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In thermodynamics, the Helmholtz free energy is a thermodynamic potential which measures the "useful" work obtainable from a closed thermodynamic system at a constant temperature. For such a system, the negative of the difference in the Helmholtz energy is equal to the maximum amount of work extractable from a thermodynamic process in which temperature is held constant. Under these conditions, it is minimized at equilibrium. Wikipedia does not have an article with this exact name. ... This article needs to be cleaned up to conform to a higher standard of quality. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... In thermodynamics and molecular chemistry, the enthalpy or heat content (denoted as Δ or ΔH, or rarely as χ) is a quotient or description of thermodynamic potential of a system, which can be used to calculate the useful work obtainable from a closed thermodynamic system under constant conditions. ... In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the useful work obtainable from a closed thermodynamic system at a constant temperature and pressure. ... Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... In thermodynamics, four quantities, measured in units of energy, are called thermodynamic potentials: where T = temperature, S = entropy, p = pressure, V = volume Differential definitions The following differential relations hold for the four potentials: If we write the above four equations generally as Then it is seen that yielding expressions for... In thermodynamics, a closed system, as contrasted with an isolated system, can exchange heat and work, but not matter, with its surroundings. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ...

The Helmholtz free energy was developed by Hermann von Helmholtz and is denoted by the letter A  (from the German "Arbeit" or work), or the letter F . The letter A  is preferred by IUPAC and will be used here. The Helmholtz free energy also goes by the name Helmholtz energy or Helmholtz function among others. Hermann Ludwig Ferdinand von Helmholtz (August 31, 1821 – September 8, 1894) was a German physician and physicist. ... The International Union of Pure and Applied Chemistry (IUPAC) is an international non-governmental organization devoted to the advancement of chemistry. ...

The Helmholtz energy is defined as:

where

• A  is the Helmholtz energy (SI: joules, CGS: ergs),
• U  is the internal energy of the system (SI: joules, CGS: ergs),
• T  is the absolute temperature (kelvins),
• S  is the entropy (SI: joules per kelvin, CGS: ergs per kelvin).

Cover of brochure The International System of Units. ... The joule (symbol: J) is the SI (metric) unit of energy, which is defined as the potential to do work. ... An erg is the unit of energy and mechanical work in the centimetre-gram-second (CGS) system of units, symbol erg. Its name is derived from the Greek word meaning work. The erg is a small unit, equal to a force of one dyne exerted for a distance of one... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... For other uses of the term entropy, see Entropy (disambiguation) The thermodynamic entropy S, often simply called the entropy in the context of thermodynamics, is a measure of the amount of energy in a physical system that cannot be used to do work. ...

History

The quantity called "free energy" is essentially a more advanced and accurate replacement for the antiquated term “affinity” used by chemists to describe the “force” that caused chemical reactions. The term affinity, as used in chemical relation, dates back to at least the time of Albertus Magnus in 1250. Look up affinity in Wiktionary, the free dictionary. ... A chemical reaction occurs when vapours of hydrogen chloride and ammonia meet to form a cloud of a new substance, ammonium chloride Chemical reaction is a process that results in the interconversion of chemical substances [1]. The substance or substances initially involved in a chemical reaction are called reactants. ... Albertus Magnus (1193? – November 15, 1280), also known as Saint Albert the Great and Albert of Cologne, was a Dominican friar who became famous for his comprehensive knowledge and advocacy for the peaceful coexistence of science and religion. ...

From the 1998 textbook Modern Thermodynamics by Nobelist and chemical engineering 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. Ilya Prigogine (January 25, 1917 – May 28, 2003) was a Belgian physicist and chemist noted for his work on dissipative structures, complex systems, and irreversibility. ...

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. Sir Isaac Newton, FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, alchemist, and natural philosopher, regarded by many as the greatest figure in the history of science. ...

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. Marcellin Berthelot Marcellin (or Marcelin) Pierre Eugène Berthelot (October 25, 1827 - March 18, 1907) was a French chemist and politician. ... Hans Peter Jörgen Julius Thomsen (February 16, 1826 – February 13, 1909) was a Danish chemist. ... The standard enthalpy change of reaction (denoted ΔH° or ΔHo)is the enthalpy change that occurs in a system when one mole of matter is transformed by a chemical reaction under standard conditions. ... In thermochemistry, the principle of maximum work was a postulate put forward in 1875 by the French chemist Marcellin Berthelot which stated that chemical reactions will tend to evolve in such a manner so as to assemble or dissasemble chemical species so to yield the maximum amount of chemical energy... In physics, heat, symbolized by Q, is defined as energy in transit. ...

In addition to this, in 1780 Antoine Lavoisier and Pierre-Simon Laplace laid the foundations of thermochemistry by showing that the heat evolved 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 evolved 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 known as Hess's 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. Antoine-Laurent de Lavoisier (August 26, 1743 – May 8, 1794) the father of modern chemistry, was a French nobleman prominent in the histories of chemistry, finance, biology, and economics. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In the thermodynamics and physical chemistry, thermochemistry is the study of the heat evolved or absorbed in chemical reactions. ... The specific heat capacity (symbol c or s, also called specific heat) of a substance is defined as heat capacity per unit mass. ... This article or section does not cite its references or sources. ... Germain Henri Hess (1802-08-07 - 1850-11-30) was a Swiss chemist and doctor who formulated Hesss Law, an early principle of thermochemistry. ... A visual representation of the law in action. ... In the history of science, the theory of heat or mechanical theory of heat was a theory, introduced predominately in 1824 by the French physicist Sadi Carnot, that heat and mechanical work are equivalent. ... Conservation of energy states that the total amount of energy (often expressed as the sum of kinetic energy and potential energy) in an isolated system remains constant. ...

Based on these and other ideas, Berthelot and Danish chemist Julius Thomsen, as well as others, considered the heat evolved 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, English physicist James Joule showed that one 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 was a precursory form of the first law of thermodynamics. Hans Peter Jörgen Julius Thomsen (February 16, 1826 – February 13, 1909) was a Danish chemist. ... James Prescott Joule (December 24, 1818–October 11, 1889) was an English physicist, born in Salford, near Manchester. ... The first law of thermodynamics, a generalized expression of the law of the conservation of energy, states: // Description Essentially, the First Law of Thermodynamics declares that energy is conserved for a closed system, with heat and work being the forms of energy transfer. ...

By 1865, the German physicist Rudolf Clausius had showed 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 than later changed the name to entropy. Thus, the heat used to transform the working body of molecule 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. Rudolf Clausius - physicist and mathematician Rudolf Julius Emanuel Clausius (January 2, 1822 – August 24, 1888), was a German physicist and mathematician. ... Combustion or burning is a complex sequence of chemical reactions between a fuel and an oxidant accompanied by the production of heat or both heat and light in the form of either a glow or flames. ... This article needs to be cleaned up to conform to a higher standard of quality. ... The Carnot cycle is a particular thermodynamic cycle, modeled on the Carnot heat engine, studied by Nicolas Léonard Sadi Carnot in the 1820s and expanded upon by Benoit Paul Émile Clapeyron in the 1830s and 40s. ... see also: Entropy (disambiguation) Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ...

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: Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American mathematical physicist who contributed much of the theoretical foundation that led to the development of chemical thermodynamics and was one of the founders of vector analysis. ... Volume is how much space a thing has. ... see also: Entropy (disambiguation) Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... In thermodynamics and chemistry, chemical potential, symbolized by μ, is a term introduced in 1876 by the American mathematical physicist (Willard Gibbs and his partner Lauren Berkley), which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given...

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 υ is the volume of the body. In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... see also: Entropy (disambiguation) Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... Volume is how much space a thing has. ...

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 evolved 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 A at T = constant, V = constant), whilst the heat evolved is usually a measure of the diminution of the total energy of the system (Internal energy). Thus, G or A is the amount of energy “free” for work under the given conditions. Hermann Ludwig Ferdinand von Helmholtz (August 31, 1821 – September 8, 1894) was a German physician and physicist. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of...

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. Lewis in the Berkeley Lab Gilbert Newton Lewis (October 23, 1875-March 23, 1946) was a famous American physical chemist. ... Merle Randall was an American physical chemist famous for his work, over the period of 25 years, in measuring free energy calculations of compounds with Gilbert N. Lewis. ...

Where did the attachment ‘free’ come from ?

In the 18th and 19th centuries, the theory of heat, i.e. that heat is a form of energy having relation to vibratory motion, was beginning to supplant both the caloric theory, i.e. that heat is a fluid, and the four element theory in which heat was the lightest of the four elements. Many textbooks and teaching articles during these centuries presented these theories side by side. Similarly, during these years, heat was beginning to be distinguished into different classification categorize, such as “free heat”, “combined heat”, “radiant heat”, specific heat, heat capacity, “absolute heat”, “latent caloric”, “free” or “perceptible” caloric (calorique sensible), among others. In the history of science, the theory of heat was a term used during the 18th and 19th centuries to describe a number of laws, relations, and experimental phenomenon in relation to heat; those such as thermometry, calorimetry, combustion, specific heat, and discussions as to the quantity of heat released... The caloric theory is an obsolete scientific theory that heat consists of a fluid called caloric that flows from hotter to colder bodies. ... Chinese Wood (木) | Fire (火) Earth (土) | Metal (金) | Water (æ°´) Hinduism and Buddhism Vayu / Pavan — Air / Wind Agni / Tejas — Fire Akasha — Aether Prithvi / Bhumi — Earth Ap / Jala — Water Many ancient philosophies used a set of archetypal classical elements to explain patterns in nature. ... In physics, heat, symbolized by Q, is defined as energy in transit. ... The specific heat capacity (symbol c or s, also called specific heat) of a substance is defined as heat capacity per unit mass. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

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. ^[1] 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 indicated that some latent caloric had become “free” or perceptible. Pierre-Simon Laplace Pierre-Simon Laplace (March 23, 1749 – March 5, 1827) was a French mathematician and astronomer, the discoverer of the Laplace transform and Laplaces equation. ... Antoine-Laurent de Lavoisier (August 26, 1743 - May 8, 1794) was a French nobleman prominent in the histories of chemistry, finance, biology, and economics. ... Vis Viva is the principle that the difference between the aggregate work of the accelerating forces of a system and that of the retarding forces is equal to one half the vis viva accumulated or lost in the system while the work is being done. ...

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 A (or G) determines the amount of energy ‘free’ for work under the given conditions. ^[2] Sadi Carnot may refer to: Nicolas Léonard Sadi Carnot (1796-1832), French physicist Marie François Sadi Carnot (1837-1894), president of the third French Republic This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Hermann Ludwig Ferdinand von Helmholtz (August 31, 1821 – September 8, 1894) was a German physician and physicist. ... In thermodynamics, thermodynamic work is a generalisation of the concept of mechanical work in mechanics. ...

Thus, in traditional usage, the term “free” was included in Gibbs free energy, i.e. for systems at constant pressure and temperature, and Helmholtz free energy, i.e. for systems at constant volume and temperature, to mean ‘available in the form of useful work.’^[3]

The current (since 1988) recommended usage is to omit the word "free", in accordance with the IUPAC guidelines ^[4]. However, many instances of the traditional form persisting in the literature are still to be found. The International Union of Pure and Applied Chemistry (IUPAC) is an international non-governmental organization devoted to the advancement of chemistry. ...

Non-viscous fluids

From the first law of thermodynamics we have: The first law of thermodynamics, a generalized expression of the law of the conservation of energy, states: // Description Essentially, the First Law of Thermodynamics declares that energy is conserved for a closed system, with heat and work being the forms of energy transfer. ...

where U is the internal energy, δQ is the energy added by heating and δW = PdV is the work done by the system. From the second law of thermodynamics, for a reversible process we may say that δQ = TdS. Differentiating the expression for A  we have: The second law of thermodynamics is an expression of the universal law of increasing entropy. ... In thermodynamics, a reversible process (or reversible cycle if the process is cyclic) is a process that can be reversed by means of infinitesimal changes in some property of the system. ...

For a process which is not reversible, the entropy will be smaller than its equilibrium value so we may say that, in general,

It is seen that if a thermodynamic process is isothermal (i.e. occurs at constant temperature), then dT = 0  and thus

The negative of the change in the Helmholtz energy is the maximum work attainable from the system in an isothermal process. In more mathematical terms, the integral of -dA over any isotherm in state space is the maximum work attainable from the system.

If, in addition the volume is held constant as well, the above equation becomes:

with the equality holding at equilibrium. It is seen that the Helmholtz energy for a general system in which the temperature and volume are held constant will continuously decrease to its minimum value, which it maintains at equilbrium.

In a more general form, the first law describes the internal energy with additional terms involving the chemical potential and the number of particles of various types. The differential statement for dA is then: In thermodynamics and chemistry, chemical potential, symbolized by μ, is a term introduced in 1876 by the American mathematical physicist (Willard Gibbs and his partner Lauren Berkley), which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given...

where μ_i is the chemical potential for an i-type particle, and N_i is the number of such particles. With this definition, we may say that the negative of the Helmholtz energy is the maximum amount of work energy available from a system in which the initial and final states have the same temperature and number of particles. Further generalizations will add even more terms whose extensive differential term must be set to zero in order for the interpretation of the Helmholtz energy to hold.

Generalized Helmholtz energy

In the more general case, the mechanical term (pdV) must be replaced by the product of the volume times the stress times an infinitesimal strain (Landau & Lifshitz 1986): Figure 1  Stress tensor A mature tree trunk may support a greater force than a fine steel wire but intuitively we feel that steel is stronger than wood. ... Look up strain in Wiktionary, the free dictionary. ...

where σ_ij is the stress tensor, and is the strain tensor. In the case of linear elastic materials which obey Hooke's Law, the stress is related to the strain by: Look up Elastic in Wiktionary, the free dictionary. ... Hookes law accurately models the physical properties of common mechanical springs at small extensions. ...

where we are now using Einstein notation for the tensors, in which repeated indices in a product are summed. We may integrate the expression for dA to obtain the Helmholtz energy: In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae. ...

See also

• work content - for applications to chemistry
• Statistical Mechanics

[this page details the Helmholtz energy from the point of view of thermal and statistical physics.]

moved from free energy In thermodynamic analysis of chemical reactions, the term free energy denotes either of two related concepts of importance expressing the total amount of energy which is used up or released during a chemical reaction. ... Chemistry (from Greek χημεία khemeia[1] meaning alchemy) is the science of matter at the atomic to molecular scale, dealing primarily with collections of atoms, such as molecules, crystals, and metals. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Statistical physics, one of the fundamental theories of physics, uses methods of statistics in solving physical problems. ... The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ...

References