The second law of thermodynamics is a physical law based on universal empirical observation concerning heat and energy interconversions. A simple statement of the law is that heat always flows spontaneously from hotter to colder regions of matter (or 'downhill' in terms of the temperature gradient). Another statement is: "Not all heat can be converted into work in a cyclic process." These are informal definitions, however; more formal definitions appear below. The second law of thermodynamics establishes the concept of entropy as a physical property of a thermodynamic system. It predicts whether processes are forbidden despite obeying the requirement of conservation of energy as expressed in the first law of thermodynamics and provides necessary criteria for spontaneous processes. For example, the first law allows the process of a cup falling off a table and breaking on the floor, as well as allowing the reverse process of the cup fragments coming back together and 'jumping' back onto the table, while the second law allows the former and denies the latter. The second law may be formulated by the observation that the entropy of isolated systems left to spontaneous evolution cannot decrease, as they always tend toward a state of thermodynamic equilibrium where the entropy is highest at the given internal energy. An increase in the combined entropy of system and surroundings accounts for the irreversibility of natural processes, often referred to in the concept of the arrow of time. Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics provides a microscopic explanation of the law in terms of probability distributions of the states of large assemblies of atoms or molecules. The second law has been expressed in many ways. Its first formulation, which preceded the proper definition of entropy and was based on caloric theory, is Carnot's theorem, formulated by the French scientist Sadi Carnot, who in 1824 showed that the efficiency of conversion of heat to work in a heat engine has an upper limit. The first rigorous definition of the second law based on the concept of entropy came from German scientist Rudolf Clausius in the 1850s and included his statement that heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time. The second law of thermodynamics allows the definition of the concept of thermodynamic temperature, but this has been formally delegated to the zeroth law of thermodynamics.
Introduction
The first law of thermodynamics provides the definition of the internal energy of a thermodynamic system, and expresses its change for a closed system in terms of work and heat. It can be linked to the law of conservation of energy. Conceptually, the first law describes the fundamental principle that systems do not consume or 'use up' energy, that energy is neither created nor destroyed, but is simply converted from one form to another. The second law is concerned with the direction of natural processes. It asserts that a natural process runs only in one sense, and is not reversible. That is, the state of a natural system itself can be reversed, but not without increasing the entropy of the system's surroundings, that is, both the state of the system plus the state of its surroundings cannot be together, fully reversed, without implying the destruction of entropy. For example, when a path for conduction or radiation is made available, heat always flows spontaneously from a hotter to a colder body. Such phenomena are accounted for in terms of entropy change. A heat pump can reverse this heat flow, but the reversal process and the original process, both cause entropy production, thereby increasing the entropy of the system's surroundings. If an isolated system containing distinct subsystems is held initially in internal thermodynamic equilibrium by internal partitioning by impermeable walls between the subsystems, and then some operation makes the walls more permeable, then the system spontaneously evolves to reach a final new internal thermodynamic equilibrium, and its total entropy, S {\displaystyle S} , increases. In a reversible or quasi-static, idealized process of transfer of energy as heat to a closed thermodynamic system of interest, (which allows the entry or exit of energy – but not transfer of matter), from an auxiliary thermodynamic system, an infinitesimal increment ( d S {\displaystyle \mathrm {d} S} ) in the entropy of the system of interest is defined to result from an infinitesimal transfer of heat ( δ Q {\displaystyle \delta Q} ) to the system of interest, divided by the common thermodynamic temperature ( T ) {\displaystyle (T)} of the system of interest and the auxiliary thermodynamic system:
d S = δ Q T (closed system; idealized, reversible process) . {\displaystyle \mathrm {d} S={\frac {\delta Q}{T}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{(closed system; idealized, reversible process)}}.}
Different notations are used for an infinitesimal amount of heat ( δ ) {\displaystyle (\delta )} and infinitesimal change of entropy ( d ) {\displaystyle (\mathrm {d} )} because entropy is a function of state, while heat, like work, is not. For an actually possible infinitesimal process without exchange of mass with the surroundings, the second law requires that the increment in system entropy fulfills the inequality
d S > δ Q T surr (closed system; actually possible, irreversible process). {\displaystyle \mathrm {d} S>{\frac {\delta Q}{T_{\text{surr}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{(closed system; actually possible, irreversible process).}}}
This is because a general process for this case (no mass exchange between the system and its surroundings) may include work being done on the system by its surroundings, which can have frictional or viscous effects inside the system, because a chemical reaction may be in progress, or because heat transfer actually occurs only irreversibly, driven by a finite difference between the system temperature (T) and the temperature of the surroundings (Tsurr). The equality still applies for pure heat flow (only heat flow, no change in chemical composition and mass),
d S = δ Q T (actually possible quasistatic irreversible process without composition change). {\displaystyle \mathrm {d} S={\frac {\delta Q}{T}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{(actually possible quasistatic irreversible process without composition change).}}}
which is the basis of the accurate determination of the absolute entropy of pure substances from measured heat capacity curves and entropy changes at phase transitions, i.e. by calorimetry. The zeroth law of thermodynamics in its usual short statement allows recognition that two bodies in a relation of thermal equilibrium have the same temperature, especially that a test body has the same temperature as a reference thermometric body. For a body in thermal equilibrium with another, there are indefinitely many empirical temperature scales, in general respectively depending on the properties of a particular reference thermometric body. The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body.
Various statements of the law The second law of thermodynamics may be expressed in many specific ways, the most prominent classical statements being the statement by Rudolf Clausius (1854), the statement by Lord Kelvin (1851), and the statement in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. The Clausius and the Kelvin statements have been shown to be equivalent.
Carnot's principle The historical origin of the second law of thermodynamics was in Sadi Carnot's theoretical analysis of the flow of heat in steam engines (1824). The centerpiece of that analysis, now known as a Carnot engine, is an ideal heat engine fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. It represents the theoretical maximum efficiency of a heat engine operating between any two given thermal or heat reservoirs at different temperatures. Carnot's principle was recognized by Carnot at a time when the caloric theory represented the dominant understanding of the nature of heat, before the recognition of the first law of thermodynamics, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, Carnot's analysis is physically equivalent to the second law of thermodynamics, and remains valid today. Some samples from his book are:
... wherever there exists a difference of temperature, motive power can be produced. The production of motive power is then due in steam engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body ... The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric. In modern terms, Carnot's principle may be stated more precisely:
The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.
Clausius statement In 1850, the German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics by examining the relation between heat transfer and work. His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:
Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time. The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other. Heat cannot spontaneously flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat is transferred from cold to hot, but only when forced by an external agent, the refrigeration system. Understanding the Clausius statement requires prior definition of the concepts of 'hot versus cold' (temperature) and 'heat' (heat interaction). Within the Gyftopoulos-Beretta formulation of the general principles of thermodynamics, where temperature and heat are defined after the definitions of energy and entropy, the Clausius statement emerges as a rigorous consequence (theorem) of the first law and the Hatsopoulos-Keenan statement of the second law (see below).
Kelvin statements Lord Kelvin expressed the second law in several wordings.
It is impossible for any system to operate in a thermodynamic cycle and deliver a net amount of energy by work to its surroundings while receiving energy by heat transfer from a single thermal reservoir.The Kelvin–Planck statement does not rule out the possibility of a system developing a net amount of work from a heat transfer drawn from a single reservoir. It only denies this possibility if the system undergoes a thermodynamic cycle.
Equivalence of the Clausius and the Kelvin statements
Suppose there is an engine violating the Kelvin statement: i.e., one that drains heat and converts it completely into work (the drained heat is fully converted to work) in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the right figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat Δ Q = Q ( 1 η − 1 ) {\textstyle \Delta Q=Q\left({\frac {1}{\eta }}-1\right)} from the cooler reservoir to the hotter one, which violates the Clausius statement. This is a consequence of the first law of thermodynamics, as for the total system's energy to remain the same; Input + Output = 0 ⟹ ( Q + Q c ) − Q η = 0 {\textstyle {\text{Input}}+{\text{Output}}=0\implies (Q+Q_{c})-{\frac {Q}{\eta }}=0} , so therefore Q c = Q ( 1 η − 1 ) {\textstyle Q_{c}=Q\left({\frac {1}{\eta }}-1\right)} , where (1) the sign convention of heat is used in which heat entering into (leaving from) an engine is positive (negative) and (2) Q η {\displaystyle {\frac {Q}{\eta }}} is obtained by the definition of efficiency of the engine when the engine operation is not reversed. Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.
Planck's proposition Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.
It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the production of work and cooling of a heat reservoir.
Relation between Kelvin's statement and Planck's proposition It is almost customary in textbooks to speak of the "Kelvin–Planck statement" of the law, as for example in the text by ter Haar and Wergeland. This version, also known as the heat engine statement, of the second law states that
It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.
Planck's statement Max Planck stated the second law as follows.
Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged. Rather like Planck's statement is that of George Uhlenbeck and G. W. Ford for irreversible phenomena.
... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.
Principle of Carathéodory Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:
In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S. With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, δ Q = T d S {\displaystyle \delta Q=TdS} . Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.
Planck's principle In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle
The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant. This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible." Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy. A statement that in a sense is complementary to Planck's principle is made by Claus Borgnakke and Richard E. Sonntag. They do not offer it as a full statement of the second law:
... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system. Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.
Hatsopoulos-Keenan statement In 1965, G. N. Hatsopoulos and Joseph Keenan published their textbook Principles of General Thermodynamics, which restates the second law of thermodynamics in terms of the existence of stable equilibrium states. Their formulation of the second law of thermodynamics states that:
When an isolated system performs a process after the removal of a series of internal constraints, it will reach a unique state of stable equilibrium: this state of equilibrium is independent of the order in which the constraints are removed. This shows that the second law of thermodynamics can be stated in terms of the existence of stable equilibrium states. The Hatsopoulos-Keenan statement of the Second Law entails the Clausius, Kelvin-Planck, and Carathéodory statements of the Second Law, and has provided a basis to extend the traditional definition of entropy to the non-equilibrium domain. Hatsopoulus and Keenan maintained that it is unlikely to identify a satisfactory definition of heat without a prior statement of the second law. The Hatsopoulos–Keenan statement of the second law has been developed into a rigorous formulation of the general axiomatic foundations of equilibrium and nonequilibrium, classical, statistical, and quantum thermodynamics by G. N. Hatsopoulos, E. P. Gyftopoulos, E. Zanchini, and G. P. Beretta.
Relating the second law to the definition of temperature The second law has been shown to be equivalent to the internal energy U defined as a convex function of the other extensive properties of the system. That is, when a system is described by stating its internal energy U, an extensive variable, as a function of its entropy S, volume V, and mol number N, i.e. U = U (S, V, N), then the temperature is equal to the partial derivative of the internal energy with respect to the entropy (essentially equivalent to the first TdS equation for V and N held constant):
T = ( ∂ U ∂ S ) V , N {\displaystyle T=\left({\frac {\partial U}{\partial S}}\right)_{V,N}}
Second law statements, such as the Clausius inequality, involving radiative fluxes The Clausius inequality, as well as some other statements of the second law, must be re-stated to have general applicability for all forms of heat transfer, i.e. scenarios involving radiative fluxes. For example, the integrand (đQ/T) of the Clausius expression applies to heat conduction and convection, and the case of ideal infinitesimal blackbody radiation (BR) transfer, but does not apply to most radiative transfer scenarios and in some cases has no physical meaning whatsoever. Consequently, the Clausius inequality was re-stated so that it is applicable to cycles with processes involving any form of heat transfer. The entropy transfer with radiative fluxes ( δ S NetRad \delta S_{\text{NetRad}} ) is taken separately from that due to heat transfer by conduction and convection ( δ Q C C \delta Q_{CC} ), where the temperature is evaluated at the system boundary where the heat transfer occurs. The modified Clausius inequality, for all heat transfer scenarios, can then be expressed as
∫ cycle ( δ Q C C T b + δ S NetRad ) ≤ 0 {\displaystyle \int _{\text{cycle}}\left({\frac {\delta Q_{CC}}{T_{b}}}+\delta S_{\text{NetRad}}\right)\leq 0}
In a nutshell, the Clausius inequality is saying that when a cycle is completed, the change in the state property S will be zero, so the entropy that was produced during the cycle must have transferred out of the system by heat transfer. The δ \delta (or đ) indicates a path dependent integration. Due to the inherent emission of radiation from all matter, most entropy flux calculations involve incident, reflected and emitted radiative fluxes. The energy and entropy of unpolarized blackbody thermal radiation, is calculated using the spectral energy and entropy radiance expressions derived by Max Planck using equilibrium statistical mechanics:
K ν = 2 h c 2 ν 3 exp ( h ν k T ) − 1 , L ν = 2 k ν 2 c 2 ( ( 1 + c 2 K ν 2 h ν 3 ) ln ( 1 + c 2 K ν 2 h ν 3 ) − ( c 2 K ν 2 h ν 3 ) ln ( c 2 K ν 2 h ν 3 ) ) {\displaystyle {\begin{aligned}K_{\nu }&={\frac {2h}{c^{2}}}{\frac {\nu ^{3}}{\exp \left({\frac {h\nu }{kT}}\right)-1}},\\L_{\nu }&={\frac {2k\nu ^{2}}{c^{2}}}((1+{\frac {c^{2}K_{\nu }}{2h\nu ^{3}}})\ln(1+{\frac {c^{2}K_{\nu }}{2h\nu ^{3}}})-({\frac {c^{2}K_{\nu }}{2h\nu ^{3}}})\ln({\frac {c^{2}K_{\nu }}{2h\nu ^{3}}}))\end{aligned}}}
where c is the speed of light, k is the Boltzmann constant, h is the Planck constant, ν is frequency, and the quantities Kv and Lv are the energy and entropy fluxes per unit frequency, area, and solid angle. In deriving this blackbody spectral entropy radiance, with the goal of deriving the blackbody energy formula, Planck postulated that the energy of a photon was quantized (partly to simplify the mathematics), thereby starting quantum theory. A non-equilibrium statistical mechanics approach has also been used to obtain the same result as Planck, indicating it has wider significance and represents a non-equilibrium entropy. A plot of Kv versus frequency (v) for various values of temperature (T) gives a family of blackbody radiation energy spectra, and likewise for the entropy spectra. For non-blackbody radiation (NBR) emission fluxes, the spectral entropy radiance Lv is found by substituting Kv spectral energy radiance data into the Lv expression (noting that emitted and reflected entropy fluxes are, in general, not independent). For the emission of NBR, including graybody radiation (GR), the resultant emitted entropy flux, or radiance L, has a higher ratio of entropy-to-energy (L/K), than that of BR. That is, the entropy flux of NBR emission is farther removed from the conduction and convection q/T result, than that for BR emission. This observation is consistent with Max Planck's blackbody radiation energy and entropy formulas and is consistent with the fact that blackbody radiation emission represents the maximum emission of entropy for all materials with the same temperature, as well as the maximum entropy emission for all radiation with the same energy radiance.
Generalized conceptual statement of the second law principle Second law analysis is valuable in scientific and engineering analysis in that it provides a number of benefits over energy analysis alone, including the basis for determining energy quality (exergy content), understanding fundamental physical phenomena, and improving performance evaluation and optimization. As a result, a conceptual statement of the principle is very useful in engineering analysis. Thermodynamic systems can be categorized by the four combinations of either entropy (S) up or down, and uniformity (Y) – between system and its environment – up or down. This 'special' category of processes, category IV, is characterized by movement in the direction of low disorder and low uniformity, counteracting the second law tendency towards uniformity and disorder.
The second law can be conceptually stated as follows: Matter and energy have the tendency to reach a state of uniformity or internal and external equilibrium, a state of maximum disorder (entropy). Real non-equilibrium processes always produce entropy, causing increased disorder in the universe, while idealized reversible processes produce no entropy and no process is known to exist that destroys entropy. The tendency of a system to approach uniformity may be counteracted, and the system may become more ordered or complex, by the combination of two things, a work or exergy source and some form of instruction or intelligence. Where 'exergy' is the thermal, mechanical, electric or chemical work potential of an energy source or flow, and 'instruction or intelligence', although subjective, is in the context of the set of category IV processes. Consider a category IV example of robotic manufacturing and assembly of vehicles in a factory. The robotic machinery requires electrical work input and instructions, but when completed, the manufactured products have less uniformity with their surroundings, or more complexity (higher order) relative to the raw materials they were made from. Thus, system entropy or disorder decreases while the tendency towards uniformity between the system and its environment is counteracted. In this example, the instructions, as well as the source of work may be internal or external to the system, and they may or may not cross the system boundary. To illustrate, the instructions may be pre-coded and the electrical work may be stored in an energy storage system on-site. Alternatively, the control of the machinery may be by remote operation over a communications network, while the electric work is supplied to the factory from the local electric grid. In addition, humans may directly play, in whole or in part, the role that the robotic machinery plays in manufacturing. In this case, instructions may be involved, but intelligence is either directly responsible, or indirectly responsible, for the direction or application of work in such a way as to counteract the tendency towards disorder and uniformity. There are also situations where the entropy spontaneously decreases by means of energy and entropy transfer. When thermodynamic constraints are not present, spontaneously energy or mass, as well as accompanying entropy, may be transferred out of a system in a progress to reach external equilibrium or uniformity in intensive properties of the system with its surroundings. This occurs spontaneously because the energy or mass transferred from the system to its surroundings results in a higher entropy in the surroundings, that is, it results in higher overall entropy of the system plus its surroundings. Note that this transfer of entropy requires dis-equilibrium in properties, such as a temperature difference. One example of this is the cooling crystallization of water that can occur when the system's surroundings are below freezing temperatures. Unconstrained heat transfer can spontaneously occur, leading to water molecules freezing into a crystallized structure of reduced disorder (sticking together in a certain order due to molecular attraction). The entropy of the system decreases, but the system approaches uniformity with its surroundings (category III). On the other hand, consider the refrigeration of water in a warm environment. Due to refrigeration, as heat is extracted from the water, the temperature and entropy of the water decreases, as the system moves further away from uniformity with its warm surroundings or environment (category IV). The main point, take-away, is that refrigeration not only requires a source of work, it requires designed equipment, as well as pre-coded or direct operational intelligence or instructions to achieve the desired refrigeration effect.
Corollaries
Perpetual motion of the second kind
Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.
Carnot's theorem Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:
All irreversible heat engines between two heat reservoirs are less efficient than a Carnot engine operating between the same reservoirs. All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs. In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realize the Carnot cycle's reversibility and was condemned to be less efficient. Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.
Clausius inequality The Clausius theorem (1854) states that in a cyclic process
∮ δ Q T surr ≤ 0. {\displaystyle \oint {\frac {\delta Q}{T_{\text{surr}}}}\leq 0.}
The equality holds in the reversible case and the strict inequality holds in the irreversible case, with Tsurr as the temperature of the heat bath (surroundings) here. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.
Thermodynamic temperature
For an arbitrary heat engine, the efficiency is:
where Wn is the net work done by the engine per cycle, qH > 0 is the heat added to the engine from a hot reservoir, and qC = −|qC| < 0 is waste heat given off to a cold reservoir from the engine. Thus the efficiency depends only on the ratio |qC| / |qH|. Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, any reversible heat engine operating