The Lewis inequalities for natural processes and unnatural process in the 1933 notation of English chemical thermodynamicist Edward Guggenheim. [2]
In inequalities, the Lewis inequality states that no actual earth-bound, freely going, natural process or reaction is thermodynamically possible unless the following relation holds:

which states that the excess of the value of the Gibbs free energy of the system in its final state Gf less the value of the Gibbs free energy of the system in its initial state Gi:

$\Delta G = G_f - G_i \,$

must be a negative value, i.e. be less than zero, where:

$G = U + PV - TS \,$

meaning that the reaction process shows a diminution in free energy over the extent of the reaction.

History
The above simplified outlined was first given focus by American physical chemist Gilbert Lewis in 1923 and referred to by others as the Lewis free energy, albeit based on the more rigorous 1876 work of American engineer Willard Gibbs.

Earth-bound
The term "earth-bound" is short for reaction processes that occur at constant temperature (isothermal) and constant pressure (isobaric), which approximate those on the surface of the earth, in a given window of time.

Freely going
The term "freely going" refers to reactions that "run freely", in Lewis' words, "like the combustion of fuel, or the action of an acid upon a metal". This signification is done to contrast with reactions that are "harnessed" in some way for the production of useful work $w' \,$, or "net work", as Lewis calls it, in which case the following "universal rule", in Lewis' words, holds:

$- \Delta G > w' \,$

whereby, although, in practice, different processes differ greatly in their degree of irreversibility, if any isothermal-isobaric process is to occur with finite velocity, it is necessary that this inequality exists. The model reaction given by Lewis for harnessed useful work is that of a galvanic cell or battery, in which an electrochemical reaction occurs inside the battery, whereby if the two electrodes are connected to an external motor or other electrical system, in such a way as to utilize the electrical energy which is available, an amount of work will be done. This amount of "net work" $w' \,$, according to Lewis, will be the electrical work w less the pressure volume work PΔV , i.e. the work done against the constant pressure of the atmosphere:

$w' = w - P \Delta V \,$

In the more general case where the reactions are not harnessed in some way for the production of work (the example of the "harnessed reaction" here being the internal battery reactions electrically connected to an external motor or other electrical system), the reactions are those of a system not subjected to external forces, except constant pressure exerted by the environment and in such cases the net work is equal to zero:

$w' = 0 \,$

whereby with substitution:

$- \Delta G > 0 \,$

and then after multiplying the negative sign through, which reverses the inequality, we have the following "universal rule" for freely going isothermal isobaric reactions not harnessed in some way for the production of useful external net work:

then, in Lewis' words, "we know that the reaction in the direction indicated, is thermodynamically possible." This is the Lewis inequality for a natural process. If, conversely, the measure of Gibbs free energy change for for the process is positive:

$\Delta G > 0 \,$

then, in Lewis' words, "we know that the reaction in the direction indicated, is thermodynamically impossible." This is the Lewis inequality for an unnatural process.

Notation and etymology
The formulas given above were first expressed in now famous capital delta symbol notation (Δ) in 1923 by American physical chemist Gilbert Lewis, albeit he used the German symbol notation ‘F’, from the German freie energie, a 1882 term introduced by Hermann Helmholtz, albeit for the isothermal-isochoric potential). The symbol G, in honor of Willard Gibbs, was introduced in 1933 by English chemical thermodynamicist Edward Guggenheim. This is summarized below:

The notation and symbol usage in chemical thermodynamics, prior to Guggenheim's uniformity of the notation system, was chaotic and varied, to say the least, as evidenced by both Guggenheim's (1933) and Theophile de Donder's (1936) historical usage characteristic function notation tables. The central point to note is that in 1933 the symbol G was assigned to replace F in the following formula:

$F = E + PV - TS \,$

by Guggenheim, as the new universal symbol for Gibbs free energy, or the thermodynamic function of the internal energy E (or U modern) plus the pressure volume work energy (PV) less the entropic energy (TS). This adoption, however, was not immediate as evidenced by the fact that in 1950 American chemical thermodynamicist Frederick Rossini, a student Lewis was people were still sometimes calling the above function the "Gibbs' or Lewis' free energy". Rossini defines the inequality as such (using the symbol δu) as the notation for what he calls "useful energy" a synonym for free energy (Helmholtz) or available energy (Gibbs). In his own words: "the algebraic sign of the useful energy obtainable from the system serves to tell us whether the given process is one in which the system is moving toward or away from equilibrium". He explains this as such: [5]

 Rossini notation (1950) Modern notation Description δu < 0 ΔG < 0 If the value of δu is negative, useful energy is obtainable from the system and we know the change is a naturally occurring one in the direction toward the state of equilibrium. δu > 0 ΔG > 0 If the value of δu is positive, useful energy is required to be supplied to the given system to bring about the desired change and we know that the change is an unnatural one in the direction away from equilibrium.

On this combined notation recommendation logic (Guggenheim + Rossini), it would seem intuitive to keep to the naming of the function ‘E + PV – TS’, by standard convention, as Gibbs free energy, symbol ‘G’, but to call the inequality ‘dG’ (or ΔG), by the name ‘Lewis inequality’, as follows:

 Differential change State change Name dG < 0 ΔG < 0 Lewis inequality for a natural process (or spontaneous process), one in which the change will occur naturally or spontaneously and in which useful energy is obtainable from the system. dG > 0 ΔG > 0 Lewis inequality for an unnatural process (or spontaneous process), one in which useful energy is required to be supplied to the given system to bring about the desired change.

on the premise that (a) the function was commonly known as the Lewis free energy, (b) Lewis was the one who did the most work to center focus on the development and application of this key function, particularly by bring it into the hands of the working chemist, (c) Lewis spent at least two decades doing some of the first free energy calculations and measurements in laboratory, (d) Lewis has been cited as the one whose efforts resulted in the replacement of the name affinity with free energy in the English speaking world. In simple terms, this one inequality, of Gibbs' 700 equations, is the one that Lewis deciphered from the Rosetta stone and brought into the light for the chemists of the world to begin to use.

Affinity and work
The negative of the Gibbs free energy change:

$-\Delta G \,$

according to English chemical thermodynamicist James Partington, "is a measure of the electrical work, chemical work, or internal work done by the system." This can be expressed in terms of the chemical affinities or forces operating among the reactants in the system, in their interactions, bond formations, and bond dissolutions, on the premise that the correct measure of the affinity (A) of a reaction is the diminution of free energy (–ΔG). This can be expressed formulaically, according to Partington, as such: [3]

$A = -\Delta G \,$

Interestingly, Partington comments on this relation between the free energy and chemical affinity that:

“The consequences of this statement for chemistry are as yet hardly realized, but with the further progress of science it is to be expected that empirical results will more and more give way to exact quantitative laws.”

Lewis goes on to state, similar to Willard Gibbs' 1876 description of the two force-functions, that:

“We may think of the quantity ‘– ΔF ’ as the driving force of a reaction; where, in a thermodynamic sense, a system is stable when no process can occur with a diminution in free energy.”

Lewis introduce here not only the notion that free energy change is the driving force of earth-bound natural processes, as was described in 1718 by Isaac Newton in terms of respective values of the forces of chemical affinity A as being the driving force behind reactions, as historically outlined below:

 Newton notation (1718) Lewis notation (1923) Partington notation (1924) Modern notation (1933) $A > 0 \,$ $A = - \Delta F \,$ $A = -\Delta Z \,$ $A = - \Delta G \,$
 The position of G1 is such that, in the words of Gilbert Lewis (1923), "no further process can occur with a diminution in free energy", and is thus representative of a state of maximal stability; whereas the position of G1 could decrease further in free energy, to the position of state one, and is thus not maximally stable.

but also introduces the notion of "stability", i.e. that when the variation of free energy for the process or reaction reaches its lowest value, or, as Lewis puts it, when "no further process can occur with a diminution in free energy", the system will be said to be stable, as would be indicated graphically by the lowest point of free energy on the reaction coordinate, as depicted adjacent.

Human chemistry
This logic was first applied to human chemical reactions in 1995 by American chemical engineer Libb Thims. [4]

Derivation
The derivation prior to this originated in the various Gibbs inequalities (1876) which in turn are based on the famous Clausius inequality (1856).

Constant volume reactions
Reaction processes governed by the Lewis inequality, to note, can be contrasted with laboratory "volume-bound" processes that occur at occur at constant temperature (isothermal) and constant volume (isochoric), in which cases the thermodynamic relation which decreases is the Helmholtz free energy function.

References
1. Lewis, Gilbert N. and Randall, Merle. (1923). Thermodynamics and the Free Energy of Chemical Substances (pgs. 160-61). McGraw-Hill Book Co., Inc.
2. Guggenheim, Edward, A. (1933). Modern Thermodynamics by the Methods of Willard Gibbs (pg. 17). London: Methuen & Co.
3. Partington, James R. (1924). Chemical Thermodynamics: An Introduction to General Thermodynamics and its Applications to Chemistry (pg. 120, 146-47). D. Van Nostrand.
4. (a) Thims, Libb. (2007). Human Chemistry (Volume One). Morrisville, NC: LuLu.
(b) Thims, Libb. (2007). Human Chemistry (Volume Two). Morrisville, NC: LuLu.
5. Rossini, Frederick D. (1950). Chemical Thermodynamics (pg. 122). John Wiley & Sons, Inc.