What Thermodynamic Condition Must Be Met for a State of Equilibrium to Exist

State of Thermodynamic system(due south) where no net macroscopic menstruum of affair or energy occurs

Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal country of a single thermodynamic system, or a relation between several thermodynamic systems connected by more than or less permeable or impermeable walls. In thermodynamic equilibrium there are no internet macroscopic flows of matter or of energy, within a system or between systems. In a system that is in its ain state of internal thermodynamic equilibrium, no macroscopic change occurs.

Systems in mutual thermodynamic equilibrium are simultaneously in mutual thermal, mechanical, chemical, and radiative equilibria. Systems tin can exist in one kind of mutual equilibrium, while not in others. In thermodynamic equilibrium, all kinds of equilibrium hold at once and indefinitely, until disturbed by a thermodynamic functioning. In a macroscopic equilibrium, perfectly or well-nigh perfectly balanced microscopic exchanges occur; this is the physical explanation of the notion of macroscopic equilibrium.

A thermodynamic system in a state of internal thermodynamic equilibrium has a spatially uniform temperature. Its intensive properties, other than temperature, may exist driven to spatial inhomogeneity by an unchanging long-range force field imposed on it by its surroundings.

In systems that are at a state of non-equilibrium at that place are, by dissimilarity, net flows of affair or free energy. If such changes can be triggered to occur in a organisation in which they are non already occurring, the organization is said to be in a meta-stable equilibrium.

Though not a widely named "law," it is an axiom of thermodynamics that there be states of thermodynamic equilibrium. The 2d police force of thermodynamics states that when an isolated body of textile starts from an equilibrium state, in which, portions of it are held at different states by more or less permeable or impermeable partitions, and a thermodynamic performance removes or makes the partitions more permeable, then it spontaneously reaches its own, new state of internal thermodynamic equilibrium, and this is accompanied by an increase in the sum of the entropies of the portions.

Overview [edit]

Classical thermodynamics deals with states of dynamic equilibrium. The state of a organization at thermodynamic equilibrium is the i for which some thermodynamic potential is minimized (in the absence of an practical voltage),^[1] or for which the entropy (Due south) is maximized, for specified weather condition. One such potential is the Helmholtz costless energy (A), for a closed arrangement at abiding book and temperature (controlled by a heat bath):

${\displaystyle A=U-TS}$

Another potential, the Gibbs complimentary energy (G), is minimized at thermodynamic equilibrium in a closed system at constant temperature and pressure level, both controlled by the environs:

${\displaystyle M=U-TS+PV}$

where T denotes the absolute thermodynamic temperature, P the pressure, S the entropy, V the volume, and U the internal energy of the system. In other words, ${\displaystyle \Delta G=0}$ is a necessary status for chemic equilibrium under these weather (in the absence of an applied voltage).

Thermodynamic equilibrium is the unique stable stationary state that is approached or eventually reached every bit the system interacts with its environment over a long time. The to a higher place-mentioned potentials are mathematically constructed to be the thermodynamic quantities that are minimized under the particular weather in the specified surroundings.

Conditions [edit]

• For a completely isolated organisation, South is maximum at thermodynamic equilibrium.
• For a closed system at controlled abiding temperature and volume, A is minimum at thermodynamic equilibrium.
• For a closed arrangement at controlled constant temperature and pressure without an practical voltage, Grand is minimum at thermodynamic equilibrium.

The various types of equilibriums are accomplished as follows:

• Two systems are in thermal equilibrium when their temperatures are the same.
• Two systems are in mechanical equilibrium when their pressures are the same.
• Two systems are in diffusive equilibrium when their chemic potentials are the same.
• All forces are balanced and there is no significant external driving strength.

Relation of exchange equilibrium between systems [edit]

Ofttimes the environment of a thermodynamic organisation may as well be regarded every bit another thermodynamic system. In this view, one may consider the organization and its surroundings every bit two systems in mutual contact, with long-range forces also linking them. The enclosure of the arrangement is the surface of contiguity or boundary between the two systems. In the thermodynamic formalism, that surface is regarded equally having specific backdrop of permeability. For example, the surface of contiguity may be supposed to exist permeable simply to heat, allowing energy to transfer only every bit heat. Then the two systems are said to be in thermal equilibrium when the long-range forces are unchanging in time and the transfer of energy as heat between them has slowed and eventually stopped permanently; this is an example of a contact equilibrium. Other kinds of contact equilibrium are defined by other kinds of specific permeability.^[2] When 2 systems are in contact equilibrium with respect to a particular kind of permeability, they have common values of the intensive variable that belongs to that detail kind of permeability. Examples of such intensive variables are temperature, force per unit area, chemical potential.

A contact equilibrium may exist regarded also as an commutation equilibrium. There is a zero balance of rate of transfer of some quantity between the two systems in contact equilibrium. For case, for a wall permeable simply to oestrus, the rates of improvidence of internal energy every bit heat between the two systems are equal and contrary. An adiabatic wall betwixt the two systems is 'permeable' only to energy transferred as work; at mechanical equilibrium the rates of transfer of energy as work between them are equal and opposite. If the wall is a simple wall, then the rates of transfer of book across information technology are also equal and opposite; and the pressures on either side of information technology are equal. If the adiabatic wall is more complicated, with a sort of leverage, having an area-ratio, then the pressures of the two systems in exchange equilibrium are in the changed ratio of the volume exchange ratio; this keeps the zero balance of rates of transfer every bit work.

A radiative commutation can occur between two otherwise separate systems. Radiative exchange equilibrium prevails when the two systems have the same temperature.^[3]

Thermodynamic state of internal equilibrium of a organisation [edit]

A collection of matter may exist entirely isolated from its environment. If it has been left undisturbed for an indefinitely long time, classical thermodynamics postulates that it is in a land in which no changes occur within information technology, and there are no flows within it. This is a thermodynamic state of internal equilibrium.^{[4] [5]} (This postulate is sometimes, but not often, called the "minus first" constabulary of thermodynamics.^[half-dozen] One textbook^[7] calls information technology the "zeroth law", remarking that the authors recall this more befitting that title than its more customary definition, which evidently was suggested by Fowler.)

Such states are a principal business organization in what is known as classical or equilibrium thermodynamics, for they are the simply states of the system that are regarded as well divers in that subject. A system in contact equilibrium with another system can by a thermodynamic performance be isolated, and upon the issue of isolation, no modify occurs in it. A organization in a relation of contact equilibrium with some other system may thus also be regarded as being in its own land of internal thermodynamic equilibrium.

Multiple contact equilibrium [edit]

The thermodynamic ceremonial allows that a system may accept contact with several other systems at once, which may or may not as well have mutual contact, the contacts having respectively different permeabilities. If these systems are all jointly isolated from the residuum of the world those of them that are in contact so reach respective contact equilibria with one another.

If several systems are free of adiabatic walls betwixt each other, but are jointly isolated from the rest of the world, then they reach a state of multiple contact equilibrium, and they take a mutual temperature, a total internal free energy, and a total entropy.^{[8] [9] [10] [11]} Amongst intensive variables, this is a unique property of temperature. It holds even in the presence of long-range forces. (That is, there is no "forcefulness" that tin can maintain temperature discrepancies.) For example, in a organisation in thermodynamic equilibrium in a vertical gravitational field, the pressure on the superlative wall is less than that on the bottom wall, merely the temperature is the same everywhere.

A thermodynamic functioning may occur as an event restricted to the walls that are inside the surroundings, direct affecting neither the walls of contact of the organisation of interest with its environment, nor its interior, and occurring within a definitely limited time. For example, an immovable adiabatic wall may be placed or removed within the surroundings. Consequent upon such an operation restricted to the surroundings, the system may be for a fourth dimension driven away from its own initial internal land of thermodynamic equilibrium. Then, according to the 2nd law of thermodynamics, the whole undergoes changes and somewhen reaches a new and final equilibrium with the surroundings. Following Planck, this consequent train of events is called a natural thermodynamic process.^[12] It is allowed in equilibrium thermodynamics only because the initial and concluding states are of thermodynamic equilibrium, even though during the procedure there is transient divergence from thermodynamic equilibrium, when neither the organisation nor its surroundings are in well defined states of internal equilibrium. A natural process proceeds at a finite rate for the main office of its course. Information technology is thereby radically different from a fictive quasi-static 'procedure' that proceeds infinitely slowly throughout its course, and is fictively 'reversible'. Classical thermodynamics allows that even though a process may accept a very long time to settle to thermodynamic equilibrium, if the main part of its form is at a finite rate, then it is considered to exist natural, and to be field of study to the second law of thermodynamics, and thereby irreversible. Engineered machines and artificial devices and manipulations are permitted within the surroundings.^{[13] [14]} The allowance of such operations and devices in the surroundings but not in the organization is the reason why Kelvin in ane of his statements of the second law of thermodynamics spoke of "inanimate" agency; a system in thermodynamic equilibrium is inanimate.^[15]

Otherwise, a thermodynamic operation may directly affect a wall of the system.

It is frequently user-friendly to suppose that some of the surrounding subsystems are so much larger than the organisation that the process tin affect the intensive variables only of the surrounding subsystems, and they are then called reservoirs for relevant intensive variables.

Local and global equilibrium [edit]

It is useful to distinguish between global and local thermodynamic equilibrium. In thermodynamics, exchanges within a arrangement and between the system and the exterior are controlled by intensive parameters. Equally an case, temperature controls rut exchanges. Global thermodynamic equilibrium (GTE) means that those intensive parameters are homogeneous throughout the whole system, while local thermodynamic equilibrium (LTE) means that those intensive parameters are varying in space and fourth dimension, merely are varying so slowly that, for any point, one can presume thermodynamic equilibrium in some neighborhood about that point.

If the description of the arrangement requires variations in the intensive parameters that are too large, the very assumptions upon which the definitions of these intensive parameters are based will pause down, and the system will be in neither global nor local equilibrium. For example, information technology takes a certain number of collisions for a particle to equilibrate to its surroundings. If the average altitude it has moved during these collisions removes it from the neighborhood it is equilibrating to, it will never equilibrate, and there will be no LTE. Temperature is, by definition, proportional to the average internal energy of an equilibrated neighborhood. Since there is no equilibrated neighborhood, the concept of temperature doesn't hold, and the temperature becomes undefined.

It is important to note that this local equilibrium may employ only to a certain subset of particles in the arrangement. For instance, LTE is usually applied just to massive particles. In a radiating gas, the photons being emitted and absorbed by the gas doesn't demand to be in a thermodynamic equilibrium with each other or with the massive particles of the gas in lodge for LTE to exist. In some cases, information technology is not considered necessary for free electrons to be in equilibrium with the much more massive atoms or molecules for LTE to exist.

Every bit an case, LTE will exist in a glass of h2o that contains a melting ice cube. The temperature inside the glass tin can be divers at any signal, but it is colder near the ice cube than far away from it. If energies of the molecules located near a given point are observed, they will be distributed according to the Maxwell–Boltzmann distribution for a certain temperature. If the energies of the molecules located nigh another bespeak are observed, they will be distributed according to the Maxwell–Boltzmann distribution for another temperature.

Local thermodynamic equilibrium does not require either local or global stationarity. In other words, each small locality need not have a constant temperature. However, it does require that each pocket-sized locality alter slowly enough to practically sustain its local Maxwell–Boltzmann distribution of molecular velocities. A global non-equilibrium country can be stably stationary simply if it is maintained by exchanges between the arrangement and the outside. For example, a globally-stable stationary state could exist maintained inside the glass of water by continuously adding finely powdered ice into it in order to compensate for the melting, and continuously draining off the meltwater. Natural transport phenomena may lead a system from local to global thermodynamic equilibrium. Going back to our example, the diffusion of estrus will pb our glass of water toward global thermodynamic equilibrium, a state in which the temperature of the glass is completely homogeneous.^[16]

Reservations [edit]

Conscientious and well informed writers near thermodynamics, in their accounts of thermodynamic equilibrium, often enough brand provisos or reservations to their statements. Some writers exit such reservations only implied or more or less unstated.

For example, one widely cited author, H. B. Callen writes in this context: "In actuality, few systems are in absolute and true equilibrium." He refers to radioactive processes and remarks that they may take "cosmic times to complete, [and] generally can exist ignored". He adds "In practice, the benchmark for equilibrium is round. Operationally, a organization is in an equilibrium state if its properties are consistently described by thermodynamic theory!"^[17]

J.A. Beattie and I. Oppenheim write: "Insistence on a strict interpretation of the definition of equilibrium would dominion out the application of thermodynamics to practically all states of real systems."^[18]

Some other author, cited by Callen equally giving a "scholarly and rigorous treatment",^[19] and cited by Adkins as having written a "archetype text",^[twenty] A.B. Pippard writes in that text: "Given long enough a supercooled vapour will eventually condense, ... . The time involved may be so enormous, however, perhaps 10¹⁰⁰ years or more, ... . For most purposes, provided the rapid change is not artificially stimulated, the systems may be regarded every bit being in equilibrium."^[21]

Another writer, A. Münster, writes in this context. He observes that thermonuclear processes often occur then slowly that they can exist ignored in thermodynamics. He comments: "The concept 'absolute equilibrium' or 'equilibrium with respect to all imaginable processes', has therefore, no physical significance." He therefore states that: "... nosotros can consider an equilibrium simply with respect to specified processes and divers experimental conditions."^[22]

According to L. Tisza: "... in the word of phenomena near accented cipher. The absolute predictions of the classical theory become particularly vague because the occurrence of frozen-in nonequilibrium states is very mutual."^[23]

Definitions [edit]

The nearly general kind of thermodynamic equilibrium of a arrangement is through contact with the surround that allows simultaneous passages of all chemic substances and all kinds of free energy. A system in thermodynamic equilibrium may motion with uniform acceleration through space but must not change its shape or size while doing so; thus it is defined by a rigid volume in space. It may lie within external fields of force, determined by external factors of far greater extent than the system itself, so that events within the organisation cannot in an observable amount affect the external fields of force. The system can exist in thermodynamic equilibrium only if the external force fields are uniform, and are determining its compatible dispatch, or if information technology lies in a not-uniform force field but is held stationary in that location by local forces, such as mechanical pressures, on its surface.

Thermodynamic equilibrium is a primitive notion of the theory of thermodynamics. According to P.G. Morse: "Information technology should be emphasized that the fact that there are thermodynamic states, ..., and the fact that there are thermodynamic variables which are uniquely specified by the equilibrium country ... are non conclusions deduced logically from some philosophical first principles. They are conclusions ineluctably drawn from more than two centuries of experiments."^[24] This means that thermodynamic equilibrium is not to be defined solely in terms of other theoretical concepts of thermodynamics. M. Bailyn proposes a cardinal police force of thermodynamics that defines and postulates the existence of states of thermodynamic equilibrium.^[25]

Textbook definitions of thermodynamic equilibrium are often stated advisedly, with some reservation or other.

For instance, A. Münster writes: "An isolated system is in thermodynamic equilibrium when, in the system, no changes of country are occurring at a measurable rate." At that place are two reservations stated here; the arrangement is isolated; whatsoever changes of state are immeasurably slow. He discusses the second proviso past giving an account of a mixture oxygen and hydrogen at room temperature in the absence of a goad. Münster points out that a thermodynamic equilibrium state is described by fewer macroscopic variables than is whatsoever other state of a given system. This is partly, but not entirely, because all flows within and through the system are nada.^[26]

R. Haase'south presentation of thermodynamics does not outset with a restriction to thermodynamic equilibrium because he intends to allow for not-equilibrium thermodynamics. He considers an arbitrary system with fourth dimension invariant properties. He tests it for thermodynamic equilibrium by cutting it off from all external influences, except external force fields. If later on insulation, nix changes, he says that the system was in equilibrium.^[27]

In a section headed "Thermodynamic equilibrium", H.B. Callen defines equilibrium states in a paragraph. He points out that they "are adamant by intrinsic factors" within the arrangement. They are "terminal states", towards which the systems evolve, over fourth dimension, which may occur with "glacial slowness".^[28] This statement does non explicitly say that for thermodynamic equilibrium, the system must be isolated; Callen does not spell out what he means past the words "intrinsic factors".

Another textbook writer, C.J. Adkins, explicitly allows thermodynamic equilibrium to occur in a arrangement which is non isolated. His system is, even so, closed with respect to transfer of thing. He writes: "In general, the approach to thermodynamic equilibrium will involve both thermal and piece of work-like interactions with the environs." He distinguishes such thermodynamic equilibrium from thermal equilibrium, in which only thermal contact is mediating transfer of energy.^[29]

Another textbook author, J.R. Partington, writes: "(i) An equilibrium country is one which is independent of time." But, referring to systems "which are just apparently in equilibrium", he adds : "Such systems are in states of ″false equilibrium.″" Partington's statement does not explicitly land that the equilibrium refers to an isolated system. Like Münster, Partington likewise refers to the mixture of oxygen and hydrogen. He adds a proviso that "In a true equilibrium state, the smallest change of any external condition which influences the state will produce a small alter of state ..."^[30] This proviso ways that thermodynamic equilibrium must be stable confronting modest perturbations; this requirement is essential for the strict significant of thermodynamic equilibrium.

A pupil textbook by F.H. Crawford has a section headed "Thermodynamic Equilibrium". It distinguishes several drivers of flows, and then says: "These are examples of the plain universal tendency of isolated systems toward a land of consummate mechanical, thermal, chemical, and electrical—or, in a unmarried word, thermodynamic—equilibrium."^[31]

A monograph on classical thermodynamics by H.A. Buchdahl considers the "equilibrium of a thermodynamic system", without actually writing the phrase "thermodynamic equilibrium". Referring to systems closed to exchange of matter, Buchdahl writes: "If a system is in a terminal status which is properly static, it volition be said to exist in equilibrium."^[32] Buchdahl's monograph also discusses amorphous glass, for the purposes of thermodynamic clarification. Information technology states: "More precisely, the drinking glass may be regarded as beingness in equilibrium so long as experimental tests show that 'slow' transitions are in consequence reversible."^[33] It is not customary to make this proviso role of the definition of thermodynamic equilibrium, just the converse is normally assumed: that if a body in thermodynamic equilibrium is subject to a sufficiently slow process, that procedure may exist considered to be sufficiently nearly reversible, and the torso remains sufficiently nearly in thermodynamic equilibrium during the process.^[34]

A. Münster carefully extends his definition of thermodynamic equilibrium for isolated systems by introducing a concept of contact equilibrium. This specifies particular processes that are immune when considering thermodynamic equilibrium for non-isolated systems, with special concern for open up systems, which may gain or lose matter from or to their surroundings. A contact equilibrium is between the system of interest and a system in the surroundings, brought into contact with the organisation of interest, the contact existence through a special kind of wall; for the remainder, the whole joint organisation is isolated. Walls of this special kind were too considered by C. Carathéodory, and are mentioned by other writers also. They are selectively permeable. They may be permeable only to mechanical work, or only to rut, or only to some particular chemical substance. Each contact equilibrium defines an intensive parameter; for example, a wall permeable just to heat defines an empirical temperature. A contact equilibrium can exist for each chemical constituent of the organisation of interest. In a contact equilibrium, despite the possible exchange through the selectively permeable wall, the system of interest is invariable, as if it were in isolated thermodynamic equilibrium. This scheme follows the full general rule that "... we tin consider an equilibrium merely with respect to specified processes and divers experimental conditions."^[22] Thermodynamic equilibrium for an open up system means that, with respect to every relevant kind of selectively permeable wall, contact equilibrium exists when the respective intensive parameters of the organisation and surroundings are equal.^[2] This definition does not consider the virtually general kind of thermodynamic equilibrium, which is through unselective contacts. This definition does not simply state that no current of thing or energy exists in the interior or at the boundaries; but it is uniform with the following definition, which does so country.

M. Zemansky also distinguishes mechanical, chemical, and thermal equilibrium. He and then writes: "When the conditions for all three types of equilibrium are satisfied, the system is said to be in a state of thermodynamic equilibrium".^[35]

P.M. Morse writes that thermodynamics is concerned with "states of thermodynamic equilibrium". He too uses the phrase "thermal equilibrium" while discussing transfer of energy as heat betwixt a body and a rut reservoir in its surroundings, though non explicitly defining a special term 'thermal equilibrium'.^[36]

J.R. Waldram writes of "a definite thermodynamic state". He defines the term "thermal equilibrium" for a arrangement "when its observables have ceased to change over fourth dimension". But soon below that definition he writes of a slice of glass that has not yet reached its "full thermodynamic equilibrium state".^[37]

Considering equilibrium states, Thousand. Bailyn writes: "Each intensive variable has its ain type of equilibrium." He then defines thermal equilibrium, mechanical equilibrium, and material equilibrium. Accordingly, he writes: "If all the intensive variables go uniform, thermodynamic equilibrium is said to be." He is non here considering the presence of an external force field.^[38]

J.G. Kirkwood and I. Oppenheim define thermodynamic equilibrium equally follows: "A organization 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 affair or energy exists in its interior or at its boundaries with the surroundings." It is axiomatic that they are not restricting the definition to isolated or to airtight systems. They do not discuss the possibility of changes that occur with "glacial slowness", and keep beyond the fourth dimension menstruum allotted for experimentation. They annotation that for 2 systems in contact, there exists a small-scale subclass of intensive properties such that if all those of that pocket-sized subclass are respectively equal, and so all respective intensive properties are equal. States of thermodynamic equilibrium may be defined by this subclass, provided another conditions are satisfied.^[39]

Characteristics of a state of internal thermodynamic equilibrium [edit]

Homogeneity in the absenteeism of external forces [edit]

A thermodynamic system consisting of a single phase in the absence of external forces, in its own internal thermodynamic equilibrium, is homogeneous.^[40] This means that the material in any pocket-sized volume element of the system can exist interchanged with the cloth of whatsoever other geometrically congruent book chemical element of the system, and the issue is to exit the system thermodynamically unchanged. In general, a strong external forcefulness field makes a organisation of a single phase in its ain internal thermodynamic equilibrium inhomogeneous with respect to some intensive variables. For example, a relatively dense component of a mixture tin be concentrated by centrifugation.

Uniform temperature [edit]

Such equilibrium inhomogeneity, induced by external forces, does not occur for the intensive variable temperature. According to Due east.A. Guggenheim, "The most important conception of thermodynamics is temperature."^[41] Planck introduces his treatise with a brief business relationship of heat and temperature and thermal equilibrium, and and then announces: "In the following nosotros shall deal chiefly with homogeneous, isotropic bodies of whatsoever class, possessing throughout their substance the aforementioned temperature and density, and discipline to a uniform pressure level interim everywhere perpendicular to the surface."^[40] Every bit did Carathéodory, Planck was setting aside surface effects and external fields and anisotropic crystals. Though referring to temperature, Planck did non there explicitly refer to the concept of thermodynamic equilibrium. In dissimilarity, Carathéodory'due south scheme of presentation of classical thermodynamics for closed systems postulates the concept of an "equilibrium state" following Gibbs (Gibbs speaks routinely of a "thermodynamic land"), though non explicitly using the phrase 'thermodynamic equilibrium', nor explicitly postulating the beingness of a temperature to ascertain it.

The temperature within a organisation in thermodynamic equilibrium is uniform in space too every bit in time. In a system in its own land of internal thermodynamic equilibrium, there are no cyberspace internal macroscopic flows. In detail, this ways that all local parts of the system are in common radiative commutation equilibrium. This ways that the temperature of the system is spatially uniform.^[iii] This is then in all cases, including those of non-uniform external forcefulness fields. 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.^{[42] [43] [44] [45] [46] [47]} Considerations of kinetic theory or statistical mechanics as well support this argument.^{[48] [49] [fifty] [51] [52] [53] [54]}

In order that a organization may be in its own internal state of thermodynamic equilibrium, it is of course necessary, but non sufficient, that it be in its ain internal state of thermal equilibrium; it is possible for a arrangement to reach internal mechanical equilibrium before it reaches internal thermal equilibrium.^[55]

Number of existent variables needed for specification [edit]

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 ascertain united states that are the points of the manifold of equilibria.^[8] In the words of Prigogine and Defay (1945): "It is a matter of experience that when we accept specified a sure number of macroscopic properties of a system, then all the other properties are fixed."^{[56] [57]} 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 organisation. Every bit noted above, J.G. Kirkwood and I. Oppenheim indicate out that a country of thermodynamic equilibrium may exist divers by a special subclass of intensive variables, with a definite number of members in that subclass.

If the thermodynamic equilibrium lies in an external force field, information technology is merely the temperature that tin in full general be expected to be spatially uniform. Intensive variables other than temperature volition in full general exist non-uniform if the external force field is non-zero. In such a case, in general, additional variables are needed to describe the spatial non-uniformity.

Stability against pocket-sized perturbations [edit]

As noted higher up, J.R. Partington points out that a state of thermodynamic equilibrium is stable against small transient perturbations. Without this condition, in general, experiments intended to study systems in thermodynamic equilibrium are in severe difficulties.

Approach to thermodynamic equilibrium within an isolated system [edit]

When a body of material starts from a non-equilibrium land of inhomogeneity or chemical non-equilibrium, and is so isolated, information technology spontaneously evolves towards its own internal state of thermodynamic equilibrium. Information technology is not necessary that all aspects of internal thermodynamic equilibrium exist reached simultaneously; some can exist established earlier others. For example, in many cases of such development, internal mechanical equilibrium is established much more chop-chop than the other aspects of the eventual thermodynamic equilibrium.^[55] Another case is that, in many cases of such evolution, thermal equilibrium is reached much more rapidly than chemical equilibrium.^[58]

Fluctuations within an isolated system in its ain internal thermodynamic equilibrium [edit]

In an isolated arrangement, thermodynamic equilibrium by definition persists over an indefinitely long time. In classical physics it is oftentimes convenient to ignore the effects of measurement and this is assumed in the present business relationship.

To consider the notion of fluctuations in an isolated thermodynamic organisation, a convenient example is a system specified by its extensive land variables, internal energy, volume, and mass composition. By definition they are time-invariant. By definition, they combine with fourth dimension-invariant nominal values of their conjugate intensive functions of land, inverse temperature, pressure divided by temperature, and the chemical potentials divided by temperature, and so as to exactly obey the laws of thermodynamics.^[59] Only the laws of thermodynamics, combined with the values of the specifying extensive variables of state, are not sufficient to provide knowledge of those nominal values. Further information is needed, namely, of the constitutive properties of the system.

It may be admitted that on repeated measurement of those conjugate intensive functions of state, they are found to have slightly unlike values from time to fourth dimension. Such variability is regarded as due to internal fluctuations. The different measured values average to their nominal values.

If the system is truly macroscopic as postulated by classical thermodynamics, and then the fluctuations are too small to detect macroscopically. This is called the thermodynamic limit. In effect, the molecular nature of matter and the quantal nature of momentum transfer have vanished from sight, as well small to see. According to Buchdahl: "... there is no identify within the strictly phenomenological theory for the idea of fluctuations about equilibrium (see, however, Department 76)."^[60]

If the system is repeatedly subdivided, eventually a system is produced that is small enough to exhibit obvious fluctuations. This is a mesoscopic level of investigation. The fluctuations are and so directly dependent on the natures of the diverse walls of the system. The precise choice of independent land variables is so important. At this stage, statistical features of the laws of thermodynamics get apparent.

If the mesoscopic arrangement is further repeatedly divided, somewhen a microscopic system is produced. Then the molecular character of matter and the quantal nature of momentum transfer get of import in the processes of fluctuation. One has left the realm of classical or macroscopic thermodynamics, and one needs quantum statistical mechanics. The fluctuations can become relatively dominant, and questions of measurement become important.

The statement that 'the organisation is its own internal thermodynamic equilibrium' may be taken to mean that 'indefinitely many such measurements have been taken from time to time, with no trend in time in the various measured values'. Thus the statement, that 'a arrangement is in its own internal thermodynamic equilibrium, with stated nominal values of its functions of country conjugate to its specifying state variables', is far far more informative than a statement that 'a set of single simultaneous measurements of those functions of state have those same values'. This is considering the single measurements might have been made during a slight fluctuation, abroad from another set up of nominal values of those cohabit intensive functions of state, that is due to unknown and different constitutive properties. A single measurement cannot tell whether that might be so, unless there is as well noesis of the nominal values that belong to the equilibrium state.

Thermal equilibrium [edit]

An explicit distinction between 'thermal equilibrium' and 'thermodynamic equilibrium' is made past B. C. European union. He considers 2 systems in thermal contact, one a thermometer, the other a system in which there are several occurring irreversible processes, entailing non-cipher fluxes; the two systems are separated past a wall permeable only to heat. He considers the case in which, over the time calibration of involvement, it happens that both the thermometer reading and the irreversible processes are steady. Then at that place is thermal equilibrium without thermodynamic equilibrium. European union proposes consequently that the zeroth police force of thermodynamics can be considered to apply even when thermodynamic equilibrium is not nowadays; also he proposes that if changes are occurring then fast that a steady temperature cannot be defined, then "information technology is no longer possible to describe the process past means of a thermodynamic ceremonial. In other words, thermodynamics has no significant for such a procedure."^[61] This illustrates the importance for thermodynamics of the concept of temperature.

Thermal equilibrium is achieved when 2 systems in thermal contact with each other finish to have a internet exchange of energy. It follows that if two systems are in thermal equilibrium, then their temperatures are the same.^[62]

Thermal equilibrium occurs when a system'south macroscopic thermal observables have ceased to change with time. For example, an ideal gas whose distribution role has stabilised to a specific Maxwell–Boltzmann distribution would be in thermal equilibrium. This outcome allows a single temperature and force per unit area to be attributed to the whole system. For an isolated trunk, information technology is quite possible for mechanical equilibrium to be reached earlier thermal equilibrium is reached, just eventually, all aspects of equilibrium, including thermal equilibrium, are necessary for thermodynamic equilibrium.^[63]

Non-equilibrium [edit]

A system's internal state of thermodynamic equilibrium should exist distinguished from a "stationary country" in which thermodynamic parameters are unchanging in fourth dimension but the system is not isolated, then that there are, into and out of the system, not-zilch macroscopic fluxes which are abiding in time.^[64]

Non-equilibrium thermodynamics is a co-operative of thermodynamics that deals with systems that are not in thermodynamic equilibrium. Most systems establish in nature are not in thermodynamic equilibrium considering they are changing or tin can exist triggered to change over time, and are continuously and discontinuously subject to flux of matter and energy to and from other systems. The thermodynamic written report of non-equilibrium systems requires more general concepts than are dealt with past equilibrium thermodynamics.^[65] Many natural systems yet today remain beyond the scope of currently known macroscopic thermodynamic methods.

Laws governing systems which are far from equilibrium are also debatable. One of the guiding principles for these systems is the maximum entropy product principle.^{[66] [67]} It states that a non-equilibrium system evolves such as to maximize its entropy product.^{[68] [69]}

See as well [edit]

Thermodynamic models

• Non-random 2-liquid model (NRTL model) - Phase equilibrium calculations
• UNIQUAC model - Phase equilibrium calculations
• Time crystal

Topics in control theory

• Steady state
• Transient state
• Coefficient diagram method
• Command reconfiguration
• Feedback
• H infinity
• Hankel singular value
• Krener's theorem
• Atomic number 82-lag compensator
• Minor loop feedback
• Multi-loop feedback
• Positive systems
• Radial footing role
• Root locus
• Betoken-flow graphs
• Stable polynomial
• State space representation
• Underactuation
• Youla–Kucera parametrization
• Markov concatenation approximation method

Other related topics

• Automation and remote control
• Bond graph
• Control technology
• Control–feedback–abort loop
• Controller (control theory)
• Cybernetics
• Intelligent control
• Mathematical system theory
• Negative feedback amplifier
• People in systems and command
• Perceptual control theory
• Systems theory
• Time scale calculus

General references [edit]

• Cesare Barbieri (2007) Fundamentals of Astronomy. First Edition (QB43.iii.B37 2006) CRC Press ISBN 0-7503-0886-9, ISBN 978-0-7503-0886-i
• Hans R. Griem (2005) Principles of Plasma Spectroscopy (Cambridge Monographs on Plasma Physics), Cambridge University Press, New York ISBN 0-521-61941-half-dozen
• C. Michael Hogan, Leda C. Patmore and Harry Seidman (1973) Statistical Prediction of Dynamic Thermal Equilibrium Temperatures using Standard Meteorological Data Bases, Second Edition (EPA-660/2-73-003 2006) United States Environmental Protection Agency Office of Enquiry and Evolution, Washington, D.C. [1]
• F. Mandl (1988) Statistical Physics, Second Edition, John Wiley & Sons

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Cited bibliography [edit]

• Adkins, C.J. (1968/1983). Equilibrium Thermodynamics, third edition, McGraw-Hill, London, ISBN 0-521-25445-0.
• Bailyn, Thou. (1994). A Survey of Thermodynamics, American Plant of Physics Press, New York, ISBN 0-88318-797-3.
• Beattie, J.A., Oppenheim, I. (1979). Principles of Thermodynamics, Elsevier Scientific Publishing, Amsterdam, ISBN 0-444-41806-7.
• Boltzmann, Fifty. (1896/1964). Lectures on Gas Theory, translated by Due south.G. Castor, University of California Press, Berkeley.
• Buchdahl, H.A. (1966). The Concepts of Classical Thermodynamics, Cambridge University Press, Cambridge UK.
• Callen, H.B. (1960/1985). Thermodynamics and an Introduction to Thermostatistics, (1st edition 1960) 2d edition 1985, Wiley, New York, ISBN 0-471-86256-8.
• Carathéodory, C. (1909). Untersuchungen über dice Grundlagen der Thermodynamik, Mathematische Annalen, 67: 355–386. A translation may be institute here. Also a mostly reliable translation is to be plant at Kestin, J. (1976). The 2nd Law of Thermodynamics, Dowden, Hutchinson & Ross, Stroudsburg PA.
• Chapman, S., Cowling, T.Chiliad. (1939/1970). The Mathematical Theory of Non-compatible gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, tertiary edition 1970, Cambridge Academy Press, London.
• Crawford, F.H. (1963). Heat, Thermodynamics, and Statistical Physics, Rupert Hart-Davis, London, Harcourt, Caryatid & Earth, Inc.
• de Groot, S.R., Mazur, P. (1962). Non-equilibrium Thermodynamics, North-Kingdom of the netherlands, Amsterdam. Reprinted (1984), Dover Publications Inc., New York, ISBN 0486647412.
• Denbigh, K.Chiliad. (1951). Thermodynamics of the Steady Land, Methuen, London.
• Eu, B.C. (2002). Generalized Thermodynamics. The Thermodynamics of Irreversible Processes and Generalized Hydrodynamics, Kluwer Bookish Publishers, Dordrecht, ISBN i-4020-0788-4.
• Fitts, D.D. (1962). Nonequilibrium thermodynamics. A Phenomenological Theory of Irreversible Processes in Fluid Systems, McGraw-Hill, New York.
• Gibbs, J.West. (1876/1878). On the equilibrium of heterogeneous substances, Trans. Conn. Acad., iii: 108–248, 343–524, reprinted in The Nerveless 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 ane, pp. 55–353.
• Griem, H.R. (2005). Principles of Plasma Spectroscopy (Cambridge Monographs on Plasma Physics), Cambridge University Press, New York ISBN 0-521-61941-6.
• Guggenheim, E.A. (1949/1967). Thermodynamics. An Advanced Handling for Chemists and Physicists, fifth revised edition, North-Holland, Amsterdam.
• Haase, R. (1971). Survey of Fundamental Laws, chapter 1 of Thermodynamics, pages 1–97 of volume i, ed. West. Jost, of Concrete Chemical science. An Advanced Treatise, ed. H. Eyring, D. Henderson, W. Jost, Academic Press, New York, lcn 73–117081.
• Kirkwood, J.G., Oppenheim, I. (1961). Chemical Thermodynamics, McGraw-Loma Book Company, New York.
• Landsberg, P.T. (1961). Thermodynamics with Quantum Statistical Illustrations, Interscience, New York.
• Lieb, Eastward. H.; Yngvason, J. (1999). "The Physics and Mathematics of the Second Law of Thermodynamics". Phys. Rep. 310 (ane): 1–96. arXiv:cond-mat/9708200. Bibcode:1999PhR...310....1L. doi:x.1016/S0370-1573(98)00082-9. S2CID 119620408.
• Levine, I.Due north. (1983), Physical Chemistry, second edition, McGraw-Colina, New York, ISBN 978-0072538625.
• Maxwell, J.C. (1867). "On the dynamical theory of gases". Phil. Trans. Roy. Soc. London. 157: 49–88.
• Morse, P.M. (1969). Thermal Physics, second edition, Due west.A. Benjamin, Inc, New York.
• Münster, A. (1970). Classical Thermodynamics, translated by E.S. Halberstadt, Wiley–Interscience, London.
• Partington, J.R. (1949). An Avant-garde Treatise on Physical Chemical science, book one, Fundamental Principles. The Properties of Gases, Longmans, Green and Co., London.
• Pippard, A.B. (1957/1966). The Elements of Classical Thermodynamics, reprinted with corrections 1966, Cambridge University Press, London.
• Planck. M. (1914). The Theory of Oestrus Radiation, a translation by Masius, Chiliad. of the 2nd German edition, P. Blakiston's Son & Co., Philadelphia.
• Prigogine, I. (1947). Étude Thermodynamique des Phénomènes irréversibles, Dunod, Paris, and Desoers, Liège.
• Prigogine, I., Defay, R. (1950/1954). Chemical Thermodynamics, Longmans, Greenish & Co, London.
• Silbey, R.J., Alberty, R.A., Bawendi, M.G. (1955/2005). Concrete Chemical science, quaternary edition, Wiley, Hoboken NJ.
• ter Haar, D., Wergeland, H. (1966). Elements of Thermodynamics, Addison-Wesley Publishing, Reading MA.
• Thomson, Due west. (March 1851). "On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and One thousand. Regnault'due south Observations on Steam". Transactions of the Majestic Society of Edinburgh. 20 (part Ii): 261–268, 289–298. Too published in Thomson, Due west. (December 1852). "On the Dynamical Theory of Oestrus, with numerical results deduced from Mr Joule'southward equivalent of a Thermal Unit, and Thousand. Regnault's Observations on Steam". Phil. Mag. 4. IV (22): 8–21. Retrieved 25 June 2012.
• Tisza, L. (1966). Generalized Thermodynamics, M.I.T Press, Cambridge MA.
• Uhlenbeck, G.Eastward., Ford, G.W. (1963). Lectures in Statistical Mechanics, American Mathematical Social club, Providence RI.
• Waldram, J.R. (1985). The Theory of Thermodynamics, Cambridge University Press, Cambridge Great britain, ISBN 0-521-24575-3.
• Zemansky, M. (1937/1968). Heat and Thermodynamics. An Intermediate Textbook, fifth edition 1967, McGraw–Colina Book Visitor, New York.

External links [edit]

• Breakdown of Local Thermodynamic Equilibrium George W. Collins, The Fundamentals of Stellar Astrophysics, Affiliate xv
• Thermodynamic Equilibrium, Local and otherwise lecture by Michael Richmond
• Not-Local Thermodynamic Equilibrium in Cloudy Planetary Atmospheres Paper past R. Due east. Samueison quantifying the effects due to non-LTE in an atmosphere
• Local Thermodynamic Equilibrium

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Source: https://en.wikipedia.org/wiki/Thermodynamic_equilibrium

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