# Goethe-Helmholtz equation

 A version of the Goethe-Helmholtz equation designed (2009) by American electrochemical engineer Libb Thims for the 200th anniversary of the publication of Goethe's 1809 Elective Affinities, his self-defined "best book". [1]
In equations, the Goethe-Helmholtz equation, an Hmolpedia-assigned name, refers to any equation, of differing formats, relating the affinities of an isothermal-isobaric reaction to the change in the Gibbs free energy of the system; the most common version of which is:

$A = - \Delta G \,$

where A is the chemical affinity or driving forces, of the isothermal-isobaric reaction, and ΔG is the change in the Gibbs free energy of the reaction or process.

Etymology
The name “Goethe-Helmholtz equation” is an Hmolpedia assigned name, done in honor of German polymath Johann Goethe who first applied “affinities” as sets of regulating principles to human interactions and affairs (1796-1832), viewing human interactions as human chemical reactions, and who in defense of his theory, famously commented (Dec 1809), emphatically, in response to a women who reproached him in the street, that:

“The principle illustrated in the book is true and not immoral.”

and German physician-physicist Hermann Helmholtz, the last universal genius, following Goethe, who famously proved ("On the Thermodynamics of Chemical Processes", 1882) that free energy is the true measure of affinity—hence, Gibbs free energy is the true measure of the affinities of human existence.

Goethe | human chemistry
In 1796, German polymath Johann Goethe, in his "Third Lecture on Anatomy", outlined the first inklings of his new theory that human behaviors, interactions, relationships, and reactions are governed by chemical forces or affinities:

“To facilitate our comprehension of the concept of organic existence, let us first take a look at mineral structures. Minerals, whose varied components are so solid and unchanging, do not seem to hold to any limits or order when then combine, although laws do determine these conditions. Different components can be easily separated and recombined into new combinations. These combinations can again be taken apart, and the mineral we thought destroyed can soon be restored to its original perfection.

The main characteristic of minerals that concerns us here is the indifference their components show toward the form of their combination, that is, their coordination or subordination. There are, by nature, stronger or weaker bonds between these components, and when they evidence themselves, they resemble attractions between human beings. This is why chemists speak of elective affinities, even though the forces that move mineral components [or humans] one way or another and create mineral structures are often purely external in origin, which by no means implies that we deny them the delicate portion of nature’s vital inspiration that is their due.”

 The arrow represents the "elective affinity", external in origin to the reactants and products, humans or chemicals, as Goethe stated in 1796.
The depiction of this, in the mind of Goethe, at this time, is shown adjacent; a model which takes it cues from Scottish chemist-physician William Cullen 1757 Glasgow University lecture notes, wherein he pioneered the use of diagrams as chemical ‘equations’ and was the first to diagram the force of affinity using arrow notion, which he explains as follows: [2]

“The dart → between them expresses the elective attraction; when I put a dart with the tail to one substance and the point to another, I mean that the substance to which the tail is directed unites with the one to which the point is directed more strongly than it does with the one united to it in the crotchet {” .

Using this chemical equation reaction model, as told through Swedish chemist Torbern Bergman (A Dissertation on Elective Attractions, 1775), Goethe presented the final version of his theory in the form of coded gestalt in his 1809 physical chemistry based novella Elective Affinities (see: Goethe timeline), about which he latter commented that "the principles in it are true" and that it was greatest work.

To exemplify, using the depicted Bergman-style example reaction (above), the Goethe-Cullen conception equates to the model wherein, for example, the reaction: AB + C AC + B (in modern terms), where AB and AC, technically called "dihumanide molecules", are held by human chemical bonds, A≡B and A≡C, and the "force", symbolized by the dart (), is the electromagnetic force, acting "external" to the reactants (people or chemicals), in the form of an exchange force.

Thermal theory of affinity
In the middle of the 19th century, independently proposed by Danish chemist Julius Thomsen (1854) and French chemist Marcellin Berthelot (1864), the “thermal theory of affinity” was proposed, arguing that all chemical action, not due to external energy, tends to the production of the body or bodies which set free the greatest heat. [3] This theory, however, soon began to show deficiencies, such as not being able to explain bodies formed as a result of endothermic reactions.

Thermodynamic theory of affinity
In 1882, German physicist Hermann Helmholtz, in his famous “On the Thermodynamics of Chemical Processes”, combined the earlier chemical thermodynamics work of American engineer Willard Gibbs with his own electrochemical thermodynamics work and, with the following statement, effectively overthrew the thermal theory of affinity:

“Given the unlimited validity of Clausius' law, it would then be the value of the free energy, not that of the total energy resulting from heat production, which determines in which sense the chemical affinity can be active.”

and gave the following equation formulation for affinity in relation to the direction of reaction changes spontaneously occurring:

where, in modern terms, F is the Helmholtz free energy, and t is time, which states that the affinities will only be active when the system of the chemical process shows a decrease in free energy with time.

Heat theorem | Absolute zero
In the late 1880s, German physical chemist Walther Nernst, having an excellent grasp of the relationship between free energy, affinity, and the external work output of chemical processes, as well as their relation to the natural advancement of processes in nature, set out to solve the long-sought question of the determination and measurement of the “chemical affinity” of a reaction. In is 1893 Theoretical Chemistry from the Standpoint of Avogadro's Rule and Thermodynamics, Nernst explained that neither the “heat-toning” (heat releasing) effects of a reaction, as French chemist Marcellin Berthelot had argued, in his 1875 principle of maximum work, nor the velocity of a reaction can function as a measure of affinity, but rather only the free energy can. In summary Nernst states: [3]

“Since every chemical process, like every process of nature, can only advance without the introduction of external energy only in the sense in which it can perform work; and since also for a measure of the chemical affinity, we must presuppose the absolute condition, that every process must complete itself in the sense of the affinity—on this basis we me may without suspicion regard the maximal external work of a chemical process (i.e. the change of free energy), as the measure of affinity. Therefore the clearly defined problem of thermo-chemistry is to measure the amounts of the changes of free energy associated with chemical processes, with the greatest accuracy possible … when this problem shall be solved, then it will be possible to predict whether or not a reaction can complete itself under the respective conditions. All reactions advance only in the sense of a diminution of free energy, i.e. only in the sense of the affinity.”

Here we see the first inklings of the view that, when scaled up to the human-human reaction level, in the future, when the details of this problem are worked out, it will be possible to "predict" whether or not a given human chemical reaction (e.g. love the chemical reaction) can complete itself under the respective conditions.

In his 1906 The New Heat Theorem, Nernst had formulated his so-called "heat theorem or "third law of thermodynamics", as it eventually came to be called), showed that the Thomsen-Berthelot principle (thermal theory of heat) 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. Nernst was awarded the 1920 Nobel Prize in chemistry for this work.

Gibbs free energy | Driving force of a reaction
In 1923, American physical chemist Gilbert Lewis, former student of Nernst, published the 1923 textbook Thermodynamics and the Free Energy of Chemical Substances, wherein, 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 (using modern notation):

“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, the 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.”

This chapter subjection, in the summary words of American chemistry historian Henry Leicester (1956), resulted to replace the notion of "affinity" with the notion of "free energy" throughout the corpus of modern science.

Brussels school
In 1936, Belgian physicist Theophile de Donder, in his Thermodynamic Theory of Affinity, head of the so-called "Brussels school of thermodynamics", using a parallel albeit slightly different approach, presented a formulation where the symbol "A" for affinity as the negative partial of the partial of the Gibbs free energy per unit partial of extent of reaction for a change in a isothermal isobaric system: [4]

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

and would go on to discuss this, using coupling theory, in terms of how reactants and chemicals can be made to go, move, or react in a direction contrary to their own affinity, or in an anthropomorphic sense, as Goethe would have seen things, in a direction contrary to their own will. De Donder's approach, according to Indian-born American physical chemist Dilip Kondepudi, of his school, is said to be based on American engineer Willard Gibbs' 1876 concept of chemical potential. [5] In this sense, the definition of what is called "standard affinity", according to French thermodynamicist Pierre Perrot (A to Z of Thermodynamics, 1998), 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. [6]

Dissection
The above logic, in simplified modern "Delta", before (initial state) and after (final state), notation, simplifies to the following:

$A = - \Delta G \,$

which shows, conclusively, that the theory contained in Goethe’s 1809 Elective Affinities, or "principles", as he called them, are true, as he so rightly said, and that his novella was a modern-day treatise on the explication of Gibbs free energy changes involved in the determination of human chemical reactions. In reduced modern format, Goethe showed, over two-hundred years ago, that the affinities, or the forces of love and hate, which are balanced in stability ratios in successful relationships, of human relationships are functions of enthalpy change ΔH, entropy change ΔS, and temperature T:

which, noting that H = U + PV, equates to:

meaning that human affinity, or the affinity between two people, i.e. the force of reaction between people, according to Helmholtz and Goethe, is a function of temperature, entropy, internal energy, pressure, and volume. This means that affinity will be favored, between two people, when there is an increase in entropy, a decrease in internal energy, and a decrease of volume, of the interactive system, over time. One can expand on this equation in more detail, by noting that Rudolf Clausius defined internal energy of the system, which in this case concerns a system of interacting people or human molecules, as the sum of the vis viva and the ergal, or U = T + J, which referring to a change Δ over time is:

Subsequently, the measure of the affinity between two people, wherein A > 0 for spontaneously favored relationships (or ΔG < 0, thermodynamic sense, i.e. according to the spontaneity criterion), is expressed by the equation:

meaning that in addition to an entropy increase, ΔS > 0, or transformational content increase (meaning that heat was transformed in to system internal evolution work), which signifies that the body or boundaried system of the relationship has transformed or evolved over time, and a volume decrease, ΔV < 0, associated with favorable relationships, meaning that spatial movements of the pair come together over time rather than diverging (simplified by saying that two homes become one), a vis viva decrease, ΔT < 0, and an ergal decrease, ΔJ < 0, will also be associated with favored relationships. A decrease in vis viva seems to make intuitive sense, in that a couple nearing their golden wedding anniversary will invariably tend to have less kinetic energy then a newly minted couple, in the sense that the former's daily movements will often be slow and tending to be confined to the kitchen, bathroom, and television room. The subject of an ergal decrease over time as associated with stable relationships requires more thought. In formulaic terms, ergal change over time is defined as:

Thus, a negative ergal change, which be the above reasoning is associated with favored relationships, implies that the ergal at the start of the relationship, Ji, must be greater than the ergal at the end of the relationship, Jf. In the terminology of William Hamilton, this is expressed by saying that the was a decrease in the force function of the relationship over time or in the terminology of William Rankine, this is expressed by saying that there was a decrease in the potential energy of the relationship over time.

Said another way, Goethe showed that human relationships (affinity relationships) are governed by the laws of chemical thermodynamics.

Coupling
With the publication of German-born American biochemist Fritz Lipmann’s 1941 paper “Metabolic Generation and Utilization of Phosphate Bond Energy”, which showed that, in nature, endergonic reactions are coupled to exergonic reactions, we know that human chemical reactions between people are coupled to each other, as Goethe showed in his novella, or that bond energy released from some certain energy dense human bonds, acts to drive less energetic human reactions that wound not otherwise go on their own. Thus, wherein affinity equates to free energy G and elective affinity reaction equates to chemical reaction in modern terms, Goethe pioneered the science of human chemical thermodynamics, two-hundred years ahead of its time.

References
1. Elective Affinities T-Shirt (by Libb Thims) – Zazzle.com.
2. Crosland, M. P. (1959). “The use of diagrams as chemical ‘equations’ in the lecture notes of William Cullen and Joseph Black.” Annals of Science, Vol 15, Num 2, June.
3. (a) Sprague, John T. (1892). Electricity: its Theory, Sources, and Applications (pg. 305). E. & F.N.
(b) Thims, Libb. (2007). Human Chemistry (Volume Two) (thermal theory of affinity, pg. 434). Morrisville, NC: LuLu.
4. De Donder, Theophile. (1936). Thermodynamic Theory of Affinity: A Book of Principles (pg. 2). Oxford University Press.
5. 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.
6. Perrot, Pierre. (1998). A to Z of Thermodynamics (standard affinity, pg. 284). Oxford University Press.
7. (a) Nernst, Walther. (1893). Theoretical Chemistry from the Standpoint of Avogadro's Rule and Thermodynamics (Theoretische Chemie vom Standpunkte der Avogadroschen Regel und der Thermodynamik). Stuttgart, F. Enke, 1893 [5th edition, 1923].
(b) Nernst, Walther. (1895). Theoretical Chemistry: from the Standpoint of Avogadro’s Rule & Thermodynamics (697-pages) (section: The Measure of Affinity, pgs. 586-88). MacMillan and Co.
(c) Nernst, Walther. (1904). Theoretical Chemistry: from the Standpoint of Avogadro’s Rule & Thermodynamics (771-pages). MacMillan and Co.