Thermal equilibrium

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When a system with a higher temperature is in contact with a lower temperature system, heat is transferred to the lower temperature system. Systems are in thermal equilibrium when they reach the same constant temperature. Thermal equilibrium follows Zeroth's Law of Thermodynamics. In physics, the phrase thermal equilibrium is used to describe the sometimes in the common parlance of physical discourse, and sometimes as a specialized technical term in thermodynamics.

As common parlance, the phrase refers to steady states of temperature, which may be spatial or temporal. The meaning varies from occasion to occasion, as with all ordinary language usages.

Thermal equilibrium as a technical term in thermodynamics can also be used in two senses. One sense is that of thermal equilibrium within a system for itself. The other sense is that of a relation between the respective physical states of two bodies. Thermal equilibrium in a system for itself means that the temperature within the system is spatially and temporally uniform. Thermal equilibrium as a relation between the physical states of two bodies means that there is actual or implied thermal connection between them, through a path that is permeable only to heat, and that no energy is transferred through that path. This technical sense is concerned with the theory of the definition of temperature.

Specialized technical senses[edit]

Thermal equilibrium of an isolated body[edit]

Thermal equilibrium of a body in itself refers to the body when it is isolated. The background is that no heat enters or leaves it, and that it is allowed unlimited time to settle under its own intrinsic characteristics. When it is completely settled, so that its temperature is spatially and temporally uniform, it is in its own thermal equilibrium. It is not implied that it is necessarily in other kinds of internal equilibrium. For example, it is possible that a body might reach internal thermal equilibrium but not be in internal chemical equilibrium; glass is an example.

The relation of thermal equilibrium between two thermally connected bodies[edit]

The relation of thermal equilibrium is an instance of a contact equilibrium between two bodies. This means that it refers to transfer through a selectively permeable partition, the contact path.[1] For the relation of thermal equilibrium, the contact path is permeable only to heat; it does not permit the passage of matter or work. According to Lieb and Yngvason, the essential meaning of the relation of thermal equilibrium includes that it is reflexive and symmetric. It is not included in the essential meaning whether it is or is not transitive. After discussing the semantics of the definition, they postulate a substantial physical axiom, that they call the "zeroth law of thermodynamics", that thermal equilibrium is a transitive relation. They comment that the equivalence classes of systems so established are called isotherms.[2]

Thermal contact[edit]

Heat can flow into or out of a closed system by way of thermal conduction or of thermal radiation to or from a thermal reservoir, and when this process is effecting net transfer of heat, the system's temperature can be changing. While the transfer of energy as heat continues, the system is not in thermal equilibrium.

Change of internal state of an isolated system[edit]

If an isolated system is left long enough, it will reach a state of thermal equilibrium in itself, in which its temperature will be uniform throughout, but not necessarily a state of thermodynamic equilibrium, if there is some structural barrier that can prevent some possible processes in the system from reaching equilibrium; glass is an example. An isolated system can change its temperature or its spatial distribution of temperature by changing the state of its materials. A rod of iron, initially prepared to be hot at one end and cold at the other, when isolated, will change so that its temperature becomes uniform all along its length; during the process, the rod is not in thermal equilibrium until its temperature is uniform. A very tall adiabatically isolating vessel with rigid walls initially containing a thermally heterogeneous distribution of material, left for a long time under the influence of a steady gravitational field, along its tall dimension, due to an outside body such as the earth, will settle to a state of uniform temperature though not of uniform pressure or density, and is then in internal thermal equilibrium and even in thermodynamic equilibrium.[3][4][5][6][7][8][9][10][11] A system prepared as a mixture of petrol vapour and air can be ignited by a spark and produce carbon dioxide and water; if this happens in an isolated system, it will increase the temperature of the system, and during the increase, the system is not in thermal equilibrium; but eventually the system will settle to a uniform temperature. In a system prepared as a block of ice floating in a bath of hot water, and then isolated, the ice can melt; during the melting, the system is not in thermal equilibrium; but eventually its temperature will become uniform. Such changes in isolated systems are irreversible in the sense that while such a change will occur spontanteously whenever the system is prepared in the same way, the reverse change will never occur spontanteously within the isolated system; this is a large part of the content of the second law of thermodynamics. Truly isolated systems hardly occur in nature, and nearly always are artificially prepared.

Bodies prepared with separately uniform temperatures, then put into purely thermal communication with each other[edit]

If bodies are prepared with separately uniform temperatures, and are then put into purely thermal communication with each other, by conductive or radiative pathways, they will be in thermal equilibrium with each other just when they have the same temperature, and then there will be no net transfer of heat between them; but if initially they do not have the same temperature, heat will flow from the hotter to the colder, by whatever pathway, conductive or radiative, is available, and this flow will continue until thermal equilibrium is reached and then they will have the same temperature.[12][13]

One form of thermal equilibrium is radiative exchange equilibrium.[14][15] Two bodies, each with its own uniform temperature, in solely radiative connection, no matter how far apart, or what partially obstructive, reflective, or refractive, obstacles lie in their path of radiative exchange, not moving relative to one another, will exchange thermal radiation, in net the hotter transferring energy to the cooler, and will exchange equal and opposite amounts just when they are at the same temperature. In this situation, Kirchhoff's law of equality of radiative emissivity and absorptivity and the Helmholtz reciprocity principle are in play.

Distinctions between thermal and thermodynamic equilibria[edit]

There is an important distinction between thermal and thermodynamic equilibrium. According to Münster (1970), in states of thermodynamic equilibrium, the state variables of a system do not change at a measurable rate. Moreover, "The proviso 'at a measurable rate' implies that we can consider an equilibrium only with respect to specified processes and defined experimental conditions." Also, a state of thermodynamic equilibrium can be described by fewer macroscopic variables than any other state of a given body of matter. A single isolated body can start in a state which is not one of thermodynamic equilibrium, and can change till thermodynamic equilibrium is reached. Thermal equilibrium is a relation between two bodies or closed systems, in which transfers are allowed only of energy and take place through a partition permeable to heat, and in which the transfers have proceeded till the states of the bodies cease to change.[16]

An explicit distinction between 'thermal equilibrium' and 'thermodynamic equilibrium' is made by C.J. Adkins. He allows that two systems might be allowed to exchange heat but be constrained from exchanging work; they will naturally exchange heat till they have equal temperatures, and reach thermal equilibrium, but in general will not be in thermodynamic equilibrium. They can reach thermodynamic equilibrium when they are allowed also to exchange work.[17]

Another explicit distinction between 'thermal equilibrium' and 'thermodynamic equilibrium' is made by B. C. Eu. He considers two systems in thermal contact, one a thermometer, the other a system in which several irreversible processes are occurring. He considers the case in which, over the time scale of interest, it happens that both the thermometer reading and the irreversible processes are steady. Then there is thermal equilibrium without thermodynamic equilibrium. Eu proposes consequently that the zeroth law of thermodynamics can be considered to apply even when thermodynamic equilibrium is not present; also he proposes that if changes are occurring so fast that a steady temperature cannot be defined, then "it is no longer possible to describe the process by means of a thermodynamic formalism. In other words, thermodynamics has no meaning for such a process."[18]


  1. ^ Münster, A. (1970). Classical Thermodynamics, translated by E.S. Halberstadt, Wiley–Interscience, London, p.49.
  2. ^ Lieb, E.H., Yngvason, J. (1999). The physics and mathematics of the second law of thermodynamics, Physics Reports, 314: 1–96, p. 55–56.
  3. ^ Maxwell, J.C. (1867). On the dynamical theory of gases, Phil. Trans. Roy. Soc. London, 157: 49–88.
  4. ^ Gibbs, J.W. (1876/1878). On the equilibrium of heterogeneous substances, Trans. Conn. Acad., 3: 108-248, 343-524, reprinted in The Collected Works of J. Willard Gibbs, Ph.D, LL. D., edited by W.R. Longley, R.G. Van Name, Longmans, Green & Co., New York, 1928, volume 1, pages 55-353, particularly pages 144-150.
  5. ^ Boltzmann, L. (1896/1964). Lectures on Gas Theory, translated by S.G. Brush, University of California Press, Berkeley, p. 143.
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  8. ^ ter Haar, D., Wergeland, H. (1966). Elements of Thermodynamics, Addison-Wesley Publishing, Reading MA, pp. 127–130.
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  10. ^ Román, F.L., White, J.A., Velasco, S. (1995). Microcanonical single-particle distributions for an ideal gas in a gravitational field, Eur. J. Phys., 16: 83–90.
  11. ^ Velasco, S., Román, F.L., White, J.A. (1996). On a paradox concerning the temperature distribution of an ideal gas in a gravitational field, Eur. J. Phys., 17: 43–44.
  12. ^ Maxwell, J.C. (1872). Theory of Heat, third edition, Longmans, Green, London, Chapters 1, 2.
  13. ^ Planck, M. (1897/1903). Treatise on Thermodynamics, translated by A. Ogg, Longmans, Green & Co., London, pages 1–2.
  14. ^ Prevost, P. (1791). Mémoire sur l'equilibre du feu. Journal de Physique (Paris), vol. 38 pp. 314-322.
  15. ^ Planck, M. (1914). The Theory of Heat Radiation, second edition translated by M. Masius, P. Blakiston's Son and Co., Philadelphia, p. 40.
  16. ^ Münster, A. (1970). Classical Thermodynamics, Wiley–Interscience, London, ISBN 0-471-62430-6, pp. 6, 22, 52.
  17. ^ Adkins, C.J. (1968/1975). Equilibrium Thermodynamics, second edition, McGraw-Hill, London, ISBN 0–07–084057–1, page 7.
  18. ^ Eu, B.C. (2002). Generalized Thermodynamics. The Thermodynamics of Irreversible Processes and Generalized Hydrodynamics, Kluwer Academic Publishers, Dordrecht, ISBN 1–4020–0788–4, page 13.