Thermal equilibrium
Encyclopedia
Thermal equilibrium is a theoretical physical concept, used especially in theoretical texts, that means that all temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

s of interest are unchanging in time and uniform in space. When the temperatures of interest are just those in the different parts of one body, the concept also requires that any flow of heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

by thermal conduction or by thermal radiation
Thermal radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation....

into or out of one part of the body be balanced by a flow of heat in the opposite sense into or out of another part of the body. When the temperatures of interest belong to several bodies, the concept also requires that flows of heat between each pair of bodies balance to a zero net flow, but it allows the several bodies to gain or lose heat to several external reservoirs provided that their total rate of inflow from all reservoirs is equal to their total rate of outflow to all reservoirs and that each flow is unchanging in time. For some situations, the definition of transfer of heat can be problematic.

Some writers use the term thermal equilibrium in a different sense. They mean by it that the spatial temperature distribution of the body is not necessarily uniform, and indeed is likely to be non-uniform, but is maintained unvarying in time, by flows of energy; for example they mean that there is spatially distributed radiative cooling of the body and equal and opposite spatially distributed energy addition by condensation of water vapour, just so as on average to keep the spatial distribution of temperature time-invariant.

Thermal equilibrium does not mean the same as thermodynamic equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...

, because the latter requires that there be equilibrium of all kinds, not only thermal, and that there be no flow of any kind, in the system of interest.

## Transfer of heat by conduction and radiation between systems

Heat can flow
Heat transfer
Heat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phase-change transfer...

into or out of a closed system
Closed system
-In physics:In thermodynamics, a closed system can exchange energy , but not matter, with its surroundings.In contrast, an isolated system cannot exchange any of heat, work, or matter with the surroundings, while an open system can exchange all of heat, work and matter.For a simple system, with...

by way of thermal conduction or of thermal radiation
Thermal radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter. All matter with a temperature greater than absolute zero emits 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, and if so, the system is not in thermal equilibrium.

For thermodynamics, it is considered that convection transports internal energy by bulk flow; this is considered not to be transfer of heat as such.

## Change of internal state of an isolated system

If an isolated system
Isolated system
In the natural sciences an isolated system, as contrasted with an open system, is a physical system without any external exchange. If it has any surroundings, it does not interact with them. It obeys in particular the first of the conservation laws: its total energy - mass stays constant...

is left long enough, it will reach a state of thermal equilibrium, 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. 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 isolating vessel 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, will settle to a state of uniform temperature though not of uniform pressure or density, and is then in thermal equilibrium and even of thermodynamic equilibrium. 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
Second law of thermodynamics
The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...

. Truly isolated systems hardly occur in nature, and nearly always are artificially prepared.

## Thermal equilibrium between bodies prepared with separately uniform temperatures, then put into purely thermal communication with each other

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.

One form of thermal equilibrium is radiative exchange equilibrium. 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
In thermodynamics, Kirchhoff's law of thermal radiation, or Kirchhoff's law for short, is a general statement equating emission and absorption in heated objects, proposed by Gustav Kirchhoff in 1859, following from general considerations of thermodynamic equilibrium and detailed balance.An object...

and the Helmholtz reciprocity
Helmholtz reciprocity
The Helmholtz reciprocity principle describes how a ray of light and its reverse ray encounter matched optical adventures, such as reflections, refractions, and absorptions in a passive medium, or at an interface....

principle are in play.

## Theoretical foundations

It is scientifically permissible, and perhaps unavoidably necessary, to start a project of reasoning with several mutually coherent and dependent primitive presupposed concepts. The concept of thermal equilibrium is thus coherent with the concepts of temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

and of heat transfer
Heat transfer
Heat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phase-change transfer...

. These three concepts hardly make physical sense without each other. They were considered coordinately before and during and after the developments of calorimetry
Calorimetry
Calorimetry is the science of measuring the heat of chemical reactions or physical changes. Calorimetry is performed with a calorimeter. The word calorimetry is derived from the Latin word calor, meaning heat...

and of thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

, for example by Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

and by Planck
Max Planck
Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...

.

Instead of relying on this triple of jointly defined physical concepts, some writers, motivated by a desire for axiomatic parsimony and precision or mathematical elegance, prefer to define thermal equilibrium by relying on a presupposed notion of thermodynamic equilibrium, in which all mechanically measurable properties of a body have become stationary, and one infers that consequently the otherwise undefined thermal properties also are stationary. Carathéodory
Constantin Carathéodory
Constantin Carathéodory was a Greek mathematician. He made significant contributions to the theory of functions of a real variable, the calculus of variations, and measure theory...

is an example of such writers, as seen in his 1909 article. This approach leaves one at the mercy of the questions of what is meant physically by "all mechanically measurable properties" and what is meant by saying that they "have become stationary", and still relying on some concept that allows bodies not 'thermally connected' to be put into 'thermal connection'. It also leaves a person, who does not know in advance the notions of heat transfer and temperature, reliant on the assumptions, such as conservation of energy, and on the mathematical development, of the theory of thermodynamics.