Equations of Elective Affinities
Note: prerequisites to a proper understanding and reading of the "equations" section of the project are that the reader has previously taken courses in (a) partial differential equations, (b) chemical thermodynamics and (c) and is familiar with the subject of physical chemistry.
Decipherment | Equations
The most complex issue, that is in great need of addressing, is the discernment of the chemical thermodynamic "equations" that quantify each chapter of the novella. Famously, in 1882, German physicist Hermann Helmholtz, in his "On the Thermodynamics of Chemical Processes", proved that "free energy" is the true measure of chemical affinity and gave the following formulation for chemical affinity:
meaning that the affinities A are "active" only in the sense in which the free energy differentials dF of the reacting system decrease with differentials of time dt. The century to follow saw purifications of this logic, significantly in the works of: Nernst (1906), Lewis (1923), De Donder (1936), and to some extent Perrot (1998) in his dictionary descriptions of the relations of affinities and free energies. In short, the human interactions of each chapter of Goethe's novella need to be explained in terms of partial differential equations of Gibbs free energy or human free energies per extent of reaction. Over 130 years have now passed since Helmholtz's famous free energy = affinity proof, but the literary world has not yet caught up to this fact, in short the world of both cultures has not yet received the memo or as Hjalmar Boyesen, in his The Life of Goethe (1885), put it:
“It is difficult to overestimate the value of Goethe’s work to humanity. The bequest which he left to the world in his writings, and in the whole intellectual result of his life, is not as yet appreciated at its full worth; because, intellectually, the world has not yet caught up to him.”
The focus of the equation decipherment section of the project will be make sure the world receives the memo.
Human molecule perspective | 21st century
Another salient point that will be followed in the "equation decipherment" section of the EA:IAD project, will be that of the point of view presented with the 2002 publication and description of Sterner-Elser “human molecular formula”, according to which humans are now defined as "abstract molecules" that react with each other and the surrounding animate and inanimate world in a way that resembles large scale surface-attached chemical reactions (see: human chemical reaction theory), which are governed by the laws, science, and principles of chemical thermodynamics, which thus defines the subject of human chemical thermodynamics.
In plain speak, (a) humans are now defined as "abstract molecules" (human molecules), which as of 2011 is a standard thermodynamics textbook definition (see: human molecular formula), and (b) the "elective affinities" (or chemical affinities) that Goethe speaks of in his novella and in his correspondences (see: Goethe timeline: 1809-1832), as governing the relationships, are now quantified by Gibbs free energy, which in the summary words of American-born Canadian biophysical chemist Julie Forman-Kay, from her 1999 article “The ‘Dynamics’ in the Thermodynamics of Binding” operate such that:
“Whether two molecules [or human molecules] will bind [or debind] is determined [destiny] by the free energy change [human free energy] of the interaction, composed of both enthalpic and entropic terms.”
In short, this means that the interactions (bondings and debondings) in each of Goethe's chapter have to be deciphered in terms of "enthalpy" and "entropy" changes, a very complex task, to say the least. This is the objective that will occupy the equation decipherment section of the EA:IAD project.
Background Count Physical chemist/Chemical physicist 7 Chemical engineer 6 Thermodynamicist (Chemical/Statistical/Mechanical) 5 Chemist 4 Sociologist/Economist/Mathematician 4 Physicist 4 Physicist-Engineer 2 Physical biologist/Physiologist 2 Engineer (Materials science/Civil) 2 Philosopher 2 Psychologist 1 The academic background of each the above 39 human free energy theorists, which shows that 46 percent (or 13 individuals), the dominate portion of human free energy theorists are either: physical chemists, chemical physicists, or thermodynamicists.
Strangely, in spite of the fact that there are over 500+ human thermodynamic theorists, of which 40 are human free energy theorizers, the only known chemical thermodynamic "equation work" done on Goethe's great novella seems to have been that of American electrochemical engineer Libb Thims as discussed in his 2007 Human Chemistry chapter ten "Goethe's Affinities" and chapter eleven "Affinity and Free Energy".  Moreover, the above affinity-free energy equations, shown in various alternative formulations, seem to have been remained an "unnamed" equation, that is until its 2011 Hmolpedia coining as the "Goethe-Helmholtz equation".
The general reason for this dearth of activity on the chemical thermodynamics of Elective Affinities, is that, in large part, the modern physical scientist, i.e. chemical engineer, physical chemist, or chemical physicist, is only generally familiar with the term and concept of "free energy" and not with "affinity", which is a result of the fact that with the publication of Gilbert Lewis' 1923 Thermodynamics and the Free Energy of Chemical Substances, the word "affinity" was replaced, in the famous 1956 words of chemistry historian Henry Leicester, throughout the English-speaking world, with the term "free energy", as such the words: affinity, affinity chemistry, elective affinity, etc., are foreign terms to most modern physical scientists; resultantly, only those willing to dig into prolonged decade-long historical research into the subject will become aware of the deep and strong bridge that connects the two descriptions: affinities and free energies. To exemplify, it took Libb Thims exactly eleven years to track down Goethe and his Elective Affinities after becoming curious, in 1995, about the methodology of applying free energy prediction to human interactions (see: Thims history). Beyond this, the two cultures divide only further exasperates the situation.
Enthalpy | Heat content | Physical beauty
As far as is known, Libb Thims and German chemist Volker Wiskamp aside, almost no connection has been made, up to this point, between enthalpy and the chapters of Goethe's novella, a term, known formerly as "heat content" (Heike Kamerlingh-Onnes, 1909), that would best seem to be representative in Goethe's discussion of human "heats", e.g. sexual heat, social heat, frictional heat, etc., and human beauties that occur throughout the novella, one example of which are as follows: 
“Beauty acts with far greater force on both inner and outer senses, so that he who beholds it is exempt from evil and feels in harmony with himself and with the world.”
To give a visual of what exactly we are talking about here, the following is 1 to 10 scale "Hot or Not" depiction of morphed photos (based on an original set of about 30 real photos), per each scale demarcation, created in 2005 by imaging researcher Pierre Tourigny: 
The lower end of the scale (1-range), according average ratings are, from the physical attractiveness (outer beauty) point of view, considered less-beautiful, whereas, conversely, the photos at the higher end of the scale (10-range) are considered the most-beautiful, in terms of outer beauty. The last female, a morph of all 30 women, is deemed very attractive, nearly in ranking to that of the perfect 10 female by virtue of the averageness effect, i.e. that people in possession of "average" to the mean physical traits tend to be well-represented in the current generation, and hence by logical extrapolations, into future generations as well.
Those near the 10-point range, no doubt, possibly in the character of Ottilie, is what Goethe meant when he states in the novella:
“Beauty is everywhere a welcome guest.”
The modern-day perspective on the nature of the relationship between physical attractiveness (outer beauty), neurological attractiveness (inner beauty), heat, and enthalpy (itself composed of internal energy and pressure volume work energy), is indeed a very complex subject, to say the least, some of which has only recently been touched on in the 2007 work of Libb Thims.  The equations decoding sections of the EA:IAD project will attempt to further elaborate and dissect this complex puzzle.
To corrobrate the above statement by Goethe, i.e. that “beauty is everywhere a welcome guest”, a 2012 survey of 25 men, by Thims, who where shown the above photos, via a sliding ruler hot-to-cold smartphone scale (namely the continuous horizontal scale shown on the Hmolpedia hot article page), and asked to stop at the lowest female that they, while in an inhibition-lowered inebriated state, would have sex with, if approached by her, while at at a party, found that the average answer was the female at 6.04, below which sex would not occur, with a range of 3.4 to 9 of bottom-barrel female.  In short, every male stated that they would have sex with female 9.0 or above, thus lending credence to Goethe's statement that "beauty [at 9.0 or above] [in regards to willingness to have sex] is everywhere a welcome guest."
Historical thermodynamic connections
The following sections gives an outline of the known thinkers, Thims aside, to have made connections between Goethe's novella and thermodynamics, which seem to be mostly entropy connections, in a verbal sense.
In 1991, American Goethean scholar Julie Reahard competed her PhD dissertation-turned-book Chaos Theory, Hermeneutics, and Goethe’s Die Wahlverwandtschaften on the subject of analyzing Goethe's Elective Affinities in terms of chaos theory. She mentions entropy in passing on one page, in her discussion of the brain imaging chaos theories of American mathematical physicist Mitchell Feigenbaum and his studies of Goethe's color theory in relation to Newton's color theory: 
“It seems only right, then, that, we turn the work of Feigenbaum and the other ‘chaos’ theoreticians like him, who see ‘order in chaos’, back upon Goethe in order to illuminate his meanings. I must emphasize these thinkers’ common ability to see ‘order in chaos’ because Goethe’s thought has much in common with this branch of chaos theory and little in common with the branch equating chaos with entropy and entropy with a complexity that is rich in information.”
Here, Reahard, in her attempt to explain Goethe’s Elective Affinities in terms of 1990s chaos theory models, gives way to the view that she is relatively a green theorist attempting to make sense of chaos theory and or complexity theory in the late 20th century, a field largely dominated by Prigoginean models of thermodynamics and of order being created past the bifurcation point. This, however, is a model, which to note has fallen out of flavor as of the 2010s, that has little to do with what Goethe had in mind in the writing of his Elective Affinities. Prigogine, although a citer of Goethe’s Elective Affinities (see: footnote 2.5), was against human free energy models, commenting, for example, in his 1977 Nobel Lecture that Helmholtz free energy does not explain or apply to the structures, such as towns and trees, that we see around us.  In any event, it is interesting here to see someone from the humanities side of the two cultures attempting to discern Goethe's hidden "one idea", that he is said to have implanted into his novella, in terms of modern "comprehensive approaches", as she says.
British playwright Tom Stoppard, in his 1993 play Arcadia, a modern day remake attempt of Elective Affinities, that takes place via a juxtaposition of the years 1809 (the year of publication of Elective Affinities) and 1989, makes connection between entropy and affinity, through storyline, via his mentions of the second law, heat, and sexual energy, etc. To exemplify, the following is from act one scene one: 
Thomasina: You can't stir things apart.
Septimus: No more you can, time must needs run backward, and since it will not, we must stir our way onward mixing as we go, disorder out of disorder into disorder until pink is complete, unchanging and unchangeable, and we are done with it forever. This is known as free will or self-determination.
Septimus: “if everything from the furthest planet to the smallest atom of our brain acts according to Newton's law of motion, what becomes of free will?”
Stoppard latter goes on to bring in the second law and heat death into the storyline:
Valentine: “We are all doomed? It’s called the second law of thermodynamics.”This is also mixed in with discussion of heat and sex; and to some extent, Stoppard attempts to explain sex via equations, but he chooses the equation of Fermat's last theorem, i.e. that no three positive integers a, b, and c can satisfy the equation:
for any integer value of n greater than two, for some reason. Stoppard's characters also discuss the Laplace's demon model of determinism.
In 1992, German solid state thermodynamicist and socio-economic physicist Jurgen Mimkes began working on a "society as a many particle system" model to explain various sociological economic phenomena, such as migrations, war, wealth, happiness. His first major publication was his 1995 "Binary Alloys as a Model for a Multicultural Society" Journal of Thermal Analysis article, in which he outlined a physics-based integration and segregation model.
In his 2000 article "Society as a Many Particle System", Mimkes begins to give some historical background, and comments how Goethe and his Elective Affinities was the first major thinker to use chemistry to explain human separations and coalitions. In his many articles, chapters, and talks, in the decade to follow, he frequently opened to an historical introduction of how Empedocles of Acragas (495-435BC) was the first to introduce the physical chemistry-based model of how aggression in countries at war, like Bosnia, could be explained in terms of how atoms and people like and attract each other, dislike or repel each other, or are indifferent; and comments how Goethe discussed this idea in his Die Wahlverwandtschaften, and supposedly how Wolfgang Weidlich (1971) is somehow the more recent advocator of this Empedocles-Goethe model, for his sociodynamics explanations using modern thermodynamics. 
This, however, is about as far as Mimkes goes, the rest of his rather statistical physics-based slanted derivation spent on an attempt to found socio-economic dynamics on a variant of the Lagrangian formulation. In his 2004 essay “Interactions of Heterogeneous Agents in Stochastic Socio-economic Systems”, to exemplify, Mimkes does, to his credit, discuss both free energy and entropy, although not enthalpy (as almost nobody, Thims aside discusses this), but his human thermodynamic variable assignments are very tenuous at best, almost guesses it would seem? The gist of his derivation is that firstly, he states that in circa 1780s Joseph Lagrange “formulated a principle that applies to stochastic systems with interactions (constraints)”, which Mimkes defines or rather introduces, without derivation, as follows: 
L = E + T ln P → Max
where L is the Lagrange function, E is the constraints of the system, i.e. the interactions of the agents, the function “ln P” the entropy of the system, according to the Boltzmann-Planck entropy formulation, T what Mimkes calls the Lagrange factor, according to which he states that the “stochastic system will be stable if the Lagrange function is a maximum.” He then states the following:
“The meaning of the functions L, E, and T will depend on the type of systems investigated: in atomic systems E will be determined by the energy of atomic interactions, L will be the free energy, and T the mean energy (temperature).”
which seems to imply that Mimkes has the Helmholtz free energy, shown below, in mind?
H = E – TS
But in this case, the following form of the equatoin will be true, on spontaneous progression to the most stable state:
E – TS → min
which seems to contradict Mimkes’ formulation, which amounts to:
E + TS → max
In any event, Mimkes then jumps to the extrapolation:
“In social systems, E will be given by the emotions of social interactions, L will be the common happiness and T the tolerance of disorder.”
“In economic systems, E will be the property value or price of goods, L the common profit, and T the average wealth or standard of living (Mimkes 2000).”
He then concludes with the following three axioms, which he seems to think are correct:
“Atomic systems will be stable if the negative free energy L is at maximum.”
“Social systems will be stable if the common happiness L is at maximum.”
“Economic systems will be stable if the common profit is at maximum.”
Here, although this is an interesting first stab at the elective affinities problem, there are numerous issues, too many to list at this point, the first of which however, is his usage of “negative free energy”, which is a type of convoluted thermodynamics gone awry, similar to “negative entropy” and its tumultuous fall from grace.
French supramolecular chemist Jean-Marie Lehn, in his 1995 Supramolecular Chemistry is well-aware of Goethe’s Elective Affinities, on one hand, and also well-aware that enthalpy and entropy determinants are the factors of supramolecular bonding, but does not seem to be able to put the two pieces of the puzzle together. In fact, on the second page of his textbook (page 2) he opens to Goethe in the following way: 
“Supramolecular chemistry is a sort of molecular sociology! Non-covalent bonds define the inter-component bond, the action and reaction, in brief, the behavior of the molecular individuals and populations: their social structure as an ensemble of individuals having its own organization; their stability and their fragility; their tendency to associate or to isolate themselves; their selectivity, their ‘elective affinities’ and class structure, their ability to recognize each other; their dynamics, fluidity or rigidity or arrangements and of castes, tensions, motions and reorientations; their mutual action and their transformations by each other.”
Then, in his later chapter subsection (page 51) on supramolecular thermodynamics, he explains receptor-substrate bonding to form a supermolecule as such:
“A receptor-substrate supermolecule is characterized by its geometric (structure, conformation), its thermodynamic (stability, enthalpy and entropy of formation) and its kinetic (rates of formation and of dissociation) features.”
Here, we see an example of one “not able to see the forest amid the trees”. Lehn clearly knows about tree one (Goethe and his Elective Affinities) and tree two (that entropy and enthalpy determines whether or not a receptor molecule and substrate molecule will bond to form a supermolecule), but is not able to place the two trees together at once so to see the forest (that entropy and enthalpy determinants are the factors in the formation of human molecular couples and social structures).
In 2004, Japanese chemical engineer Tominaga Keii, in his chemical thermodynamics chapter, devotes a section, entitled "Chemical Affinity in 1806", to Goethe's novel, as what seems to be an amusement diversion to himself, but does no equation derivation and comments it a somewhat odd turn of phrase that Goethe's Elective Affinities "added no scientific value"; very strange indeed?
In 2004, writer Jang-Hyok An, in his Goethe’s Elective Affinities and the Other of Reason, has a chapter section loosely translated as the “entropy of the thing”, wherein the discusses elective affinities, entropy, and heat death, but the main citation seems to be Jeremy Rifkin, which is mostly garbage writing, intermixed with discussions on chaos theory. 
1. (a) Thims, Libb. (2007). Human Chemistry (Volume One). Morrisville, NC: LuLu.
(b) Thims, Libb. (2007). Human Chemistry (Volume Two) (§10: Goethe’s Affinities, pgs. 371-422; §11: Affinity and Free Energy, pgs. 423-468). Morrisville, NC: LuLu.
2. Stoppard, Tom. (1993). Arcadia (heat, 6+ pgs; sexual energy, pg. 33; atom, 2+ pgs; second law, pg. 65; quote: "Ah. The attraction that Newton left out. All the way back to the apple in the garden.", pg. 74). London: Faber and Faber.
3. Mimkes, Jurgen, Hillebrand, M., Frund, F, Denk, Ch., and Willis, G. (2004). “Interactions of Heterogeneous Agents in Stochastic Socio-economic Systems”, in The Complex Dynamics of Economic Interactions: Essays in Economics and Econophysics (pg. 381-) (editors: Mauro Gallegati A.P. Kirman). Springer.
4. (a) Lehn, Jean-Marie. (1995). Supramolecular Chemistry (molecular sociology, pg. 2; §4.5: Supramolecular Dynamics, pgs. 51-53). New York: VCH.
(b) Thims, Libb. (2007). Human Chemistry (Volume Two) (Lehn, pgs. 450-51). Morrisville, NC: LuLu.
5. Mimkes, Jürgen. (2000). “Society as a many-particle System” (abs), Journal Thermal Analysis, 60(3):1055-69.
6. Stoppard, Tom. (1993). Arcadia (disorder, pg. 5; second law, pg. 65). Samuel French, Inc.
7. (a) Reahard, Julie A. (1997). Aus einem unbekannten Zentrum, zu einer nicht erkennbaren Grenze: Chaos Theory, Hermeneutics, and Goethe’s Die Wahlverwandtschaften (entropy, pg. 20). Rodopi.
(b) Mitchell Feigenbaum – Wikipedia.
8. Prigogine, Ilya. (1977). “Time, Structure and Fluctuations”, Nobel Lecture, Dec. 08.
9. Hot or Not – Wikipedia.
10. Thims, Libb. (2012). "Lowest Female to Have Sex With" (N=25), IoHT research project, May 30-31.
11. Note: in 2011, following the posting of Libb Thims "Sexual Heat | Pop Quiz" video, some have vigorously objected to enthalpy = physical attractiveness model, including: Ryan Grannell and Lubos Motl to name a two.
12. (a) An, Jang-Hyok. (2004). Goethe’s Elective Affinities and the Other of Reason: the Micro and Macro of Constellation of Otherness as atopic Counter Instance for Identity Constraint (Goethes"Wahlverwandtschaften" und das Andere der Vernunft: Die Mikro- und Makrokonstellation der Andersheit als atopische Gegeninstanz zum Identitätszwang) (§2.7: The order of discourse (as ordinal instance) or the entropy of the thing (Die Ordnung des Diskurses (als Ordnungsinstanz) oder die Entropie der Ding), pgs. 78-80). Königshausen & Neumann.
(b) Found via Google search: “Entropie, Goethe, Wahlverwandtschaften”.
● Gladyshev, Georgi P. (2012). “Love: the State of Living Organism” (Russian → English), WordPress.com, Mar. 9.
● Gladyshev, Georgi P. (2012). “On the Thermodynamics of Feelings and ‘Goethe’s Affinity’” (Russian → English), WordPress.com, Sep. 24.