In equations, the following formula:

$A = - \Delta G \,$

is called the "affinity-free energy equation" or "Goethe-Helmholtz equation" and is the relation between the chemical affinity A and Gibbs free energy change ΔG a closed isothermal-isobaric reacting system. The free energy affinity equation is the main governing equation of human existence, to the affect that the sum of affinities of all humans in a system will be equal to the negative Gibbs free energy change of that system on going between two states of existence. In terms of enthalpy and entropy, this equation reduced to:

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

which states that affinity equals the change in the entropic energy (or bound energy) "TΔS", i.e. the quantity of the product of the absolute temperature of the system T times the entropy change ΔS, less the change in the enthalpic energy "ΔH" (enthalpy change) of the system. If the enthalpy function is expanded:

$A= T\Delta S - \Delta U - P\Delta V \,$

we see that not only is affinity a function of bound energy, but also internal energy and pressure-volume work energy.

Etymology
The name Goethe-Helmholtz equation is an "Hmolpedia assigned name"; premised on the fact that a prior established name for this equation, in the literature, is generally lacking or nonexistent. In this direction, the name "Goethe-Helmholtz equation" has been assigned to this relation, on the premise that German polymath Johann Goethe was the first to explain how chemical affinity A applies to human affairs (1809), in standard physical chemistry format, and German physicist Hermann Helmholtz was the first to prove that free energy change (isochoric-isobaric: Helmholtz free energy (ΔF) or isothermal-isobaric: Gibbs free energy (ΔG)) is the true measure of chemical affinity (in his disproof of the Thomsen-Berthelot principle). This should not be confused with either the Helmholtz equation (acoustics) or Gibbs-Helmholtz equation (electrochemical thermodynamics).

Humans
Greek philosopher Empedocles was the first to state that chemical affinity applies to humans, in love and war, with his various circa 450BC chemistry aphorisms.

The first to formulate a "standard model of human existence", structured on standard physical chemistry, in particular the logic of affinity reactions, was German polymath Johann Goethe as found in his 1809 novella Elective Affinities.

The first to state the relation between affinity and Gibbs free energy is the governing equation of human relationships, in both intimate and social terms, was American electrochemical engineering Libb Thims in his 2007 Human Chemistry. [1]

In lay terms, assuming all human interactions to be chemical reactions, pure in simple, and assuming that the sum of human attractions (desires) and repulsions (aversions) to the forces of chemical affinities, in the words of Ilya Prigogine:

“All chemical reactions drive the system to a state of equilibrium in which the affinities of the reactions vanish.”

Progogine summarizes this by the following three inequality rules: [4]

Some of this logic has been quantified in terms of "Gottman stability ratio" of attractions and repulsions for successful marriage reactions, in the 1990s studies of American mathematical psychologist John Gottman.

Thomsen-Berthelot principle
Independently, in 1854 and 1864, chemists Julius Thomsen and Marcellin Berthelot posited that heat is the driving force of chemical reactions and thus the measure of chemical affinity. This came to be know as the Thomsen-Berthelot principle.

Chemical thermodynamics
The proof that free energy, and not heat, is the "true" measure of chemical affinity, was first given by German physicist Hermann Helmholtz in his 1882 “On the Thermodynamics of Chemical Processes”, in which he worked out the derivation specifically for the isochoric-isobaric free energy or Helmholtz free energy, as it has come to be called. [2] In modern notation, Helmholtz showed that:

$A = - \Delta F \,$

or in expanded form:

$A = T \Delta S - \Delta U \,$

which states that affinity A equals the change in the entropic energy (or bound energy) "TΔS" (entropy change) less the change in the internal energy "ΔU" (internal energy change) of the system.

Absolute zero
In his 1906, German physical chemist Walther Nernst, in his heat theorem (or third law of thermodynamics as it has came to be called), showed that the Thomsen-Berthelot principle is only true at absolute zero, in the sense that as the temperature approaches zero, entropy change becomes zero:

$\lim_{T\rightarrow 0} \Delta S = 0$

and when substituted into the Helmholtz formula for affinity (above):

$A = - \Delta U \,$

at which point heat, or rather the heat released form the internal energy (bond energy) chances "ΔU" of the chemical reactions, becomes the true measure of chemical affinity A.

Gibbs free energy
The first to work out the proof that free energy is the measure of chemical affinity specifically for isobaric-isothermal reacting systems, in which case the free energy is Gibbs free energy, the type of reactions that freely occur on the surface of the earth, such as do human chemical reactions, seems to have been Belgian mathematical physicist Theophile de Donder, sometime beginning in 1920, culminating in his 1936 Thermodynamic Theory of Affinity. De Donder stated the relation in terms of variation in the extent of reaction: [3]

$A=-\left(\frac{\partial G}{\partial \xi}\right)_{p,T}$

which states that the affinity A of a closed isothermal-isobaric reacting system is equal to the negative of the variation of the Gibbs free energy ∂G per variation in the extent of reaction ∂ξ.

De Donder's formulation of chemical affinity is said to be based on American engineer Willard Gibbs' 1876 concept of chemical potential. [4] The definition of what is called "standard affinity", according to French thermodynamicist Pierre Perrot, is:

$A^{\circ} = - \Delta_r G^{\circ} \,$

which according to Perrot, equates to:

$A^{\circ} = \sum_{i=1}^k - \nu_i \mu^{\circ}_i \,$

such that the affinity of the chemical reaction is calls "standard" when each constituent is taken in its standard state. [5]

Thermodynamic coupling
Add: affinity coupling (De Donder, pg. 113);
Add: free energy coupling

References
1. (a) Thims, Libb. (2007). Human Chemistry (Volume One) (pg. xvi). Morrisville, NC: LuLu.
(b) Thims, Libb. (2007). Human Chemistry (Volume Two) (pg. 443). Morrisville, NC: LuLu.
2. Helmholtz, Hermann. (1882). “On the Thermodynamics of Chemical Processes”, in: Physical Memoirs Selected and Translated from Foreign Sources, 1: 43-97. Physical Society of London, Taylor and Francis, 1888.
3. (a) De Donder, Théophile. (1920). Lecons de Thermodynamique et de Chemie Physique (Lessons of Thermodynamics and Physical Chemistry), (pg. 117, formula 318). Paris: Gauthier-Villars.
(b) De Donder, T. (1922). “Article”. Bull. Ac. Roy. Belg. (Cl. Sc.) (5) 7: 197-205.
(c) De Donder, T. (1927). “L’ Affinite”, Paris: Gauthier-Villars.
(e) De Donder, Theophile and Van Rysselberghe Pierre. (1936). Thermodynamic Theory of Affinity: A Book of Principles, (pg. 2).Oxford University Press.
4. Kondepudi, Dilip and Prigogine, Ilya. (1998). Modern Thermodynamics – from Heat Engines to Dissipative Structures (4.1: Chemical potential and Affinity: the Driving Force of Chemical Reactions, pgs. 103-13). New York: John Wiley & Sons.
5. Perrot, Pierre. (1998). A to Z of Thermodynamics (Standard affinity, pg. 284). Oxford University Press.