Gibbs free energy

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Thermodynamic potentials
Internal energy	U(S,V)
Helmholtz free energy	A(T,V) = U − TS
Enthalpy	H(S,P) = U + PV
Gibbs free energy	G(T,P) = U + PV − TS
edit

In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the "useful" work obtainable from an isothermal, isobaric thermodynamic system. When a system evolves from a well-defined initial state to a well-defined final state, the Gibbs free energy ΔG equals the work exchanged by the system with its surroundings, less the work of the pressure forces, during a reversible transformation of the system from the same initial state to the same final state.^[1] The Gibbs free energy was developed in the 1870s by the American mathematical physicist Willard Gibbs.

Contents 1 Definitions 1.1 Synonyms 2 Overview 3 History 4 Why is the attachment ‘free’ so important? 5 Useful identities 6 Derivation of 7 Gibbs free energy change of formation 8 What is Life? 9 See also 10 References 11 External links

[edit] Definitions

The Gibbs free energy, for closed systems, is defined as:

The Gibbs free energy, for open systems, is defined as:

Here, in SI units, G is the Gibbs energy (joules), H is the enthalpy (joules), T is the temperature (kelvins), and S is the entropy (joules per kelvin), V is volume, P is pressure. If chemical species flow into or out of the system, the chemical work -μ dn_i, where μ is the chemical potential, associated with the transport of each ith component must be accounted for. In open systems, aside from mechanical work, a system may in addition perform numerous other types of work. The quantity fdl is contractile work energy associated with a thermodynamic system that is a contractile fiber which shortens by an amount -dl under a force f. If a quantity of energy -de is given off by a system at an electrical potential Ψ, the electrical work associated with this is -Ψde. Other work terms are added on per system requirements.^[2]

Each quantity in the equations above can be divided by the amount of substance, measured in moles, to form molar Gibbs energy. The Gibbs 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.

[edit] Synonyms

The Gibbs free energy, in addition, goes by various names such as: Gibbs energy, Gibbs energy function, free energy, free enthalpy, thermodynamic potential at constant pressure, and others. In isothermal, isobaric systems, Gibbs free energy can be thought of as a "dynamic" quantity, it that it is a representative measure of the competing effects of the enthalpic and entropic driving forces involved in a thermodynamic process.

[edit] Overview

In a simple manner, with respect to STP reacting systems, a general rule of thumb is:

Hence, out of this general natural tendency, a quantitative measure as to how near or far a potential reaction is from this minimum is when the calculated energetics of the process indicate that the change in Gibbs free energy ΔG is negative. Essentially, this means that such a reaction will be favored and will release energy. The energy released equals the maximum amount of work that can be performed as a result of the chemical reaction. Conversely, if conditions indicated a positive ΔG, then energy--in the form of work--would have to be added to the reacting system to make the reaction go.

[edit] History

The quantity called "free energy" is essentially a more advanced and accurate replacement for the term “affinity” used by chemists, of olden days, 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.

From the 1998 textbook Modern Thermodynamics by Nobelist and chemical engineering professor Ilya Prigogine’s we find: “as motion was explained by the Newtonian concept of force, chemists wanted a similar concept of ‘driving force’ for chemical change? Why do chemical reactions occur, and why do they stop at certain points? Chemists called the ‘force’ that caused chemical reactions affinity, but it lacked a clear definition.

During the entire 18th century, the dominant view in regards to heat and light was that put forward by Isaac Newton, called the “Newtonian hypothesis”, which stated that light and heat are forms of matter attracted or repelled by other forms of matter, with forces analogous to gravitation or to chemical affinity.

In the 19th century, the French chemist Marcellin Berthelot and the Danish chemist Julius Thomsen had attempted to quantify affinity using heats of reaction. In 1875, after quantifying the heats of reaction for a large number of compounds, Berthelot proposed the “principle of maximum work” in which all chemical changes occurring without intervention of outside energy tend toward the production of bodies or of a system of bodies which liberate heat.

In addition to this, in 1780 Antoine Lavoisier and Pierre-Simon Laplace laid the foundations of thermochemistry by showing that the heat 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' law. With the advent of the mechanical theory of heat in the early 19th century, Hess’s law came to be viewed as a consequence of the law of conservation of energy.

Based on these and other ideas, Berthelot and 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, the English physicist James Joule showed that he could raise the temperature of water by turning a paddle wheel in it, thus showing that heat and mechanical work were equivalent or proportional to each other, i.e. approximately, . This statement came to be known as the mechanical equivalent of heat and was a precursory form of the first law of thermodynamics.

By 1865, the German physicist Rudolf Clausius had 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 molecules from one state to the next cannot be used to do external work, e.g. to push the piston. Clausius defined this transformation heat as dQ = TdS.

In 1873, Willard Gibbs published A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces in which he introduced the preliminary outline of the principles of his new equation able to predict or estimate the tendencies of various natural processes to ensue when bodies or systems are brought into contact. By studying the interactions of homogeneous substances in contact, i.e. bodies, being in composition part solid, part liquid, and part vapor, and by using a three-dimensional volume-entropy-internal energy graph, Gibbs was able to determine three states of equilibrium, i.e. "necessarily stable", "neutral", and "unstable", and whether or not changes will ensue. In 1876, Gibbs built on this framework by introducing the concept of chemical potential so to take into account chemical reactions and states of bodies which are chemically different from each other. In his own words, to summarize his results in 1873, Gibbs states:

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 υ is the volume of the body.

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 F 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 F is the amount of energy “free” for work under the given conditions.

Up until this point, the general view had been such that: “all chemical reactions drive the system to a state of equilibrium in which the affinities of the reactions vanish”. Over the next 60 years, the term affinity came to be replaced with the term free energy. According to chemistry historian Henry Leicester, the influential 1923 textbook Thermodynamics and the Free Energy of Chemical Reactions by Gilbert N. Lewis and Merle Randall led to the replacement of the term “affinity” by the term “free energy” in much of the English-speaking world.

[edit] Why is the attachment ‘free’ so important?

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 categories, 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 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. ^[3] 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.

During the early 19th century, the concept of perceptible or free caloric began to be referred to as “free heat” or heat set free. In 1824, for example, the French physicist Sadi Carnot, in his famous “Reflections on the Motive Power of Fire”, speaks of quantities of heat ‘absorbed or set free’ in different transformations. In 1882, the German physicist and physiologist Hermann von Helmholtz coined the phrase ‘free energy’ for the expression E – TS, in which the change in F (or G) determines the amount of energy ‘free’ for work under the given conditions. ^[4]

In modern use, we attach the term “free” to Gibbs free energy, i.e. for systems at constant pressure and temperature, or to Helmholtz free energy, i.e. for systems at constant volume and temperature, to mean ‘available in the form of useful work.’^[1] With reference to the Gibbs free energy, we add the qualification that it is the energy free for non-volume work.^[5]

To note, some books do not include the attachment “free”, referring to G as simply Gibbs energy. This influence is the result of a 1988 IUPAC meeting designed to unified terminologies among the USA, Europe, and other countries, in which descriptive ‘free’ was supposedly banished.^[6] This ruling, however, is still far from accepted and the majority of published articles and books still use the descriptive ‘free’ for both historical, informative, and for clarification reasons.

[edit] Useful identities

for constant temperature

and rearranging gives

which relates the electrical potential of a reaction to the equilibrium coefficient for that reaction.

where

ΔG = change in Gibbs energy, ΔH = change in enthalpy, T = absolute temperature, ΔS = change in entropy, R = gas constant, ln = natural logarithm, K = equilibrium constant, Q = reaction quotient, n = number of electrons/mole product, F = Faraday constant (coulombs/mole), and ΔE = electrical potential of the reaction. Moreover, we also have:

which relates the equilibrium constant with Gibbs energy.

[edit] Derivation of

To derive the Gibbs free energy equation for an isolated system, let S_tot be the total entropy of the isolated system, that is, a system which cannot exchange heat or mass with its surroundings. According to the second law of thermodynamics:

and if then the process is reversible. The heat transfer Q vanishes for an adiabatic system. Any adiabatic process is reversible and isentropic .

Now consider diabatic systems, having internal entropy S_int. Such a system is thermally connected to its surroundings, which have entropy S_ext. The entropy form of the second law does not apply directly to the diabatic system, it only applies to the closed system formed by both the system and its surroundings. Therefore a process is possible if

.

We will try to express the left side of this equation entirely in terms of state functions. ΔS_ext is defined as:

Temperature T is the same for two systems in thermal equilibrium. By the zeroth law of thermodynamics, if a system is in thermal equilibrium with a second and a third system, the latter two are in equilibrium as well. Also, Δq_rev is heat transferred to the system, so -Δq_rev is heat transferred to the surroundings, and −ΔQ/T is entropy gained by the surroundings. We now have:

Multiply both sides by T:

ΔQ is heat transferred to the system; if the process is now assumed to be isobaric, then Δq_p = ΔH:

ΔH is the enthalpy change of reaction (for a chemical reaction at constant pressure and temperature). Then

for a possible process. Let the change ΔG in Gibbs energy be defined as

(eq.1)

Notice that it is not defined in terms of any external state functions, such as ΔS_ext or ΔS_tot. Then the second law becomes, which also tells us about the spontaneity of the reaction:

favored reaction (Spontaneous)

reversible reaction (Equilibrium)

disfavored reaction (Nonspontaneous)

Gibbs energy G itself is defined as

(eq.2)

but notice that to obtain equation (2) from equation (1) we must assume that T is constant. Thus, Gibbs energy is most useful for thermochemical processes at constant temperature and pressure: both isothermal and isobaric. Such processes don't move on a P-T diagram, such as phase change of a pure substance, which takes place at the saturation pressure and temperature. Chemical reactions, however, do undergo changes in chemical potential, which is a state function. Thus, thermodynamic processes are not confined to the two dimensional P-V diagram. There is a third dimension for n, the quantity of gas. Naturally for the study of explosive chemicals, the processes are not necessarily isothermal and isobaric. For these studies, Helmholtz free energy is used.

If a closed system (Δq_rev = 0) is at constant pressure (Δq_rev = ΔH), then

Therefore the Gibbs free energy of a closed system is:

and if then this implies that , back to where we started the derivation of ΔG.

[edit] Gibbs free energy change of formation

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 degrees Celsius and 100 kilopascals). Its symbol is ΔG_f^O.

All elements in their standard states (oxygen gas, graphite, etc.) have 0 standard Gibbs free energy change of formation, as there is no change involved.

ΔG = ΔG˚ + RTlnK

At equilibrium, ΔG=0 so the equation becomes ΔG˚= -RTlnK

[edit] What is Life?

Main article: Entropy and life

To a good approximation, one of the most oft-quoted references in thermodynamics is Erwin Schrödinger’s 1944 postulate that an organism keeps itself alive or aloof by feeding on negative entropy from its environment. From the famous chapter six “Order, Disorder and Entropy” of his book What is Life?, Schrödinger asks: “what is the characteristic feature of life? and “when is a piece of matter said to be alive?” To answer these questions, Schrödinger turns to thermodynamics. Life, according to Schrödinger, avoids a decay to maximum entropy, or thermodynamic equilibrium, which Schrödinger equates with death, by feeding on negative entropy. Specifically, according to Schrödinger, an organism avoids decay by eating, drinking, breathing, and in the case of plants assimilating, a process called metabolism.

In the past, Schrödinger states, this process would have been considered an exchange of matter or energy, such that organisms stay alive by exchanging energy. He uses the example of how caloric values are printed in certain menus in the United States or Germany, but states that these caloric energy exchange values are useless in trying to quantify life. He then asks “what then is that precious something contained in our food which keeps us from death?” The answer, according to Schrödinger, is that because according the second law of thermodynamics an organism continually produces “positive entropy” it must continually draw in “negative entropy” from its environment to stay alive. Or, specifically “the essential thing in metabolism is that the organism succeeds in freeing itself from all the entropy it cannot help producing while alive.”

These suppositions, because they were intended for a lay audience, however, met with great opposition in the physics community. In later editions of his book, Schrödinger attached a note to chapter six explaining his use of the term “negative entropy”. He states “the remarks on negative entropy have met with doubt and opposition from physicist colleagues. Let me say first, that if I had been catering for them alone I should have let the discussion turn on free energy instead. It is the more familiar notion in this context. But this highly technical term seemed too linguistically near to energy for making the average reader alive to the contrast between the two things.”

[edit] See also

• Enthalpy
• Entropy
• Free energy
• Helmholtz free energy
• Thermodynamics
• Willard Gibbs

[edit] References

[edit] External links

• Gibbs Free Energy - Eric Weissteins World of Physics
• Gibbs Free Energy - Chemistry Gateway
• Gibbs Free Energy - Illinois State University
• Entropy and Gibbs Free Energy - www.2ndlaw.com
• Gibbs Free Energy - Georgia State University
• Gibbs Free Energy Java Applet - University of California, Berkeley
• Gibbs Free Energy - Florida State University
• IUPAC definition