German physicist Hermann Helmholtz's 1882 paper "On the Thermodynamics of Chemical Processes", which showed that free energy is the measure of affinity; it is one of the founding papers of chemical thermodynamics.

In famous publications, “On the Thermodynamics of Chemical Processes” or Ueber Die Thermodynamik Chemischer Vorgange is a three-part, 55-page address given to the Berlin Academy on February 02 and July 27 of 1882 by German physicist Hermann Helmholtz, which integrated thermodynamics into affinity chemistry, thus being a founding paper in chemical thermodynamics, which acted to disprove the Berthelot-Thomsen principle. [1] The following is a noted summary quotation from this address by Helmholtz: [2]

“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.”

In this paper, Helmholtz defines the magnitude of entropy |S| as the measure of atomic-molecular disorganization. [1] The first English translation seems to be the 1888 translation by the Physical Society of London. The most-cited aspect of the paper is that Helmholtz showed that the true measure of affinity is free energy, not heat as it had previously been assumed. It seems that Helmholtz had plans to expand on this paper into a book, but this aim was never completed.

Notes
The terms free energy and bound energy were introduced in this paper.

Overview
In 1880, Helmholtz became the director of the Institute of Physics in Berlin. Between 1881 and 1884, Helmholtz attacked the question of how to integrate energy conservation and Maupertuis's principle of least action to describe thermodynamic and chemical processes. [3]

The usual approach to mathematical description of a physical system is to use differential equations. Differential equations describe the change in variable elements of the system over time. Using functions to describe the variables, the differential calculus establishes the rates of change of the functions over time by determining the change in a variable at any one point. For instance, Newtonian mechanics uses the calculus to describe the velocity and position of any given particle of a system at any given point in time.

While differential equations differentiate between distinct points in time, “action” principles are integral principles, that is, they measure the change in a variable from one state to another, which is an “action.” To analyze a physical system using an action principle, it is only necessary to know two states of the system. For instance, if you throw a ball in the air and want to know how much energy it expends to reach its highest point, you need only know the initial state, the ball in your hand, and its final state, the highest point of its path through the air.

The Lagrangian equation, in its most simple case, is the kinetic energy of a system minus the potential energy: L = T − U. The Hamiltonian equations also evaluate potential and kinetic energy in a system, but use an integral sum of the momenta of the elements of the system, H = T + U. The Hamiltonian gives a minimum value for a function over any path of an action given the initial and end states of the action. The Hamiltonian expression of the momenta of the system can be derived from the Lagrangian, and vice versa, using a Legendre transform. The Hamiltonian function yields the derivative of the Lagrangian: in the simplest case, the Lagrangian deals with the velocity of a particle and the Hamiltonian with the momenta of the particle. The Legendre equation transforms the Lagrangian into a function of the Hamiltonian, its derivative.

The equations for a system containing heat as a variable contain entropy as a variable quantity. Entropy is an inconvenient variable, difficult to control for and hold constant as one can hold temperature, pressure, and volume constant. The Legendre transform allows a researcher to convert equations containing entropy into equations expressed only in terms of temperature, pressure, and volume. The Legendre transform can be applied correctly only under certain conditions, which must be specified.

In 1882, Helmholtz gave an address, “The Thermodynamics of Chemical Processes,” at the Berlin Academy. Up until Helmholtz's address, chemical reactions had been explained by “chemical forces” or “affinities” between chemical substances, measured quantitatively by the heat developed during a chemical reaction. Gustave Coriolis had clarified the notion of work as the product of force over distance in 1821, and this notion was in common use by the late 19th century. In his address, Helmholtz “proved that affinity was not given by the heat evolved in a chemical reaction but rather by the maximum work produced when the reaction was carried out reversibly”. [4] However, while kinetic and mechanical energy can be converted into heat in every case, only in restricted cases can heat be converted into kinetic and mechanical energy. Hence, the equations describing chemical processes involving heat could not always be reversed. These are the conditions under which the Legendre transform could not be applied.

Helmholtz proposed the notion of a “free energy” to account for cases involving heat and entropy. The Helmholtz free energy is defined as F ⇔ E − TS, where E is energy, T is temperature, and S is entropy. The free energy equation yields a quantity, F, that is independent of heat and entropy. Many equations involving F and not T or S are fully reversible, and so Helmholtz's work allowed for the application of the Hamiltonian to many chemical processes. Hence while “Helmholtz was neither the sole nor the most important contributor” to theoretical chemistry, “his thermodynamic theory of 1882–1883 was the pioneering work on which much of the new theoretical chemistry rested”. [4]

References
1. (a) 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.
(b) Koenigsberger, Leo. (1902). Hermann von Helmholtz ("The Thermodynamics of Chemical Processes", pgs. 335-39; 1883: Note on an Introduction to Thermodynamics, pgs 340-43) trans. Frances A. Welby, preface by Lord Kelvin. Oxford at the Clarendon Press.
2. Helmholtz, Helmholtz v. (1882). "Die Thermodynamic Chemischer Vorgange," SB, pg. 23, pg. 22-29, in Wissenschaftlich Abhandlundgen von Hermann von Helmholtz. 3 vols. Leipzig: J.A. Barth, 1882-95.)
3. Patton, Lydia. (2008). “Herman von Helmholtz: 5. Thermodynamics, the least action principle, and free energies: 1881-1887”, Stanford Encyclopedia of Philosophy.
4. Kragh, Helge. (1993). “Between Physics and Chemistry: Helmholtz’s Route to a Theory of Chemical Thermodynamics”, In: Hermann von Helmholtz and the Foundations of Nineteenth-Century Science (pgs. 405-06), by David Cahan, Berkeley: University of California Press.

Further reading
● Laidler, Keith J. (1993). The World of Physical Chemistry (The Thermodynamics of Helmholtz, pgs. 112-14). Oxford University Press.
● Campisi, Michele. (2005). “On the Mechanical Foundations of Thermodynamics: The generalized Helmholtz theorem,” Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics, 36(2): 276–290.