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In physics, heating is spontaneous transfer of energy from one body to another, other than by work or transfer of matter. Heat is said to pass from the hotter to the colder of the two bodies.[1][2][3][4][5][6] The transfer can occur by conduction, radiation, and convective circulation.[7][8][9] Heat is a central concept in thermodynamics and statistical mechanics, and is also important in chemistry, engineering, and other disciplines.

In a heat engine, the internal energy supplied by a hot body can be harnessed in order to do work. This always requires the discharge of waste energy as heat to another body which is cold; the colder it is, the more efficient the process. In a heat pump, external energy is harnessed to transfer energy from a colder body to an already hotter target body. In a refrigeration, external energy is harnessed to further lower the temperature of an already cold target body, with discharge of waste energy as heat to an already hotter one. According to the second law of thermodynamics, heat pumps and refrigerators require consumption of energy as work. For refrigeration, a working substance at one stage of its cycle is made colder than the target body. Such engines necessarily function cyclically, requiring thermodynamic operations as well as natural, spontaneous thermodynamic processes.[10]

Kinetic theory explains transfers of energy as heat as macroscopic manifestations of the motions and interactions of microscopic constituents such as molecules and photons.

The quantity of energy transferred as heat is a scalar expressed in an energy unit such as the joule (J) (SI), with a sign that is customarily positive when a transfer adds to the energy of a system. It can be measured by calorimetry,[11] or determined by calculations based on other quantities, relying on the first law of thermodynamics. In calorimetry, latent heat changes a system's state without temperature change, while sensible heat changes its temperature, leaving some other state variable(s) unchanged.

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.


Scottish 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 Sir 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:

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.[13] This was the way of the historical pioneers of thermodynamics.[14][15]

Transfers of energy as heat between two bodies[edit]

Referring to conduction, 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."[16]

Referring to radiation, 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."[17]

Maxwell writes that convection as such "is not a purely thermal phenomenon".[18] 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 transferring energy as heat between source and destination bodies, because it transfers only energy and not matter from the source to the destination body.[19]

Practical operating devices that harness transfers of energy as heat[edit]

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.

Heat engine[edit]

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[edit]

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 sees energy taken in by the working body from the cold reservoir. Then the work reservoir does work on the working body, adding more to its internal energy. The hot working body passes heat to the hot reservoir, but still remains hot. Then the working body is made cool by allowing it to expand without doing work on another body. It is now cool enough to be passed to contact the cold reservoir to start another cycle. The device has transported energy from a colder to a hotter reservoir, in a convective circulation, but this is not regarded as being by an inanimate agency, because the work reservoir is depleted by an amount of energy that is eventually deposited in the hot reservoir, and the work is supplied by thermodynamic operations, not just by simple thermodynamic processes. 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.

Microscopic view of heat[edit]

In the kinetic theory, heat is explained in terms of the microscopic motions and interactions of constituent particles, such as electrons, atoms, and molecules.[20] 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.[21][22] Precise and detailed versions of it were developed in the nineteenth century.[23]

Notation and units[edit]

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 for algebraic purposes. 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 \dot{Q}. This should not be confused with a time derivative of a function of state (which can also be written with the dot notation) since heat is not a function of state.[24] 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[edit]

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.[25][26]

A calculation of quantity of heat transferred can rely on a hypothetical quantity of energy transferred as adiabatic work and on the first law of thermodynamics. Such calculation is the primary approach of many theoretical studies of quantity of heat transferred.[27][28][29]

Internal energy and enthalpy[edit]

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 ΔU of the system is equal to the amount of heat Q supplied to the system minus the amount of work 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.

\Delta U = Q - W \, .

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 ΔU is defined and measured solely by processes of adiabatic work:

 Q = \Delta U + W.

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:

 Q = \Delta U + W_\text{boundary} + W_\text{isochoric}.

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

The internal energy, 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 dU. The symbol for exact differentials is the lowercase letter d.

In contrast, neither of the infintestimal increments δQ nor 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, 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, 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 T, the increment of heat δQ and the temperature T form the exact differential

 \mathrm{d}S =\frac{\delta Q}{T},

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

 \mathrm{d}V =\frac{\delta W}{P},

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

\mathrm{d}U = T\mathrm{d}S - P\mathrm{d}V.

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


If V is constant

T\mathrm{d}S=\mathrm{d}U\,\,\,\,\,\,\,\,\,\,\,\,(V\,\, \text{constant)}

and if P is constant

T\mathrm{d}S=\mathrm{d}H\,\,\,\,\,\,\,\,\,\,\,\,(P\,\, \text{constant)}

with H the enthalpy defined by


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


The internal energy representation and the enthalpy representation are 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.[32][33]

Heat added to a body at constant pressure[edit]

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

\Delta H=\Delta U + \Delta(PV)\,.

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

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

Consequently, by substitution one has

\Delta H=Q - P\, \Delta V + P\, \Delta V
=Q\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text{at constant pressure.}

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'.[34] It is sometimes also called the heat function.[35]

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

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

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

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


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

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


Rudolf Clausius

In 1856, German physicist Rudolf Clausius, referring to closed systems, in which transfer of matter does 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 at the temperature T, has the equivalence-value:"[36][37]

 {} \frac {Q}{T}.

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

 \Delta S = \frac {Q}{T}\,\,\,\,\,\,\,\,\,\,\,\,\,(1)

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, ΔS, of entropy in the system may be considered to consist of two parts, an increment, ΔS that matches, or 'compensates', the change, −ΔS, of entropy in the surroundings, and a further increment, ΔS′′ that may be considered to be 'generated' or 'produced' in the system, and is said therefore to be 'uncompensated'. Thus

 \Delta S = \Delta S^\prime+\Delta S^{\prime\prime} .

This may also be written

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

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

 \Delta S_{\mathrm{overall}} = \Delta S^\prime+\Delta S^{\prime\prime}-\Delta S^\prime=\Delta S^{\prime\prime} .

This may also be written

 \Delta S_{\mathrm{overall}} = \Delta S_{\mathrm{compensated}}+\Delta S_{\mathrm{uncompensated}}+\Delta S_{\mathrm{surroundings}}=\Delta S_{\mathrm{uncompensated}} .

It is then said that an amount of entropy 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

 \Delta S_{\mathrm{overall}}>0 .

For purposes of mathematical analysis of transfers, one thinks of fictive processes that are called 'reversible', with the temperature 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 δQ (an inexact differential) is analyzed as a quantity T dS, with dS (an exact differential):

 T\,\mathrm{d}S=\delta Q .

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 dSuncompensated > 0. The quantity T dSuncompensated was termed by Clausius the "uncompensated heat", though that does not accord with present-day terminology. Then one has

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

This leads to the statement

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

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 T. Then one writes

 \mathrm{d}S =\mathrm{d}S_{\mathrm e}+\mathrm{d}S_{\mathrm i}\,,

where by definition

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

The second law for a natural process asserts that

 \mathrm d S_{\mathrm i}>0 .[38][39][40][41]

Latent and sensible heat[edit]

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.[42][quotations 1] 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 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.[43][44] 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.[45]

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 Wikipedia article on calorimetry.

Specific heat[edit]

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) to change the system temperature by one 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, hotness, and temperature[edit]

According to Baierlein, a system's hotness is its tendency to transfer energy as heat. All physical systems are capable of heating or cooling others.[46] 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.[47][48][49] If a physical system is 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.[50][51][52] 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.

Heat transfer in engineering[edit]

A red-hot iron rod from which heat transfer to the surrounding environment will be primarily through 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 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 (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.

See also[edit]


  1. ^ Born, M. (1949), p. 31.
  2. ^ Pippard, A.B. (1957/1966), p. 16.
  3. ^ Landau, L., Lifshitz, E.M. (1958/1969), p. 43
  4. ^ Callen, H.B. (1960/1985), pp. 18–19.
  5. ^ Reif, F. (1965), pp. 67, 73.
  6. ^ Bailyn, M. (1994), p. 82.
  7. ^ Guggenheim, E.A. (1949/1967), p. 8.
  8. ^ Planck. M. (1914).
  9. ^ Chandrasekhar, S. (1961).
  10. ^ Shavit, A., Gutfinger, C. (1995), Chapter 7, pp. 108–114, 223–229.
  11. ^ Maxwell, J.C. (1871), Chapter III.
  12. ^ Maxwell, J.C. (1871), p. 7.
  13. ^ Planck, M. (1903).
  14. ^ Partington, J.R. (1949).
  15. ^ Truesdell, C. (1980), page 15.
  16. ^ Partington, J.R. (1949), p. 118.
  17. ^ Maxwell, J.C. (1871), p. 10.
  18. ^ Maxwell, J.C. (1871), p. 11.
  19. ^ Chandrasekhar, S. (1961).
  20. ^ Kittel, C. Kroemer, H. (1980).
  21. ^ Bacon, F. (1620).
  22. ^ Partington, J.R. (1949), page 131.
  23. ^ Partington, J.R. (1949), pages 132–136.
  24. ^ Baierlein, R. (1999), p. 21.
  25. ^ Maxwell J.C. (1872), p. 54.
  26. ^ Planck (1927), Chapter 3.
  27. ^ Carathéodory, C. (1909).
  28. ^ Bryan, G.H. (1907), p. 47.
  29. ^ Callen, H.B. (1985), Section 1-8.
  30. ^ Callen, H.B., (1985), Section 2-3, pp. 40–42.
  31. ^ a b Adkins, C.J. (1983), p. 101.
  32. ^ a b Callen, H.B. (1985), p. 147.
  33. ^ Adkins, C.J. (1983), pp. 100–104.
  34. ^ Adkins, C.J. (1968/1983), p. 46.
  35. ^ Bailyn, M. (1994), p. 208.
  36. ^ Clausius, R. (1854).
  37. ^ Clausius, R. (1865), pp. 125–126.
  38. ^ De Groot, S.R., Mazur, P. (1962), p. 20.
  39. ^ Kondepudi, D, Prigogine, I. (1998), p. 82.
  40. ^ Kondepudi, D. (2008), p. 114.
  41. ^ Lebon, g., Jou, D., Casas-Vásquez, J. (2008), p. 41.
  42. ^ Joule J.P. (1884).
  43. ^ Perrot, P. (1998).
  44. ^ Clark, J.O.E. (2004).
  45. ^ Ritter, M.E. (2006).
  46. ^ Baierlein, R. (1999), p. 349.
  47. ^ Adkins, C.J. (1968/1983), p. 34.
  48. ^ Pippard, A.B. (1957/1966), p. 18.
  49. ^ Haase, R. (1971), p. 7.
  50. ^ Mach, E. (1900), section 5, pp. 48–49, section 22, pages 60–61.
  51. ^ Truesdell, C. (1980).
  52. ^ Serrin, J. (1986), especially p. 6.


  1. ^ "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." Joule, J.P. (1884).

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