A depiction of a van't Hoff equilibrium box, according to American chemist Clayton Gearhart, of the type said to have been used by German physical chemist Walther Nernst in 1893 to calculate the available work A of an isothermal reversible reaction in which reactants entered on the left and products exited on the right, the entire process mediated by moving pistons and semi-permeable membranes. [4]
In thermodynamics, a van't Hoff equilibrium box, or “equilibrium box” or “van’t Hoff box”, is a vessel of unchanging volume (fact check?) in which various substances taking part in a given reaction are present in the equilibrium state, wherein the walls of the vessel are suitably permeable to given substances, but impermeable to others. [1] The “equilibrium box idea”, supposedly, can only be applied to reactions taking place in a homogeneous system, such as gases or dilute solutions. [1] The aim of the operation of the van’t Hoff equilibrium box is to produce useful work arising from the following scheme:

reaction → energy → heat → useful work

without violation of the second law of thermodynamics, even at constant temperature, using heat evolved by a reaction. [2]

History
The equilibrium box was invented by Danish physical chemist Jacobus van't Hoff in circa 1886 and supposedly is the basis for the equation that relates free energy change (ΔG or ΔF) to the equilibrium constant (k) of a chemical reaction. [3] The following is a 2003 summary of van't Hoff and his equilibrium box conception by American biochemist, science historian, and physiologist Robert Root-Bernstein: [5]

“When I turned to the history of chemistry in graduate school, it was perhaps inevitable that I should therefore have been drawn to the man who best combined both physical chemistry and stereochemistry in his work, Jacobus van’t Hoff. The beauty of three-dimensional chemical forms and their interactions still intrigues me and occupies my daily research, but the most intense aesthetic experience I have ever had in science (outside of the experience of my own rare illuminations) came when I read van’t Hoff’s original derivations of his equations describing the thermodynamic properties of solutions. I was struck first by the brilliance of the analogy he created between the adiabatic cycle that Clausius had imagined for pistons working on a gas and the equivalent cycle that van’t Hoff mentally devised using pistons working on osmotic pressure by means of semipermeable membranes. Beyond that – far beyond that – was the experience of reading the equations he then derived describing the equivalent of PV = nRT for solutions (van’t Hoff 1887). It was the most brilliant, insightful poem I had ever read!

Van’t Hoff’s poem [“The Role of Osmotic Pressure in the Analogy between Solutions and Gases”, 1887] deriving the laws of solutions wasted not a symbol. Each one made numerous connections to existing principles so that each line became a nexus of meaning. His symbols, like a magic key, opened a door from the mansion of gas thermodynamics onto the vista of an entirely unexpected estate that nature had somehow hidden from everyone else. Order from disorder, sense from confusion, imagination, insight, surprise – van’t Hoff had it all. I suddenly understood a comment that van’t Hoff’s contemporary, Max Planck, had made in his Autobiography, to the effect that he was drawn to science when he encountered the first law of thermodynamics in high school. It appeared to him to be ‘like a sacred commandment […] sublime’ (Planck 1949). I suddenly appreciated viscerally how one could shudder before the majestic beauty of unexpected comprehension (Chandrasekhar 1987).”

The derivation supposedly done by German physical chemist Walther Nernst in his 1893 Theoretical Chemistry. Nernst, following the work of others, especially van't Hoff, is said to have started with the following equation:

$A - U = T \frac{dA}{dT} \,$

which reoccurs frequently in 1880-1930 German thermodynamics.
 A 1928 diagrams of an Equilibrium Boxes from English physical chemist John Butler's Fundamentals of Chemical Thermodynamics, which can be used to deduce the equilibrium constant. [4]

To note, this last equation, in modern thermodynamic potential notation, supposedly, can be written as:

$\Delta F - \Delta U = T \Bigg ( \frac{\partial \Delta F}{\partial T} \Bigg )_V \,$

In any event, using this equation, Nernst was able to show that for an isothermal chemical reaction, occurring in an "equilibrium box", that:

$A = RT \ln K \,$

where K is the equilibrium constant as defined in the circa 1865 work of Norwegian mathematicians Cato Guldberg and Peter Waage and their "law of mass action". [4] To connect this last equation to free energy, the 1882 Goethe-Helmholtz equation:

$A = - \Delta G \,$

can then be substituted in to yield:

$\Delta G = - RT \ln K \,$

which is the standard modern day equation relating the equilibrium constant to free energy change.

Christopher Hirata
Relationship physics

References
1. Lewis, William C.M. (1920). A System of Physical Chemistry (section: Van’t Hoff’s ‘Equilibrium Box”, pg. 103). Longmans, Green.
2. Bazhin, N.M. and Parmon, V.N. (2007). “Conversion of Chemical Reaction Energy into Useful Work in the Van’t Hoff Equilibrium Box” (abs), Journal of Chemical Education. 84(6): 1053.
3. Laidler, Keith. (1993). The World of Physical Chemistry (section: thermodynamics of electrochemical cells, pgs. 218-19). Oxford University Press.
4. Gearhart, Clayton A. (2011). “Walther Nernst, Max Planck, Albert Einstein, and the Third Law of Thermodynamics”, 18-pgs. St. John’s University.
Butler, John A.V. (1928). The Fundamentals of Chemical Thermodynamics: Part 1: Elementary Theory and Electrochemistry (pgs. 87, 93). Macmillan and Co.
5. (a) Root-Bernstein, Robert. (2003). “Sensual Chemistry: Aesthetics as a Motivation for Research”, HYLE: International Journal for Philosophy of Chemistry, 9(1):33-50.
(b) Van’t Hoff, Jacobus. (1887). “The Role of Osmotic Pressure in the Analogy between Solutions and Gases” (“Die Rolle des osmotischen Drucks in der Analogie zwischen Lösungen und Gasen”, Zeitschrift fur physikalische Chemie, 1:481-508; English translation by James Walker in The Foundations of the Theory of Dilute Solutions (no. 19, pgs. 5-42), London: Alembic Club, 1929.
(c) Planck, Max. (1949). Scientific Autobiography and Other Papers (translator: F. Gaynor) (pg. 14). New York: Philosophical Library.
(d) Chandrasekhar, Subrahmanyan. (1987). Truth and Beauty. Aesthetics and Motivations in Science. University of Chicago Press.