# warmth

Physical size
Surname warmth
Formula symbol ${\ displaystyle Q}$
Size and
unit system
unit dimension
SI J  = kg · m 2 · s -2 L 2 · M · T −2
cgs erg L 2 · M · T −2
Heat is transported in different ways: by conduction (in the horseshoe), by convection (in the rising hot air) and by thermal radiation (visible through the glow of the red embers)

The physical quantity heat records part of the energy that is absorbed or emitted by a thermodynamic system . The other part is the physical work . According to the first law of thermodynamics, both together cause a change in the internal energy of the system. The work is defined as that portion of the transferred energy that is associated with a change in external parameters , e.g. B. with the reduction of the volume when compressing a gas. The remaining part is the heat. It leaves the external parameters unchanged and instead increases the entropy of the system, which for example reduces its internal order, e.g. B. when melting an ice cube. Like all energies, heat is specified in the international system in the unit of measurement joule and is usually denoted by the symbol . ${\ displaystyle Q}$

Between two systems with different temperatures, heat always flows from the higher to the lower temperature. The heat can be transported by conduction , radiation or convection .

## overview

The physical term “heat” differs significantly from the colloquial use of the word “heat”. In everyday language, this usually refers to the property of a body that makes it “warm” and thus describes a certain state. This is best expressed physically by the term thermal energy . In this sense, the word component “heat” is also used in numerous technical terms for historical reasons (e.g. heat capacity , heat content, etc.).

The quantity that is called “heat” in physics is not a state quantity . Rather, it is used to describe processes in which the state of the system changes. It is therefore a process variable . According to the first law of thermodynamics, the change in its internal energy for each system is equal to the sum of the heat supplied and the work done on the system. Conversely, this means that the heat supplied exactly the increase of internal energy minus the work done corresponds: . How much energy is transferred in total depends only on the initial and final state; however, the division into work and heat can be different - depending on the course of the process. ${\ displaystyle Q}$${\ displaystyle \ Delta U}$${\ displaystyle W}$${\ displaystyle Q = \ Delta UW}$

Since energy is a conservation quantity , heat and work cannot arise in the system itself, but rather describe the energy transport across the system boundaries. If only two systems are involved in the process, then one system releases just as much work or heat as the other absorbs. Therefore the quantities heat, work and change of the internal energy have the same values ​​for both systems, only with opposite signs. A process in which no heat is transferred is called adiabatic ; a process in which only heat is transferred is called work-tight. An example of this would be the isochoric change of state .

When two systems exchange heat with each other, it always flows from high to low temperature. Often the lower of the two temperatures increases and the higher of the two decreases, but there are also exceptions, e.g. B. Ice at 0 ° C is converted into water at 0 ° C by adding heat.

A machine that continuously or periodically absorbs heat and does work is called a heat engine . For reasons of principle, the energy absorbed through heat cannot be completely released as work, but has to be partially dissipated again as waste heat (for more details see the 2nd law of thermodynamics ).

In the fundamental explanation of thermodynamic phenomena by statistical mechanics , every system consists of a large number of individual particles in more or less ordered motion. Heat is only linked to the amount of disordered movement. If a radiation field belongs to the system under consideration, heat refers to the energy that is randomly distributed over the various possible wave forms (see thermal radiation ). In the image of the energy levels, the particles are distributed to all different levels and change between them in a statistically fluctuating manner, but in the state of equilibrium the average occupation number of each level remains the same and is determined in the form of a statistical distribution . Addition of heat shifts this distribution curve to higher energy, while work done on the system raises the energies of the individual energy levels.

## Development of the concept of heat

As far as the technical and scientific field is concerned, “heat” is and was colloquially used on the one hand as an expression of an increased temperature , on the other hand for the associated energies and energy flows, which were initially referred to as the amount of heat. The distinction between the two aspects was already prepared by the nominalists in the 14th century, i.e. before the beginning of modern natural sciences. Regarding temperature, reliable thermometers were developed in the 17th and 18th centuries . However, the amount of heat was only considered more closely after the equilibrium temperatures were investigated after mixing substances with different starting temperatures with the help of calorimeters from 1750 . The amount of heat was later given its own physical dimension with the unit calorie , defined in the form (but modified several times): "1 calorie is the heat supply that increases the temperature of 1 g of water by 1 ° C." This resulted in a conservation law ( "Heat given off = heat absorbed"), which is still valid today if no work is being done.

Up until around 1850, two doctrines were opposed to the interpretation of what heat was about: One explanation was based on a hypothetical “heat substance”, which Antoine de Lavoisier last named calorique (caloricum) . The heat material is immortal, unreachable, imponderable, penetrates every piece of matter and determines its "heat content" by its amount and the temperature by its concentration. The expressions “amount of heat”, “heat energy” and “ specific heat ” come from the context of this thermal substance theory. On the other hand, as early as the 13th century by Roger Bacon and from the 17th century a. a. a mechanical theory of heat proposed by Johannes Kepler , Francis Bacon , Robert Boyle , Daniel Bernoulli : Heat is a movement of small particles of matter hidden from the eyes. In fact, in 1798 Benjamin Thompson (later Lord Rumford) observed while drilling cannon barrels that any amount of heat generated by drilling was solely due to mechanical work. Thompson could even have estimated the approximate value of the mechanical heat equivalent from this. However, a precise measurement was only achieved by James Prescott Joule around 1850.

The fact that, conversely, heat can also be a source of mechanical work was known from the first steam engines since the beginning of the 18th century. The attempts to explain the thermal substance theory culminated in Sadi Carnot's finding in 1824 that the work that can be gained from the supply of heat is limited for fundamental reasons, because the heat absorbed at high temperatures must be released again at low temperatures. The efficiency that can ideally be achieved does not depend on the design of the machine, but solely on the two temperatures and is always below 100%. Carnot argued entirely on the basis of the thermal substance theory, but also gave a value for the mechanical heat equivalent, but his writings were initially forgotten.

Decisive for the refutation of the thermal substance theory was the knowledge published by Rudolf Clausius in 1850 that the relationship between heat and work is a mutual transformation, i.e. i.e., heat is consumed when work is gained and vice versa. When converting work into heat, Clausius relied on the aforementioned observation by Thompson and other findings on frictional heat. When converting heat into work, he relied on the increased heat requirement when heating a gas, if it can also expand, and on a key experiment carried out by Joule in 1844: When compressed, compressed air performs mechanical work precisely when it is in the environment Removes heat, so it cools down. This ultimately enabled the mechanical theory of heat to prevail.

The realization that heat is energy paved the way for the law of conservation of energy , which Hermann von Helmholtz first formulated in general in 1847. In the further development of the concept of heat, the concept of energy moved into the center.

Despite the refutation of the thermal mass theory, Carnot's discovery that the extraction of work from heat is limited by the temperature difference remained valid. Rudolf Clausius succeeded in deriving the concept of another quantity-like quantity from this, which must always flow when heat is transferred. In 1865 he called this quantity entropy . In many respects the entropy corresponds to the caloricum postulated in the thermal substance theory. However, the law of conservation at the time assumed for the Caloricum does not apply to entropy: Entropy cannot be destroyed, but it can be created. For example, this happens in the case of heat conduction from high to low temperatures.

There is no particular form of energy associated with heat, but rather the property of transporting entropy. The current definition of heat, which also corresponds to the definition given above in the introduction, no longer relates to temperature changes or material conversions, but is based entirely on the concept of energy. It was formulated by Max Born in 1921 after Constantin Carathéodory had brought thermodynamics into an axiomatic form in 1909. Accordingly, the actual definition is the heat in the 1st law of thermodynamics and is (s u..) If on a macroscopic system in a process, the work performed, and changes its internal energy to thereby , then the difference is the heat that this in the system has been transferred. ${\ displaystyle W}$${\ displaystyle \ Delta U}$${\ displaystyle Q = \ Delta UW}$

The two quantities heat and work are not as independent of one another as it might appear at first glance: If, for example, heat is added to the air in a balloon, this is not only expressed in an increase in temperature and entropy. The balloon inflates, its volume increases. The gas also does work due to the supply of heat (against the ambient pressure and against the elasticity of the rubber cover). Conversely, external work can also have an indirect influence on the internal parameters of the system. If you z. B. kneading a dough, you are obviously doing work. Internal friction causes the temperature of the dough to increase, as does its entropy. The work was dissipated in this process . It did increase the internal energy of the dough, but it has the same effect as the added heat. You can therefore no longer take it from the dough in the form of work. So this process is irreversible.

In the international system of units , the special heat unit calorie was abolished in 1948 and replaced by the general unit joule for energy.

## Heat transfer

### Derived quantities

If the heat transferred in a process up to the point in time , then the current heat flow is given by: ${\ displaystyle Q (t)}$${\ displaystyle t}$

${\ displaystyle {\ dot {Q}} (t) = {\ frac {\ mathrm {d} Q (t)} {\ mathrm {d} t}}}$

In the SI it has the unit watt .

If the transfer occurs through an area , then the average instantaneous heat flux density is . The current local heat flux density is the quotient of the differential heat flux and the differential area d A through which it passes: ${\ displaystyle A}$ ${\ displaystyle {\ dot {Q}} _ {A} = {\ frac {\ dot {Q}} {A}}}$${\ displaystyle {\ dot {q}} _ {A} (t)}$

${\ displaystyle {\ dot {q}} _ {A} (t) = {\ frac {\ mathrm {d} {\ dot {Q}} (t)} {\ mathrm {d} A}} = {\ frac {\ mathrm {d ^ {2}} {Q} (t)} {\ mathrm {d} A \, \ mathrm {d} t}}}$

In the case of heat transfer by convection, the heat can be related to the flowing mass , or in the case of steady flow, the heat flow to the mass flow : ${\ displaystyle m}$ ${\ displaystyle {\ dot {m}}}$

${\ displaystyle {q} _ {m} = {\ frac {Q} {m}} = {\ frac {\ dot {Q}} {\ dot {m}}}}$

In the SI, this specific heat flow has the unit J / kg (joule per kilogram ), but must not be confused with the specific heat capacity .

### Conduction

Are two systems with different temperatures by a common surface thermally coupled flows a heat flow , which by Isaac Newton by ${\ displaystyle T_ {1}> T_ {2}}$${\ displaystyle A}$${\ displaystyle {\ dot {Q}} _ {1 \ rightarrow 2}}$

${\ displaystyle {\ dot {Q}} _ {1 \ rightarrow 2} = k \, A \, (T_ {1} -T_ {2})}$

given is. The strength of the thermal coupling at the system boundary is described by the heat transfer coefficient . ${\ displaystyle k}$

Every body emits a heat flow through electromagnetic radiation , which in this context is also referred to as radiant power. According to the Stefan-Boltzmann law (by Josef Stefan and Ludwig Boltzmann ): ${\ displaystyle {\ dot {Q}}}$

${\ displaystyle {\ dot {Q}} = \ varepsilon \, \ sigma \, A \, T ^ {4}}$

In it is

${\ displaystyle \ varepsilon}$the emissivity : the values ​​are between 0 (perfect mirror) and 1 (ideal black body ),
${\ displaystyle \ sigma = 5 {,} 67 \ cdot 10 ^ {- 8} ~ \ mathrm {\ frac {W} {m ^ {2} \, K ^ {4}}}}$the Stefan-Boltzmann constant ,
${\ displaystyle A}$ the surface of the radiating body,
${\ displaystyle T}$the absolute temperature of the radiating body.

The heat transfer to a second body comes about because it - at least partially - absorbs the incident radiation . The degree of absorption is again between 0 (perfect mirror) and 1 (ideal black body). Two bodies shine on each other through the facing parts of the surface. This always results in an energy flow from the warmer to the colder surface, regardless of their nature, emission and absorption capacity.

### convection

Convective heat transfer takes place with the help of a mass transfer. It consists of three sub-processes:

1. Heat passes from a hot body by conduction to a transportable substance, whereby this z. B. is heated or evaporated;
2. the substance flows - in the simplest case without any further change of state - through space to a colder body;
3. Heat is transferred from the fabric to the colder body through thermal conduction.

In accordance with the older understanding of the term heat , the heat transfer by convection is only related to the mass transport in the middle sub-process. In the sense of the definition given in the introduction, the other two sub-steps are included.

The transferred heat flow depends on several parameters, including duration, area and strength of the thermal coupling between the warm or cold body and the transport medium as well as its flow rate.

The whole process can be further modified by having the medium doing or picking up work in transit. Is it z. B. To work through adiabatic expansion or compression, the temperature of the medium also changes. The relevant variable for the energy carried is then the enthalpy , i.e. H. the sum of inner energy and shift work . Based on it z. B. the refrigerator and the heat pump . ${\ displaystyle H = U + pV}$ ${\ displaystyle U}$ ${\ displaystyle pV}$

### Effects of heat transfer

Typically, the supply or withdrawal of heat leads to an increase or decrease in the temperature of the substance in question. The heat is approximately proportional to the change in temperature and proportional to the mass of the substance: ${\ displaystyle Q}$${\ displaystyle \ Delta T}$${\ displaystyle m}$

${\ displaystyle Q = c \, m \, \ Delta T}$

The constant of proportionality is the specific heat capacity of the substance. It is a characteristic parameter for the respective substance, which only weakly depends on the other state variables such as pressure, temperature etc. For example, the temperature of 1 kg of liquid water increases by 1 ° C if it is supplied with heat of approx. 4.2 kJ. ${\ displaystyle c}$

Bodies that expand when heated do work against the ambient pressure or intermolecular forces. A certain supply of heat then only partially benefits the thermal movement of the particles. Therefore it is associated with a lower temperature increase than without thermal expansion. A distinction is therefore made between the specific heat capacities at constant pressure and at constant volume. In the case of solid and liquid substances, the difference is usually negligible, but in the case of gases it can be up to 68% (see isentropic exponent ). With the isothermal expansion of an ideal gas , by definition, the temperature does not change at all. So here all the heat supplied is converted into expansion work. In certain cases, the work done may even exceed the heat supplied. Then the temperature of the system decreases despite the supplied heat. This is e.g. B. the case with the Joule-Thomson effect .

At certain values ​​of temperature, pressure, and possibly other parameters, substances do not react to the supply of heat with a change in temperature, but with a phase change such as evaporation , melting , sublimation etc. The energy required for this is called enthalpy of evaporation , heat of fusion or heat of sublimation . Conversely, a release of heat leads to condensation , solidification and resublimation of the substances under the same conditions . The heat per unit of the amount of substance depends heavily on which substance and which phase change it is. For example, liquid water freezes at 0 ° C under atmospheric pressure if heat of approx. 333 kJ per 1 kg is removed from it.

## Latent warmth

The phase transition of all substances between solid / liquid, liquid / gaseous or solid / gaseous takes place in both directions at constant temperature ( isothermal ). The energy used or released for the phase transition was previously called latent heat (latent = hidden). Depending on the type of phase transition, this involves the enthalpy of fusion , the enthalpy of condensation or evaporation and the enthalpy of sublimation . For example, water needs 333.5 kJ / kg to convert from 0 ° C ice to 0 ° C water and 2257 kJ / kg to turn 100 ° C into steam at 100 ° C. The energy supplied does not cause any change in temperature and is released back to the environment when the phase transition is reversed.

In addition to the heat of fusion and evaporation, the enthalpy of transformation (previously: heat of transformation) also counts as latent heat. It occurs, for example, with iron with 0.9% carbon content and temperatures around 720 ° C. As it cools, the crystal lattice flips from face-centered cubic to body-centered cubic, with heat being given off from this transformation.

Latent heat storage systems use this effect and can store large amounts of energy with a small increase in temperature.

## Heat, work, internal energy and the 1st law of thermodynamics

The first law of thermodynamics states that the internal energy of a physical system changes when the work is done on the system and the heat is supplied to it: ${\ displaystyle U}$${\ displaystyle \ Delta U}$${\ displaystyle W}$${\ displaystyle Q}$

${\ displaystyle \ Delta U = Q + W}$

(The work done by the system or the heat given off are counted negatively. In some texts, the opposite sign convention applies.) This is the total energy that the system possesses when the center of gravity is at rest and without considering potential energy in an external field. The 1st law expresses part of the law of conservation of energy . ${\ displaystyle U}$

${\ displaystyle U}$is a state variable , i.e. In other words, the value is completely determined by the current state of the system and, in particular, independent of the way in which this state was established. The process variables and depend, however, on the path taken. However, the sum of and inevitably results from the difference between the internal energies of the initial and final state. If a process leads the system back to its original state, the initial and final energy match. As a result, the difference between the heat absorbed and emitted by the system is just as great as the energy it emits through work (or vice versa). Such a process is called a circular process . This is the basis of heat engines, on the one hand, which generate mechanical work from a heat source in continuous operation, and, on the other hand, heat pumps , which use work to move heat against a temperature gradient. ${\ displaystyle W}$${\ displaystyle Q}$${\ displaystyle W}$${\ displaystyle Q}$

## Heat, entropy and the 2nd law of thermodynamics

A special feature applies to heat compared to other forms of energy transfer: heat can never pass from a colder body to a warmer one unless another related change occurs at the same time. This is the 2nd law of thermodynamics, reproduced in the words of its first formulation by Rudolf Clausius. There are numerous other formulations that are equivalent to this. One of them is that entropy can only remain constant or increase in a closed system. It is based on the state variable entropy discovered by Clausius , which is closely related to heat.

The fact that both formulations are equivalent can be seen from an ideal heat engine: it converts heat into work by guiding a working substance through a Carnot cycle . The working material absorbs the heat from a system with the temperature , does the work and transfers the waste heat to a system with the lower temperature . Since these three energy quantities are usually counted positively in this context, the 1st law applies . ${\ displaystyle Q_ {1}}$${\ displaystyle T_ {1}}$${\ displaystyle W}$ ${\ displaystyle Q_ {2}}$${\ displaystyle T_ {2}}$${\ displaystyle Q_ {1} = W + Q_ {2}}$

So only part of the heat used was used. The efficiency is accordingly . ${\ displaystyle Q_ {1}}$${\ displaystyle W = Q_ {1} -Q_ {2}}$${\ displaystyle \ eta = {\ frac {(Q_ {1} -Q_ {2})} {Q_ {1}}} = 1 - {\ frac {Q_ {2}} {Q_ {1}}}}$

The Carnot cycle is reversible, so it could also run the other way round. Then, as a heat pump, it would consume the work to absorb the heat at the low temperature and to transfer it (increased by the amount of the work done) to the high one. From the first formulation of the 2nd law it follows that any reversible cycle has the same efficiency as long as it works with the same temperatures. The exact course of the process and the choice of working material are irrelevant for this consideration. If there were a process with a higher degree of efficiency, it could be interconnected with a Carnot cycle to a combination of heat engine and heat pump, which transferred heat from the low to the high temperature after one pass without leaving any other changes. However, according to the first formulation of the 2nd main clause, this is excluded. The efficiency of the reversible cycle is also referred to as Carnot's or ideal efficiency , because with the same argumentation discovered by Carnot, one can rule out that there is even a cycle (whether reversible or not) that has a higher efficiency. ${\ displaystyle W}$${\ displaystyle Q_ {2}}$${\ displaystyle T_ {2}}$

Accordingly, the ideal efficiency is generally applicable and can therefore be determined using a single example. The example of the ideal gas as a working substance in the Carnot process results (with as absolute temperature ) ${\ displaystyle \ eta _ {\ mathrm {ideal}}}$${\ displaystyle T}$

${\ displaystyle {\ frac {Q_ {1}} {T_ {1}}} = {\ frac {Q_ {2}} {T_ {2}}},}$

From which follows:

${\ displaystyle \ eta _ {\ mathrm {ideal}} = 1 - {\ frac {T_ {2}} {T_ {1}}}}$

The generality of these equations allows the size

${\ displaystyle \ Delta S = {\ frac {Q} {T}}}$

to be regarded as the change of a new state variable . is the entropy of the system. It changes when the heat is reversibly supplied to the system at that temperature . Entropy flows with the reversibly transferred heat. ${\ displaystyle S}$${\ displaystyle S}$${\ displaystyle \ Delta S}$${\ displaystyle Q}$${\ displaystyle T}$

In the cycle, the entropy flows from the warmer system into the working substance, which in turn transfers the entropy to the colder system. Since it is true in the reversible cycle , the entire entropy is preserved. It flows without decrease from the system with the high temperature into the working material and further into the system with the low temperature. This differentiates the flow of entropy from the flow of heat, which decreases by as much as work has been done in between. ${\ displaystyle \ Delta S_ {1} = {Q_ {1} \ over T_ {1}}}$${\ displaystyle \ Delta S_ {2} = {Q_ {2} \ over T_ {2}}}$${\ displaystyle \ Delta S_ {1} = \ Delta S_ {2}}$

In a non-ideal, real cycle, the entropy flow cannot decrease, but only increase, because according to the above, this has a lower efficiency, i.e. greater waste heat. Therefore, the following applies: The working material gives off more entropy than it received , the entropy has increased overall. This applies in general: Every irreversible process gives rise to new entropy and leads to an increase in the total entropy of the system, even if no heat is supplied at all. Such processes are called dissipation . In a real heat engine, for example, some of the work that would be available to an ideal heat engine is dissipated through friction . ${\ displaystyle \ Delta S_ {2} <\ Delta S_ {1}}$

So follows from the first of the above formulations of the 2nd main clause the second. Conversely, the first also follows from the second, because the spontaneous transfer of heat to a system of higher temperature would be a process in which the overall entropy would decrease, and is therefore impossible according to the second formulation.

## Heat and work of microscopic interpretation

The simple model system of non-interacting particles allows a microscopic interpretation of heat and work. If such particles with occupation numbers are distributed over the energy levels , then the total energy is ${\ displaystyle N}$${\ displaystyle n_ {i}}$${\ displaystyle E_ {i}}$

${\ displaystyle E _ {\ mathrm {ges}} = \ sum _ {i = 1} ^ {N} n_ {i} \, E_ {i}.}$

Then there is an infinitesimal change of${\ displaystyle E _ {\ mathrm {ges}}}$

${\ displaystyle \ mathrm {d} E _ {\ mathrm {ges}} = \ sum _ {i = 1} ^ {N} E_ {i} \, \ mathrm {d} n_ {i} + \ sum _ {i = 1} ^ {N} n_ {i} \, \ mathrm {d} E_ {i} \.}$

If the particle system is in a thermodynamic state of equilibrium, then the total energy is precisely the internal energy ( ) and it can be shown that the two terms of this equation correspond to the two terms in the 1st law . The first term represents the energy supplied by a reversible change of state through heat , the second term the work done on the system . Complete differentials are indicated by and inexact differentials by . Accordingly, heat means that the total energy increases or decreases due to changed occupation numbers of the energy levels, while work with unchanged occupation numbers shifts the position of the levels. The latter thus represents the microscopic criterion for an adiabatic process. ${\ displaystyle E _ {\ mathrm {ges}} = U}$${\ displaystyle \ mathrm {d} U = \ delta Q + \ delta W}$${\ displaystyle \ delta Q = T \, \ mathrm {d} S}$${\ displaystyle \ delta W = -p \, \ mathrm {d} V}$${\ displaystyle \ mathrm {d}}$${\ displaystyle \ delta}$

Wikiquote: Warmth  - Quotes
Wiktionary: Warmth  - explanations of meanings, word origins, synonyms, translations

## Remarks

1. If a system absorbs heat once in a single process, but in the end has the same internal energy as before, the heat supplied has actually been converted into an equally large amount of work. One example is the isothermal expansion of the ideal gas . However, this is only possible as a one-time process. It can only be repeated after another process has reset the external parameters to their initial values. Work has to be done on the system and an equal amount of heat dissipated.
2. Max Born wrote in his 1921 publication: “Only after the first main clause has been established is it possible to properly introduce the term heat quantity. Chemists refer to the energy of a body as heat content, the change in energy as heat tone; this is entirely justified insofar as the change in state associated with the change in energy is mainly shown in a change in temperature. The connection to the historical concept of the amount of heat is achieved by using the energy as a caloric unit which is necessary for a certain temperature change of 1 g of water (at constant volume); this energy, expressed in mechanical measure (erg), is the heat equivalent. The 1st main clause gives information about how far it is possible to operate with heat in the traditional way as a substance, as it is e.g. B. happens when using the water calorimeter; so that the heat 'flows' (without transformation), any work must be excluded. The increase in energy of the water in the calorimeter only measures the decrease in energy of the immersed body if changes in volume (or other work-performing processes) are prevented or are insignificant by themselves. As self-evident as this restriction is after the establishment of the 1st law, it is absurd before. Now we can define the amount of heat for any process; for this it must be assumed that the energy is known as a function of the state and that the work expended in any process is to be measured, then the heat supplied in the process is Q = UU o -A. In the following, the concept of heat does not play an independent role; we only use it as a short description of the difference between the increase in energy and the work added. "
3. These and other characterizations of warmth are also discussed in G. Job: Anthologia Calorica.

## Individual evidence

1. a b Klaus Stierstadt, Günther Fischer: Thermodynamics: From Microphysics to Macrophysics (Section 4.2) . Springer, Berlin, New York 2010, ISBN 978-3-642-05097-8 ( limited preview in Google book search).
2. Friedrich Hund: History of physical terms. Volume 1, BI university pocket books, Mannheim 1978, p. 206 ff.
3. Roberto Toretti: The Philosophy of Physics . Cambridge University Press, Cambridge 1999, ISBN 0-521-56259-7 , pp. 180 ff .
4. ^ Ervin Szücs: Dialogues about technical processes. VEB Fachbuchverlag, Leipzig 1976.
5. Rudolf Clausius: About the moving force of heat and the laws which can be derived from it for the theory of heat itself . In: Annals of Physics . tape 155 , 1850, pp. 368-397 , doi : 10.1002 / andp.18501550306 .
6. Friedrich Hund: History of physical terms. Volume 2, BI university pocket books, Mannheim 1978, p. 93 ff.
7. Rudolf Clausius: About different, for the application convenient forms of the main equations of the mechanical heat theory . (also lecture to the Zurich Natural Research Society). In: Annals of Physics and Chemistry . tape 125 , 1865, pp. 353-400 .
8. Rudolf Clausius: About the second law of mechanical heat theory . Lecture given at a general meeting of the 41st Assembly of German Naturalists and Physicians in Frankfurt am Main on September 23, 1867. 1867 ( Original from Michigan State University, digitized June 29, 2007 in Google Book Search).
9. ^ William H. Cropper: Rudolf Clausius and the road to entropy . In: American Journal of Physics . tape 54 , 1986, pp. 1068-1074 , doi : 10.1119 / 1.14740 (English).
10. ^ Hugh Longbourne Callendar : Proceedings of the Royal Society of London . tape  134 , p. xxv ( snippet in Google book search - around 1911).
11. Gottfried Falk , Wolfgang Ruppel : Energy and Entropy . Springer-Verlag, 1976, ISBN 978-3-642-67900-1 .
12. Max Born: Critical considerations on the traditional representation of thermodynamics . In: Physikalische Zeitschrift . tape 22 , 1921, pp. 218-224 .
13. See Section 13.4 Latent Heat in Klaus Lüders, Robert O. Pohl (Ed.): Pohl's Introduction to Physics. Volume 1, 21st edition, Springer-Verlag, Berlin / Heidelberg 2017, ISBN 978-3-662-48662-7 .
14. Rudolf Clausius : About a modified form of the second law of the mechanical heat theory . In: Annals of Physics . tape 169 , 1854, pp. 481-506 , doi : 10.1002 / andp.18541691202 .
15. See e.g. B. Andreas Heintz: Statistical Thermodynamics, Fundamentals and Treatment of Simple Chemical Systems. Cape. 2.2 ff. PDF ( memento from September 23, 2015 in the Internet Archive ), accessed on April 20, 2015.