SECTION C(1)Existence, Definition, Measure, Relations, Properties, and Scope of Irreversibility and Reversibility.In establishing the existence of irreversibility, we can use one or both of the two general methods of approaching any physical problems (see Introduction, pp.2,3) we can approach by way of the atomic theory or by considering the behavior of aggregates in Nature. Enough has already been said in this presentation of atomic behavior and arrangements to justify the statement that irreversibility is not inherent in the elementary procedures themselves but in their irregular arrangement. The motion of each atom is by itself reversible, but their combined mean effect is to produce something irreversible.[12]This has been rigorously demonstrated by BOLTZMANN'S H-theorem for molecular physics, and when sufficiently general co-ordinates are substituted it is also available for the other domains of natural events. When we consider the behavior of aggregates we recognize at once a general, empirical law, which has also been called theone physical axiom, namely, that all natural processes are essentially irreversible. When we use this method of approach we confessedly rest entirely on experience, and then it does not make any logical difference whether we start with one particular fact or another, whether we start with a fact itself or its necessary consequence: For instance we may recognize that the universe is permanently different after a frictional event from what it was before, or we may start, as PLANCK does, by putting forward the following proposition:"It is impossible to construct an engine which will work ina complete cycle,[13]and produce no effect except the raising of a weight and the cooling of a heat reservoir."[14]Now up to this time no natural event has contradicted this theorem or its corollaries. The proof for it is cumulative, wholly experiential and therefore exactly like that for the law of conservation of energy.Returning to irreversibility, the matter for immediate discussion, we premise that it will here clarify and simplify our ideas if we considerall the participatingbodies as parts of the system experiencing the contemplated process. It is in this sense that we must understand the statement: Every natural event leaves the universe different from what it was before. Speaking very generally, we may say that in this difference lies what we call irreversibility.Now irreversibility is what really does exist, everywhere in Nature, and our idea of reversibility is only a very convenient and fruitful fiction; our conception of reversibility must, therefore, ultimately be derived from that of irreversibility."A process which can in no way be completely reversed is termedirreversible, all other processesreversible. That a process may be irreversible, it is not sufficient that it cannot be directly reversed. This is the case with many mechanical processes which are not irreversible (Seep. 32). The full requirement is, that it be impossible, even with the assistance of all agents in Nature, to restore everywhere the exact initial state when the process has once taken place."Examples of irreversible processes, which involve only heat and mechanical phenomena, may be grouped in four classes:(a) The body whose changes of state are considered is in contact with bodies whose temperature differs by a finite amountfrom its own. There is here flow of heat from the hotter to the colder body and the process is an irreversible one.(b) The body experiences resistance from friction which develops heat; it is not possible to effect completely the opposite operation of restoring the whole system to its initial state.(c) The body expands without at the same time developing an amount ofexternalenergy which is exactly equal to the work of its own elastic forces. For example, this occurs when the pressure which a body has to overcome is essentially (i.e., finitely) less than the body's own internal tension. In such a case it is not possible to bring the whole system (of which the body is a part) completely back into its initial state. Illustrations are: steam escaping from a high-pressure boiler, compressed air flowing into a vacuum tank, and a spring suddenly released from its state of high tension.(d) Two gases at the same pressure and temperature are separated by a partition. When this is suddenly removed, the two gases mix or diffuse. This too is an essentially irreversible process.Outside of chemical phenomena, we may instance still other examples of irreversible processes: flow of electricity in conductors of finite resistance, emission of heat and light radiation, and decomposition of the atoms of radio-active substances."Numerous reversible processes can at leastbe imagined, as, for instance, those consisting throughout of a succession of states of equilibrium, and therefore directly reversible in all their parts. Further, all perfectly periodic processes, e.g., an ideal pendulum or planetary motion, are reversible, for, at the end of every period the initial state is completely restored. Also, all mechanical processes with absolutely rigid bodies and incompressible liquids, as far as friction can be avoided, are reversible. By the introduction of suitable machines with absolutely unyielding connecting-rods, frictionless joints, and bearings, inextensible belts, etc., it is always possible to work the machine in such away as to bring the system completely into its initial state without leaving any change in or out of the machines, for the machines of themselves do not perform any work."Other examples of such reversible processes are: Free fall in a vacuum, propagation of light and sound waves without absorption and reflection and unchecked electrical oscillations. All the latter processes are either naturally periodic, or they can be made completely reversible by suitable devices so that no sort of change in Nature remains behind; for example, the free fall of a body by utilizing the velocity acquired to bring the body back to its original height, light and sound waves by suitably reflecting them from perfect mirrors.[12]This would seem to imply the existence of a broader principle, the properties of systems as a whole arenotnecessarily found in their parts.[13]Such an engine if it would work might be called "perpetual motion of the second kind."[14]The term perpetual is justified because such an engine would possess the most esteemed feature of perpetual motion—power production free of cost.(2)Character of Process Decided by the Limiting States"Since the decision as to whether a particular process is irreversible or reversible depends only on whether the process can in any manner whatsoever be completely reversed or not, the nature of the initial and final states, and not the intermediate steps of the process, entirely settle it. The question is, whether or not it is possible, starting from the final state, to reach the initial one in any way without any other change.... The final state of an irreversible process is evidently in some way discriminate from the initial state, while in reversible processes the two states are in certain respects equivalent.... To discriminate between the two states they must be fully characterized. Besides the chemical constitution of the systems in question, the physical conditions, viz., the state of aggregation, temperature, and pressure in both states, must be known, as is necessary for the application of the First Law.""Let us consider any process whatsoever occurring in Nature. This conducts all participating bodies from a particular initial conditionto a certain final condition. The process is either reversible or irreversible, any third possibility beingexcluded. But whether it is reversible or irreversible depends solely and only on the constitution of the two statesand, not upon the other features of the course; after statehas been attained, we must here simply answer the question whether the complete return tocan or cannot be effected in any manner whatsoever. Now if such complete return fromtois not possible then evidently statein Nature is somehow distinguished from state. Nature may be said to prefer stateto state. Reversible processes are a limiting case; here Nature manifests no preference and the passage from the one to other can take place at pleasure, in either direction. [In the common case of isentropic expansion fromto, there is no exchange of heat with the outside; external work is performed at the expense of the inner energy of the expanding body. When stateis attained we can effect a complete return toby compressing isentropically, thus consuming the external work performed on the trip fromtoand restoring the internal energy of the body.]"Now it becomes a question of finding a physical magnitude whose amount will serve as a general measure of Nature's preference for a state. This must be a magnitude which is directly determined by the state of the contemplated system, without knowing anything of the past history of the system, just as is the case when we deal with the state's energy, volume, etc. This magnitude would possess the property of growing in all irreversible processes, while in all reversible processes it would remain unchanged. The amount of its change in a process would furnish a general measure for the irreversibility of the process.""Now R. CLAUSIUS really found such a magnitude and called it entropy. Every bodily system possesses in every state a particular entropy, and this entropy designates the preference of nature for the state in question; in all the processes which occur in the system, entropy can only grow, never diminish. If we wish to consider a process in whichsaid system is subject to influences from without, we must regard the bodies exerting such influences as incorporated with the original system and then the statement will hold in the above given form."From what has gone before it is evident that the following commonly drawn conclusions are correct:An irreversible process is a passage from a less probable to a more probable state of the system.An irreversible process is a passage from a less stable to a more stable state of the system.An irreversible process is essentially a spontaneous one, inasmuch as once started it will proceed without the help of any external agency.We have in a general way reached the conclusion that entropy is both the criterion and the measure of irreversibility. But now let us become more specific and go more into certain details, namely, the common features in all irreversibility. The property of irreversibility is not inherent in the elementary occurrences themselves, but only in their irregular arrangement. Irreversibility depends only on the statistical property of a system possessing many degrees of freedom, and is therefore essentially based on mean values; in this connection we may repeat an earlier statement, the individual motions of atoms are in themselves reversible, but their result in the aggregate is not.(3)All the Irreversible Processes Stand or Fall TogetherThis is proved with the help of the theorem (p. 30) which denies the possibility of perpetual motion of the second kind.[15]The argument is this: take any case in any one of the four classes of irreversible processes given onp. 31. Now if thisselected case is in reality reversible, i.e., suppose a method were discovered ofcompletelyreversing this process and thus leave no other change whatsoever, then combining the direct course of the process with this latter reversed process, they would together constitute a cyclical process, which would effect nothing but the production of work and the absorption of an equivalent amount of heat. But this would be perpetual motion of the second kind, which to be sure is denied by the empirical theorem onp. 30. But for the sake of the argument we may just now waive said impossibility; then we would have an engine which, co-operating with any second (so-called), irreversible process, would completely restore the initial state of the whole system without leaving any other change whatsoever. Then under our definition onp. 30this second process ceases to be irreversible. The same result will obtain for any third, fourth, etc. So that the above proposition is established. "All the irreversible processes stand or fall together." If any one of them is reversible all are reversible.[16][15]At this stage we appreciate that any irreversible process is a passage from a stateof low entropy to a stateof high entropy. We may simplify our proof by considering the return passage fromtoto in part occur isothermally and in part isentropically; then external agencies must produce work and absorb an equivalent amount of heat.[16]With the help of the preceding footnote this argument can be followed through in detail for each of the cases enumerated onp. 31; only the complicated case of diffusion presents any difficulty.(4)Convenience of the Fiction, the Reversible ProcessesA reversible process we have declared to be only anideal case, a convenient and fruitful fiction which we canimagineby eliminating from an irreversible process one or more of its inevitable accompaniments like friction or heat conduction. But reversible (as well as irreversible) processes have common features. "They resemble each other more than they do any one irreversible process. This is evident from an examination of the differential equations which control them; the differential with respect to time is always of an even order, because the essential sign of time can be reversed. Then too they (in whatever domain of physics they may lie) have the common property that the Principle of Least Actioncan represent all of them completely and uniquely determines the sequence of their events." They are useful for theoretical demonstration and for the study of conditions of equilibrium.There is a certain, limited, incomplete sense in which we say that we can change from one state of equilibrium to another in a reversible manner. For example, we can, considering only the one converting (or intermediate) body, effect said change by a successive use of isentropic and isothermal change. But this ignores all but one of the participating bodies and this is not permissible if we strictly adhere to the true definition ofcompletereversible action.We must remember too that no other universal measure of irreversibility exists than entropy. "Dissipation" of energy has been put forward as such a measure, but we know already of two irreversible cases where there is no change of energy, namely, diffusion and expansion of a gas into a vacuum. [Unavailable, distributed, scattered energy are terms which could be used here, free from all objection.]But of course, the full equivalent of entropy can be substituted as auniversalmeasure of irreversibility. Onp. 27we have pointed out that thenumber of complexionsincluded in a given state can be defined as theprobability W of the state, then in a footnote, attention is called to the identity of entropy with the logarithm of this state of probability = logarithm of the number of complexions of the state. This makes entropy a function of the number of complexions, so that one may in this sense be regarded as the equivalent of the other. We may now properly speak of the number of complexions of a state as the universal measure of its irreversibility. The physical meaning of irreversibility becomes apparent when put in this form. The greater the number of complexions included in a state the more disordered is its elementary condition and the more difficult (more impossible, so to speak), is it to directly so influence the constituents of the whole that they will reverse the sequence of the mean values the aggregate tends of itself to assume. An illustrationwill help to make this clear; the irreversible case in which work (i.e., friction) is converted into heat. "For example, the direct reversal of a frictional process is impossible because this would presuppose the existence of an elementary order among adjacent, mutually interacting molecules. For then it must predominantly be the case that the collisions of each pair of molecules must bear a certain distinguishable character inasmuch as the velocities of two colliding molecules must always depend in a determinate manner on the place where they meet. Only thereby can it be attained that there will result from the collisions predominantly like directed velocities."The outcome of the whole study of irreversibility results in the briefly stated law: "There exists in Nature a quantity which changes always in the same sense in all natural processes."This boldly asserts the essential one-sidedness of Nature. The proposition stated in this general form may be correct or incorrect; but whichever it may be it will remain so independently of human experimental skill.
(1)Existence, Definition, Measure, Relations, Properties, and Scope of Irreversibility and Reversibility.
In establishing the existence of irreversibility, we can use one or both of the two general methods of approaching any physical problems (see Introduction, pp.2,3) we can approach by way of the atomic theory or by considering the behavior of aggregates in Nature. Enough has already been said in this presentation of atomic behavior and arrangements to justify the statement that irreversibility is not inherent in the elementary procedures themselves but in their irregular arrangement. The motion of each atom is by itself reversible, but their combined mean effect is to produce something irreversible.[12]
This has been rigorously demonstrated by BOLTZMANN'S H-theorem for molecular physics, and when sufficiently general co-ordinates are substituted it is also available for the other domains of natural events. When we consider the behavior of aggregates we recognize at once a general, empirical law, which has also been called theone physical axiom, namely, that all natural processes are essentially irreversible. When we use this method of approach we confessedly rest entirely on experience, and then it does not make any logical difference whether we start with one particular fact or another, whether we start with a fact itself or its necessary consequence: For instance we may recognize that the universe is permanently different after a frictional event from what it was before, or we may start, as PLANCK does, by putting forward the following proposition:
"It is impossible to construct an engine which will work ina complete cycle,[13]and produce no effect except the raising of a weight and the cooling of a heat reservoir."[14]
Now up to this time no natural event has contradicted this theorem or its corollaries. The proof for it is cumulative, wholly experiential and therefore exactly like that for the law of conservation of energy.
Returning to irreversibility, the matter for immediate discussion, we premise that it will here clarify and simplify our ideas if we considerall the participatingbodies as parts of the system experiencing the contemplated process. It is in this sense that we must understand the statement: Every natural event leaves the universe different from what it was before. Speaking very generally, we may say that in this difference lies what we call irreversibility.
Now irreversibility is what really does exist, everywhere in Nature, and our idea of reversibility is only a very convenient and fruitful fiction; our conception of reversibility must, therefore, ultimately be derived from that of irreversibility.
"A process which can in no way be completely reversed is termedirreversible, all other processesreversible. That a process may be irreversible, it is not sufficient that it cannot be directly reversed. This is the case with many mechanical processes which are not irreversible (Seep. 32). The full requirement is, that it be impossible, even with the assistance of all agents in Nature, to restore everywhere the exact initial state when the process has once taken place."
Examples of irreversible processes, which involve only heat and mechanical phenomena, may be grouped in four classes:
(a) The body whose changes of state are considered is in contact with bodies whose temperature differs by a finite amountfrom its own. There is here flow of heat from the hotter to the colder body and the process is an irreversible one.
(b) The body experiences resistance from friction which develops heat; it is not possible to effect completely the opposite operation of restoring the whole system to its initial state.
(c) The body expands without at the same time developing an amount ofexternalenergy which is exactly equal to the work of its own elastic forces. For example, this occurs when the pressure which a body has to overcome is essentially (i.e., finitely) less than the body's own internal tension. In such a case it is not possible to bring the whole system (of which the body is a part) completely back into its initial state. Illustrations are: steam escaping from a high-pressure boiler, compressed air flowing into a vacuum tank, and a spring suddenly released from its state of high tension.
(d) Two gases at the same pressure and temperature are separated by a partition. When this is suddenly removed, the two gases mix or diffuse. This too is an essentially irreversible process.
Outside of chemical phenomena, we may instance still other examples of irreversible processes: flow of electricity in conductors of finite resistance, emission of heat and light radiation, and decomposition of the atoms of radio-active substances.
"Numerous reversible processes can at leastbe imagined, as, for instance, those consisting throughout of a succession of states of equilibrium, and therefore directly reversible in all their parts. Further, all perfectly periodic processes, e.g., an ideal pendulum or planetary motion, are reversible, for, at the end of every period the initial state is completely restored. Also, all mechanical processes with absolutely rigid bodies and incompressible liquids, as far as friction can be avoided, are reversible. By the introduction of suitable machines with absolutely unyielding connecting-rods, frictionless joints, and bearings, inextensible belts, etc., it is always possible to work the machine in such away as to bring the system completely into its initial state without leaving any change in or out of the machines, for the machines of themselves do not perform any work."
Other examples of such reversible processes are: Free fall in a vacuum, propagation of light and sound waves without absorption and reflection and unchecked electrical oscillations. All the latter processes are either naturally periodic, or they can be made completely reversible by suitable devices so that no sort of change in Nature remains behind; for example, the free fall of a body by utilizing the velocity acquired to bring the body back to its original height, light and sound waves by suitably reflecting them from perfect mirrors.
[12]This would seem to imply the existence of a broader principle, the properties of systems as a whole arenotnecessarily found in their parts.
[12]This would seem to imply the existence of a broader principle, the properties of systems as a whole arenotnecessarily found in their parts.
[13]Such an engine if it would work might be called "perpetual motion of the second kind."
[13]Such an engine if it would work might be called "perpetual motion of the second kind."
[14]The term perpetual is justified because such an engine would possess the most esteemed feature of perpetual motion—power production free of cost.
[14]The term perpetual is justified because such an engine would possess the most esteemed feature of perpetual motion—power production free of cost.
(2)Character of Process Decided by the Limiting States
"Since the decision as to whether a particular process is irreversible or reversible depends only on whether the process can in any manner whatsoever be completely reversed or not, the nature of the initial and final states, and not the intermediate steps of the process, entirely settle it. The question is, whether or not it is possible, starting from the final state, to reach the initial one in any way without any other change.... The final state of an irreversible process is evidently in some way discriminate from the initial state, while in reversible processes the two states are in certain respects equivalent.... To discriminate between the two states they must be fully characterized. Besides the chemical constitution of the systems in question, the physical conditions, viz., the state of aggregation, temperature, and pressure in both states, must be known, as is necessary for the application of the First Law."
"Let us consider any process whatsoever occurring in Nature. This conducts all participating bodies from a particular initial conditionto a certain final condition. The process is either reversible or irreversible, any third possibility beingexcluded. But whether it is reversible or irreversible depends solely and only on the constitution of the two statesand, not upon the other features of the course; after statehas been attained, we must here simply answer the question whether the complete return tocan or cannot be effected in any manner whatsoever. Now if such complete return fromtois not possible then evidently statein Nature is somehow distinguished from state. Nature may be said to prefer stateto state. Reversible processes are a limiting case; here Nature manifests no preference and the passage from the one to other can take place at pleasure, in either direction. [In the common case of isentropic expansion fromto, there is no exchange of heat with the outside; external work is performed at the expense of the inner energy of the expanding body. When stateis attained we can effect a complete return toby compressing isentropically, thus consuming the external work performed on the trip fromtoand restoring the internal energy of the body.]
"Now it becomes a question of finding a physical magnitude whose amount will serve as a general measure of Nature's preference for a state. This must be a magnitude which is directly determined by the state of the contemplated system, without knowing anything of the past history of the system, just as is the case when we deal with the state's energy, volume, etc. This magnitude would possess the property of growing in all irreversible processes, while in all reversible processes it would remain unchanged. The amount of its change in a process would furnish a general measure for the irreversibility of the process."
"Now R. CLAUSIUS really found such a magnitude and called it entropy. Every bodily system possesses in every state a particular entropy, and this entropy designates the preference of nature for the state in question; in all the processes which occur in the system, entropy can only grow, never diminish. If we wish to consider a process in whichsaid system is subject to influences from without, we must regard the bodies exerting such influences as incorporated with the original system and then the statement will hold in the above given form."
From what has gone before it is evident that the following commonly drawn conclusions are correct:
An irreversible process is a passage from a less probable to a more probable state of the system.
An irreversible process is a passage from a less stable to a more stable state of the system.
An irreversible process is essentially a spontaneous one, inasmuch as once started it will proceed without the help of any external agency.
We have in a general way reached the conclusion that entropy is both the criterion and the measure of irreversibility. But now let us become more specific and go more into certain details, namely, the common features in all irreversibility. The property of irreversibility is not inherent in the elementary occurrences themselves, but only in their irregular arrangement. Irreversibility depends only on the statistical property of a system possessing many degrees of freedom, and is therefore essentially based on mean values; in this connection we may repeat an earlier statement, the individual motions of atoms are in themselves reversible, but their result in the aggregate is not.
(3)All the Irreversible Processes Stand or Fall Together
This is proved with the help of the theorem (p. 30) which denies the possibility of perpetual motion of the second kind.[15]The argument is this: take any case in any one of the four classes of irreversible processes given onp. 31. Now if thisselected case is in reality reversible, i.e., suppose a method were discovered ofcompletelyreversing this process and thus leave no other change whatsoever, then combining the direct course of the process with this latter reversed process, they would together constitute a cyclical process, which would effect nothing but the production of work and the absorption of an equivalent amount of heat. But this would be perpetual motion of the second kind, which to be sure is denied by the empirical theorem onp. 30. But for the sake of the argument we may just now waive said impossibility; then we would have an engine which, co-operating with any second (so-called), irreversible process, would completely restore the initial state of the whole system without leaving any other change whatsoever. Then under our definition onp. 30this second process ceases to be irreversible. The same result will obtain for any third, fourth, etc. So that the above proposition is established. "All the irreversible processes stand or fall together." If any one of them is reversible all are reversible.[16]
[15]At this stage we appreciate that any irreversible process is a passage from a stateof low entropy to a stateof high entropy. We may simplify our proof by considering the return passage fromtoto in part occur isothermally and in part isentropically; then external agencies must produce work and absorb an equivalent amount of heat.
[15]At this stage we appreciate that any irreversible process is a passage from a stateof low entropy to a stateof high entropy. We may simplify our proof by considering the return passage fromtoto in part occur isothermally and in part isentropically; then external agencies must produce work and absorb an equivalent amount of heat.
[16]With the help of the preceding footnote this argument can be followed through in detail for each of the cases enumerated onp. 31; only the complicated case of diffusion presents any difficulty.
[16]With the help of the preceding footnote this argument can be followed through in detail for each of the cases enumerated onp. 31; only the complicated case of diffusion presents any difficulty.
(4)Convenience of the Fiction, the Reversible Processes
A reversible process we have declared to be only anideal case, a convenient and fruitful fiction which we canimagineby eliminating from an irreversible process one or more of its inevitable accompaniments like friction or heat conduction. But reversible (as well as irreversible) processes have common features. "They resemble each other more than they do any one irreversible process. This is evident from an examination of the differential equations which control them; the differential with respect to time is always of an even order, because the essential sign of time can be reversed. Then too they (in whatever domain of physics they may lie) have the common property that the Principle of Least Actioncan represent all of them completely and uniquely determines the sequence of their events." They are useful for theoretical demonstration and for the study of conditions of equilibrium.
There is a certain, limited, incomplete sense in which we say that we can change from one state of equilibrium to another in a reversible manner. For example, we can, considering only the one converting (or intermediate) body, effect said change by a successive use of isentropic and isothermal change. But this ignores all but one of the participating bodies and this is not permissible if we strictly adhere to the true definition ofcompletereversible action.
We must remember too that no other universal measure of irreversibility exists than entropy. "Dissipation" of energy has been put forward as such a measure, but we know already of two irreversible cases where there is no change of energy, namely, diffusion and expansion of a gas into a vacuum. [Unavailable, distributed, scattered energy are terms which could be used here, free from all objection.]
But of course, the full equivalent of entropy can be substituted as auniversalmeasure of irreversibility. Onp. 27we have pointed out that thenumber of complexionsincluded in a given state can be defined as theprobability W of the state, then in a footnote, attention is called to the identity of entropy with the logarithm of this state of probability = logarithm of the number of complexions of the state. This makes entropy a function of the number of complexions, so that one may in this sense be regarded as the equivalent of the other. We may now properly speak of the number of complexions of a state as the universal measure of its irreversibility. The physical meaning of irreversibility becomes apparent when put in this form. The greater the number of complexions included in a state the more disordered is its elementary condition and the more difficult (more impossible, so to speak), is it to directly so influence the constituents of the whole that they will reverse the sequence of the mean values the aggregate tends of itself to assume. An illustrationwill help to make this clear; the irreversible case in which work (i.e., friction) is converted into heat. "For example, the direct reversal of a frictional process is impossible because this would presuppose the existence of an elementary order among adjacent, mutually interacting molecules. For then it must predominantly be the case that the collisions of each pair of molecules must bear a certain distinguishable character inasmuch as the velocities of two colliding molecules must always depend in a determinate manner on the place where they meet. Only thereby can it be attained that there will result from the collisions predominantly like directed velocities."
The outcome of the whole study of irreversibility results in the briefly stated law: "There exists in Nature a quantity which changes always in the same sense in all natural processes."
This boldly asserts the essential one-sidedness of Nature. The proposition stated in this general form may be correct or incorrect; but whichever it may be it will remain so independently of human experimental skill.