4 Challenges to the Second Law In this,its most primordial form,the second law is an injunction against perpetuum mobile of the second type (PM2).Such a device would transform heat from a heat bath into useful work,in principle,indefinitely.It formalizes the reasoning under- girding Carnot's theorem,proposed over 25 years earlier. The second most cited version,and perhaps the most natural and experientially obvious,is due to Clausius (1854)[4]: (2)Clausius-Heat No process is possible for which the sole effect is that heat flows from a reservoir at a given temperature to a reservoir at higher temperature. In the vernacular:Heat flows from hot to cold.In contradistinction to some formu- lations that follow,these two statements make claims about strictly nonequilibrium systems;as such,they cannot be considered equivalent to later equilibrium for- mulations.Also,both versions turn on the key term,sole effect,which specifies that the heat flow must not be aided by external agents or processes.Thus,for example,heat pumps and refrigerators,which do transfer heat from a cold reser- voir to a hot reservoir,do so without violating the second law since they require work input from an external source that inevitably satisfies the law. Other common (and equivalent)statements to these two include: (3)Perpetual Motion Perpetuum mobile of the second type are im- possible. and (4)Refrigerators Perfectly efficient refrigerators are impossible. The primary result of Carnot's work and the root of many second law formu- lations is Carnot's theorem [1]: (5)Carnot theorem All Carnot engines operating between the same two temperatures have the same efficiency. Carnot's theorem is occasionally but not widely cited as the second law.Usually it is deduced from the Kelvin-Planck or Clausius statements.Analysis of the Carnot cycle shows that a portion of the heat flowing through a heat engine must always be lost as waste heat,not to contribute to the overall useful heat output3.The maximum efficiency of heat engines is given by the Carnot efficiency:n1-, where Te.h are the temperatures of the colder and hotter heat reservoirs between which the heat engine operates.Since absolute zero(Te=0)is unattainable(by one version of the third law)and since Thoo for any realistic system,the Carnot efficiency forbids perfect conversion of heat into work (i.e.,n 1).Equivalent second law formulations embody this observation: 3One could say that the second law is Nature's tax on the first
4 Challenges to the Second Law In this, its most primordial form, the second law is an injunction against perpetuum mobile of the second type (PM2). Such a device would transform heat from a heat bath into useful work, in principle, indefinitely. It formalizes the reasoning undergirding Carnot’s theorem, proposed over 25 years earlier. The second most cited version, and perhaps the most natural and experientially obvious, is due to Clausius (1854) [4]: (2) Clausius-Heat No process is possible for which the sole effect is that heat flows from a reservoir at a given temperature to a reservoir at higher temperature. In the vernacular: Heat flows from hot to cold. In contradistinction to some formulations that follow, these two statements make claims about strictly nonequilibrium systems; as such, they cannot be considered equivalent to later equilibrium formulations. Also, both versions turn on the key term, sole effect, which specifies that the heat flow must not be aided by external agents or processes. Thus, for example, heat pumps and refrigerators, which do transfer heat from a cold reservoir to a hot reservoir, do so without violating the second law since they require work input from an external source that inevitably satisfies the law. Other common (and equivalent) statements to these two include: (3) Perpetual Motion Perpetuum mobile of the second type are impossible. and (4) Refrigerators Perfectly efficient refrigerators are impossible. The primary result of Carnot’s work and the root of many second law formulations is Carnot’s theorem [1]: (5) Carnot theorem All Carnot engines operating between the same two temperatures have the same efficiency. Carnot’s theorem is occasionally but not widely cited as the second law. Usually it is deduced from the Kelvin-Planck or Clausius statements. Analysis of the Carnot cycle shows that a portion of the heat flowing through a heat engine must always be lost as waste heat, not to contribute to the overall useful heat output3. The maximum efficiency of heat engines is given by the Carnot efficiency: η = 1 − Tc Th , where Tc,h are the temperatures of the colder and hotter heat reservoirs between which the heat engine operates. Since absolute zero (Tc = 0) is unattainable (by one version of the third law) and since Th = ∞ for any realistic system, the Carnot efficiency forbids perfect conversion of heat into work (i.e., η = 1). Equivalent second law formulations embody this observation: 3One could say that the second law is Nature’s tax on the first
Chapter 1:Entropy and the Second Law 5 (6)Efficiency All Carnot engines have efficiencies satisfying: 0<n<1. and. (7)Heat Engines Perfectly efficient heat engines(n=1)are impos- sible. The efficiency form is not cited in textbooks,but is suggested as valid by Koenig [9].There is disagreement over whether Carnot should be credited with the dis- covery of the second law 10.Certainly,he did not enunciate it explicitly,but he seems to have understood it in spirit and his work was surely a catalyst for later, explicit statements of it. Throughout this discussion it is presumed that realizable heat engines must operate between two reservoirs at different temperatures.(Te and Th).This con- dition is considered so stringent that it is often invoked as a litmus test for second law violators;that is,if a heat engine purports to operate at a single temperature, it violates the second law.Of course,mathematically this is no more than assert- ing n =1,which is already forbidden. Since thermodynamics was initially motivated by the exigencies of the indus- trial revolution,it is unsurprising that many of its formulations involve engines and cycles. (8)Cycle Theorem Any physically allowed heat engine,when oper- ated in a cycle,satisfies the condition [6Q =0 (1.2) if the cycle is reversible;and 6Q T <0 (1.3) if the cycle is irreversible Again,6Q is the inexact differential of heat.This theorem is widely cited in the thermodynamic literature,but is infrequently forwarded as a statement of the sec- ond law.In discrete summation form for reversible cycles (Qi/T;=0),it was proposed early on by Kelvin [5]as a statement of the second law. (9)Irreversibility All natural processes are irreversible. Irreversibility is an essential feature of natural processes and it is the essential thermodynamic characteristic defining the direction of time-e.g.,omelettes do 4It is often said that irreversibility gives direction to time's arrow.Perhaps one should say irreversibility is time's arrow [11-17]
Chapter 1: Entropy and the Second Law 5 (6) Efficiency All Carnot engines have efficiencies satisfying: 0 <η< 1. and, (7) Heat Engines Perfectly efficient heat engines (η = 1) are impossible. The efficiency form is not cited in textbooks, but is suggested as valid by Koenig [9]. There is disagreement over whether Carnot should be credited with the discovery of the second law [10]. Certainly, he did not enunciate it explicitly, but he seems to have understood it in spirit and his work was surely a catalyst for later, explicit statements of it. Throughout this discussion it is presumed that realizable heat engines must operate between two reservoirs at different temperatures. (Tc and Th). This condition is considered so stringent that it is often invoked as a litmus test for second law violators; that is, if a heat engine purports to operate at a single temperature, it violates the second law. Of course, mathematically this is no more than asserting η = 1, which is already forbidden. Since thermodynamics was initially motivated by the exigencies of the industrial revolution, it is unsurprising that many of its formulations involve engines and cycles. (8) Cycle Theorem Any physically allowed heat engine, when operated in a cycle, satisfies the condition δQ T = 0 (1.2) if the cycle is reversible; and δQ T < 0 (1.3) if the cycle is irreversible. Again, δQ is the inexact differential of heat. This theorem is widely cited in the thermodynamic literature, but is infrequently forwarded as a statement of the second law. In discrete summation form for reversible cycles ( i Qi/Ti = 0), it was proposed early on by Kelvin [5] as a statement of the second law. (9) Irreversibility All natural processes are irreversible. Irreversibility is an essential feature of natural processes and it is the essential thermodynamic characteristic defining the direction of time4 — e.g., omelettes do 4It is often said that irreversibility gives direction to time’s arrow. Perhaps one should say irreversibility is time’s arrow [11-17].
6 Challenges to the Second Law not spontaneously unscramble;redwood trees do not 'ungrow';broken Ming vases do not reassemble;the dead to not come back to life.An irreversible process is, by definition,not quasi-static (reversible);it cannot be undone without additional irreversible changes to the universe.Irreversibility is so undeniably observed as an essential behavior of the physical world that it is put forward by numerous authors in second law statements. In many thermodynamic texts,natural and irreversible are equated,in which case this formulation is tautological;however,as a reminder of the essential con- tent of the law,it is unsurpassed.In fact,it is so deeply understood by most scientists as to be superfluous. A related formulation,advanced by Koenig 9 reads: (10)Reversibility All normal quasi-static processes are reversible, and conversely. Koenig claims,"this statement goes further than the irreversibility statement] in that it supplies a necessary and sufficient condition for reversibility (and irre- versibility)."This may be true,but it is also sufficiently obtuse to be forgettable; it does not appear in the literature beyond Koenig. Koenig also offers the following orphan version 9]: (11)Free Expansion Adiabatic free expansion of a perfect gas is an irreversible process. He demonstrates that,within his thermodynamic framework,this proposition is equivalent to the statement,"If a [PM2]is possible,then free expansion of a gas is a reversible process;and conversely."Of course,since adiabatic free expansion is irreversible,it follows perpetuum mobile are logically impossible-a standard statement of the second law.By posing the second law in terms of a particu- lar physical process(adiabatic expansion),the door is opened to use any natural (irreversible)process as the basis of a second law statement.It also serves as a reminder that the second law is not only of the world and in the world,but,in an operational sense,it is the world.This formulation also does not enjoy citation outside Koenig[⑨]. A relatively recent statement is proposed by Macdonald [18.Consider a system Z,which is closed with respect to material transfers,but to which heat and work can be added or subtracted so as to change its state from A to B by an arbitrary process P that is not necessarily quasi-static.Heat (Hp)is added by a standard heat source,taken by Macdonald to be a reservoir of water at its triple point.The second law is stated: (12)Macdonald 18 It is impossible to transfer an arbitrarily large amount of heat from a standard heat source with processes terminating at a fixed state of Z.In other words,for every state B of Z
6 Challenges to the Second Law not spontaneously unscramble; redwood trees do not ‘ungrow’; broken Ming vases do not reassemble; the dead to not come back to life. An irreversible process is, by definition, not quasi-static (reversible); it cannot be undone without additional irreversible changes to the universe. Irreversibility is so undeniably observed as an essential behavior of the physical world that it is put forward by numerous authors in second law statements. In many thermodynamic texts, natural and irreversible are equated, in which case this formulation is tautological; however, as a reminder of the essential content of the law, it is unsurpassed. In fact, it is so deeply understood by most scientists as to be superfluous. A related formulation, advanced by Koenig [9] reads: (10) Reversibility All normal quasi-static processes are reversible, and conversely. Koenig claims, “this statement goes further than [the irreversibility statement] in that it supplies a necessary and sufficient condition for reversibility (and irreversibility).” This may be true, but it is also sufficiently obtuse to be forgettable; it does not appear in the literature beyond Koenig. Koenig also offers the following orphan version [9]: (11) Free Expansion Adiabatic free expansion of a perfect gas is an irreversible process. He demonstrates that, within his thermodynamic framework, this proposition is equivalent to the statement, “If a [PM2] is possible, then free expansion of a gas is a reversible process; and conversely.” Of course, since adiabatic free expansion is irreversible, it follows perpetuum mobile are logically impossible — a standard statement of the second law. By posing the second law in terms of a particular physical process (adiabatic expansion), the door is opened to use any natural (irreversible) process as the basis of a second law statement. It also serves as a reminder that the second law is not only of the world and in the world, but, in an operational sense, it is the world. This formulation also does not enjoy citation outside Koenig [9]. A relatively recent statement is proposed by Macdonald [18]. Consider a system Z, which is closed with respect to material transfers, but to which heat and work can be added or subtracted so as to change its state from A to B by an arbitrary process P that is not necessarily quasi-static. Heat (HP ) is added by a standard heat source, taken by Macdonald to be a reservoir of water at its triple point. The second law is stated: (12) Macdonald [18] It is impossible to transfer an arbitrarily large amount of heat from a standard heat source with processes terminating at a fixed state of Z. In other words, for every state B of Z
Chapter 1:Entropy and the Second Law 7 Sup[Hp P terminates at B<oo, where Sup[...is the supremum of heat for the process P. Absolute entropy is defined easily from here as the supremum of the heat Hp divided by a fiduciary temperature To,here taken to be the triple point of water (273.16 K);that is,S(B)=Sup[Hp/To:P terminates at B].Like most formu- lations of entropy and the second law,these apply strictly to closed equilibrium systems. Many researchers take equilibrium as the sine qua non for the second law. (13)Equilibrium The macroscopic properties of an isolated nonstatic system eventually assume static values. Note that here,as with many equivalent versions,the term equilibrium is purpose- fully avoided.A related statement is given by Gyftopolous and Beretta [19]: (14)Gyftopolous and Beretta Among all the states of a system with given values of energy,the amounts of constituents and the pa- rameters,there is one and only one stable equilibrium state.Moreover, starting from any state of a system it is always possible to reach a stable equilibrium state with arbitrary specified values of amounts of constituents and parameters by means of a reversible weight process. (Details of nomenclature (e.g.,weight process)can be found in $1.3.)Several aspects of these two equilibrium statements merit unpacking. Macroscopic properties (e.g.,temperature,number density,pressure)are ones that exhibit statistically smooth behavior at equilibrium.Scale lengths are critical;for example,one expects macroscopic properties for typical liq- uids at scale lengths greater than about 10-6m.At shorter scale lengths statistical fluctuations become important and can undermine the second law. This was understood as far back as Maxwell [20,21,22,23]. There are no truly isolated systems in nature;all are connected by long-range gravitational and perhaps electromagnetic forces;all are likely affected by other uncontrollable interactions,such as by neutrinos,dark matter,dark en- ergy and perhaps local cosmological expansion;and all are inevitably coupled thermally to their surroundings to some degree.Straightforward calculations show,for instance,that the gravitational influence of a minor asteroid in the Asteroid Belt is sufficient to instigate chaotic trajectories of molecules in a parcel of air on Earth in less than a microsecond.Since gravity cannot be screened,the exact molecular dynamics of all realistic systems are constantly affected in essentially unknown and uncontrollable ways.Unless one is able to model the entire universe,one probably cannot exactly model any subset of it.Fortunately,statistical arguments (e.g.,molecular chaos,ergodicity) allow thermodynamics to proceed quite well in most cases. 5Quantum mechanical entanglement,of course,further complicates this task
Chapter 1: Entropy and the Second Law 7 Sup[HP : P terminates at B] < ∞, where Sup[...] is the supremum of heat for the process P. Absolute entropy is defined easily from here as the supremum of the heat HP divided by a fiduciary temperature To, here taken to be the triple point of water (273.16 K); that is, S(B) = Sup[HP /To : P terminates at B]. Like most formulations of entropy and the second law, these apply strictly to closed equilibrium systems. Many researchers take equilibrium as the sine qua non for the second law. (13) Equilibrium The macroscopic properties of an isolated nonstatic system eventually assume static values. Note that here, as with many equivalent versions, the term equilibrium is purposefully avoided. A related statement is given by Gyftopolous and Beretta [19]: (14) Gyftopolous and Beretta Among all the states of a system with given values of energy, the amounts of constituents and the parameters, there is one and only one stable equilibrium state. Moreover, starting from any state of a system it is always possible to reach a stable equilibrium state with arbitrary specified values of amounts of constituents and parameters by means of a reversible weight process. (Details of nomenclature (e.g., weight process) can be found in §1.3.) Several aspects of these two equilibrium statements merit unpacking. • Macroscopic properties (e.g., temperature, number density, pressure) are ones that exhibit statistically smooth behavior at equilibrium. Scale lengths are critical; for example, one expects macroscopic properties for typical liquids at scale lengths greater than about 10−6m. At shorter scale lengths statistical fluctuations become important and can undermine the second law. This was understood as far back as Maxwell [20, 21, 22, 23]. • There are no truly isolated systems in nature; all are connected by long-range gravitational and perhaps electromagnetic forces; all are likely affected by other uncontrollable interactions, such as by neutrinos, dark matter, dark energy and perhaps local cosmological expansion; and all are inevitably coupled thermally to their surroundings to some degree. Straightforward calculations show, for instance, that the gravitational influence of a minor asteroid in the Asteroid Belt is sufficient to instigate chaotic trajectories of molecules in a parcel of air on Earth in less than a microsecond. Since gravity cannot be screened, the exact molecular dynamics of all realistic systems are constantly affected in essentially unknown and uncontrollable ways. Unless one is able to model the entire universe, one probably cannot exactly model any subset of it5. Fortunately, statistical arguments (e.g., molecular chaos, ergodicity) allow thermodynamics to proceed quite well in most cases. 5Quantum mechanical entanglement, of course, further complicates this task
8 Challenges to the Second Law .One can distinguish between stable and unstable static (or equilibrium)states, depending on whether they "persist over time intervals significant for some particular purpose in hand."[9].For instance,to say "Diamonds are for- ever."is to assume much.Diamond is a metastable state of carbon un- der everyday conditions;at elevated temperatures (2000 K),it reverts to graphite.In a large enough vacuum,graphite will evaporate into a vapor of carbon atoms and they,in turn,will thermally ionize into a plasma of elec- trons and ions.After 103 years,the protons might decay,leaving a tenuous soup of electrons,positrons,photons,and neutrinos.Which of these is a stable equilibrium?None or each,depending on the time scale and environ- ment of interest.By definition,a stable static state is one that can change only if its surroundings change,but still,time is a consideration.To a large degree,equilibrium is a matter of taste,time,and convenience. Gyftopoulos and Beretta emphasise one and only one stable equilibrium state.This is echoed by others,notably by Mackey who reserves this caveat for his strong form of the second law 24. Thus far,entropy has not entered into any of these second law formulations. Although,in everyday scientific discourse the two are inextricably linked,this is clearly not the case.Entropy was defined by Clausius in 1865,nearly 15 years after the first round of explicit second law formulations.Since entropy was origi- nally wrought in terms of heat and temperature,this allows one to recast earlier formulations easily.Naturally,the first comes from Clausius: (15)Clausius-Entropy [4,6]For an adiabatically isolated system that undergoes a change from one equilibrium state to another,if the thermodynamic process is reversible,then the entropy change is zero;if the process is irreversible,the entropy change is positive.Respectively, this is: 6Q =Sj -Si (1.4) and 8Q <Sy-S (1.5) Ji T Planck(1858-1947),a disciple of Clausius,refines this into what he describes as "the most general expression of the second law of thermodynamics."[8,6] (16)Planck Every physical or chemical process occurring in nature proceeds in such a way that the sum of the entropies of all bodies which participate in any way in the process is increased.In the limiting case, for reversible processes,the sum remains unchanged. Alongside the Kelvin-Planck version,these two statements have dominated the scientific landscape for nearly a century and a half.Planck's formulation implic- itly cuts the original ties between entropy and heat,thereby opening the door for
8 Challenges to the Second Law • One can distinguish between stable and unstable static (or equilibrium) states, depending on whether they “persist over time intervals significant for some particular purpose in hand.” [9]. For instance, to say “Diamonds are forever.” is to assume much. Diamond is a metastable state of carbon under everyday conditions; at elevated temperatures (∼ 2000 K), it reverts to graphite. In a large enough vacuum, graphite will evaporate into a vapor of carbon atoms and they, in turn, will thermally ionize into a plasma of electrons and ions. After 1033 years, the protons might decay, leaving a tenuous soup of electrons, positrons, photons, and neutrinos. Which of these is a stable equilibrium? None or each, depending on the time scale and environment of interest. By definition, a stable static state is one that can change only if its surroundings change, but still, time is a consideration. To a large degree, equilibrium is a matter of taste, time, and convenience. • Gyftopoulos and Beretta emphasise one and only one stable equilibrium state. This is echoed by others, notably by Mackey who reserves this caveat for his strong form of the second law [24]. Thus far, entropy has not entered into any of these second law formulations. Although, in everyday scientific discourse the two are inextricably linked, this is clearly not the case. Entropy was defined by Clausius in 1865, nearly 15 years after the first round of explicit second law formulations. Since entropy was originally wrought in terms of heat and temperature, this allows one to recast earlier formulations easily. Naturally, the first comes from Clausius: (15) Clausius-Entropy [4, 6] For an adiabatically isolated system that undergoes a change from one equilibrium state to another, if the thermodynamic process is reversible, then the entropy change is zero; if the process is irreversible, the entropy change is positive. Respectively, this is: f i δQ T = Sf − Si (1.4) and f i δQ T < Sf − Si (1.5) Planck (1858-1947), a disciple of Clausius, refines this into what he describes as “the most general expression of the second law of thermodynamics.” [8, 6] (16) Planck Every physical or chemical process occurring in nature proceeds in such a way that the sum of the entropies of all bodies which participate in any way in the process is increased. In the limiting case, for reversible processes, the sum remains unchanged. Alongside the Kelvin-Planck version, these two statements have dominated the scientific landscape for nearly a century and a half. Planck’s formulation implicitly cuts the original ties between entropy and heat, thereby opening the door for