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).

History of entropy

From Wikipedia, the free encyclopedia

The concept of entropy developed in response to the observation that a certain amount of functional energy released from combustion reactions is always lost to dissipation or friction and is thus not transformed into useful work. Early heat-powered engines such as Thomas Savery's (1698), the Newcomen engine (1712) and the Cugnot steam tricycle (1769) were inefficient, converting less than two percent of the input energy into useful work output; a great deal of useful energy was dissipated or lost. Over the next two centuries, physicists investigated this puzzle of lost energy; the result was the concept of entropy.

In the early 1850s, Rudolf Clausius set forth the concept of the thermodynamic system and posited the argument that in any irreversible process a small amount of heat energy δQ is incrementally dissipated across the system boundary. Clausius continued to develop his ideas of lost energy, and coined the term entropy.

Since the mid-20th century the concept of entropy has found application in the field of information theory, describing an analogous loss of data in information transmission systems.

Classical thermodynamic views

In 1803, mathematician Lazare Carnot published a work entitled Fundamental Principles of Equilibrium and Movement. This work includes a discussion on the efficiency of fundamental machines, i.e. pulleys and inclined planes. Lazare Carnot saw through all the details of the mechanisms to develop a general discussion on the conservation of mechanical energy. Over the next three decades, Lazare Carnot's theorem was taken as a statement that in any machine the accelerations and shocks of the moving parts all represent losses of moment of activity, i.e. the useful work done. From this Lazare drew the inference that perpetual motion was impossible. This loss of moment of activity was the first-ever rudimentary statement of the second law of thermodynamics and the concept of 'transformation-energy' or entropy, i.e. energy lost to dissipation and friction.

Lazare Carnot died in exile in 1823. During the following year Lazare's son Sadi Carnot, having graduated from the École Polytechnique training school for engineers, but now living on half-pay with his brother Hippolyte in a small apartment in Paris, wrote Reflections on the Motive Power of Fire. In this book, Sadi visualized an ideal engine in which any heat (i.e., caloric) converted into work, could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Building on his father's work, Sadi postulated the concept that "some caloric is always lost" in the conversion into work, even in his idealized reversible heat engine, which excluded frictional losses and other losses due to the imperfections of any real machine. He also discovered that this idealized efficiency was dependent only on the temperatures of the heat reservoirs between which the engine was working, and not on the types of working fluids. Any real heat engine could not realize the Carnot cycle's reversibility, and was condemned to be even less efficient. This loss of usable caloric was a precursory form of the increase in entropy as we now know it. Though formulated in terms of caloric, rather than entropy, this was an early insight into the second law of thermodynamics.

1854 definition

Rudolf Clausius - originator of the concept of "entropy"

 

In his 1854 memoir, Clausius first develops the concepts of interior work, i.e. that "which the atoms of the body exert upon each other", and exterior work, i.e. that "which arise from foreign influences [to] which the body may be exposed", which may act on a working body of fluid or gas, typically functioning to work a piston. He then discusses the three categories into which heat Q may be divided:

1. Heat employed in increasing the heat actually existing in the body.
2. Heat employed in producing the interior work.
3. Heat employed in producing the exterior work.

Building on this logic, and following a mathematical presentation of the first fundamental theorem, Clausius then presented the first-ever mathematical formulation of entropy, although at this point in the development of his theories he called it "equivalence-value", perhaps referring to the concept of the mechanical equivalent of heat which was developing at the time rather than entropy, a term which was to come into use later. He stated:

the second fundamental theorem in the mechanical theory of heat may thus be enunciated:

If two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generations of the quantity of heat Q from work at the temperature T, has the equivalence-value:

${\displaystyle {\frac {Q}{T}}}$

and the passage of the quantity of heat Q from the temperature T₁ to the temperature T₂, has the equivalence-value:

${\displaystyle Q\left({\frac {1}{T_{2}}}-{\frac {1}{T_{1}}}\right)}$

wherein T is a function of the temperature, independent of the nature of the process by which the transformation is effected.

In modern terminology, we think of this equivalence-value as "entropy", symbolized by S. Thus, using the above description, we can calculate the entropy change ΔS for the passage of the quantity of heat Q from the temperature T₁, through the "working body" of fluid, which was typically a body of steam, to the temperature T₂ as shown below: 

Diagram of Sadi Carnot's heat engine, 1824

 

If we make the assignment:

${\displaystyle S={\frac {Q}{T}}}$

Then, the entropy change or "equivalence-value" for this transformation is:

${\displaystyle \Delta S=S_{\rm {final}}-S_{\rm {initial}}\,}$

which equals:

${\displaystyle \Delta S=\left({\frac {Q}{T_{2}}}-{\frac {Q}{T_{1}}}\right)}$

and by factoring out Q, we have the following form, as was derived by Clausius:

${\displaystyle \Delta S=Q\left({\frac {1}{T_{2}}}-{\frac {1}{T_{1}}}\right)}$

1856 definition

In 1856, Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:

${\displaystyle \int {\frac {\delta Q}{T}}=-N}$

where N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. This equivalence-value was a precursory formulation of entropy.

1862 definition

In 1862, Clausius stated what he calls the "theorem respecting the equivalence-values of the transformations" or what is now known as the second law of thermodynamics, as such:

The algebraic sum of all the transformations occurring in a cyclical process can only be positive, or, as an extreme case, equal to nothing.

Quantitatively, Clausius states the mathematical expression for this theorem is as follows. Let δQ be an element of the heat given up by the body to any reservoir of heat during its own changes, heat which it may absorb from a reservoir being here reckoned as negative, and T the absolute temperature of the body at the moment of giving up this heat, then the equation:

${\displaystyle \int {\frac {\delta Q}{T}}=0}$

must be true for every reversible cyclical process, and the relation:

${\displaystyle \int {\frac {\delta Q}{T}}\geq 0}$

must hold good for every cyclical process which is in any way possible. This was an early formulation of the second law and one of the original forms of the concept of entropy.

1865 definition

In 1865, Clausius gave irreversible heat loss, or what he had previously been calling "equivalence-value", a name:

“

I propose that S be taken from the Greek words, `en-tropie' [intrinsic direction]. I have deliberately chosen the word entropy to be as similar as possible to the word energy: the two quantities to be named by these words are so closely related in physical significance that a certain similarity in their names appears to be appropriate.

”

Although Clausius did not specify why he chose the symbol "S" to represent entropy, it is arguable that Clausius chose "S" in honor of Sadi Carnot, to whose 1824 article Clausius devoted over 15 years of work and research. On the first page of his original 1850 article "On the Motive Power of Heat, and on the Laws which can be Deduced from it for the Theory of Heat", Clausius calls Carnot the most important of the researchers in the theory of heat.

Later developments

In 1876, physicist J. Willard Gibbs, building on the work of Clausius, Hermann von Helmholtz and others, proposed that the measurement of "available energy" ΔG in a thermodynamic system could be mathematically accounted for by subtracting the "energy loss" TΔS from total energy change of the system ΔH. These concepts were further developed by James Clerk Maxwell [1871] and Max Planck [1903].

Statistical thermodynamic views

In 1877, Ludwig Boltzmann developed a statistical mechanical evaluation of the entropy S, of a body in its own given macrostate of internal thermodynamic equilibrium. It may be written as:

${\displaystyle S=k_{\rm {B}}\ln \Omega \!}$

where

k_B denotes Boltzmann's constant and

Ω denotes the number of microstates consistent with the given equilibrium macrostate.

Boltzmann himself did not actually write this formula expressed with the named constant k_B, which is due to Planck's reading of Boltzmann.

Boltzmann saw entropy as a measure of statistical "mixedupness" or disorder. This concept was soon refined by J. Willard Gibbs, and is now regarded as one of the cornerstones of the theory of statistical mechanics.

Erwin Schrödinger made use of Boltzmann’s work in his book What is Life? to explain why living systems have far fewer replication errors than would be predicted from Statistical Thermodynamics. Schrödinger used the Boltzmann equation in a different form to show increase of entropy

${\displaystyle S=k_{\rm {B}}\ln D\!}$

where D is the number of possible energy states in the system that can be randomly filled with energy. He postulated a local decrease of entropy for living systems when (1/D) represents the number of states that are prevented from randomly distributing, such as occurs in replication of the genetic code.

${\displaystyle -S=k_{\rm {B}}\ln(1/D)\!}$

Without this correction Schrödinger claimed that statistical thermodynamics would predict one thousand mutations per million replications, and ten mutations per hundred replications following the rule for square root of n, far more mutations than actually occur.

Schrödinger’s separation of random and non-random energy states is one of the few explanations for why entropy could be low in the past, but continually increasing now. It has been proposed as an explanation of localized decrease of entropy in radiant energy focusing in parabolic reflectors and during dark current in diodes, which would otherwise be in violation of Statistical Thermodynamics.

Information theory

An analog to thermodynamic entropy is information entropy. In 1948, while working at Bell Telephone Laboratories electrical engineer Claude Shannon set out to mathematically quantify the statistical nature of "lost information" in phone-line signals. To do this, Shannon developed the very general concept of information entropy, a fundamental cornerstone of information theory. Although the story varies, initially it seems that Shannon was not particularly aware of the close similarity between his new quantity and earlier work in thermodynamics. In 1939, however, when Shannon had been working on his equations for some time, he happened to visit the mathematician John von Neumann. During their discussions, regarding what Shannon should call the "measure of uncertainty" or attenuation in phone-line signals with reference to his new information theory, according to one source:

“

My greatest concern was what to call it. I thought of calling it ‘information’, but the word was overly used, so I decided to call it ‘uncertainty’. When I discussed it with John von Neumann, he had a better idea. Von Neumann told me, ‘You should call it entropy, for two reasons: In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage.

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According to another source, when von Neumann asked him how he was getting on with his information theory, Shannon replied:

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The theory was in excellent shape, except that he needed a good name for "missing information". "Why don’t you call it entropy", von Neumann suggested. "In the first place, a mathematical development very much like yours already exists in Boltzmann's statistical mechanics, and in the second place, no one understands entropy very well, so in any discussion you will be in a position of advantage.

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In 1948 Shannon published his famous paper A Mathematical Theory of Communication, in which he devoted a section to what he calls Choice, Uncertainty, and Entropy. In this section, Shannon introduces an H function of the following form:

${\displaystyle H=-K\sum _{i=1}^{k}p(i)\log p(i),}$

where K is a positive constant. Shannon then states that "any quantity of this form, where K merely amounts to a choice of a unit of measurement, plays a central role in information theory as measures of information, choice, and uncertainty." Then, as an example of how this expression applies in a number of different fields, he references R.C. Tolman's 1938 Principles of Statistical Mechanics, stating that "the form of H will be recognized as that of entropy as defined in certain formulations of statistical mechanics where p_i is the probability of a system being in cell i of its phase space… H is then, for example, the H in Boltzmann's famous H theorem." As such, over the last fifty years, ever since this statement was made, people have been overlapping the two concepts or even stating that they are exactly the same. 

Shannon's information entropy is a much more general concept than statistical thermodynamic entropy. Information entropy is present whenever there are unknown quantities that can be described only by a probability distribution. In a series of papers by E. T. Jaynes starting in 1957, the statistical thermodynamic entropy can be seen as just a particular application of Shannon's information entropy to the probabilities of particular microstates of a system occurring in order to produce a particular macrostate.

Popular use

The term entropy is often used in popular language to denote a variety of unrelated phenomena. One example is the concept of corporate entropy as put forward somewhat humorously by authors Tom DeMarco and Timothy Lister in their 1987 classic publication Peopleware, a book on growing and managing productive teams and successful software projects. Here, they view energy waste as red tape and business team inefficiency as a form of entropy, i.e. energy lost to waste. This concept has caught on and is now common jargon in business schools.

In another example, entropy is the central theme in Isaac Asimov's short story The Last Question (first copyrighted in 1956). The story plays with the idea that the most important question is how to stop the increase of entropy.

Terminology overlap

When necessary, to disambiguate between the statistical thermodynamic concept of entropy, and entropy-like formulae put forward by different researchers, the statistical thermodynamic entropy is most properly referred to as the Gibbs entropy. The terms Boltzmann–Gibbs entropy or BG entropy, and Boltzmann–Gibbs–Shannon entropy or BGS entropy are also seen in the literature.