Josiah Willard Gibbs
Chemical and electrochemical thermodynamics

Gibbs's papers from the 1870s introduced the idea of expressing the internal energy U of a system in terms of the entropy S, in addition to the usual state-variables of volume V, pressure p, and temperature T. He also introduced the concept of the chemical potential μ {\displaystyle \mu } of a given chemical species, defined to be the rate of the increase in U associated with the increase in the number N of molecules of that species (at constant entropy and volume). Thus, it was Gibbs who first combined the first and second laws of thermodynamics by expressing the infinitesimal change in the internal energy, d U, of a closed system in the form:

where T is the absolute temperature, p is the pressure, d S is an infinitesimal change in entropy and d V is an infinitesimal change of volume. The last term is the sum, over all the chemical species in a chemical reaction, of the chemical potential, μ, of the i species, multiplied by the infinitesimal change in the number of moles, d N of that species. By taking the Legendre transform of this expression, he defined the concepts of enthalpy, H and Gibbs free energy, G.

This compares to the expression for Helmholtz free energy, A.

When the Gibbs free energy for a chemical reaction is negative the reaction will proceed spontaneously. When a chemical system is at equilibrium, the change in Gibbs free energy is zero. An equilibrium constant is simply related to the free energy change when the reactants are in their standard states.

Chemical potential is usually defined as partial molar Gibbs free energy.

Gibbs also obtained what later came to be known as the " Gibbs–Duhem equation ".

In an electrochemical reaction characterized by an electromotive force ℰ and an amount of transferred charge Q, Gibbs's starting equation becomes

The publication of the paper " On the Equilibrium of Heterogeneous Substances " (1874–78) is now regarded as a landmark in the development of chemistry. In it, Gibbs developed a rigorous mathematical theory for various transport phenomena, including adsorption, electrochemistry, and the Marangoni effect in fluid mixtures. He also formulated the phase rule

for the number F of variables that may be independently controlled in an equilibrium mixture of C components existing in P phases. Phase rule is very useful in diverse areas, such as metallurgy, mineralogy, and petrology. It can also be applied to various research problems in physical chemistry.
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