# Thermodynamic Potentials

## Potentials, Thermodynamic

certain functions of the volume *V*, pressure *p*, temperature *T*, entropy *S*, the number of particles in a system *N*, and other macroscopic parameters (*x _{i}*) that characterize the state of a thermodynamic system. Thermodynamic potentials include the internal energy

*U = U(S, V, N, x*enthalpy

_{i});*H = H(S, ρ, N, xi); Helmholtz*energy, which is the free energy, or isochoric-isothermal potential, designated by

*A*or

*F*, where

*F = F(V, T, N*, x

_{i}); and the Gibbs energy, or the isobaric-isothermal potential, designated by Φ or

*G*, where

*G = G(p, T, N, x*

_{i}).If the thermodynamic potentials are known as functions of the given parameters, it is possible to obtain all other parameters characterizing the system by differentiating the thermodynamic potentials, just as in mechanics it is possible to determine the components of the forces acting on a system by differentiating the system’s potential energy with respect to the corresponding coordinates. The thermodynamic potentials are interrelated by the equations *F = U - TS, H = U + pV*, and *G = F +* ρV. If one of the thermodynamic potentials is known, it is possible to determine all thermodynamic properties of a system, in particular, the equation of state. The conditions of thermodynamic equilibrium of a system and the criteria for its stability are also derived from these potentials.

The work accomplished by a thermodynamic system in a given process is determined by the decrease in the thermodynamic potential corresponding to the conditions of the process. Thus, under conditions of thermal insulation, an adiabatic process *(S* = const), the elementary work *dA* is equal to the decrease in internal energy: *dA = — dU.* In an isothermal process (T = const) *dA = — dF.* Here work is accomplished not only through internal energy but also through heat entering the system. Processes within systems, such as chemical reactions, often proceed with constant *ρ* and *T.* In this case, the elementary work of all thermodynamic forces, other than those due to pressure, is equal to the decrease in the Gibbs thermodynamic potential *G;* that is, *dA’ = - dG.*

The equality *dA = — dU* is satisfied for adiabatic processes both quasi-static, or reversible, and dynamical, or irreversible, processes. In other cases, however, the work is equal to the decrease in the thermodynamic potentials only for quasi-static processes; for dynamical processes, the work accomplished is less than the change in the thermodynamic potentials. The theoretical determination of thermodynamic potentials as functions of the corresponding variables is the main task of statistical thermodynamics.

The technique of employing thermodynamic potentials is widely used to arrive at general relations between the physical properties of mascroscopic bodies and to analyze thermodynamic processes and equilibrium conditions in physicochemical systems. The term “thermodynamic potentials” was introduced in 1884 by the French physicist P. Duhem, while J. W. Gibbs, who developed the technique of employing thermodynamic potentials, used the term “fundamental functions” in his works.

### REFERENCES

Landau, L. D., and E. M. Lifshits.*Statisticheskaia fizika*, 2nd ed. Moscow, 1964.

*(Teoreticheskaia fizika*, vol. 5.)

Leontovich, M. A.

*Vvedenie ν termodinamiku*, 2nd ed. Moscow-Leningrad, 1952.

Reif, F.

*Statisticheskaia fizika.*Moscow, 1972.

*(Berkleevskii kurs fizi-ki*, vol. 5.) (Translated from English.)

Gibbs, J. W.

*Termodinamicheskie raboty.*Moscow-Leningrad, 1950. (Translated from English.)

G. IA. MIAKISHEV