Helmholtz free energy

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.

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.

The Helmholtz energy is defined as:

$A=U-TS\,$

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).

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.

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'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.

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, $dW\propto dQ$. This 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 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.

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: $\delta (\epsilon -T\eta +P\nu )=0$ when $\delta$ 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 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.

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.

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 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 A (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 is the influence 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

From the first law of thermodynamics we have:

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

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

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

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

$=-pdV-SdT\,$

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

$dA\leq -pdV-SdT\,$

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

$dA\leq -\delta W\,$

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:

$dA\leq 0\,$

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:

$dA\leq -pdV-SdT+\sum _{i}\mu _{i}dN_{i}\,$

where $\mu _{i}$ 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

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 Template:Ref harvard:

$dA\leq V\sum _{ij}\sigma _{ij}d\varepsilon _{ij}-SdT+\sum _{i}\mu _{i}dN_{i}\,$

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

$\sigma _{ij}=C_{ijkl}\varepsilon _{kl}$

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:

$A={\frac {1}{2}}VC_{ijkl}\varepsilon _{kl}^{2}-ST+\sum _{i}\mu _{i}N_{i}\,$

$={\frac {1}{2}}V\sigma _{ij}\varepsilon _{ij}-ST+\sum _{i}\mu _{i}N_{i}\,$

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.]

References

• IUPAC definition
• Atkins' Physical Chemistry, 7th edition, by Peter Atkins and Julio de Paula, Oxford University Press
• Landau, L. D. (1986). Theory of Elasticity (Course of Theoretical Physics Volume 7). (Translated from Russian by J.B. Sykes and W.H. Reid) (Third ed. ed.). Boston, MA: Butterworth Heinemann. ISBN 0-7506-2633-X. {{cite book}}: |edition= has extra text (help); Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |languange= ignored (help)