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 IUPAC recommends the letter A  as well as the use of name Helmholtz energy; in physics, A is called the Helmholtz function or simply “free energy”. This article needs to be cleaned up to conform to a higher standard of quality. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... In thermodynamics and molecular chemistry, the enthalpy or heat content (denoted as Δ or ΔH, or rarely as χ) is a quotient or description of thermodynamic potential of a system, which can be used to calculate the useful work obtainable from a closed thermodynamic system under constant conditions. ... In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the useful work obtainable from a closed thermodynamic system at a constant temperature and pressure. ... Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... In thermodynamics, four quantities, measured in units of energy, are called thermodynamic potentials: where T = temperature, S = entropy, p = pressure, V = volume Differential definitions The following differential relations hold for the four potentials: If we write the above four equations generally as Then it is seen that yielding expressions for... In thermodynamics, a closed system, as contrasted with an isolated system, can exchange heat and work, but not matter, with its surroundings. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ... Hermann Ludwig Ferdinand von Helmholtz (August 31, 1821 – September 8, 1894) was a German physician and physicist. ... The International Union of Pure and Applied Chemistry (IUPAC) is an international non-governmental organization devoted to the advancement of chemistry. ... Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space, and time. ...

Definition

The Helmholtz energy is defined as:

where

• A  is the Helmholtz free 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).

Cover of brochure The International System of Units. ... A joule is the work done or energy required to exert a force of one newton for a distance of one metre, so the same quantity may be referred to as a newton metre or newton-metre with the symbol N·m. ... An erg is the unit of energy and mechanical work in the centimetre-gram-second (CGS) system of units, symbol erg. Its name is derived from the Greek word meaning work. The erg is a small unit, equal to a force of one dyne exerted for a distance of one... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... For other uses of the term entropy, see Entropy (disambiguation) The thermodynamic entropy S, often simply called the entropy in the context of thermodynamics, is a measure of the amount of energy in a physical system that cannot be used to do work. ...

Non-viscous fluids

From the first law of thermodynamics we have: The first law of thermodynamics, a generalized expression of the law of the conservation of energy, states: // Description Essentially, the First Law of Thermodynamics declares that energy is conserved for a closed system, with heat and work being the forms of energy transfer. ...

where U is the internal energy, δQ is the energy added by heating and δW = PdV is the work done by the system. From the second law of thermodynamics, for a reversible process we may say that δQ = TdS. Differentiating the expression for A  we have: The second law of thermodynamics is an expression of the universal law of increasing entropy. ... In thermodynamics, a reversible process (or reversible cycle if the process is cyclic) is a process that can be reversed by means of infinitesimal changes in some property of the system. ...

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

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

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:

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: In thermodynamics and chemistry, chemical potential, symbolized by μ, is a term introduced in 1876 by the American mathematical physicist (Willard Gibbs and his partner Lauren Berkley), which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given...

where μ_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 (Landau & Lifshitz 1986): Figure 1  Stress tensor A mature tree trunk may support a greater force than a fine steel wire but intuitively we feel that steel is stronger than wood. ... Look up strain in Wiktionary, the free dictionary. ...

where σ_ij is the stress tensor, and is the strain tensor. In the case of linear elastic materials which obey Hooke's Law, the stress is related to the strain by: Look up Elastic in Wiktionary, the free dictionary. ... Hookes law accurately models the physical properties of common mechanical springs at small extensions. ...

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: In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae. ...

See also

• Gibbs free energy for thermodynamics history overview and discussion of free
• work content - for applications to chemistry
• Statistical Mechanics
• This page details the Helmholtz energy from the point of view of thermal and statistical physics.

In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the useful work obtainable from a closed thermodynamic system at a constant temperature and pressure. ... moved from free energy In thermodynamic analysis of chemical reactions, the term free energy denotes either of two related concepts of importance expressing the total amount of energy which is used up or released during a chemical reaction. ... Chemistry (from Greek χημεία khemeia[1] meaning alchemy) is the science of matter at the atomic to molecular scale, dealing primarily with collections of atoms, such as gases, molecules, crystals, and metals. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Statistical physics, one of the fundamental theories of physics, uses methods of statistics in solving physical problems. ... Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space, and time. ...

References