History of thermodynamics

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The 1698 Savery Engine – the world's first commercially useful steam engine: built by Thomas Savery

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 woven with the developments of classical mechanics, quantum mechanics, magnetism, and chemical kinetics, to more distant applied fields such as meteorology, information theory, and biology (physiology), and to technological developments such as the steam engine, internal combustion engine, cryogenics and electricity generation. The development of thermodynamics both drove and was driven by atomic theory. It also, albeit in a subtle manner, motivated new directions in probability and statistics.

History

Contributions from antiquity

The ancients viewed heat as that related to fire. In 3000 BC, the ancient Egyptians viewed heat as related to origin mythologies. In the Western philosophical tradition, after much debate about the primal element among earlier pre-Socratic philosophers, Empedocles proposed a four-element theory, in which all substances derive from earth, water, air, and fire. The Empedoclean element of fire is perhaps the principal ancestor of later concepts such as phlogin and caloric. Around 500 BC, the Greek philosopher Heraclitus became famous as the "flux and fire" philosopher for his proverbial utterance: "All things are flowing." Heraclitus argued that the three principal elements in nature were fire, earth, and water. 

Heating a body, such as a segment of protein alpha helix (above), tends to cause its atoms to vibrate more, and to expand or change phase, if heating is continued; an axiom of nature noted by Herman Boerhaave in the 1700s.

Atomism is a central part of today's relationship between thermodynamics and statistical mechanics. Ancient thinkers such as Leucippus and Democritus, and later the Epicureans, by advancing atomism, laid the foundations for the later atomic theory. Until experimental proof of atoms was later provided in the 20th century, the atomic theory was driven largely by philosophical considerations and scientific intuition. 

The 5th century BC, Greek philosopher Parmenides, in his only known work, a poem conventionally titled On Nature, uses verbal reasoning to postulate that a void, essentially what is now known as a vacuum, in nature could not occur. This view was supported by the arguments of Aristotle, but was criticized by Leucippus and Hero of Alexandria. From antiquity to the Middle Ages various arguments were put forward to prove or disapprove the existence of a vacuum and several attempts were made to construct a vacuum but all proved unsuccessful. 

The European scientists Cornelius Drebbel, Robert Fludd, Galileo Galilei and Santorio Santorio in the 16th and 17th centuries were able to gauge the relative "coldness" or "hotness" of air, using a rudimentary air thermometer (or thermoscope). This may have been influenced by an earlier device which could expand and contract the air constructed by Philo of Byzantium and Hero of Alexandria.

Around 1600, the English philosopher and scientist Francis Bacon surmised: "Heat itself, its essence and quiddity is motion and nothing else." In 1643, Galileo Galilei, while generally accepting the 'sucking' explanation of horror vacui proposed by Aristotle, believed that nature's vacuum-abhorrence is limited. Pumps operating in mines had already proven that nature would only fill a vacuum with water up to a height of ~30 feet. Knowing this curious fact, Galileo encouraged his former pupil Evangelista Torricelli to investigate these supposed limitations. Torricelli did not believe that vacuum-abhorrence (Horror vacui) in the sense of Aristotle's 'sucking' perspective, was responsible for raising the water. Rather, he reasoned, it was the result of the pressure exerted on the liquid by the surrounding air. 

To prove this theory, he filled a long glass tube (sealed at one end) with mercury and upended it into a dish also containing mercury. Only a portion of the tube emptied (as shown adjacent); ~30 inches of the liquid remained. As the mercury emptied, and a partial vacuum was created at the top of the tube. The gravitational force on the heavy element Mercury prevented it from filling the vacuum.

Transition from chemistry to thermochemistry

The world's first ice-calorimeter, used in the winter of 1782-83, by Antoine Lavoisier and Pierre-Simon Laplace, to determine the heat evolved in various chemical changes; calculations which were based on Joseph Black's prior discovery of latent heat. These experiments mark the foundation of thermochemistry.

The theory of phlogiston arose in the 17th century, late in the period of alchemy. Its replacement by caloric theory in the 18th century is one of the historical markers of the transition from alchemy to chemistry. Phlogiston was a hypothetical substance that was presumed to be liberated from combustible substances during burning, and from metals during the process of rusting. Caloric, like phlogiston, was also presumed to be the "substance" of heat that would flow from a hotter body to a cooler body, thus warming it. 

The first substantial experimental challenges to caloric theory arose in Rumford's 1798 work, when he showed that boring cast iron cannons produced great amounts of heat which he ascribed to friction, and his work was among the first to undermine the caloric theory. The development of the steam engine also focused attention on calorimetry and the amount of heat produced from different types of coal. The first quantitative research on the heat changes during chemical reactions was initiated by Lavoisier using an ice calorimeter following research by Joseph Black on the latent heat of water.

More quantitative studies by James Prescott Joule in 1843 onwards provided soundly reproducible phenomena, and helped to place the subject of thermodynamics on a solid footing. William Thomson, for example, was still trying to explain Joule's observations within a caloric framework as late as 1850. The utility and explanatory power of kinetic theory, however, soon started to displace caloric and it was largely obsolete by the end of the 19th century. Joseph Black and Lavoisier made important contributions in the precise measurement of heat changes using the calorimeter, a subject which became known as thermochemistry.

Phenomenological thermodynamics

Robert Boyle. 1627-1691

• Boyle's law (1662)
• Charles's law was first published by Joseph Louis Gay-Lussac in 1802, but he referenced unpublished work by Jacques Charles from around 1787. The relationship had been anticipated by the work of Guillaume Amontons in 1702.
• Gay-Lussac's law (1802)

Birth of thermodynamics as a science

Irish physicist and chemist Robert Boyle in 1656, in coordination with English scientist Robert Hooke, built an air pump. Using this pump, Boyle and Hooke noticed the pressure-volume correlation: P.V=constant. In that time, air was assumed to be a system of motionless particles, and not interpreted as a system of moving molecules. The concept of thermal motion came two centuries later. Therefore, Boyle's publication in 1660 speaks about a mechanical concept: the air spring. Later, after the invention of the thermometer, the property temperature could be quantified. This tool gave Gay-Lussac the opportunity to derive his law, which led shortly later to the ideal gas law. But, already before the establishment of the ideal gas law, an associate of Boyle's named Denis Papin built in 1679 a bone digester, which is a closed vessel with a tightly fitting lid that confines steam until a high pressure is generated. 

Later designs implemented a steam release valve to keep the machine from exploding. By watching the valve rhythmically move up and down, Papin conceived of the idea of a piston and cylinder engine. He did not however follow through with his design. Nevertheless, in 1697, based on Papin's designs, engineer Thomas Savery built the first engine. Although these early engines were crude and inefficient, they attracted the attention of the leading scientists of the time. One such scientist was Sadi Carnot, the "father of thermodynamics", who in 1824 published Reflections on the Motive Power of Fire, a discourse on heat, power, and engine efficiency. This marks the start of thermodynamics as a modern science. 

A Watt steam engine, the steam engine that propelled the Industrial Revolution in Britain and the world

Hence, prior to 1698 and the invention of the Savery Engine, horses were used to power pulleys, attached to buckets, which lifted water out of flooded salt mines in England. In the years to follow, more variations of steam engines were built, such as the Newcomen Engine, and later the Watt Engine. In time, these early engines would eventually be utilized in place of horses. Thus, each engine began to be associated with a certain amount of "horse power" depending upon how many horses it had replaced. The main problem with these first engines was that they were slow and clumsy, converting less than 2% of the input fuel into useful work. In other words, large quantities of coal (or wood) had to be burned to yield only a small fraction of work output. Hence the need for a new science of engine dynamics was born. 

Sadi Carnot (1796-1832): the "father" of thermodynamics

Most cite Sadi Carnot's 1824 book Reflections on the Motive Power of Fire as the starting point for thermodynamics as a modern science. Carnot defined "motive power" to be the expression of the useful effect that a motor is capable of producing. Herein, Carnot introduced us to the first modern day definition of "work": weight lifted through a height. The desire to understand, via formulation, this useful effect in relation to "work" is at the core of all modern day thermodynamics.

In 1843, James Joule experimentally found the mechanical equivalent of heat. In 1845, Joule reported his best-known experiment, involving the use of a falling weight to spin a paddle-wheel in a barrel of water, which allowed him to estimate a mechanical equivalent of heat of 819 ft·lbf/Btu (4.41 J/cal). This led to the theory of conservation of energy and explained why heat can do work. 

In 1850, the famed mathematical physicist Rudolf Clausius coined the term "entropy" (das Wärmegewicht, symbolized S) to denote heat lost or turned into waste. ("Wärmegewicht" translates literally as "heat-weight"; the corresponding English term stems from the Greek τρέπω, "I turn".) 

The name "thermodynamics", however, did not arrive until 1854, when the British mathematician and physicist William Thomson (Lord Kelvin) coined the term thermo-dynamics in his paper On the Dynamical Theory of Heat.

In association with Clausius, in 1871, the Scottish mathematician and physicist James Clerk Maxwell formulated a new branch of thermodynamics called Statistical Thermodynamics, which functions to analyze large numbers of particles at equilibrium, i.e., systems where no changes are occurring, such that only their average properties as temperature T, pressure P, and volume V become important. 

Soon thereafter, in 1875, the Austrian physicist Ludwig Boltzmann formulated a precise connection between entropy S and molecular motion:

${\displaystyle S=k\log W\,}$

being defined in terms of the number of possible states [W] such motion could occupy, where k is the Boltzmann's constant. 

The following year, 1876, chemical engineer Willard Gibbs published an obscure 300-page paper titled: On the Equilibrium of Heterogeneous Substances, wherein he formulated one grand equality, the Gibbs free energy equation, which suggested a measure of the amount of "useful work" attainable in reacting systems. Gibbs also originated the concept we now know as enthalpy H, calling it "a heat function for constant pressure". The modern word enthalpy would be coined many years later by Heike Kamerlingh Onnes, who based it on the Greek word enthalpein meaning to warm.

Building on these foundations, those as Lars Onsager, Erwin Schrödinger, and Ilya Prigogine, and others, functioned to bring these engine "concepts" into the thoroughfare of almost every modern-day branch of science.

Kinetic theory

The idea that heat is a form of motion is perhaps an ancient one and is certainly discussed by Francis Bacon in 1620 in his Novum Organum. The first written scientific reflection on the microscopic nature of heat is probably to be found in a work by Mikhail Lomonosov, in which he wrote:

(..) movement should not be denied based on the fact it is not seen. Who would deny that the leaves of trees move when rustled by a wind, despite it being unobservable from large distances? Just as in this case motion remains hidden due to perspective, it remains hidden in warm bodies due to the extremely small sizes of the moving particles. In both cases, the viewing angle is so small that neither the object nor their movement can be seen.

During the same years, Daniel Bernoulli published his book Hydrodynamics (1738), in which he derived an equation for the pressure of a gas considering the collisions of its atoms with the walls of a container. He proves that this pressure is two thirds the average kinetic energy of the gas in a unit volume. Bernoulli's ideas, however, made little impact on the dominant caloric culture. Bernoulli made a connection with Gottfried Leibniz's vis viva principle, an early formulation of the principle of conservation of energy, and the two theories became intimately entwined throughout their history. Though Benjamin Thompson suggested that heat was a form of motion as a result of his experiments in 1798, no attempt was made to reconcile theoretical and experimental approaches, and it is unlikely that he was thinking of the vis viva principle. 

John Herapath later independently formulated a kinetic theory in 1820, but mistakenly associated temperature with momentum rather than vis viva or kinetic energy. His work ultimately failed peer review and was neglected. John James Waterston in 1843 provided a largely accurate account, again independently, but his work received the same reception, failing peer review even from someone as well-disposed to the kinetic principle as Davy.

Further progress in kinetic theory started only in the middle of the 19th century, with the works of Rudolf Clausius, James Clerk Maxwell, and Ludwig Boltzmann. In his 1857 work On the nature of the motion called heat, Clausius for the first time clearly states that heat is the average kinetic energy of molecules. This interested Maxwell, who in 1859 derived the momentum distribution later named after him. Boltzmann subsequently generalized his distribution for the case of gases in external fields. 

Boltzmann is perhaps the most significant contributor to kinetic theory, as he introduced many of the fundamental concepts in the theory. Besides the Maxwell–Boltzmann distribution mentioned above, he also associated the kinetic energy of particles with their degrees of freedom. The Boltzmann equation for the distribution function of a gas in non-equilibrium states is still the most effective equation for studying transport phenomena in gases and metals. By introducing the concept of thermodynamic probability as the number of microstates corresponding to the current macrostate, he showed that its logarithm is proportional to entropy.

Branches of thermodynamics

The following list gives a rough outline as to when the major branches of thermodynamics came into inception:

• Thermochemistry - 1780s
• Classical thermodynamics - 1824
• Chemical thermodynamics - 1876
• Statistical mechanics - c. 1880s
• Equilibrium thermodynamics
• Engineering thermodynamics
• Chemical engineering thermodynamics - c. 1940s
• Non-equilibrium thermodynamics - 1941
• Small systems thermodynamics - 1960s
• Biological thermodynamics - 1957
• Ecosystem thermodynamics - 1959
• Relativistic thermodynamics - 1965
• Rational thermodynamics - 1960s
• Quantum thermodynamics - 1968
• Black hole thermodynamics - c. 1970s
• Geological thermodynamics - c. 1970s
• Biological evolution thermodynamics - 1978
• Geochemical thermodynamics - c. 1980s
• Atmospheric thermodynamics - c. 1980s
• Natural systems thermodynamics - 1990s
• Supramolecular thermodynamics - 1990s
• Earthquake thermodynamics - 2000
• Drug-receptor thermodynamics - 2001
• Pharmaceutical systems thermodynamics – 2002

Ideas from thermodynamics have also been applied in other fields, for example:

• Thermoeconomics - c. 1970s

Entropy and the second law

Even though he was working with the caloric theory, Sadi Carnot in 1824 suggested that some of the caloric available for generating useful work is lost in any real process. In March 1851, while grappling to come to terms with the work of James Prescott Joule, Lord Kelvin started to speculate that there was an inevitable loss of useful heat in all processes. The idea was framed even more dramatically by Hermann von Helmholtz in 1854, giving birth to the spectre of the heat death of the universe. 

In 1854, William John Macquorn Rankine started to make use in calculation of what he called his thermodynamic function. This has subsequently been shown to be identical to the concept of entropy formulated by Rudolf Clausius in 1865. Clausius used the concept to develop his classic statement of the second law of thermodynamics the same year.

Heat transfer

The phenomenon of heat conduction is immediately grasped in everyday life. In 1701, Sir Isaac Newton published his law of cooling. However, in the 17th century, it came to be believed that all materials had an identical conductivity and that differences in sensation arose from their different heat capacities. 

Suggestions that this might not be the case came from the new science of electricity in which it was easily apparent that some materials were good electrical conductors while others were effective insulators. Jan Ingen-Housz in 1785-9 made some of the earliest measurements, as did Benjamin Thompson during the same period.

The fact that warm air rises and the importance of the phenomenon to meteorology was first realised by Edmund Halley in 1686. Sir John Leslie observed that the cooling effect of a stream of air increased with its speed, in 1804.

Carl Wilhelm Scheele distinguished heat transfer by thermal radiation (radiant heat) from that by convection and conduction in 1777. In 1791, Pierre Prévost showed that all bodies radiate heat, no matter how hot or cold they are. In 1804, Leslie observed that a matte black surface radiates heat more effectively than a polished surface, suggesting the importance of black body radiation. Though it had become to be suspected even from Scheele's work, in 1831 Macedonio Melloni demonstrated that black body radiation could be reflected, refracted and polarised in the same way as light.

James Clerk Maxwell's 1862 insight that both light and radiant heat were forms of electromagnetic wave led to the start of the quantitative analysis of thermal radiation. In 1879, Jožef Stefan observed that the total radiant flux from a blackbody is proportional to the fourth power of its temperature and stated the Stefan–Boltzmann law. The law was derived theoretically by Ludwig Boltzmann in 1884.

Absolute Zero

In 1702 Guillaume Amontons introduced the concept of absolute zero based on observations of gases. In 1810, Sir John Leslie froze water to ice artificially. The idea of absolute zero was generalized in 1848 by Lord Kelvin. In 1906, Walther Nernst stated the third law of thermodynamics.

Gibbs free energy (updated)

From Wikipedia, the free encyclopedia

In thermodynamics, the Gibbs free energy (IUPAC recommended name: Gibbs energy or Gibbs function; also known as free enthalpy to distinguish it from Helmholtz free energy) is a thermodynamic potential that can be used to calculate the maximum of reversible work that may be performed by a thermodynamic system at a constant temperature and pressure (isothermal, isobaric). The Gibbs free energy (ΔGº = ΔHº-TΔSº) (J in SI units) is the maximum amount of non-expansion work that can be extracted from a thermodynamically closed system (one that can exchange heat and work with its surroundings, but not matter); this maximum can be attained only in a completely reversible process. When a system transforms reversibly from an initial state to a final state, the decrease in Gibbs free energy equals the work done by the system to its surroundings, minus the work of the pressure forces.

The Gibbs energy (also referred to as G) is also the thermodynamic potential that is minimized when a system reaches chemical equilibrium at constant pressure and temperature. Its derivative with respect to the reaction coordinate of the system vanishes at the equilibrium point. As such, a reduction in G is a necessary condition for the spontaneity of processes at constant pressure and temperature.

The Gibbs free energy, originally called available energy, was developed in the 1870s by the American scientist Josiah Willard Gibbs. In 1873, Gibbs described this "available energy" as

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.

Overview

The reaction C_(s)^diamond → C_(s)^graphite has a negative change in Gibbs free energy and is therefore thermodynamically favorable at 25 °C and 1 atm. However, even though favorable, it is so slow that it is not observed. Whether a reaction is thermodynamically favorable does not determine its rate.

 

According to the second law of thermodynamics, for systems reacting at STP (or any other fixed temperature and pressure), there is a general natural tendency to achieve a minimum of the Gibbs free energy. 

A quantitative measure of the favorability of a given reaction at constant temperature and pressure is the change ΔG (sometimes written "delta G" or "dG") in Gibbs free energy that is (or would be) caused by the reaction. As a necessary condition for the reaction to occur at constant temperature and pressure, ΔG must be smaller than the non-PV (e.g. electrical) work, which is often equal to zero (hence ΔG must be negative). ΔG equals the maximum amount of non-PV work that can be performed as a result of the chemical reaction for the case of reversible process. If the analysis indicated a positive ΔG for the reaction, then energy — in the form of electrical or other non-PV work — would have to be added to the reacting system for ΔG to be smaller than the non-PV work and make it possible for the reaction to occur.

We can think of ∆G as the amount of "free" or "useful" energy available to do work. The equation can be also seen from the perspective of the system taken together with its surroundings (the rest of the universe). First, assume that the given reaction at constant temperature and pressure is the only one that is occurring. Then the entropy released or absorbed by the system equals the entropy that the environment must absorb or release, respectively. The reaction will only be allowed if the total entropy change of the universe is zero or positive. This is reflected in a negative ΔG, and the reaction is called exergonic. 

If we couple reactions, then an otherwise endergonic chemical reaction (one with positive ΔG) can be made to happen. The input of heat into an inherently endergonic reaction, such as the elimination of cyclohexanol to cyclohexene, can be seen as coupling an unfavourable reaction (elimination) to a favourable one (burning of coal or other provision of heat) such that the total entropy change of the universe is greater than or equal to zero, making the total Gibbs free energy difference of the coupled reactions negative. 

In traditional use, the term "free" was included in "Gibbs free energy" to mean "available in the form of useful work". The characterization becomes more precise if we add the qualification that it is the energy available for non-volume work. (An analogous, but slightly different, meaning of "free" applies in conjunction with the Helmholtz free energy, for systems at constant temperature). However, an increasing number of books and journal articles do not include the attachment "free", referring to G as simply "Gibbs 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.

History

The quantity called "free energy" is a more advanced and accurate replacement for the outdated term affinity, which was used by chemists in the earlier years of physical chemistry to describe the force that caused chemical reactions. 

In 1873, Willard Gibbs published A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces, in which he sketched the principles of his new equation that was 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 composed of 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 would ensue. Further, Gibbs stated:

If we wish to express in a single equation the necessary and sufficient condition of thermodynamic equilibrium for a substance when surrounded by a medium of constant pressure p and temperature T, this equation may be written: δ(ε − Tη + pν) = 0 when δ refers to the variation produced by any variations in the state of the parts of the body, and (when different parts of the body are in different states) in the proportion in which the body is divided between the different states. The condition of stable equilibrium is that the value of the expression in the parenthesis shall be a minimum.

In this description, as used by Gibbs, ε refers to the internal energy of the body, η refers to the entropy of the body, and ν is the volume of the body. 

Thereafter, in 1882, the German scientist Hermann von Helmholtz characterized the affinity as 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. 

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 Substances 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.

Graphical interpretation

Gibbs free energy was originally defined graphically. In 1873, American scientist Willard Gibbs published his first thermodynamics paper, "Graphical Methods in the Thermodynamics of Fluids", in which Gibbs used the two coordinates of the entropy and volume to represent the state of the body. In his second follow-up paper, "A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces", published later that year, Gibbs added in the third coordinate of the energy of the body, defined on three figures. In 1874, Scottish physicist James Clerk Maxwell used Gibbs' figures to make a 3D energy-entropy-volume thermodynamic surface of a fictitious water-like substance. Thus, in order to understand the very difficult concept of Gibbs free energy one must be able to understand its interpretation as Gibbs defined originally by section AB on his figure 3 and as Maxwell sculpted that section on his 3D surface figure. 

American scientist Willard Gibbs' 1873 figures two and three (above left and middle) used by Scottish physicist James Clerk Maxwell in 1874 to create a three-dimensional entropy, volume, energy thermodynamic surface diagram for a fictitious water-like substance, transposed the two figures of Gibbs (above right) onto the volume-entropy coordinates (transposed to bottom of cube) and energy-entropy coordinates (flipped upside down and transposed to back of cube), respectively, of a three-dimensional Cartesian coordinates; the region AB being the first-ever three-dimensional representation of Gibbs free energy, or what Gibbs called "available energy"; the region AC being its capacity for entropy, what Gibbs defined as "the amount by which the entropy of the body can be increased without changing the energy of the body or increasing its volume.

Definitions

Willard Gibbs’ 1873 available energy (free energy) graph, which shows a plane perpendicular to the axis of v (volume) and passing through point A, which represents the initial state of the body. MN is the section of the surface of dissipated energy. Qε and Qη are sections of the planes η = 0 and ε = 0, and therefore parallel to the axes of ε (internal energy) and η (entropy), respectively. AD and AE are the energy and entropy of the body in its initial state, AB and AC its available energy (Gibbs free energy) and its capacity for entropy (the amount by which the entropy of the body can be increased without changing the energy of the body or increasing its volume) respectively.

 

The Gibbs free energy is defined as

${\displaystyle G(p,T)=U+pV-TS,}$

which is the same as

${\displaystyle G(p,T)=H-TS,}$

where:

U is the internal energy (SI unit: joule),

p is pressure (SI unit: pascal),

V is volume (SI unit: m³),

T is the temperature (SI unit: kelvin),

S is the entropy (SI unit: joule per kelvin),

H is the enthalpy (SI unit: joule).

The expression for the infinitesimal reversible change in the Gibbs free energy as a function of its "natural variables" p and T, for an open system, subjected to the operation of external forces (for instance, electrical or magnetic) X_i, which cause the external parameters of the system a_i to change by an amount da_i, can be derived as follows from the first law for reversible processes:

${\displaystyle T\,\mathrm {d} S=\mathrm {d} U+p\,\mathrm {d} V-\sum _{i=1}^{k}\mu _{i}\,\mathrm {d} N_{i}+\sum _{i=1}^{n}X_{i}\,\mathrm {d} a_{i}+\cdots }$

${\displaystyle \mathrm {d} (TS)-S\,\mathrm {d} T=\mathrm {d} U+\mathrm {d} (pV)-V\,\mathrm {d} p-\sum _{i=1}^{k}\mu _{i}\,\mathrm {d} N_{i}+\sum _{i=1}^{n}X_{i}\,\mathrm {d} a_{i}+\cdots }$

${\displaystyle \mathrm {d} (U-TS+pV)=V\,\mathrm {d} p-S\,\mathrm {d} T+\sum _{i=1}^{k}\mu _{i}\,\mathrm {d} N_{i}-\sum _{i=1}^{n}X_{i}\,\mathrm {d} a_{i}+\cdots }$

${\displaystyle \mathrm {d} G=V\,\mathrm {d} p-S\,\mathrm {d} T+\sum _{i=1}^{k}\mu _{i}\,\mathrm {d} N_{i}-\sum _{i=1}^{n}X_{i}\,\mathrm {d} a_{i}+\cdots }$

where:

μ_i is the chemical potential of the ith chemical component. (SI unit: joules per particle or joules per mole)

N_i is the number of particles (or number of moles) composing the ith chemical component.

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 or for a closed, chemically reacting system where the N_i are changing. For a closed, non-reacting 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 that shortens by an amount −dl under a force f would result in a term f dl 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 −Ψ de, 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^{[clarification needed]} and entropic driving forces involved in a thermodynamic process.

Relation to other relevant parameters

 

The temperature dependence of the Gibbs energy for an ideal gas is given by the Gibbs–Helmholtz equation, and its pressure dependence is given by

${\displaystyle {\frac {G}{N}}={\frac {G^{\circ }}{N}}+kT\ln {\frac {p}{p^{\circ }}}.}$

If the volume is known rather than pressure, then it becomes

${\displaystyle {\frac {G}{N}}={\frac {G^{\circ }}{N}}+kT\ln {\frac {V^{\circ }}{V}},}$

or more conveniently as its chemical potential:

${\displaystyle {\frac {G}{N}}=\mu =\mu ^{\circ }+kT\ln {\frac {p}{p^{\circ }}}.}$

In non-ideal systems, fugacity comes into play.

Derivation

The Gibbs free energy total differential natural variables may be derived by Legendre transforms of the internal energy.

${\displaystyle \mathrm {d} U=T\,\mathrm {d} S-p\,\mathrm {d} V+\sum _{i}\mu _{i}\,\mathrm {d} N_{i}.}$

The definition of G from above is

${\displaystyle G=U+pV-TS}$.

Taking the total differential, we have

${\displaystyle \mathrm {d} G=\mathrm {d} U+p\,\mathrm {d} V+V\,\mathrm {d} p-T\,\mathrm {d} S-S\,\mathrm {d} T.}$

Replacing dU with the result from the first law gives

${\displaystyle {\begin{aligned}\mathrm {d} G&=T\,\mathrm {d} S-p\,\mathrm {d} V+\sum _{i}\mu _{i}\,\mathrm {d} N_{i}+p\,\mathrm {d} V+V\,\mathrm {d} p-T\,\mathrm {d} S-S\,\mathrm {d} T\\&=V\,\mathrm {d} p-S\,\mathrm {d} T+\sum _{i}\mu _{i}\,\mathrm {d} N_{i}.\end{aligned}}}$

The natural variables of G are then p, T, and {N_i}.

Homogeneous systems

Because S, V, and N_i are extensive variables, an Euler integral allows easy integration of dU:

${\displaystyle U=TS-pV+\sum _{i}\mu _{i}N_{i}.}$

Because some of the natural variables of G are intensive, dG may not be integrated using Euler integrals as is the case with internal energy. However, simply substituting the above integrated result for U into the definition of G gives a standard expression for G:

${\displaystyle {\begin{aligned}G&=U+pV-TS\\&=(TS-pV+\sum _{i}\mu _{i}N_{i})+pV-TS\\&=\sum _{i}\mu _{i}N_{i}.\end{aligned}}}$

This result applies to homogeneous, macroscopic systems, but not to all thermodynamic systems.

Gibbs free energy of reactions

The system under consideration is held at constant temperature and pressure, and is closed (no matter can come in or out). The Gibbs energy of any system is ${\displaystyle G=U+PV-TS}$ and an infinitesimal change in G, at constant temperature and pressure yields:

${\displaystyle dG=dU+PdV-TdS}$

By the first law of thermodynamics, a change in the internal energy U is given by

${\displaystyle dU=\delta Q+\delta W}$

where δQ is energy added as heat, and δW is energy added as work. The work done on the system may be written as δW = −PdV + δW_x, where −PdV is the mechanical work of compression/expansion done on the system and δW_x is all other forms of work, which may include electrical, magnetic, etc. Assuming that only mechanical work is done,

${\displaystyle dU=\delta Q-PdV}$

and the infinitesimal change in G is:

${\displaystyle dG=\delta Q-TdS}$

The second law of thermodynamics states that for a closed system, ${\displaystyle TdS\geq \delta Q}$, and so it follows that:

${\displaystyle dG\leq 0}$

This means that for a system which is not in equilibrium, its Gibbs energy will always be decreasing, and when it is in equilibrium (i.e. no longer changing), the infinitesimal change dG will be zero. In particular, this will be true if the system is experiencing any number of internal chemical reactions on its path to equilibrium.

Useful identities to derive the Nernst equation

During a reversible electrochemical reaction at constant temperature and pressure, the following equations involving the Gibbs free energy hold:

${\displaystyle \Delta _{\text{r}}G=\Delta _{\text{r}}G^{\circ }+RT\ln Q_{\text{r}}}$,

${\displaystyle \Delta _{\text{r}}G^{\circ }=-RT\ln K_{\text{eq}}}$ (for a system at chemical equilibrium),

${\displaystyle \Delta _{\text{r}}G=w_{\text{elec,rev}}=-nFE}$ (for a reversible electrochemical process at constant temperature and pressure),

${\displaystyle \Delta _{\text{r}}G^{\circ }=-nFE^{\circ }}$ (definition of E°),

and rearranging gives

${\displaystyle nFE^{\circ }=RT\ln K_{\text{eq}},}$

${\displaystyle nFE=nFE^{\circ }-RT\ln Q_{\text{r}}r,}$

${\displaystyle E=E^{\circ }-{\frac {RT}{nF}}\ln Q_{\text{r}},}$

which relates the cell potential resulting from the reaction to the equilibrium constant and reaction quotient for that reaction (Nernst equation),

where

Δ_rG = Gibbs free energy change per mole of reaction,

Δ_rG° = Gibbs free energy change per mole of reaction for unmixed reactants and products at standard conditions (i.e. 298K, 100kPa, 1M of each reactant and product),

R = gas constant,

T = absolute temperature,

ln = natural logarithm,

Q_r = reaction quotient (unitless),

K_eq = equilibrium constant (unitless),

w_elec,rev = electrical work in a reversible process (chemistry sign convention),

n = number of moles of electrons transferred in the reaction,

F = Faraday constant = 96485 C/mol (charge per mole of electrons),

E = cell potential,

E° = standard cell potential.

Moreover, we also have:

${\displaystyle K_{\text{eq}}=e^{-{\frac {\Delta _{\text{r}}G^{\circ }}{RT}}},}$

${\displaystyle \Delta _{\text{r}}G^{\circ }=-RT(\ln K_{\text{eq}})=-2.303\,RT(\log _{10}K_{\text{eq}}),}$

which relates the equilibrium constant with Gibbs free energy. This implies that at equilibrium

${\displaystyle Q_{\text{r}}=K_{\text{eq}}}$ and ${\displaystyle \Delta _{\text{r}}G=0}$

The second law of thermodynamics and metabolism

A chemical reaction will (or can) proceed spontaneously if the change in the total entropy of the universe that would be caused by the reaction is nonnegative. As discussed in the overview, if the temperature and pressure are held constant, the Gibbs free energy is a (negative) proxy for the change in total entropy of the universe. It is "negative" because S appears with a negative coefficient in the expression for G, so the Gibbs free energy moves in the opposite direction from the total entropy. Thus, a reaction with a positive Gibbs free energy will not proceed spontaneously. However, in biological systems (among others), energy inputs from other energy sources (including the Sun and exothermic chemical reactions) are "coupled" with reactions that are not entropically favored (i.e. have a Gibbs free energy above zero). Taking into account the coupled reactions, the total entropy in the universe increases. This coupling allows endergonic reactions, such as photosynthesis and DNA synthesis, to proceed without decreasing the total entropy of the universe. Thus biological systems do not violate the second law of thermodynamics.

Standard energy change of formation

Table of selected substances

Substance (State)	ΔfG° (kJ/mol)	ΔfG° (kcal/mol)
NO(g)	87.6	20.9
NO2(g)	51.3	12.3
N2O(g)	103.7	24.78
H2O(g)	−228.6	−54.64
H2O(l)	−237.1	−56.67
CO2(g)	−394.4	−94.26
CO(g)	−137.2	−32.79
CH4(g)	−50.5	−12.1
C2H6(g)	−32.0	−7.65
C3H8(g)	−23.4	−5.59
C6H6(g)	129.7	29.76
C6H6(l)	124.5	31.00

The standard Gibbs free energy of formation of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of that substance from its component elements, at their standard states (the most stable form of the element at 25 °C and 100 kPa). Its symbol is Δ_fG˚. 

All elements in their standard states (diatomic oxygen gas, graphite, etc.) have standard Gibbs free energy change of formation equal to zero, as there is no change involved.

Δ_fG = Δ_fG˚ + RT ln Q_f ;

Q_f is the reaction quotient.

At equilibrium, Δ_fG = 0, and Q_f = K, so the equation becomes Δ_fG˚ = −RT ln K (where K is the equilibrium constant).