Heat

About|a mode of transfer of energy, from a hotter to a colder body|other uses pp-move|small=yes Use dmy dates|date=August 2013

In physics, heat is energy as it spontaneously passes between a system and its surroundings, other than as Work (thermodynamics)|work or with the transfer of matter. In thermodynamics, finer detail of the process of transfer is in principle unspecified. When there is a suitable physical pathway, heat transfer occurs from a hotter to a colder body.<ref>Max Born|Born, M. (1949), p. 31.</ref><ref>Brian Pippard|Pippard, A.B. (1957/1966), p. 16.</ref><ref>Lev Landau|Landau, L., Evgeny Lifshitz|Lifshitz, E.M. (1958/1969), p. 43</ref><ref>Herbert Callen|Callen, H.B. (1960/1985), pp. 18–19.</ref><ref>Reif, F. (1965), pp. 67, 73.</ref><ref>Bailyn, M. (1994), p. 82.</ref> The transfer can be direct, by contact, from the source to the destination body, as in Heat conduction|conduction combined with Thermal radiation|radiation, or remote, as in radiation without conduction; or indirect, by conduction and radiation through a thick solid wall, or by way of an intermediate fluid body, as in Convective heat transfer|convective circulation.<ref>harvnb|Guggenheim, E.A. (1949/1967)|page=8</ref><ref>Planck. M. (1914)</ref><ref name="Chandrasekhar 1961"/>

Originally, quantity of heat transferred was measured by how much it changed the states of participating bodies, for example, as amount of ice melted, or change in temperature, without work or matter transfer.<ref>James Clerk Maxwell|Maxwell, J.C. (1871), Chapter III.</ref> Such measurement is possible because many bodies, over most temperature ranges, expand reversibly on being heated. This is called thermal expansion.

Historically, when the First law of thermodynamics|first and Second law of thermodynamics|second laws of thermodynamics had been established, it came to be regarded by physicists as more rational to define quantity of heat transferred in terms of equivalent work. Thus, for the sake of logical development, the concept of temperature was reserved for definition in terms of the second law, segregated from the statement of the first law.

Kinetic theory of gases|Kinetic theory explains heat as a Macroscopic scale|macroscopic manifestation of the motions and interactions of microscopic constituents such as molecules and photons.

In calorimetry, sensible heat is defined with respect to a specific chosen Thermodynamics|state variable of the system, such as pressure or volume. Sensible heat transfer causes change of temperature of the system while leaving the chosen state variable unchanged. Heat transfer that occurs with the system at constant temperature and that does change that particular state variable is called latent heat with respect to that variable. For infinitesimal changes, the total incremental heat transfer is then the sum of the latent and sensible heat increments. This is a basic paradigm for thermodynamics, and was important in the historical development of the subject.

The quantity of energy transferred as heat is a Scalar (physics)|scalar expressed in an energy unit such as the joule (J) (International System of Units|SI), with a sign that is customarily positive when a transfer adds to the energy of a system.

File:171879main LimbFlareJan12 lg.jpg|300px|thumb|right|The Sun and Earth form an ongoing example of a heating process. Some of the Sun's thermal radiation strikes and heats the Earth. Compared to the Sun, Earth has a much lower temperature and so sends far less thermal radiation back to the Sun. The heat of this process can be quantified by the net amount, and direction (Sun to Earth), of energy it transferred in a given period of time.

History

Physicist James Clerk Maxwell, in his 1871 classic Theory of Heat, was one of many who began to build on the already established idea that heat has something to do with matter in motion. This was the same idea put forth by Benjamin Thompson in 1798, who said he was only following up on the work of many others. One of Maxwell's recommended books was Heat as a Mode of Motion, by John Tyndall. Maxwell outlined four stipulations for the definition of heat:

• It is something which may be transferred from one body to another, according to the second law of thermodynamics.
• It is a measurable quantity, and so can be treated mathematically.
• It cannot be treated as a material substance, because it may be transformed into something that is not a material substance, e.g., mechanical work.
• Heat is one of the forms of energy.<ref>James Clerk Maxwell|Maxwell, J.C. (1871), p. 7.</ref>

From empirically based ideas of heat, and from other empirical observations, the notions of internal energy and of entropy can be derived, so as to lead to the recognition of the first and second laws of thermodynamics.<ref>Planck, M. (1903).</ref> This was the way of the historical pioneers of thermodynamics.<ref>Partington, J.R. (1949).</ref><ref>Clifford Truesdell|Truesdell, C. (1980), page 15.</ref>

Transfers of energy as heat between two bodies

Referring to conduction, J.R. Partington|Partington writes: "If a hot body is brought in conducting contact with a cold body, the temperature of the hot body falls and that of the cold body rises, and it is said that a quantity of heat has passed from the hot body to the cold body."<ref>J.R. Partington|Partington, J.R. (1949), p. 118.</ref>

Referring to radiation, James Clerk Maxwell|Maxwell writes: "In Radiation, the hotter body loses heat, and the colder body receives heat by means of a process occurring in some intervening medium which does not itself thereby become hot."<ref>James Clerk Maxwell|Maxwell, J.C. (1871), p. 10.</ref>

Maxwell writes that convection as such "is not a purely thermal phenomenon".<ref>James Clerk Maxwell|Maxwell, J.C. (1871), p. 11.</ref> In thermodynamics, convection in general is regarded as transport of internal energy. If, however, the convection is enclosed and circulatory, then it may be regarded as an intermediary that transfers energy as heat between source and destination bodies, because it transfers only energy and not matter from the source to the destination body.<ref name="Chandrasekhar 1961">Subrahamanyan Chandrasekhar|Chandrasekhar, S. (1961).</ref>

Practical operating devices that harness transfers of energy as heat

In accordance with the first law for closed systems, energy transferred solely as heat enters one body and leaves another, changing the internal energies of each. Transfer, between bodies, of energy as work is a complementary way of changing internal energies. Though it is not logically rigorous from the viewpoint of strict physical concepts, a common form of words that expresses this is to say that heat and work are interconvertible.

Cyclically operating engines, that use only heat and work transfers, have two thermal reservoirs, a hot and a cold one. They may be classified by the range of operating temperatures of the working body, relative to those reservoirs. In a heat engine, the working body is at all times colder than the hot reservoir and hotter than the cold reservoir. In a sense, it uses heat transfer to produce work. In a heat pump, the working body, at stages of the cycle, goes both hotter than the hot reservoir, and colder than the cold reservoir. In a sense, it uses work to produce heat transfer.

Heat engine

In classical thermodynamics, a commonly considered model is the heat engine. It consists of four bodies: the working body, the hot reservoir, the cold reservoir, and the work reservoir. A cyclic process leaves the working body in an unchanged state, and is envisaged as being repeated indefinitely often. Work transfers between the working body and the work reservoir are envisaged as reversible, and thus only one work reservoir is needed. But two thermal reservoirs are needed, because transfer of energy as heat is irreversible. A single cycle sees energy taken by the working body from the hot reservoir and sent to the two other reservoirs, the work reservoir and the cold reservoir. The hot reservoir always and only supplies energy and the cold reservoir always and only receives energy. The second law of thermodynamics requires that no cycle can occur in which no energy is received by the cold reservoir. Heat engines achieve higher efficiency when the difference between initial and final temperature is greater.

Heat pump or refrigerator

Another commonly considered model is the heat pump or refrigerator. Again there are four bodies: the working body, the hot reservoir, the cold reservoir, and the work reservoir. A single cycle starts with the working body colder than the cold reservoir, and then energy is taken in as heat by the working body from the cold reservoir. Then the work reservoir does work on the working body, adding more to its internal energy, making it hotter than the hot reservoir. The hot working body passes heat to the hot reservoir, but still remains hotter than the cold reservoir. Then, by allowing it to expand without doing work on another body and without passing heat to another body, the working body is made colder than the cold reservoir. It can now accept heat transfer from the cold reservoir to start another cycle.

The device has transported energy from a colder to a hotter reservoir, but this is not regarded as being by an inanimate agency; rather, it is regarded as by the harnessing of work . This is because work is supplied from the work reservoir, not just by a simple thermodynamic process, but by a cycle of thermodynamic operations and processes, which may be regarded as directed by an animate or harnessing agency. Accordingly, the cycle is still in accord with the second law of thermodynamics. The efficiency of a heat pump is best when the temperature difference between the hot and cold reservoirs is least.

Functionally, such engines are used in two ways, distinguishing a target reservoir and a resource or surrounding reservoir. A heat pump transfers heat, to the hot reservoir as the target, from the resource or surrounding reservoir. A refrigerator transfers heat, from the cold reservoir as the target, to the resource or surrounding reservoir. The target reservoir may be regarded as leaking: when the target leaks hotness to the surroundings, heat pumping is used; when the target leaks coldness to the surroundings, refrigeration is used. The engines harness work to overcome the leaks.

Macroscopic view of quantity of energy transferred as heat

According to Planck, there are three main conceptual approaches to heat. One is the microscopic or kinetic theory approach. Also there are two macroscopic approaches. One is the approach through the law of conservation of energy taken as prior to thermodynamics, with a mechanical analysis of processes, for example in the work of Helmholtz. This mechanical view is taken as currently customary in this article. The other macroscopic approach is the thermodynamic one, which admits heat as a primitive concept, which contributes, by scientific induction<ref>Jaakko Hintikka|Hintikka, J. (1988), p. 180.</ref> to knowledge of the law of conservation of energy.<ref>Max Planck|Planck, M. (1897/1903), p. viii.</ref>

Bailyn also distinguishes the two macroscopic approaches as the mechanical and the thermodynamic.<ref>Bailyn, M. (1994), pp. 65, 79.</ref> The thermodynamic view was taken by the founders of thermodynamics in the nineteenth century. It regards quantity of energy transferred as heat as a primitive concept coherent with a primitive concept of temperature, measured primarily by calorimetry. A calorimeter is a body in the surroundings of the system, with its own temperature and internal energy; when it is connected to the system by a path for heat transfer, changes in it measure heat transfer. The mechanical view was pioneered by Helmholtz and developed and used in the twentieth century, largely through the influence of Max Born.<ref>Max Born|Born, M.(1949), Lecture math|V.</ref> It regards quantity of heat transferred as heat as a derived concept, defined for closed systems as quantity of heat transferred by mechanisms other than work transfer, the latter being regarded as primitive for thermodynamics, defined by macroscopic mechanics. According to Born, the transfer of internal energy between open systems that accompanies transfer of matter "cannot be reduced to mechanics".<ref>Max Born|Born, M. (1949), p. 44.</ref> It follows that there is no well-founded definition of quantities of energy transferred as heat or as work associated with transfer of matter.

Nevertheless, for the thermodynamical description of non-equilibrium processes, it is desired to consider the effect of a temperature gradient established by the surroundings across the system of interest when there is no physical barrier or wall between system and surroundings, that is to say, when they are open with respect to one another. The impossibility of a mechanical definition in terms of work for this circumstance does not alter the physical fact that a temperature gradient causes a diffusive flux of internal energy, a process that, in the thermodynamic view, might be proposed as a candidate concept for transfer of energy as heat.

In this circumstance, it may be expected that there may also be active other drivers of diffusive flux of internal energy, such as gradient of chemical potential which drives transfer of matter, and gradient of electric potential which drives electric current and iontophoresis; such effects usually interact with diffusive flux of internal energy driven by temperature gradient, and such interactions are known as cross-effects.<ref>De Groot, S.R., Mazur, P. (1962), p. 30.</ref>

If cross-effects that result in diffusive transfer of internal energy were also labeled as heat transfers, they would sometimes violate the rule that pure heat transfer occurs only down a temperature gradient, never up one. They would also contradict the principle that all heat transfer is of one and the same kind, a principle founded on the idea of heat conduction between closed systems. One might to try to think narrowly of heat flux driven purely by temperature gradient as a conceptual component of diffusive internal energy flux, in the thermodynamic view, the concept resting specifically on careful calculations based on detailed knowledge of the processes and being indirectly assessed. In these circumstances, if perchance it happens that no transfer of matter is actualized, and there are no cross-effects, then the thermodynamic concept and the mechanical concept coincide, as if one were dealing with closed systems. But when there is transfer of matter, the exact laws by which temperature gradient drives diffusive flux of internal energy, rather than being exactly knowable, mostly need to be assumed, and in many cases are practically unverifiable. Consequently, when there is transfer of matter, the calculation of the pure 'heat flux' component of the diffusive flux of internal energy rests on practically unverifiable assumptions.<ref>Denbigh, K.G. (1951), p. 56.</ref><ref group=quotations>Denbigh states in a footnote that he is indebted to correspondence with Edward A. Guggenheim|Professor E.A. Guggenheim and with Professor N.K. Adam. From this, Denbigh concludes "It seems, however, that when a system is able to exchange both heat and matter with its environment, it is impossible to make an unambiguous distinction between energy transported as heat and by the migration of matter, without already assuming the existence of the 'heat of transport'." Denbigh K.G. (1951), p. 56.</ref><ref>Fitts, D.D. (1962), p. 28.</ref> This is a reason to think of heat as a specialized concept that relates primarily and precisely to closed systems, and applicable only in a very restricted way to open systems.

In many writings in this context, the term "heat flux" is used when what is meant is therefore more accurately called diffusive flux of internal energy; such usage of the term "heat flux" is a residue of older and now obsolete language usage that allowed that a body may have a "heat content".<ref>Gyarmati, I. (1970), p. 68.</ref>

Microscopic view of heat

In the kinetic theory of gases|kinetic theory, heat is explained in terms of the microscopic motions and interactions of constituent particles, such as electrons, atoms, and molecules.<ref name="Kittel & Kroemer">Kittel, C. Kroemer, H. (1980).</ref> Heat transfer arises from temperature gradients or differences, through the diffuse exchange of microscopic kinetic and potential particle energy, by particle collisions and other interactions. An early and vague expression of this was made by Francis Bacon.<ref>Bacon, F. (1620).</ref><ref>Partington, J.R. (1949), page 131.</ref> Precise and detailed versions of it were developed in the nineteenth century.<ref>Partington, J.R. (1949), pages 132–136.</ref>

In statistical mechanics, for a closed system (no transfer of matter), heat is the energy transfer associated with a disordered, microscopic action on the system, associated with jumps in occupation numbers of the energy levels of the system, without change in the values of the energy levels themselves.<ref>Reif (1965), pp.67-68</ref> It is possible for macroscopic thermodynamic work to alter the occupation numbers without change in the values of the system energy levels themselves, but what distinguishes transfer as heat is that the transfer is entirely due to disordered, microscopic action, including radiative transfer. A Microstate (statistical mechanics)#Microscopic definitions of thermodynamic concepts|mathematical definition can be formulated for small increments of quasi-static adiabatic work in terms of the statistical distribution of an ensemble of microstates.

Notation and units

As a form of energy heat has the unit joule (J) in the International System of Units (SI). However, in many applied fields in engineering the British thermal unit (BTU) and the calorie are often used. The standard unit for the rate of heat transferred is the watt (W), defined as joules per second.

The total amount of energy transferred as heat is conventionally written as Q (from Quantity) for algebraic purposes.<ref>Jensen, W.B. (2010).</ref> Heat released by a system into its surroundings is by convention a negative quantity (Q < 0); when a system absorbs heat from its surroundings, it is positive (Q > 0). Heat transfer rate, or heat flow per unit time, is denoted by <math>\dot{Q}</math>. This should not be confused with a time derivative of a Functions of state|function of state (which can also be written with the dot notation) since heat is not a function of state.<ref name="Baierlein1999">Baierlein, R. (1999), p. 21.</ref> Heat flux is defined as rate of heat transfer per unit cross-sectional area, resulting in the unit watts per square metre.

Estimation of quantity of heat

Quantity of heat transferred can measured by calorimetry, or determined through calculations based on other quantities.

Calorimetry is the empirical basis of the idea of quantity of heat transferred in a process. The transferred heat is measured by changes in a body of known properties, for example, temperature rise, change in volume or length, or phase change, such as melting of ice.<ref>Maxwell J.C. (1872), p. 54.</ref><ref>Planck (1927), Chapter 3.</ref>

A calculation of quantity of heat transferred can rely on a hypothetical quantity of energy transferred as Adiabatic process|adiabatic work and on the first law of thermodynamics. Such calculation is the primary approach of many theoretical studies of quantity of heat transferred.<ref name="Carathéodory 1909">Constantin Carathéodory|Carathéodory, C. (1909).</ref><ref name="Bryan 47">Bryan, G.H. (1907), p. 47.</ref><ref>Herbert Callen|Callen, H.B. (1985), Section 1-8.</ref>

Internal energy and enthalpy

For a closed system (a system from which no matter can enter or exit), one version of the first law of thermodynamics states that the change in internal energy math|ΔU of the system is equal to the amount of heat math|Q supplied to the system minus the amount of work (thermodynamics)|work math|W done by system on its surroundings. The foregoing sign convention for work is used in the present article, but an alternate sign convention, followed by IUPAC, for work, is to consider the work performed on the system by its surroundings as positive. This is the convention adopted by many modern textbooks of physical chemistry, such as those by Peter Atkins and Ira Levine, but many textbooks on physics define work as work done by the system.

<math>\Delta U = Q - W \, . </math>

This formula can be re-written so as to express a definition of quantity of energy transferred as heat, based purely on the concept of adiabatic work, if it is supposed that math|ΔU is defined and measured solely by processes of adiabatic work:

<math> Q = \Delta U + W.</math>

The work done by the system includes boundary work (when the system increases its volume against an external force, such as that exerted by a piston) and other work (e.g. shaft work performed by a compressor fan), which is called isochoric work:

<math> Q = \Delta U + W_\text{boundary} + W_\text{isochoric}.</math>

In this Section we will neglect the "other-" or isochoric work contribution.

The internal energy, math|U, is a state function. In cyclical processes, such as the operation of a heat engine, state functions of the working substance return to their initial values upon completion of a cycle.

The differential, or infinitesimal increment, for the internal energy in an infinitesimal process is an exact differential math|dU. The symbol for exact differentials is the lowercase letter math|d.

In contrast, neither of the infinitesimal increments math|δQ nor math|δW in an infinitesimal process represents the state of the system. Thus, infinitesimal increments of heat and work are inexact differentials. The lowercase Greek letter delta, math|δ, is the symbol for inexact differentials. The integral of any inexact differential over the time it takes for a system to leave and return to the same thermodynamic state does not necessarily equal zero.

As recounted below, in the section headed #Entropy|Entropy, the second law of thermodynamics observes that if heat is supplied to a system in which no irreversible processes take place and which has a well-defined temperature math|T, the increment of heat math|δQ and the temperature math|T form the exact differential

<math> \mathrm{d}S =\frac{\delta Q}{T},</math>

and that math|S, the entropy of the working body, is a function of state. Likewise, with a well-defined pressure, math|P, behind the moving boundary, the work differential, math|δW, and the pressure, math|P, combine to form the exact differential

<math> \mathrm{d}V =\frac{\delta W}{P},</math>

with math|V the volume of the system, which is a state variable. In general, for homogeneous systems,

<math>\mathrm{d}U = T\mathrm{d}S - P\mathrm{d}V.</math>

Associated with this differential equation is that the internal energy may be considered to be a function math|U (S,V) of its Thermodynamic potential|natural variables math|S and math|V. The internal energy representation of the fundamental thermodynamic relation is written

<math>U=U(S,V).</math><ref>Herbert Callen|Callen, H.B., (1985), Section 2-3, pp. 40–42.</ref><ref name="Adkins 1983 101">Adkins, C.J. (1983), p. 101.</ref>

If math|V is constant

<math>T\mathrm{d}S=\mathrm{d}U\,\,\,\,\,\,\,\,\,\,\,\,(V\,\, \text{constant)}</math>

and if math|P is constant

<math>T\mathrm{d}S=\mathrm{d}H\,\,\,\,\,\,\,\,\,\,\,\,(P\,\, \text{constant)}</math>

with math|H the enthalpy defined by

<math>H=U+PV.</math>

The enthalpy may be considered to be a function math|H (S,P) of its natural variables math|S and math|P. The enthalpy representation of the fundamental thermodynamic relation is written

<math>H=H(S,P).</math><ref name="Adkins 1983 101"/><ref name="Callen 147">Herbert Callen|Callen, H.B. (1985), p. 147.</ref>

The internal energy representation and the enthalpy representation are Legendre transformation|partial Legendre transforms of one another. They contain the same physical information, written in different ways. Like the internal energy, the enthalpy stated as a function of its natural variables is a thermodynamic potential and contains all thermodynamic information about a body.<ref name="Callen 147"/><ref>Adkins, C.J. (1983), pp. 100–104.</ref>

Heat added to a body at constant pressure

If a quantity math|Q of heat is added to a body while it does expansion work math|W on its surroundings, one has

<math>\Delta H=\Delta U + \Delta(PV)\,.</math>

If this is constrained to happen at constant pressure with math|ΔP = 0, the expansion work math|W done by the body is given by math|W = P ΔV; recalling the first law of thermodynamics, one has

<math>\Delta U=Q - W=Q - P\, \Delta V \text{ and }\Delta (PV) = P\, \Delta V \,.</math>

Consequently, by substitution one has

<math>\Delta H=Q - P\, \Delta V + P\, \Delta V</math>

<math>=Q\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text{at constant pressure.}</math>

In this scenario, the increase in enthalpy is equal to the quantity of heat added to the system. Since many processes do take place at constant pressure, or approximately at atmospheric pressure, the enthalpy is therefore sometimes given the misleading name of 'heat content'.<ref>Adkins, C.J. (1968/1983), p. 46.</ref> It is sometimes also called the heat function.<ref>Bailyn, M. (1994), p. 208.</ref>

In terms of the natural variables math|S and P of the state function math|H, this process of change of state from state 1 to state 2 can be expressed as

<math>\Delta H=\int_{S_1}^{S_2} \left(\frac{\partial H}{\partial S}\right)_P \mathrm dS +\int_{P_1}^{P_2} \left(\frac{\partial H}{\partial P}\right)_S \mathrm dP</math>

<math>=\int_{S_1}^{S_2} \left(\frac{\partial H}{\partial S}\right)_P \mathrm dS\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text{at constant pressure.}</math>

It is known that the temperature math|T(S, P) is identically stated by

<math>\left(\frac{\partial H}{\partial S}\right)_P \equiv T(S,P)\,.</math>

Consequently,

<math>\Delta H=\int_{S_1}^{S_2} T(S,P) \mathrm dS\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text{at constant pressure.}</math>

In this case, the integral specifies a quantity of heat transferred at constant pressure.

Entropy

main|Entropy File:Clausius-1.jpg|thumb|120px|Rudolf Clausius In 1856, German physicist Rudolf Clausius, referring to closed systems, in which transfers of matter do not occur, defined the second fundamental theorem (the second law of thermodynamics) in the mechanical theory of heat (thermodynamics): "if two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generations of the quantity of heat Q from work (thermodynamics)|work at the temperature T, has the equivalence-value:"<ref>Rudolf Clausius|Clausius, R. (1854).</ref><ref>Rudolf Clausius|Clausius, R. (1865), pp. 125–126.</ref>

<math> {} \frac {Q}{T}.</math>

In 1865, he came to define the entropy symbolized by S, such that, due to the supply of the amount of heat Q at temperature T the entropy of the system is increased by

<math> \Delta S = \frac {Q}{T}\,\,\,\,\,\,\,\,\,\,\,\,\,(1)</math>

In a transfer of energy as heat without work being done, there are changes of entropy in both the surroundings which lose heat and the system which gains it. The increase, math|ΔS, of entropy in the system may be considered to consist of two parts, an increment, math|ΔS′ that matches, or 'compensates', the change, math|−ΔS′, of entropy in the surroundings, and a further increment, math|ΔS′′ that may be considered to be 'generated' or 'produced' in the system, and is said therefore to be 'uncompensated'. Thus

<math> \Delta S = \Delta S^\prime+\Delta S^{\prime\prime} .</math>

This may also be written

<math> \Delta S_{\mathrm{system = \Delta S_{\mathrm{compensated+\Delta S_{\mathrm{uncompensated\,\,\,\,\text{with}\,\,\,\,\Delta S_{\mathrm{compensated=-\Delta S_{\mathrm{surroundings.</math>

The total change of entropy in the system and surroundings is thus

<math> \Delta S_{\mathrm{overall = \Delta S^\prime+\Delta S^{\prime\prime}-\Delta S^\prime=\Delta S^{\prime\prime} .</math>

This may also be written

<math> \Delta S_{\mathrm{overall = \Delta S_{\mathrm{compensated+\Delta S_{\mathrm{uncompensated+\Delta S_{\mathrm{surroundings=\Delta S_{\mathrm{uncompensated .</math>

It is then said that an amount of entropy math|ΔS′ has been transferred from the surroundings to the system. Because entropy is not a conserved quantity, this is an exception to the general way of speaking, in which an amount transferred is of a conserved quantity.

The second law of thermodynamics observes that in a natural transfer of energy as heat, in which the temperature of the system is different from that of the surroundings, it is always so that

<math> \Delta S_{\mathrm{overall>0 .</math>

For purposes of mathematical analysis of transfers, one thinks of fictive processes that are called 'reversible', with the temperature math|T of the system being hardly less than that of the surroundings, and the transfer taking place at an imperceptibly slow speed.

Following the definition above in formula (1), for such a fictive 'reversible' process, a quantity of transferred heat math|δQ (an inexact differential) is analyzed as a quantity math|T dS, with math|dS (an exact differential):

<math> T\,\mathrm{d}S=\delta Q .</math>

This equality is only valid for a fictive transfer in which there is no production of entropy, that is to say, in which there is no uncompensated entropy.

If, in contrast, the process is natural, and can really occur, with irreversibility, then there is entropy production, with math|dS_{uncompensated} > 0. The quantity math|T dS_{uncompensated} was termed by Clausius the "uncompensated heat", though that does not accord with present-day terminology. Then one has

<math> T\,\mathrm{d}S=\delta Q+T\,\mathrm{d}S_{\mathrm{uncompensated > \delta Q .</math>

This leads to the statement

<math> T\,\mathrm{d}S \geq \delta Q \quad{\rm{(second\,\,law)\,.</math>

which is the second law of thermodynamics for closed systems.

In non-equilibrium thermodynamics that approximates by assuming the hypothesis of local thermodynamic equilibrium, there is a special notation for this. The transfer of energy as heat is assumed to take place across an infinitesimal temperature difference, so that the system element and its surroundings have near enough the same temperature math|T. Then one writes

<math> \mathrm{d}S =\mathrm{d}S_{\mathrm e}+\mathrm{d}S_{\mathrm i}\,,</math>

where by definition

<math> \delta Q = T\,\mathrm{d}S_{\mathrm e}\,\,\,\,\,\text{and}\,\,\,\,\,\mathrm{d}S_{\mathrm i}\equiv\mathrm{d}S_{\mathrm {uncompensated.</math>

The second law for a natural process asserts that

<math> \mathrm d S_{\mathrm i}>0 .</math><ref>De Groot, S.R., Peter Mazur|Mazur, P. (1962), p. 20.</ref><ref>Kondepudi, D, Ilya Prigogine|Prigogine, I. (1998), p. 82.</ref><ref>Kondepudi, D. (2008), p. 114.</ref><ref>Lebon, g., Jou, D., Casas-Vásquez, J. (2008), p. 41.</ref>

Latent and sensible heat

File:Black Joseph.jpg|thumb||120px|Joseph Black In an 1847 lecture entitled On Matter, Living Force, and Heat, James Prescott Joule characterized the terms latent heat and sensible heat as components of heat each affecting distinct physical phenomena, namely the potential and kinetic energy of particles, respectively.<ref>James Prescott Joule|Joule J.P. (1884).</ref><ref group=quotations>"Heat must therefore consist of either living force or of attraction through space. In the former case we can conceive the constituent particles of heated bodies to be, either in whole or in part, in a state of motion. In the latter we may suppose the particles to be removed by the process of heating, so as to exert attraction through greater space. I am inclined to believe that both of these hypotheses will be found to hold good,—that in some instances, particularly in the case of sensible heat, or such as is indicated by the thermometer, heat will be found to consist in the living force of the particles of the bodies in which it is induced; whilst in others, particularly in the case of latent heat, the phenomena are produced by the separation of particle from particle, so as to cause them to attract one another through a greater space." James Prescott Joule|Joule, J.P. (1884).</ref> He described latent energy as the energy possessed via a distancing of particles where attraction was over a greater distance, i.e. a form of potential energy, and the sensible heat as an energy involving the motion of particles or what was known as a living force. At the time of Joule kinetic energy either held 'invisibly' internally or held 'visibly' externally was known as a living force.

Latent heat is the heat released or absorbed by a chemical substance or a thermodynamic system during a change of state of matter|state that occurs without a change in temperature. Such a process may be a phase transition, such as the melting of ice or the boiling of water.<ref name="Perrot" >Perrot, P. (1998).</ref><ref>Clark, J.O.E. (2004).</ref> The term was introduced around 1750 by Joseph Black as derived from the Latin latere (to lie hidden), characterizing its effect as not being directly measurable with a thermometer.

Sensible heat, in contrast to latent heat, is the heat transferred to a thermodynamic system that has as its sole effect a change of temperature.<ref>Ritter, M.E. (2006).</ref>

Both latent heat and sensible heat transfers increase the internal energy of the system to which they are transferred.

Consequences of Black's distinction between sensible and latent heat are examined in the calorimetry|Wikipedia article on calorimetry.

Specific heat

Specific heat, also called specific heat capacity, is defined as the amount of energy that has to be transferred to or from one unit of mass (kilogram) or amount of substance (mole (unit)|mole) to change the system temperature by one degree (temperature)|degree. Specific heat is a physical property, which means that it depends on the substance under consideration and its state as specified by its properties.

The specific heats of monatomic gases (e.g., helium) are nearly constant with temperature. Diatomic gases such as hydrogen display some temperature dependence, and triatomic gases (e.g., carbon dioxide) still more.

Relation between heat and temperature

Before the discovery of the laws of thermodynamics, quantity of energy transferred as heat was measured by changes in the states of the participating bodies.

Some general rules, with important exceptions, that will be indicated noted in following paragraphs of this section, can be stated as follows.

Most bodies, over most temperature ranges, expand on being heated. Mostly, heating a body at a constant volume increases the pressure it exerts on its constraining walls, and increases its temperature. Also mostly, heating a body at a constant pressure increases its volume, and increases its temperature.

Beyond this, most substances have three ordinarily recognized states of matter, solid, liquid, and gas, and a fourth less obviously recognized one, Plasma (physics)|plasma. Many have further, more finely differentiated, states of matter, such as for example, glass, and liquid crystal. In many cases, at fixed temperature and pressure, a substance can exist in several distinct states of matter in what might be viewed as the same 'body'. For example, ice may float in a glass of water. Then the ice and the water are said to constitute two Phase (matter)|phases within the 'body'. Definite Phase rule|rules are known, telling how distinct phases may coexist in a 'body'. Mostly, at a fixed pressure, there is a definite temperature at which heating causes a solid to melt or evaporate, and a definite temperature at which heating causes a liquid to evaporate. In such cases, cooling has the reverse effects.

All of these, the commonest cases, fit with a rule that heating can be measured by changes of state of a body. Such cases supply what are called Thermometer|thermometric bodies, that allow the definition of empirical temperatures. Before 1848, all temperatures were defined in this way. There was thus a tight link, apparently logically determined, between heat and temperature, though they were recognized as conceptually thoroughly distinct, especially by Joseph Black in the later eighteenth century.

There are important exceptions. They break the obviously apparent link between heat and temperature. They make it clear that empirical definitions of temperature are contingent on the peculiar properties of particular thermometric substances, and are thus precluded from the title 'absolute'. For example, Properties of water#Physics and chemistry|water contracts on being heated near 277 K. It cannot be used as a thermometric substance near that temperature. Also, over a certain temperature range, ice contracts on heating. Moreover, many substances can exist in metastable states, such as with negative pressure, that survive only transiently and in very special conditions. Such facts, sometimes called 'anomalous', are some of the reasons for the thermodynamic definition of absolute temperature.

In the early days of measurement of high temperatures, another factor was important, and used by Josiah Wedgwood in his Pyrometer#History|pyrometer. The temperature reached in a process was estimated by the shrinkage of a sample of clay. The higher the temperature, the more the shrinkage. This was the only available more or less reliable method of measurement of temperatures above 1000 °C. But such shrinkage is irreversible. The clay does not expand again on cooling. That is why it could be used for the measurement. But only once. It is not a thermometric material in the usual sense of the word.

Nevertheless, the Temperature#Thermodynamic approach to temperature|thermodynamic definition of absolute temperature does make essential use of the concept of heat, with proper circumspection.

Relation between hotness and temperature

According to Denbigh, the property of hotness is a concern of thermodynamics that should be defined without reference to the concept of heat. Consideration of hotness leads to the concept of empirical temperature.<ref>Denbigh, K. (1981), p. 9.</ref> All physical systems are capable of heating or cooling others.<ref>Baierlein, R. (1999), p. 349.</ref> This does not require that they have thermodynamic temperatures. With reference to hotness, the comparative terms hotter and colder are defined by the rule that heat flows from the hotter body to the colder.<ref>Adkins, C.J. (1968/1983), p. 34.</ref><ref>Pippard, A.B. (1957/1966), p. 18.</ref><ref>Haase, R. (1971), p. 7.</ref>

If a physical system is inhomogeneous or very rapidly or irregularly changing, for example by turbulence, it may be impossible to characterize it by a temperature, but still there can be transfer of energy as heat between it and another system. If a system has a physical state that is regular enough, and persists long enough to allow it to reach thermal equilibrium with a specified thermometer, then it has a temperature according to that thermometer. An empirical thermometer registers degree of hotness for such a system. Such a temperature is called empirical.<ref>Ernst Mach|Mach, E. (1900), section 5, pp. 48–49, section 22, pages 60–61.</ref><ref>Clifford Truesdell|Truesdell, C. (1980).</ref><ref>James Serrin|Serrin, J. (1986), especially p. 6.</ref> For example, Truesdell writes about classical thermodynamics: "At each time, the body is assigned a real number called the temperature. This number is a measure of how hot the body is."<ref>Clifford Truesdell|Truesdell, C. (1969), p. 6.</ref>

Physical systems that are too turbulent to have temperatures may still differ in hotness. A physical system that passes heat to another physical system is said to be the hotter of the two. More is required for the system to have a thermodynamic temperature. Its behavior must be so regular that its empirical temperature is the same for all suitably calibrated and scaled thermometers, and then its hotness is said to lie on the one-dimensional hotness manifold. This is part of the reason why heat is defined following Carathéodory and Born, solely as occurring other than by work or transfer of matter; temperature is advisedly and deliberately not mentioned in this now widely accepted definition.

This is also the reason why the zeroth law of thermodynamics is stated explicitly. If three physical systems, A, B, and C are each not in their own states of internal thermodynamic equilibrium, it is possible that, with suitable physical connections being made between them, A can heat B and B can heat C and C can heat A. In non-equilibrium situations, cycles of flow are possible. It is the special and uniquely distinguishing characteristic of internal thermodynamic equilibrium that this possibility is not open to thermodynamic systems (as distinguished amongst physical systems) which are in their own states of internal thermodynamic equilibrium; this is the reason why the zeroth law of thermodynamics needs explicit statement. That is to say, the relation 'is not colder than' between general non-equilibrium physical systems is not transitive, whereas, in contrast, the relation 'has no lower a temperature than' between thermodynamic systems in their own states of internal thermodynamic equilibrium is transitive. It follows from this that the relation 'is in thermal equilibrium with' is transitive, which is one way of stating the zeroth law.

Just as temperature may undefined for a sufficiently inhomogeneous system, so also may entropy be undefined for a system not in its own state of internal thermodynamic equilibrium. For example, 'the temperature of the solar system' is not a defined quantity. Likewise, 'the entropy of the solar system' is not defined in classical thermodynamics. It has not been possible to define non-equilibrium entropy, as a simple number for a whole system, in a clearly satisfactory way.<ref>Lieb, E.H., Yngvason, J. (2003), page 190.</ref>

Rigorous definition of quantity of energy transferred as heat

It is sometimes convenient to have a rigorous definition of quantity of energy transferred as heat. Such a definition is customarily based on the work of Constantin Carathéodory|Carathéodory (1909), referring to processes in a closed system, as follows.<ref name="Carathéodory 1909"/><ref>Adkins, C.J. (1968/1983).</ref><ref>Münster, A. (1970).</ref><ref>Pippard, A.B. (1957).</ref><ref>Fowler, R., Guggenheim, E.A. (1939).</ref><ref>Buchdahl, H.A. (1966).</ref>

The internal energy math|U_X of a body in an arbitrary state math|X can be determined by amounts of work adiabatically performed by the body on its surrounds when it starts from a reference state math|O. Such work is assessed through quantities defined in the surroundings of the body. It is supposed that such work can be assessed accurately, without error due to friction in the surroundings; friction in the body is not excluded by this definition. The adiabatic performance of work is defined in terms of adiabatic walls, which allow transfer of energy as work, but no other transfer, of energy or matter. In particular they do not allow the passage of energy as heat. According to this definition, work performed adiabatically is in general accompanied by friction within the thermodynamic system or body. On the other hand, according to Carathéodory (1909), there also exist non-adiabatic walls, which are postulated to be "permeable only to heat", and are called diathermal.

For the definition of quantity of energy transferred as heat, it is customarily envisaged that an arbitrary state of interest math|Y is reached from state math|O by a process with two components, one adiabatic and the other not adiabatic. For convenience one may say that the adiabatic component was the sum of work done by the body through volume change through movement of the walls while the non-adiabatic wall was temporarily rendered adiabatic, and of isochoric adiabatic work. Then the non-adiabatic component is a process of energy transfer through the wall that passes only heat, newly made accessible for the purpose of this transfer, from the surroundings to the body. The change in internal energy to reach the state math|Y from the state math|O is the difference of the two amounts of energy transferred.

Although Carathéodory himself did not state such a definition, following his work it is customary in theoretical studies to define the quantity of energy transferred as heat, math|Q, to the body from its surroundings, in the combined process of change to state math|Y from the state math|O, as the change in internal energy, math|ΔU_Y, minus the amount of work, math|W, done by the body on its surrounds by the adiabatic process, so that math|Q = ΔU_Y − W.

In this definition, for the sake of conceptual rigour, the quantity of energy transferred as heat is not specified directly in terms of the non-adiabatic process. It is defined through knowledge of precisely two variables, the change of internal energy and the amount of adiabatic work done, for the combined process of change from the reference state math|O to the arbitrary state math|Y. It is important that this does not explicitly involve the amount of energy transferred in the non-adiabatic component of the combined process. It is assumed here that the amount of energy required to pass from state math|O to state math|Y, the change of internal energy, is known, independently of the combined process, by a determination through a purely adiabatic process, like that for the determination of the internal energy of state math|X above. The rigour that is prized in this definition is that there is one and only one kind of energy transfer admitted as fundamental: energy transferred as work. Energy transfer as heat is considered as a derived quantity. The uniqueness of work in this scheme is considered to guarantee rigor and purity of conception. The conceptual purity of this definition, based on the concept of energy transferred as work as an ideal notion, relies on the idea that some frictionless and otherwise non-dissipative processes of energy transfer can be realized in physical actuality. The second law of thermodynamics, on the other hand, assures us that such processes are not found in nature.

Heat, temperature, and thermal equilibrium regarded as jointly primitive notions

Before the rigorous mathematical definition of heat based on Carathéodory's 1909 paper, recounted just above, historically, heat, temperature, and thermal equilibrium were presented in thermodynamics textbooks as jointly primitive notions.<ref>Lieb, E.H., Yngvason, J. (1999), p. 10.</ref> Carathéodory introduced his 1909 paper thus: "The proposition that the discipline of thermodynamics can be justified without recourse to any hypothesis that cannot be verified experimentally must be regarded as one of the most noteworthy results of the research in thermodynamics that was accomplished during the last century." Referring to the "point of view adopted by most authors who were active in the last fifty years", Carathéodory wrote: "There exists a physical quantity called heat that is not identical with the mechanical quantities (mass, force, pressure, etc.) and whose variations can be determined by calorimetric measurements." James Serrin introduces an account of the theory of thermodynamics thus: "In the following section, we shall use the classical notions of heat, work, and hotness as primitive elements, ... That heat is an appropriate and natural primitive for thermodynamics was already accepted by Carnot. Its continued validity as a primitive element of thermodynamical structure is due to the fact that it synthesizes an essential physical concept, as well as to its successful use in recent work to unify different constitutive theories."<ref>Serrin, J. (1986), p. 5 .</ref><ref>Owen, D.R. (1984), pp. 43–45.</ref> This traditional kind of presentation of the basis of thermodynamics includes ideas that may be summarized by the statement that heat transfer is purely due to spatial non-uniformity of temperature, and is by conduction and radiation, from hotter to colder bodies. It is sometimes proposed that this traditional kind of presentation necessarily rests on "circular reasoning"; against this proposal, there stands the rigorously logical mathematical development of the theory presented by Truesdell and Bharatha (1977).<ref>Clifford Truesdell|Truesdell, C., Bharatha, S. (1977).</ref>

This alternative approach to the definition of quantity of energy transferred as heat differs in logical structure from that of Carathéodory, recounted just above.

This alternative approach admits calorimetry as a primary or direct way to measure quantity of energy transferred as heat. It relies on temperature as one of its primitive concepts, and used in calorimetry.<ref>James Clerk Maxwell|Maxwell, J.C. (1871), p. v.</ref> It is presupposed that enough processes exist physically to allow measurement of differences in internal energies. Such processes are not restricted to adiabatic transfers of energy as work. They include calorimetry, which is the commonest practical way of finding internal energy differences.<ref>Peter Atkins|Atkins, P., de Paula, J. (1978/2010), p. 54.</ref> The needed temperature can be either empirical or absolute thermodynamic.

In contrast, the Carathéodory way recounted just above does not use calorimetry or temperature in its primary definition of quantity of energy transferred as heat. The Carathéodory way regards calorimetry only as a secondary or indirect way of measuring quantity of energy transferred as heat. As recounted in more detail just above, the Carathéodory way regards quantity of energy transferred as heat in a process as primarily or directly defined as a residual quantity. It is calculated from the difference of the internal energies of the initial and final states of the system, and from the actual work done by the system during the process. That internal energy difference is supposed to have been measured in advance through processes of purely adiabatic transfer of energy as work, processes that take the system between the initial and final states. By the Carathéodory way it is presupposed as known from experiment that there actually physically exist enough such adiabatic processes, so that there need be no recourse to calorimetry for measurement of quantity of energy transferred as heat. This presupposition is essential but is explicitly labeled neither as a law of thermodynamics nor as an axiom of the Carathéodory way. In fact, the actual physical existence of such adiabatic processes is indeed mostly supposition, and those supposed processes have in most cases not been actually verified empirically to exist.<ref>Brian Pippard|Pippard, A.B. (1957/1966), p. 15.</ref>

Heat transfer in engineering

File:Hot metalwork.jpg|thumb|right|225px|A red-hot iron rod from which heat transfer to the surrounding environment will be primarily through Thermal radiation|radiation. The discipline of heat transfer, typically considered an aspect of mechanical engineering and chemical engineering, deals with specific applied methods by which thermal energy in a system is generated, or converted, or transferred to another system. Although the definition of heat implicitly means the transfer of energy, the term heat transfer encompasses this traditional usage in many engineering disciplines and laymen language.

Heat transfer includes the mechanisms of heat conduction, thermal radiation, and mass transfer.

In engineering, the term convection|convective heat transfer is used to describe the combined effects of conduction and fluid flow. From the thermodynamic point of view, heat flows into a fluid by diffusion to increase its energy, the fluid then transfers (advection|advects) this increased internal energy (not heat) from one location to another, and this is then followed by a second thermal interaction which transfers heat to a second body or system, again by diffusion. This entire process is often regarded as an additional mechanism of heat transfer, although technically, "heat transfer" and thus heating and cooling occurs only on either end of such a conductive flow, but not as a result of flow. Thus, conduction can be said to "transfer" heat only as a net result of the process, but may not do so at every time within the complicated convective process.

Although distinct physical laws may describe the behavior of each of these methods, real systems often exhibit a complicated combination which are often described by a variety of complex mathematical methods.