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Thermodynamic potentials
Internal energy	$U (S, V)$
Helmholtz free energy	$A (T, V) = U - T S$
Enthalpy	$H (S, p) = U + p V$
Gibbs free energy	$G (T, p) = H - T S$
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In thermodynamics, the term thermodynamic free energy refers to the amount of work that can be extracted from a system, and is helpful in engineering applications. It is a subtraction of the entropy of a system multiplied by a reference temperature (giving the "unusable energy") from the total energy, yielding a thermodynamic state function which represents the "useful energy".

[edit] Overview

In short, free energy is that portion of any First-Law energy that is available for doing thermodynamic work; i.e., work mediated by thermal energy. Since free energy is subject to irreversible loss in the course of such work^[1] and First-Law energy is always conserved, it is evident that free energy is an expendable, Second-Law kind of energy that can make things happen within finite amounts of time. The free energy functions are Legendre transforms of the internal energy. For processes involving a system at constant pressure p and temperature T, the Gibbs free energy is the most useful because, in addition to subsuming any entropy change due merely to heat flux, it does the same for the pdV work needed to "make space for additional molecules" produced by various processes. (Hence its utility to solution-phase chemists, including biochemists.) The Helmholtz free energy has a special theoretical importance since it is proportional to the logarithm of the partition function for the canonical ensemble in statistical mechanics. (Hence its utility to physicists; and to gas-phase chemists and engineers, who do not want to ignore pdV work.)

The (historically earlier) Helmholtz free energy is defined as A = U − TS, where U is the internal energy, T is the absolute temperature, and S is the entropy. Its change is equal to the amount of reversible work done on, or obtainable from, a system at constant T. Thus its appellation "work content", and the designation A from Arbeit, the German word for work. Since it makes no reference to any quantities involved in work (such as p and V), the Helmholtz function is completely general: its decrease is the maximum amount of work which can be done by a system, and it can increase at most by the amount of work done on a system.

The Gibbs free energy G = H − TS, where H is the enthalpy. (H = U + pV, where p is the pressure and V is the volume.)

There has been historical controversy:

In physics, “free energy” most often refers to the Helmholtz free energy, denoted by F.
In chemistry, “free energy” most often refers to the Gibbs free energy.

Since both fields use both functions, a compromise has been suggested, using A to denote the Helmholtz function, with G for the Gibbs function. While A is preferred by IUPAC, F is sometimes still in use, and the correct free energy function is often implicit in manuscripts and presentations.

[edit] Application

The experimental usefulness of these functions is restricted to conditions where certain variables (T, and V or external p) are held constant, although they also have theoretical importance in deriving Maxwell relations. Work other than pdV may be added, e.g., for electrochemical cells, or f ˑdx work in elastic materials and in muscle contraction. Other forms of work which must sometimes be considered are stress-strain, magnetic, as in adiabatic demagnetization used in the approach to absolute zero, and work due to electric polarization. These are described by tensors.

In most cases of interest there are internal degrees of freedom and processes, such as chemical reactions and phase transitions, which create entropy. Even for homogeneous "bulk" materials, the free energy functions depend on the (often suppressed) composition, as do all proper thermodynamic potentials (extensive functions), including the internal energy.

Name	Definition	Natural variables
Helmholtz free energy	$A=U-TS\,$	$~~~~~T,V,\{N_i\}\,$
Gibbs free energy	$G=U+pV-TS\,$	$~~~~~T,p,\{N_i\}\,$

N_i is the number of molecules (alternatively, moles) of type i in the system. If these quantities do not appear, it is impossible to describe compositional changes. The differentials for reversible processes are (assuming only pV work)

$\mathrm{d}A = - p\,\mathrm{d}V - S\mathrm{d}T + \sum_i \mu_i \,\mathrm{d}N_i\,$

$\mathrm{d}G = V\mathrm{d}P - S\mathrm{d}T + \sum_i \mu_i \,\mathrm{d}N_i\,$

where μ_i is the chemical potential for the i-th component in the system. The second relation is especially useful at constant T and p, conditions which are easy to achieve experimentally, and which approximately characterize living creatures.

$(\mathrm{d}G)_{T,p} = \sum_i \mu_i \,\mathrm{d}N_i\,$

Any decrease in the Gibbs function of a system is the upper limit for any isothermal, isobaric work that can be captured in the surroundings, or it may simply be dissipated, appearing as T times a corresponding increase in the entropy of the system and/or its surrounding.

[edit] What does the term ‘free’ mean?

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. In a similar manner, 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, whereas the other component, the latent caloric, could not.^[2] 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, indicating 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.^[3]

Thus, in traditional use, the term “free” was attached 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.’^[4] With reference to the Gibbs free energy, we add the qualification that it is the energy free for non-volume work.^[5]

An increasing number of books and journal articles do not include the attachment “free”, referring to G as simply Gibbs energy (and likewise for the Helmholtz energy). This is the result of a 1988 IUPAC meeting to set unified terminologies for the international scientific community, in which the adjective ‘free’ was supposedly banished.^[6] This standard, however, has not yet been universally adopted, and many published articles and books still include the descriptive ‘free’.

[edit] References

^ Stoner, Clinton D. (2000). Inquiries into the Nature of Free Energy and Entropy in Respect to Biochemical Thermodynamics. Entropy Vol. 2.
^ Mendoza, E. (1988). Reflections on the Motive Power of Fire – and other Papers on the Second Law of Thermodynamics by E. Clapeyron and R. Carnot. Dover Publications, Inc.. ISBN 0-486-44641-7.
^ Baierlein, Ralph (2003). Thermal Physics. Cambridge University Press. ISBN 0-521-65838-1.
^ Perrot, Pierre (1998). A to Z of Thermodynamics. Oxford University Press. ISBN 0-19-856552-6.
^ Reiss, Howard (1965). Methods of Thermodynamics. Dover Publications. ISBN 0-486-69445-3.
^ International Union of Pure and Applied Chemistry Commission on Atomspheric Chemistry (1990). "Glossary of Atmospheric Chemistry Terms (Recommendations 1990)". Pure Appl. Chem. 62: 2167–2219. doi:10.1351/pac199062112167. http://www.iupac.org/publications/pac/1990/pdf/6211x2167.pdf. Retrieved 2006-12-28. International Union of Pure and Applied Chemistry Commission on Physicochemical Symbols Terminology and Units (1993). Quantities, Units and Symbols in Physical Chemistry (2nd Edition). Oxford: Blackwell Scientific Publications. pp. 48. ISBN 0-632-03583-8. http://www.iupac.org/publications/books/gbook/green_book_2ed.pdf. Retrieved 2006-12-28. International Union of Pure and Applied Chemistry Commission on Quantities and Units in Clinical Chemistry; International Federation of Clinical Chemistry Committee on Quantities and Units (1996). "Glossary of Terms in Quantities and Units in Clinical Chemistry (IUPAC-IFCC Recommendations 1996)". Pure Appl. Chem. 68: 957–1000. doi:10.1351/pac199668040957. http://www.iupac.org/publications/pac/1996/pdf/6804x0957.pdf. Retrieved 2006-12-28.

[edit] See also

[0] Stoner, Clinton D. (2000). Inquiries into the Nature of Free Energy and Entropy in Respect to Biochemical Thermodynamics. Entropy Vol. 2.

[1] Mendoza, E. (1988). Reflections on the Motive Power of Fire – and other Papers on the Second Law of Thermodynamics by E. Clapeyron and R. Carnot. Dover Publications, Inc.. ISBN 0-486-44641-7.

[2] Baierlein, Ralph (2003). Thermal Physics. Cambridge University Press. ISBN 0-521-65838-1.

[Perrot-3] Perrot, Pierre (1998). A to Z of Thermodynamics. Oxford University Press. ISBN 0-19-856552-6.

[4] Reiss, Howard (1965). Methods of Thermodynamics. Dover Publications. ISBN 0-486-69445-3.

[5] International Union of Pure and Applied Chemistry Commission on Atomspheric Chemistry (1990). "Glossary of Atmospheric Chemistry Terms (Recommendations 1990)". Pure Appl. Chem. 62: 2167–2219. doi:10.1351/pac199062112167. http://www.iupac.org/publications/pac/1990/pdf/6211x2167.pdf. Retrieved 2006-12-28. International Union of Pure and Applied Chemistry Commission on Physicochemical Symbols Terminology and Units (1993). Quantities, Units and Symbols in Physical Chemistry (2nd Edition). Oxford: Blackwell Scientific Publications. pp. 48. ISBN 0-632-03583-8. http://www.iupac.org/publications/books/gbook/green_book_2ed.pdf. Retrieved 2006-12-28. International Union of Pure and Applied Chemistry Commission on Quantities and Units in Clinical Chemistry; International Federation of Clinical Chemistry Committee on Quantities and Units (1996). "Glossary of Terms in Quantities and Units in Clinical Chemistry (IUPAC-IFCC Recommendations 1996)". Pure Appl. Chem. 68: 957–1000. doi:10.1351/pac199668040957. http://www.iupac.org/publications/pac/1996/pdf/6804x0957.pdf. Retrieved 2006-12-28.

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Contents

[edit] Overview

[edit] Application

[edit] What does the term ‘free’ mean?

[edit] References

[edit] See also