). An Introduction to Thermal Physics (pg. 150). Addison Wesley Longman.
11. Shepherd, Dante. (2019). “Fugacity” (artist: Joan Cooke), Science Comic, North Eastern University.