In thermodynamics, the **Gibbs fundamental equation**, also called the combined law of thermodynamics, states that the change in energy of a system can always be written as the product of an intensive and an extensive parameter. This takes the form of: [1]

where *Xi *is an intensive quantity, such as pressure or temperature, and *Yi* is an extensive quantity, such as volume. The expression on the right side of this equation is called a Pfaffian form.

History

The Gibbs equation, was first derived by American engineer Willard Gibbs in the first two pages to his 1873 "Graphical Methods in the Thermodynamics of Fluids" as: [2]

dε = tdη - pdv

where, according to Gibbs, *ε* is the energy, *t* the temperature, *η *the entropy, *p* the pressure, and *v *the volume of the body in a state.

Advanced systems

When a system does work or has work done on it the system internal energy is effected, such as when it has a volume d*V* change due the action of a pressure *P*, i.e. pressure-volume work (d*W* = *p*d*V*), an expansion of a material section or elongation d*l* in response to the application of a force *F*, i.e. elongation work (d*W* = *F*d*l*), does work (*μ*d*n*) by transporting a certain number of atoms or molecules d*n* against a concentration gradient, where *μ* is the chemical potential, i.e. transport work (d*W* = *μ*d*n*), or, among many other possible examples, does work in the action of charge transport in which an amount of charge d*q* is transported against an electric potential *ψ*, i.e. electrical work (d*W* = *ψ*d*q*), then the Gibbs fundamental equation becomes: [3]

The application of this type of quantification of internal energy of human social systems, e.g. how does one quantify the transport of a singe human molecule across the boundary of a system or how human pressures, temperatures, or volumes are quantified, etc., is one of the most advanced topics in human thermodynamics. [4]

References

1. Schmitz, John E.J. (2007). *The Second Law of Life: Energy, Technology, and the Future of Earth as We Know It, *(pg. 165).* *William Andrew Publishing.

2. Gibbs, J. Willard. (1873). "Graphical Methods in the Thermodynamics of Fluids", *Transactions of the Connecticut Academy, *I. pp. 309-342, April-May.

3. Glaser, Roland. (2000). *Biophysics,* (pgs. 110-11). Springer.

4. (a) Thims, Libb. (2007). *Human Chemistry (Volume One)*, (preview). (Index: "human thermodynamics", pgs. x, 14, 74, 79, 107, 110, 204, 273, 315). Morrisville, NC: LuLu.

(b) Thims, Libb. (2007). *Human Chemistry (Volume Two)*, (preview), (Ch. 16: "Human Thermodynamics", pgs. 653-702). Morrisville, NC: LuLu.