In a system that is in its own state of internal thermodynamic equilibrium, not only is there an absence of macroscopic change, but there is an “absence of any tendency toward change on a macroscopic scale.”[1] Systems in mutual thermodynamic equilibrium are simultaneously in mutual thermal, mechanical, chemical, and radiative equilibria.
Its intensive properties, other than temperature, may be driven to spatial inhomogeneity by an unchanging long-range force field imposed on it by its surroundings.
One such potential is the Helmholtz free energy (A), for a closed system at constant volume and temperature (controlled by a heat bath): Another potential, the Gibbs free energy (G), is minimized at thermodynamic equilibrium in a closed system at constant temperature and pressure, both controlled by the surroundings: where T denotes the absolute thermodynamic temperature, P the pressure, S the entropy, V the volume, and U the internal energy of the system.
Thermodynamic equilibrium is the unique stable stationary state that is approached or eventually reached as the system interacts with its surroundings over a long time.
If it has been left undisturbed for an indefinitely long time, classical thermodynamics postulates that it is in a state in which no changes occur within it, and there are no flows within it.
[7] One textbook[8] calls it the "zeroth law", remarking that the authors think this more befitting that title than its more customary definition, which apparently was suggested by Fowler.)
Consequent upon such an operation restricted to the surroundings, the system may be for a time driven away from its own initial internal state of thermodynamic equilibrium.
For example, one widely cited writer, H. B. Callen writes in this context: "In actuality, few systems are in absolute and true equilibrium."
Beattie and I. Oppenheim write: "Insistence on a strict interpretation of the definition of equilibrium would rule out the application of thermodynamics to practically all states of real systems.
He discusses the second proviso by giving an account of a mixture oxygen and hydrogen at room temperature in the absence of a catalyst.
[29] This statement does not explicitly say that for thermodynamic equilibrium, the system must be isolated; Callen does not spell out what he means by the words "intrinsic factors".
It distinguishes several drivers of flows, and then says: "These are examples of the apparently universal tendency of isolated systems toward a state of complete mechanical, thermal, chemical, and electrical—or, in a single word, thermodynamic—equilibrium.
It states: "More precisely, the glass may be regarded as being in equilibrium so long as experimental tests show that 'slow' transitions are in effect reversible.
This scheme follows the general rule that "... we can consider an equilibrium only with respect to specified processes and defined experimental conditions.
But shortly below that definition he writes of a piece of glass that has not yet reached its "full thermodynamic equilibrium state".
Kirkwood and I. Oppenheim define thermodynamic equilibrium as follows: "A system is in a state of thermodynamic equilibrium if, during the time period allotted for experimentation, (a) its intensive properties are independent of time and (b) no current of matter or energy exists in its interior or at its boundaries with the surroundings."
They do not discuss the possibility of changes that occur with "glacial slowness", and proceed beyond the time period allotted for experimentation.
In general, a strong external force field makes a system of a single phase in its own internal thermodynamic equilibrium inhomogeneous with respect to some intensive variables.
"[42] Planck introduces his treatise with a brief account of heat and temperature and thermal equilibrium, and then announces: "In the following we shall deal chiefly with homogeneous, isotropic bodies of any form, possessing throughout their substance the same temperature and density, and subject to a uniform pressure acting everywhere perpendicular to the surface.
In contrast, Carathéodory's scheme of presentation of classical thermodynamics for closed systems postulates the concept of an "equilibrium state" following Gibbs (Gibbs speaks routinely of a "thermodynamic state"), though not explicitly using the phrase 'thermodynamic equilibrium', nor explicitly postulating the existence of a temperature to define it.
In a thought experiment, Reed A. Howald conceived of a system called "The Fizz Keeper"[43]consisting of a cap with a nozzle that can re-pressurize any standard bottle of carbonated beverage.
Nitrogen and oxygen, which air are mostly made out of, would keep getting pumped in, which would slow down the rate at which the carbon dioxide fizzles out of the system.
To come to this conclusion, he also appeals to Henry's Law, which states that gases dissolve in direct proportion to their partial pressures.
By influencing the partial pressure on the top of a closed system, this would help slow down the rate of fizzing out of carbonated beverages which is governed by thermodynamic equilibrium.
The equilibria of carbon dioxide and other gases would not change, however the partial pressure on top would slow down the rate of dissolution extending the time a gas stays in a particular state.
For an externally imposed gravitational field, this may be proved in macroscopic thermodynamic terms, by the calculus of variations, using the method of Langrangian multipliers.
[57] In his exposition of his scheme of closed system equilibrium thermodynamics, C. Carathéodory initially postulates that experiment reveals that a definite number of real variables define the states that are the points of the manifold of equilibria.
"[58][59] As noted above, according to A. Münster, the number of variables needed to define a thermodynamic equilibrium is the least for any state of a given isolated system.
If the thermodynamic equilibrium lies in an external force field, it is only the temperature that can in general be expected to be spatially uniform.
This is because the single measurements might have been made during a slight fluctuation, away from another set of nominal values of those conjugate intensive functions of state, that is due to unknown and different constitutive properties.