In thermodynamics, heat is a type of energy transfer in which energy flows from a warmer substance or object to a colder one. It can be defined as the total amount of transferred energy excluding any macroscopic work that was done and any transfer of part of the object itself. Transfer of energy as heat can occur through direct contact, through a barrier that is impermeable to matter (as in conduction), by radiation between separated bodies, by way of an intermediate fluid (as in convective circulation), or by a combination of these. By contrast to work, heat involves the stochastic (random) motion of particles (such as atoms or molecules) that is equally distributed among all degrees of freedom, while work is confined to one or more specific degrees of freedom such as those of the center of mass.

Since heat (like work) is a quantity of energy being transferred between two bodies by certain processes, neither body "has" a definite amount of heat (much like a body in itself doesn't "have" work). In contrast, a body indeed has properties (state functions) such as temperature and internal energy. Energy exchanged as heat during a given process changes the (internal) energy of each body by equal and opposite amounts. The sign of the quantity of heat indicates the direction of the transfer, for example from system A to system B; negation indicates energy flowing in the opposite direction.

Although heat flows spontaneously from a hotter body to a cooler one, it is possible to construct a heat pump or refrigeration system that does work to increase the difference in temperature between two systems. In contrast, a heat engine reduces an existing temperature difference to do work on another system.

Historically, many energy units for measurement of heat have been used. The standards-based unit in the International System of Units (SI) is the joule (J). Heat is measured by its effect on the states of interacting bodies, for example, by the amount of ice melted or a change in temperature. The quantification of heat via the temperature change of a body is called calorimetry, and is widely used in practice. In calorimetry, sensible heat is defined with respect to a specific chosen state variable of the system, such as pressure or volume. Sensible heat causes a change of the temperature of the system while leaving the chosen state variable unchanged. Heat transfer that occurs at a constant system temperature but changes the state variable is called latent heat with respect to the variable. For infinitesimal changes, the total incremental heat transfer is then the sum of the latent and sensible heat.


The concept of heat has been important since pre-history. The earliest notions connected heat to origin mythologies. Heat, as "fire", was one of the ancient classical elements. The notion of heat as a conserved self-repelling fluid that permeated matter developed as the respected caloric theory until rendered obsolete by Thompson's 1798 mechanical theory of heat.

Physicist James Clerk Maxwell, in his 1871 classic Theory of Heat, was one of many who began to build on the theory that heat has 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.

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.

Transfers of energy as heat between two bodies

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."

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."

Maxwell writes that convection as such "is not a purely thermal phenomenon". 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.

Practical operating devices that harness transfers of energy as heat

In accordance with the first law for closed systems, energy transferred solely as heat leaves one body and enters 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 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. The other two are 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 in this article as currently customary for thermodynamic theory. The other macroscopic approach is the thermodynamic one, which admits heat as a primitive concept, which contributes, by scientific induction to knowledge of the law of conservation of energy. This view is widely taken as the practical one, quantity of heat being measured by calorimetry.

Bailyn also distinguishes the two macroscopic approaches as the mechanical and the thermodynamic. 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. 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". 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.

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. 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".

Microscopic view of heat

In the kinetic theory, heat is explained in terms of the microscopic motions and interactions of constituent particles, such as electrons, atoms, and molecules. The immediate meaning of the kinetic energy of the constituent particles is not as heat. It is as a component of internal energy. In microscopic terms, heat is a transfer quantity, and is described by a transport theory, not as steadily localized kinetic energy of particles. 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. Precise and detailed versions of it were developed in the nineteenth century.

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. 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 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 one joule per second.

The total amount of energy transferred as heat is conventionally written as Q (from Quantity) 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 Q?{\displaystyle {\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. 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 be 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.

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.

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 ?U 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.

?U=Q-W.{\displaystyle \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 ?U is defined and measured solely by processes of adiabatic work:

Q=?U+W.{\displaystyle 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=?U+Wboundary+Wisochoric.{\displaystyle 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 infinitesimal increments ? Q nor ?W 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, ?, 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 ? Q and the temperature T form the exact differential

dS=?QT,{\displaystyle \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, ?W, and the pressure, P, combine to form the exact differential

dV=?WP,{\displaystyle \mathrm {d} V={\frac {\delta W}{P}},}

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

dU=TdS-PdV.{\displaystyle \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

U=U(S,V).{\displaystyle U=U(S,V).}

If V is constant

TdS=dU(Vconstant){\displaystyle T\mathrm {d} S=\mathrm {d} U\,\,\,\,\,\,\,\,\,\,\,\,(V\,\,{\text{constant)}}}

and if P is constant

TdS=dH(Pconstant){\displaystyle T\mathrm {d} S=\mathrm {d} H\,\,\,\,\,\,\,\,\,\,\,\,(P\,\,{\text{constant)}}}

with H the enthalpy defined by

H=U+PV.{\displaystyle H=U+PV.}

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

H=H(S,P).{\displaystyle H=H(S,P).}

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.

Heat added to a body at constant pressure

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

?H=?U+?(PV).{\displaystyle \Delta H=\Delta U+\Delta (PV)\,.}

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

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

Consequently, by substitution one has

?H=Q-P?V+P?V{\displaystyle \Delta H=Q-P\,\Delta V+P\,\Delta V}
=Qat constant pressure.{\displaystyle =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'. It is sometimes also called the heat function.

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

?H=?S1S2(?H?S)PdS+?P1P2(?H?P)SdP{\displaystyle \Delta H=\int _{S_{1}}^{S_{2}}\left({\frac {\partial H}{\partial S}}\right)_{P}\mathrm {d} S+\int _{P_{1}}^{P_{2}}\left({\frac {\partial H}{\partial P}}\right)_{S}\mathrm {d} P}
=?S1S2(?H?S)PdSat constant pressure.{\displaystyle =\int _{S_{1}}^{S_{2}}\left({\frac {\partial H}{\partial S}}\right)_{P}\mathrm {d} S\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{at constant pressure.}}}

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

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


?H=?S1S2T(S,P)dSat constant pressure.{\displaystyle \Delta H=\int _{S_{1}}^{S_{2}}T(S,P)\mathrm {d} S\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{at constant pressure.}}}

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


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 at the temperature T, has the equivalence-value:"

QT.{\displaystyle {}{\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

?S=QT(1){\displaystyle \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, ?S, of entropy in the system may be considered to consist of two parts, an increment, ?S? that matches, or 'compensates', the change, -?S?, of entropy in the surroundings, and a further increment, ?S?? that may be considered to be 'generated' or 'produced' in the system, and is said therefore to be 'uncompensated'. Thus

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

This may also be written

?Ssystem=?Scompensated+?Suncompensatedwith?Scompensated=-?Ssurroundings.{\displaystyle \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

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

This may also be written

?Soverall=?Scompensated+?Suncompensated+?Ssurroundings=?Suncompensated.{\displaystyle \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 ?S? 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.

From the second law of thermodynamics follows that in a spontaneous transfer of heat, in which the temperature of the system is different from that of the surroundings:

?Soverall>0.{\displaystyle \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 rate.

Following the definition above in formula (1), for such a fictive reversible process, a quantity of transferred heat ? Q (an inexact differential) is analyzed as a quantity T dS, with dS (an exact differential):

TdS=?Q.{\displaystyle 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

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

This leads to the statement

TdS>=?Q(secondlaw).{\displaystyle 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

dS=dSe+dSi,{\displaystyle \mathrm {d} S=\mathrm {d} S_{\mathrm {e} }+\mathrm {d} S_{\mathrm {i} }\,,}

where by definition

?Q=TdSeanddSi?dSuncompensated.{\displaystyle \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

dSi>0.{\displaystyle \mathrm {d} S_{\mathrm {i} }>0.}

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