Gibbs energy is also the chemical potential that is minimized when a system reaches equilibrium at constant pressure and temperature. As such, it is a convenient criterion of spontaneity for processes with constant pressure and temperature.
The Gibbs free energy, originally called available energy, was developed in the 1870s by the American mathematical physicist Josiah Gibbs. In 1873, in a footnote, Gibbs defined what he called the “available energy” of a body as such:
The greatest amount of mechanical work which can be obtained from a given quantity of a certain substance in a given initial state, without increasing its total volume or allowing heat to pass to or from external bodies, except such as at the close of the processes are left in their initial condition.
The initial state of the body, according to Gibbs, is supposed to be such that "the body can be made to pass from it to states of dissipated energy by reversible processes". In his 1876 magnum opus On the Equilibrium of Heterogeneous Substances, a graphical analysis of multi-phase chemical systems, he engaged his thoughts on chemical free energy in full.
The Gibbs free energy is defined as:
which is the same as:
The expression for the infinitesimal reversible change in the Gibbs free energy, for an open system, subjected to the operation of external forces Xi, which cause the external parameters of the system ai to change by an amount dai, is given by:
This is one form of Gibbs fundamental equation. In the infinitesimal expression, the term involving the chemical potential accounts for changes in Gibbs free energy resulting from an influx or outflux of particles. In other words, it holds for an open system. For a closed system, this term may be dropped.
Any number of extra terms may be added, depending on the particular system being considered. Aside from mechanical work, a system may in addition perform numerous other types of work. For example, in the infinitesimal expression, the contractile work energy associated with a thermodynamic system that is a contractile fiber which shortens by an amount -dl under a force f would result in a term fdl being added. If a quantity of charge -de is acquired by a system at an electrical potential Ψ, the electrical work associated with this is -Ψde, which would be included in the infinitesimal expression. Other work terms are added on per system requirements.
Each quantity in the equations above can be divided by the amount of substance, measured in moles, to form molar Gibbs free energy. The Gibbs free energy is one of the most important thermodynamic functions for the characterization of a system. It is a factor in determining outcomes such as the voltage of an electrochemical cell, and the equilibrium constant for a reversible reaction. In isothermal, isobaric systems, Gibbs free energy can be thought of as a "dynamic" quantity, in that it is a representative measure of the competing effects of the enthalpic and entropic driving forces involved in a thermodynamic process.
if the volume is known rather than pressure then it becomes:
or more conveniently as its chemical potential:
In non-ideal systems, fugacity comes into play.
The definition of from above is
Taking the total differential, we have
Replacing with the result from the first law gives
The natural variables of are then , , and . Because some of the natural variables are intensive, may not be integrated using Euler integrals as is the case with internal energy. However, simply substituting the result for into the definition of gives a standard expression for G:
Every system seeks to achieve a minimum of free energy.
Hence, out of this general natural tendency, a quantitative measure as to how near or far a potential reaction is from this minimum is when the calculated energetics of the process indicate that the change in Gibbs free energy ΔG is negative. Essentially, this means that such a reaction will be favored and will release energy. The energy released equals the maximum amount of work that can be performed as a result of the chemical reaction. Conversely, if conditions indicated a positive ΔG, then energy--in the form of work--would have to be added to the reacting system to make the reaction go.
From the 1998 textbook Modern Thermodynamics by Nobel Laureate and chemistry professor Ilya Prigogine we find: "As motion was explained by the Newtonian concept of force, chemists wanted a similar concept of ‘driving force’ for chemical change? Why do chemical reactions occur, and why do they stop at certain points? Chemists called the ‘force’ that caused chemical reactions affinity, but it lacked a clear definition."
During the entire 18th century, the dominant view in regards to heat and light was that put forward by Isaac Newton, called the “Newtonian hypothesis”, which stated that light and heat are forms of matter attracted or repelled by other forms of matter, with forces analogous to gravitation or to chemical affinity.
In the 19th century, the French chemist Marcellin Berthelot and the Danish chemist Julius Thomsen had attempted to quantify affinity using heats of reaction. In 1875, after quantifying the heats of reaction for a large number of compounds, Berthelot proposed the “principle of maximum work” in which all chemical changes occurring without intervention of outside energy tend toward the production of bodies or of a system of bodies which liberate heat.
In addition to this, in 1780 Antoine Lavoisier and Pierre-Simon Laplace laid the foundations of thermochemistry by showing that the heat given out in a reaction is equal to the heat absorbed in the reverse reaction. They also investigated the specific heat and latent heat of a number of substances, and amounts of heat given out in combustion. Similarly, in 1840 Swiss chemist Germain Hess formulated the principle that the evolution of heat in a reaction is the same whether the process is accomplished in one-step or in a number of stages. This is known as Hess' law. With the advent of the mechanical theory of heat in the early 19th century, Hess’s law came to be viewed as a consequence of the law of conservation of energy.
Based on these and other ideas, Berthelot and Thomsen, as well as others, considered the heat given out in the formation of a compound as a measure of the affinity, or the work done by the chemical forces. This view, however, was not entirely correct. In 1847, the English physicist James Joule showed that he could raise the temperature of water by turning a paddle wheel in it, thus showing that heat and mechanical work were equivalent or proportional to each other, i.e. approximately, . This statement came to be known as the mechanical equivalent of heat and was a precursory form of the first law of thermodynamics.
By 1865 the German physicist Rudolf Clausius had shown that this equivalence principle needed amendment. That is, one can use the heat derived from a combustion reaction in a coal furnace to boil water, and use this heat to vaporize steam, and then use the enhanced high pressure energy of the vaporized steam to push a piston. Thus, we might naively reason that one can entirely convert the initial combustion heat of the chemical reaction into the work of pushing the piston. Clausius showed, however, that we need to take into account the work that the molecules of the working body, i.e. the water molecules in the cylinder, do on each other as they pass or transform from one step of or state of the engine cycle to the next, e.g. from (P1,V1) to (P2,V2). Clausius originally called this the “transformation content” of the body, and then later changed the name to entropy. Thus, the heat used to transform the working body of molecules from one state to the next cannot be used to do external work, e.g. to push the piston. Clausius defined this transformation heat as dQ = TdS.
In 1873, Willard Gibbs published A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces in which he introduced the preliminary outline of the principles of his new equation able to predict or estimate the tendencies of various natural processes to ensue when bodies or systems are brought into contact. By studying the interactions of homogeneous substances in contact, i.e. bodies, being in composition part solid, part liquid, and part vapor, and by using a three-dimensional volume-entropy-internal energy graph, Gibbs was able to determine three states of equilibrium, i.e. "necessarily stable", "neutral", and "unstable", and whether or not changes will ensue. In 1876, Gibbs built on this framework by introducing the concept of chemical potential so to take into account chemical reactions and states of bodies which are chemically different from each other. In his own words, to summarize his results in 1873, Gibbs states:
Hence, in 1882, after the introduction of these arguments by Clausius and Gibbs, the German scientist Hermann von Helmholtz stated, in opposition to Berthelot and Thomas’ hypothesis that chemical affinity is a measure of the heat of reaction of chemical reaction as based on the principle of maximal work, that affinity is not the heat given out in the formation of a compound but rather it is the largest quantity of work which can be gained when the reaction is carried out in a reversible manner, e.g. electrical work in a reversible cell. The maximum work is thus regarded as the diminution of the free, or available, energy of the system (Gibbs free energy G at T = constant, P = constant or Helmholtz free energy F at T = constant, V = constant), whilst the heat given out is usually a measure of the diminution of the total energy of the system (Internal energy). Thus, G or F is the amount of energy “free” for work under the given conditions.
Up until this point, the general view had been such that: “all chemical reactions drive the system to a state of equilibrium in which the affinities of the reactions vanish”. Over the next 60 years, the term affinity came to be replaced with the term free energy. According to chemistry historian Henry Leicester, the influential 1923 textbook Thermodynamics and the Free Energy of Chemical Reactions by Gilbert N. Lewis and Merle Randall led to the replacement of the term “affinity” by the term “free energy” in much of the English-speaking world.
In 1780, for example, Laplace and Lavoisier stated: “In general, one can change the first hypothesis into the second by changing the words ‘free heat, combined heat, and heat released’ into ‘vis viva, loss of vis viva, and increase of vis viva.’” In this manner, the total mass of caloric in a body, called absolute heat, was regarded as a mixture of two components; the free or perceptible caloric could affect a thermometer while the other component, the latent caloric, could not. The use of the words “latent heat” implied a similarity to latent heat in the more usual sense; it was regarded as chemically bound to the molecules of the body. In the adiabatic compression of a gas the absolute heat remained constant by the observed rise of temperature, indicating that some latent caloric had become “free” or perceptible.
During the early 19th century, the concept of perceptible or free caloric began to be referred to as “free heat” or heat set free. In 1824, for example, the French physicist Sadi Carnot, in his famous “Reflections on the Motive Power of Fire”, speaks of quantities of heat ‘absorbed or set free’ in different transformations. In 1882, the German physicist and physiologist Hermann von Helmholtz coined the phrase ‘free energy’ for the expression E – TS, in which the change in F (or G) determines the amount of energy ‘free’ for work under the given conditions.
Thus, in traditional use, the term “free” was attached to Gibbs free energy, i.e. for systems at constant pressure and temperature, or to Helmholtz free energy, i.e. for systems at constant volume and temperature, to mean ‘available in the form of useful work.’ With reference to the Gibbs free energy, we add the qualification that it is the energy free for non-volume work.
An increasing number of books and journal articles do not include the attachment “free”, referring to G as simply Gibbs energy (and likewise for the Helmholtz energy). This is the result of a 1988 IUPAC meeting to set unified terminologies for the international scientific community, in which the adjective ‘free’ was supposedly banished. This standard, however, has not yet been universally adopted, and many published articles and books still include the descriptive ‘free’.
Now consider diabatic systems, having internal entropy Sint. Such a system is thermally connected to its surroundings, which have entropy Sext. The entropy form of the second law does not apply directly to the diabatic system, it only applies to the closed system formed by both the system and its surroundings. Therefore a process is possible if
We will try to express the left side of this equation entirely in terms of state functions. ΔSext is defined as:
Temperature T is the same for two systems in thermal equilibrium. By the zeroth law of thermodynamics, if a system is in thermal equilibrium with a second and a third system, the latter two are in equilibrium as well. Also, is heat transferred to the system, so is heat transferred to the surroundings, and −ΔQ/T is entropy gained by the surroundings. We now have:
ΔQ is heat transferred to the system; if the process is now assumed to be isobaric, then ΔQp = ΔH:
Gibbs free energy G itself is defined as
If a closed system (ΔQ = 0) is at constant pressure (ΔQ = ΔH), then
Therefore the Gibbs free energy of a closed system is:
and if then this implies that , back to where we started the derivation of ΔG.
and rearranging gives
which relates the electrical potential of a reaction to the equilibrium coefficient for that reaction.
ΔG = change in Gibbs free energy, ΔH = change in enthalpy, T = absolute temperature, ΔS = change in entropy, R = gas constant, ln = natural logarithm, K = equilibrium constant, Q = reaction quotient, n = number of electrons per mole product, F = Faraday constant (coulombs per mole), and ΔE = electrical potential of the reaction. Moreover, we also have:
which relates the equilibrium constant with Gibbs free energy.