thermodynamic free energy

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The thermodynamic free energy is a concept useful in the
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
of chemical or thermal processes in
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more specializ ...
and science. The change in the free energy is the maximum amount of
work Work may refer to: * Work (human activity) Work or labor (or labour in British English) is intentional activity people perform to support the needs and wants of themselves, others, or a wider community. In the context of economics, work ...
that a
thermodynamic system A thermodynamic system is a body of matter and/or radiation, confined in space by walls, with defined permeabilities, which separate it from its surroundings. The surroundings may include other thermodynamic systems, or physical systems that are ...
can perform in a process at constant temperature, and its sign indicates whether the process is thermodynamically favorable or forbidden. Since free energy usually contains
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
, it is not absolute but depends on the choice of a zero point. Therefore, only relative free energy values, or changes in free energy, are physically meaningful. The free energy is a thermodynamic
state function In the Thermodynamics#Equilibrium thermodynamics, thermodynamics of equilibrium, a state function, function of state, or point function for a thermodynamic system is a Function (mathematics), mathematical function relating several state variables ...
, like the
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
,
enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant p ...
, and
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
. The free energy is the portion of any first-law energy that is available to perform thermodynamic
work Work may refer to: * Work (human activity) Work or labor (or labour in British English) is intentional activity people perform to support the needs and wants of themselves, others, or a wider community. In the context of economics, work ...
at constant
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...
, ''i.e.'', work mediated by
thermal energy The term "thermal energy" is used loosely in various contexts in physics and engineering. It can refer to several different well-defined physical concepts. These include the internal energy or enthalpy of a body of matter and radiation; heat, de ...
. Free energy is subject to irreversible loss in the course of such work. Since first-law energy is always conserved, it is evident that free energy is an expendable, second-law kind of energy. Several free energy functions may be formulated based on system criteria. Free energy functions are Legendre transforms of the
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
. The
Gibbs free energy In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work (physics), work that may be performed by a closed system, thermodynamically closed system a ...
is given by , where ''H'' is the
enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant p ...
, ''T'' is the
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from Kinetic theory of gases, kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by William Thomson, 1st Baron Kelvin, Kelvin ...
, and ''S'' is the
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
. , where ''U'' is the internal energy, ''p'' is the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
, and ''V'' is the volume. ''G'' is the most useful for processes involving a system at constant
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
''p'' and
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...
''T'', because, in addition to subsuming any entropy change due merely to
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
, a change in ''G'' also excludes the work needed to "make space for additional molecules" produced by various processes. Gibbs free energy change therefore equals work not associated with system expansion or compression, at constant temperature and pressure. (Hence its utility to solution-
phase Phase or phases may refer to: Science *State of matter In physics, a state of matter is one of the distinct forms in which matter can exist. Four states of matter are observable in everyday life: solid, liquid, gas, and Plasma (physics), p ...
chemists, including biochemists.) The, historically earlier,
Helmholtz free energy In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work (thermodynamics), work obtainable from a closed system, closed thermodynamic system at a constant temperature (Isotherma ...
is defined in contrast as . Its change is equal to the amount of reversible work done on, or obtainable from, a system at constant ''T''. Thus its appellation "work content", and the designation ''A'' from ''Arbeit'', the German word for work. Since it makes no reference to any quantities involved in work (such as ''p'' and ''V''), the Helmholtz function is completely general: its decrease is the maximum amount of work which can be done ''by'' a system at constant temperature, and it can increase at most by the amount of work done ''on'' a system isothermally. The Helmholtz free energy has a special
theoretical A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or resear ...
importance since it is proportional to the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
of the partition function for the
canonical ensemble In statistical mechanics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of ene ...
in
statistical mechanics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scien ...
. (Hence its utility to
physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate caus ...
s; and to
gas Gas is one of the four fundamental states of matter In physics, a state of matter is one of the distinct forms in which matter can exist. Four states of matter are observable in everyday life: solid, liquid, gas, and Plasma (physics), pl ...
-phase chemists and engineers, who do not want to ignore work.) Historically, the term 'free energy' has been used for either quantity. In
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
, ''free energy'' most often refers to the Helmholtz free energy, denoted by ''A'' (or ''F''), while in
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, properties ...
, ''free energy'' most often refers to the Gibbs free energy. The values of the two free energies are usually quite similar and the intended free energy function is often implicit in manuscripts and presentations.

# Meaning of "free"

The basic definition of "energy" is a measure of a body's (in thermodynamics, the system's) ability to cause change. For example, when a person pushes a heavy box a few metres forward, that person exerts mechanical energy, also known as work, on the box over a distance of a few meters forward. The mathematical definition of this form of energy is the product of the force exerted on the object and the distance by which the box moved (). Because the person changed the stationary position of the box, that person exerted energy on that box. The work exerted can also be called "useful energy", because energy was converted from one form into the intended purpose, i.e. mechanical utilisation. For the case of the person pushing the box, the energy in the form of internal (or potential) energy obtained through metabolism was converted into work in order to push the box. This energy conversion, however, was not straightforward: while some internal energy went into pushing the box, some was diverted away (lost) in the form of heat (transferred thermal energy). For a reversible process, heat is the product of the absolute temperature $T$ and the change in entropy $S$ of a body (entropy is a measure of disorder in a system). The difference between the change in internal energy, which is $\Delta U$, and the energy lost in the form of heat is what is called the "useful energy" of the body, or the work of the body performed on an object. In thermodynamics, this is what is known as "free energy". In other words, free energy is a measure of work (useful energy) a system can perform at constant temperature. Mathematically, free energy is expressed as: free energy $A=U-TS$ This expression has commonly been interpreted to mean that work is extracted from the internal energy $U$ while $TS$ represents energy not available to perform work. However, this is incorrect. For instance, in an isothermal expansion of an ideal gas, the internal energy change is $\Delta U=0$ and the expansion work $w=-T\Delta S$ is derived exclusively from the $TS$ term supposedly not available to perform work. But it is noteworthy that the derivative form of the free energy: $dA=-SdT-PdV$ (for Helmholtz free energy) does indeed indicate that a spontaneous change in a non-reactive system's free energy (NOT the internal energy) comprises the available energy to do work (compression in this case) $-PdV$ and the unavailable energy $-SdT$. Similar expression can be written for the Gibbs free energy change. In the 18th and 19th centuries, the
theory of heat The history of thermodynamics is a fundamental strand in the history of physics, the history of chemistry, and the history of science in general. Owing to the relevance of thermodynamics in much of science and technology, its history is finely wov ...
, i.e., that heat is a form of energy having relation to vibratory motion, was beginning to supplant both the
caloric theory The caloric theory is an obsolete scientific theory that heat consists of a self-repellent fluid called caloric that flows from hotter bodies to colder bodies. Caloric was also thought of as a weightless gas that could pass in and out of pores in ...
, i.e., that heat is a fluid, and the
four element theory Classical elements typically refer to earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large list of largest lakes and seas in the Solar System, volumes of water can be foun ...
, in which heat was the lightest of the four elements. In a similar manner, during these years,
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
was beginning to be distinguished into different classification categories, such as “free heat”, “combined heat”, “radiant heat”,
specific heat In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
,
heat capacity Heat capacity or thermal capacity is a physical quantity, physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The International System of Units, SI unit of heat ca ...
, “absolute heat”, “latent caloric”, “free” or “perceptible” caloric (''calorique sensible''), among others. In 1780, for example,
Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
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 ''Vis viva'' (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) aroun ...
, 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, whereas 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 but the observed rise in temperature implied 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 Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association ...
coined the phrase ‘free energy’ for the expression , in which the change in ''A'' (or ''G'') determines the amount of
energy In physics, energy (from Ancient Greek: wikt:ἐνέργεια#Ancient_Greek, ἐνέργεια, ''enérgeia'', “activity”) is the physical quantity, quantitative physical property, property that is #Energy transfer, transferred to a phy ...
‘free’ for
work Work may refer to: * Work (human activity) Work or labor (or labour in British English) is intentional activity people perform to support the needs and wants of themselves, others, or a wider community. In the context of economics, work ...
under the given conditions, specifically constant temperature. Thus, in traditional use, the term “free” was attached to Gibbs free energy for systems at constant pressure and temperature, or to Helmholtz free energy for systems at constant temperature, to mean ‘available in the form of useful work.’ With reference to the Gibbs free energy, we need to add the qualification that it is the energy free for non-volume work and compositional changes. 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 The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
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’.

# Application

Just like the general concept of energy, free energy has a few definitions suitable for different conditions. In physics, chemistry, and biology, these conditions are thermodynamic parameters (temperature $T$, volume $V$, pressure $p$, etc.). Scientists have come up with several ways to define free energy. The mathematical expression of Helmholtz free energy is: :$A = U-TS$ This definition of free energy is useful for gas-phase reactions or in physics when modeling the behavior of isolated systems kept at a constant volume. For example, if a researcher wanted to perform a combustion reaction in a bomb calorimeter, the volume is kept constant throughout the course of a reaction. Therefore, the heat of the reaction is a direct measure of the free energy change, $q=\Delta U$. In solution chemistry, on the other hand, most chemical reactions are kept at constant pressure. Under this condition, the heat $q$ of the reaction is equal to the enthalpy change $\Delta H$ of the system. Under constant pressure and temperature, the free energy in a reaction is known as Gibbs free energy $G$. :$G = H-TS$ These functions have a minimum in chemical equilibrium, as long as certain variables ($T$, and $V$ or $p$) are held constant. In addition, they also have theoretical importance in deriving Maxwell relations. Work other than may be added, e.g., for
electrochemical Electrochemistry is the branch of physical chemistry concerned with the relationship between Electric potential, electrical potential difference, as a measurable and quantitative phenomenon, and identifiable chemical change, with the potential dif ...
cells, or work in elastic materials and in
muscle Skeletal muscles (commonly referred to as muscles) are Organ (biology), organs of the vertebrate muscular system and typically are attached by tendons to bones of a skeleton. The muscle cells of skeletal muscles are much longer than in the other ...
contraction. Other forms of work which must sometimes be considered are stress- strain,
magnetic Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
, as in adiabatic de
magnetization In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or Diam ...
used in the approach to
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibration ...
, and work due to electric
polarization Polarization or polarisation may refer to: Mathematics *Polarization of an Abelian variety, in the mathematics of complex manifolds *Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...
. These are described by
tensor In mathematics, a tensor is an mathematical object, algebraic object that describes a Multilinear map, multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as Vect ...
s. In most cases of interest there are internal
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variable Dependent and independent variables are variables in mathematical modeling, statistical model A statistical model is a mathematical model that emb ...
and processes, such as
chemical reaction A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the pos ...
s and
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic State of ...
s, which create entropy. Even for homogeneous "bulk" materials, the free energy functions depend on the (often suppressed) composition, as do all proper
thermodynamic potentials A thermodynamic potential (or more accurately, a thermodynamic potential energy)ISO/IEC 80000-5, Quantities an units, Part 5 - Thermodynamics, item 5-20.4 Helmholtz energy, Helmholtz functionISO/IEC 80000-5, Quantities an units, Part 5 - Thermod ...
( extensive functions), including the internal energy. $N_i$ is the number of molecules (alternatively, moles) of type $i$ in the system. If these quantities do not appear, it is impossible to describe compositional changes. The differentials for processes at uniform pressure and temperature are (assuming only $pV$ work): :$\mathrm dA = - p\,\mathrm dV - S\,\mathrm dT + \sum_i \mu_i \,\mathrm dN_i\,$ :$\mathrm dG = V\,\mathrm dp - S\,\mathrm dT + \sum_i \mu_i \,\mathrm dN_i\,$ where ''μ''''i'' is the
chemical potential In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potent ...
for the ''i''th component in the system. The second relation is especially useful at constant $T$ and $p$, conditions which are easy to achieve experimentally, and which approximately characterize living creatures. Under these conditions, it simplifies to :$\left(\mathrm dG\right)_ = \sum_i \mu_i \,\mathrm dN_i\,$ Any decrease in the Gibbs function of a system is the upper limit for any
isothermal In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, mea ...
, isobaric work that can be captured in the surroundings, or it may simply be
dissipated In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quan ...
, appearing as $T$ times a corresponding increase in the entropy of the system and/or its surrounding. An example is surface free energy, the amount of increase of free energy when the area of surface increases by every unit area. The
path integral Monte Carlo Path integral Monte Carlo (PIMC) is a quantum Monte Carlo method used to solve quantum statistical mechanics problems numerically within the path integral formulation The path integral formulation is a description in quantum mechanics ...
method is a numerical approach for determining the values of free energies, based on quantum dynamical principles.

## Work and free energy change

For a reversible isothermal process, Δ''S'' = ''q''rev/''T'' and therefore the definition of ''A'' results in : $\Delta A = \Delta U - T \Delta S = \Delta U - q_\text = w_\text$ (at constant temperature) This tells us that the change in free energy equals the reversible or maximum work for a process performed at constant temperature. Under other conditions, free-energy change is not equal to work; for instance, for a reversible adiabatic expansion of an ideal gas, Importantly, for a heat engine, including the
Carnot cycle A Carnot cycle is an ideal thermodynamic cycle proposed by French physicist Nicolas Léonard Sadi Carnot, Sadi Carnot in 1824 and expanded upon by others in the 1830s and 1840s. By Carnot's theorem (thermodynamics), Carnot's theorem, it provides ...
, the free-energy change after a full cycle is zero, while the engine produces nonzero work. It is important to note that for heat engines and other thermal systems, the free energies do not offer convenient characterizations; internal energy and enthalpy are the preferred potentials for characterizing thermal systems.

## Free energy change and spontaneous processes

According to the
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
, for any process that occurs in a closed system, the inequality of Clausius, Δ''S'' > ''q''/''T''surr, applies. For a process at constant temperature and pressure without non-''PV'' work, this inequality transforms into $\Delta G < 0$. Similarly, for a process at constant temperature and volume, $\Delta A < 0$. Thus, a negative value of the change in free energy is a necessary condition for a process to be spontaneous; this is the most useful form of the second law of thermodynamics in chemistry. In chemical equilibrium at constant ''T'' and ''p'' without electrical work, d''G'' = 0.

# History

The quantity called "free energy" is a more advanced and accurate replacement for the outdated term ''affinity'', which was used by chemists in previous years to describe the ''force'' that caused
chemical reaction A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the pos ...
s. The term affinity, as used in chemical relation, dates back to at least the time of
Albertus Magnus Albertus Magnus (c. 1200 – 15 November 1280), also known as Saint Albert the Great or Albert of Cologne, was a German Dominican Order, Dominican friar, philosopher, scientist, and Bishop in the Catholic Church, bishop. Later Canonization ...
. From the 1998 textbook ''Modern Thermodynamics'' Chapter 4, Section 1, Paragraph 2 (page 103) by Nobel Laureate and chemistry professor
Ilya Prigogine Viscount A viscount ( , for male) or viscountess (, for female) is a title A title is one or more words used before or after a person's name, in certain contexts. It may signify either generation, an official position, or a professiona ...
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 with regard to heat and light was that put forth by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
, called the ''Newtonian hypothesis'', which states 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 Pierre Eugène Marcellin Berthelot (; 25 October 1827 – 18 March 1907) was a French chemist and Republican Union (France), Republican politician noted for the Thomsen-Berthelot principle, ThomsenBerthelot principle of thermochemistry. He synt ...
and the Danish chemist
Julius Thomsen The gens Julia (''gēns Iūlia'', ) was one of the most prominent patrician (ancient Rome), patrician families in ancient Rome. Members of the gens attained the highest dignities of the state in the earliest times of the Roman Republic, Republic ...
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 thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
. In addition to this, in 1780
Antoine Lavoisier Antoine-Laurent de Lavoisier ( , ; ; 26 August 17438 May 1794), When reduced without charcoal, it gave off an air which supported respiration and combustion in an enhanced way. He concluded that this was just a pure form of common air and th ...
and
Pierre-Simon Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
laid the foundations of
thermochemistry Thermochemistry is the study of the heat energy which is associated with chemical reaction A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical subst ...
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 In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
and
latent heat Latent heat (also known as latent energy or heat of transformation) is energy released or absorbed, by a body or a thermodynamic system, during a constant-temperature process — usually a Phase transition#Modern classifications, first-order phase ...
of a number of substances, and amounts of heat given out in combustion. In a similar manner, in 1840 Swiss chemist
Germain Hess Germain Henri Hess (russian: Герман Иванович Гесс, German Ivanovich Gess; 7 August 1802 – 30 November 1850) was a Swiss-Russian chemist and doctor who formulated Hess's law, an early principle of thermochemistry Thermoch ...
formulated the principle that the evolution of heat in a reaction is the same whether the process is accomplished in one-step process or in a number of stages. This is known as Hess' law. With the advent of the
mechanical theory of heat The history of thermodynamics is a fundamental strand in the history of physics Physics is a branch of science whose primary objects of study are matter and energy. Discoveries of physics find applications throughout the natural sciences and i ...
in the early 19th century, Hess's law came to be viewed as a consequence of the law of
conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be Conservation law, ''conserved'' over time. This law, first proposed and tested by Émilie du Chât ...
. 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 In the history of science, the mechanical equivalent of heat states that Motion (physics), motion and heat are mutually interchangeable and that in every case, a given amount of work (thermodynamics), work would generate the same amount of heat, p ...
and was a precursory form of the
first law of thermodynamics The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and Work (thermodynamics), thermodynamic work for a syst ...
. By 1865, the German physicist
Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist and mathematician and is considered one of the central founding fathers of the science of thermodynamics. By his restatement of Nicolas Léonard Sadi Ca ...
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 must 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 State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * The State (newspaper), ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, U ...
of the
engine cycle A Carnot cycle is an ideal thermodynamic cycle A thermodynamic cycle consists of a linked sequence of thermodynamic processes that involve transfer of heat and work into and out of the system, while varying pressure, temperature, and other ...
to the next, e.g., from ($P_1,V_1$) to ($P_2,V_2$). Clausius originally called this the “transformation content” of the body, and then later changed the name to
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
. 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 Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics Thermodynamics i ...
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 Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
-
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
-
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
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 In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potent ...
so to take into account chemical reactions and states of bodies that are chemically different from each other. In his own words, to summarize his results in 1873, Gibbs states: In this description, as used by Gibbs, ''ε'' refers to the
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
of the body, ''η'' refers to the
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
of the body, and ''ν'' is the
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
of the body. Hence, in 1882, after the introduction of these arguments by Clausius and Gibbs, the German scientist
Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association ...
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 In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work (physics), work that may be performed by a closed system, thermodynamically closed system a ...
''G'' at ''T'' = constant, ''P'' = constant or
Helmholtz free energy In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work (thermodynamics), work obtainable from a closed system, closed thermodynamic system at a constant temperature (Isotherma ...
''A'' 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 The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
). Thus, ''G'' or ''A'' 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.

*
Energy In physics, energy (from Ancient Greek: wikt:ἐνέργεια#Ancient_Greek, ἐνέργεια, ''enérgeia'', “activity”) is the physical quantity, quantitative physical property, property that is #Energy transfer, transferred to a phy ...
*
Exergy In thermodynamics, the exergy of a System (thermodynamics), system is the maximum useful Mechanical work, work possible during a Thermodynamic process, process that brings the system into Thermodynamic equilibrium, equilibrium with a heat reservo ...
*
Second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
*
Superconductivity Superconductivity is a set of physical properties observed in certain materials where electrical resistance The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its Multiplicative inverse, ...
* Merle Randall

# References

{{DEFAULTSORT:Thermodynamic Free Energy Energy (physics) State functions