
Left: American engineer Willard Gibbs' 1873 graph of the of the energy ε verses the entropy η of a body showing the available energy of a body, section AB, at an entropy of D, a type of energy that would famously be labeled as "free energy" by German physicist Hermann Helmholtz in 1882. [1] Right: Irish physicist James Maxwell's 1875 addition of the volume axis, shown perpendicular to the plane here, and used sunlight and shadows, to make a three-dimensional thermodynamic surface.

Left: American engineer Willard Gibbs' 1873 graph of the of the energy ε verses the entropy η of a body showing the available energy of a body, section AB, at an entropy of D, a type of energy that would famously be labeled as "free energy" by German physicist Hermann Helmholtz in 1882. [1] Right: Irish physicist James Maxwell's 1875 addition of the volume axis, shown perpendicular to the plane here, and used sunlight and shadows, to make a three-dimensional thermodynamic surface.

In thermodynamics, available energy is the greatest amount of mechanical work that can be obtained from a system or body, with a given quantity of substance, in a given initial state, without increasing its total volume or allowing heat to pass to or from external bodies, except such as at the close of the processes are left in their initial condition. In this definition, the initial state of the body is supposed to be such that the body can be made to pass from it to states of dissipated energy by reversible processes.

Overview
In 1873, American engineer Willard Gibbs, in his "A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces", introduced the definition of available energy; although, to note, it may have had an earlier etymology, as Gibbs seems to refer to an earlier use when he states “this [definition] has been called the available energy of the body”. [1] Gibbs' available energy came to be called “free energy” by German physicist Hermann Helmholtz in 1882. [2]

The graph made by Gibbs to illustrate his ideas, pictured adjacent and below, shows a plane perpendicular to the axis of V (volume) and passing through point A, which represents the "initial state" of the body. Qε and Qη are sections of the planes η = 0 and ε = 0, and therefore parallel to the axes of ε (internal energy) and η (entropy), respectively. Other sections, according to Gibbs, are:

MN = the section of the surface of dissipated energy.
AD = the energy of the body in its initial state.
AE = the entropy of the body in its initial state.
AB = its available energy.
AC = its capacity for entropy.

Here, Gibbs defines "capacity for entropy" as the amount by which the entropy of the body can be increased without changing the energy of the body or increasing its volume.

Mis-extrapolations | Humanities
As to the extrapolation of Gibbs’ conception of available energy in relation to human activity and in the thermodynamics of human social systems, the translation has often been lost in transit. To cite a classic example of mistranslation, in the 1971 book The Entropy Law and the Economic Process, Romanian mathematician Nicholas Georgescu-Roegen used references such as Webster’s Collegiate Dictionary to define entropy as the measure of unavailable energy in a closed thermodynamic system and the goes off the rails by stating that in the thermodynamical operation of a steam engine, where the heat from the burning coal flows into the boiler and out through the escaping steam:

“At the beginning, the chemical energy of the coal is free, in the sense that it is available to us for producing some mechanical work. In the process, however, the free energy loses this quality, bit by bit. Ultimately, it always dissipates completely into the whole system where it becomes bound energy, that is, energy which we can no longer use for the same purpose.”

On this and other ill-contrived statements, Georgescu-Roegen, in his book, went on to build entropy-base theory of economics. This book soon thereafter seeded the careers of many burgeoning economists, such as American writer Jeremy Rifkin, who came to incorrectly view Georgescu-Roegen as a foremost thermodynamicist. This logic can now often be found in modern publications, by those arguing that we are depleting our free energies by the consumption of our oil-gas-coal natural fuels. This, however, is a huge mis-extrapolation of what constitutes “free energy” in a chemical system. [3]

References
1. Gibbs, J. Willard. (1873). "A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces" (pgs. 49-50), Transactions of the Connecticut Academy, II. pp.382-404, Dec.
2. Helmholtz, Hermann. (1882). “Die Thermodynamik Chemischer Vorgänge (The Thermodynamics of Chemical Operations”, SB: 22-39, in Wissenschaftliche Abhandlungen von Hermann von Helmholtz, 3 vols. Leipzig: J.A. barth, 1882-95. 2:958-78.
3. (a) Thims, Libb. (2007). Human Chemistry (Volume One), (preview), (Google books). Morrisville, NC: LuLu.
(b) Thims, Libb. (2007). Human Chemistry (Volume Two), (preview), (Google books). Morrisville, NC: LuLu.

Further reading
● Burges, Edward A. (1959). Available Energy and the Second Law Analysis (available energy, 34+ pgs). Academic Press.