The Helmholtz energy is a thermodynamic potential which measures the "useful" work obtainable from constant temperature, constant volume thermodynamic systems. It is sometimes known as the "work content", and many sources use the superceded term "Helmholtz free energy". For a simple system, with a fixed number of particles, 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.

The Helmholtz 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 energy is defined as:

$A=U-TS\backslash ,$

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).

Why is the attachment ‘free’ so important?

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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 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. 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.

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.’ With reference to the Gibbs free energy, we add the qualification that it is the energy free for non-volume work.

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 between the USA, Europe, and other countries, in which descriptive ‘free’ was supposedly banished. 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.

Non-viscous fluids

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From the first law of thermodynamics we have:

$dU\; =\; \backslash delta\; Q\; -\; \backslash delta\; W\backslash ,$

where $U$ is the internal energy, $\backslash delta\; Q$ is the energy added by heating and $\backslash delta\; W=PdV$ is the work done by the system. From the second law of thermodynamics, for a reversible process we may say that $\backslash delta\; Q=TdS$. Differentiating the expression for A  we have:

$dA\; =\; dU\; -\; (TdS\; +\; SdT)\backslash ,$

$=\; (TdS\; -\; pdV)\; -\; TdS\; -\; SdT\backslash ,$

$=\; -\; pdV\; -\; SdT\backslash ,$

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

$dA\; \backslash le\; -\; pdV\; -\; SdT\backslash ,$

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

$dA\; \backslash le\; -\backslash delta\; W\backslash ,$

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:

$dA\; \backslash le\; 0\backslash ,$

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:

$dA\; \backslash le\; -\; pdV\; -\; SdT\; +\; \backslash sum\_i\; \backslash mu\_i\; dN\_i\backslash ,$

where $\backslash mu\_i$ 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

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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 :

$dA\; \backslash le\; V\backslash sum\_\{ij\}\backslash sigma\_\{ij\}d\backslash varepsilon\_\{ij\}\; -\; SdT\; +\; \backslash sum\_i\; \backslash mu\_i\; dN\_i\backslash ,$

where $\backslash sigma\_\{ij\}$ is the stress tensor, and $\backslash varepsilon\_\{ij\}$ is the strain tensor. In the case of linear elastic materials which obey Hooke's Law, the stress is related to the strain by:

$\backslash sigma\_\{ij\}=C\_\{ijkl\}\backslash varepsilon\_\{kl\}$

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:

$A\; =\; \backslash frac\{1\}\{2\}VC\_\{ijkl\}\backslash varepsilon\_\{kl\}^2\; -\; ST\; +\; \backslash sum\_i\; \backslash mu\_i\; N\_i\backslash ,$

$=\; \backslash frac\{1\}\{2\}V\backslash sigma\_\{ij\}\backslash varepsilon\_\{ij\}\; -\; ST\; +\; \backslash sum\_i\; \backslash mu\_i\; N\_i\backslash ,$

See also

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• work content - for applications to chemistry
• Statistical Mechanics

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

References

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Free energy

Helmholtz-Energie | Énergie libre | Energia libera di Helmholtz | אנרגיה חופשית של הלמהולץ | ヘルムホルツエネルギー | Energia swobodna | Свободная энергия Гельмгольца | Helmholtz fria energi | 亥姆霍茨自由能