PRESENTATION OF SECOND LAW THEORY. II.* A PROOF OF THE SECOND LAW FOR SYSTEMS COMPOSED OF PERFECT GASES T. H. HAZLEHURST, JR..LEHIOHUNNERSITY.BETHLEHEM, PENNSYLVANIA The purpose of this article i s lo show that the complete Second Law relationship AS 2 0 may be obtained for a system of perfect gases m'thout the use of any of the "principles" from which the law i s usually derived. The basic assumplions are (1) the First Law, (2) the defining equations of a perfect gas, and (3) that portion of mechanics which i s usually assumed in thermodynamics, thut is, the portion which indicates that the work done by a n expanding gas is given by sw = paw.
.
. .
. . .
.
Definition of Entropy * A perfect gas is defined by the equations pu = RT; E
=
const. X T .
These equations are valid for one gram-molecular weight of the gas. The ., the molal heat capacity constant of the energy equation is obviously C of the gas at constant volume. Thus for a perfect gas: pdu = RT d lnv;
dE = U T .
Using the First Law in the form 6q = dE
+ 6w = dE + pduY*
and substituting the values of the quantities on the right, 657 = C d T + R T d l n u .
Now although 69 is not a perfect Merential it may evidently be transformed into one by the multiplying factor 1/T, since 6q/T = C d T / T 4-R du/u and the right side is a perfect diierential because d/dv(C./T) = 0 = d/dT(R/v).
Hence 69/T defines the differential of a point function, S , which may be named entropy. Change in S during Reversible Changes of State Consider a perfect gas passing reversibly from state A to state B along any fixed path. Entropy having been defined to the extent of an additive * Cj. HAZLEHURST, J. CHEM. EDUC., 8,498 (March, 1931).
"The substitution of for 6w automatically assumes that theonly workdoneis mechanical work of expansion. Electrical, magnetic, surface, and chemical forces are ignored on the ground that they are essentially foreign to the motion of a perfect gas. 1087
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constant in the previous section, we may represent the momentary state of the system by a point in a temperature-entropy diagram, values of entropy being assigned relative to an arbitrary standard state. (See Figure 1.) Suppose the path representing the given change of state in this diagram to be AEFB. Subdivide this path into many small portions and let the. 7th subdivision be represented by the arc, EF, during the traversal of which the gas will do a quantity of work 6w, and absorb from its surroundings a quantity of heat Through each point of division draw the corresponding isothermal and adiabatic of the gas. It is evident that adiabatics, being defined as paths along which 69 = 0, must be identical with isentropics along which dS = 0. The required isothermals and gdiabatics will then be, as drawn, segments of horizontal and vertical lines, respectively. Now let us replace the original path by another formed of the mutually intersected portions OF the isentropics and isothermals. For example, replace the arc, EF, by the broken line, EGF. Let the heat which would be absorbed by the gas in traversing EG be a$. Of course there would be no heat exchanged with the surroundings during the traversal of GF. In this way each arc of the original cycle is replaced by a broken line alone the isothermal segment of which heat is absorbed. It is evident that, in general, the heat taken in as the gas moves through the states E t o F via E F will be different from that taken in as i t moves from E to F via EGF, that is, 6 g , ~ 6 q : . In fact, these quantities of heat are, respectively,
-
F
f TdS and It
G JTdS E
and these are equal to the areas, EFFIE' and EGF'E', respectively. The difference is the triangular area, EFG. But, as the points of division be-
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come more and more numerous and closer and closer together, it is obvious that, EF 0, as 69, EG X (EE' f GF/2) 0, 69,' EG X EE' 0, and 69, - 69,' EG X GF/2 0 X 0.
--- - -
so that the difference between the two quantities of heat is an infinitesimal of the second order while the quantities themselves are infinitesimals of the first order only. Consequently, in the limit, when these points of subdivision are infinitely close together, the differencebetween the areas under the two cycles becomes zero and 6qr = 8q:. The same is naturally true of the corresponding quantities of work since in the reversible .cycle EFG the total work done must equal the total heat taken in, which is zero. Of these two thermodynamically identical paths, the one represented by the smooth curve, AEFG, and the other by a very large number of very small isothermal and isentropic steps, the latter is the more convenient for the present purpose since it has the advantage of allowing the heat taken in a t each step to be absorbed isothermally. If we suppose the heat to be furnished by, say, a perfect gas confined in a cylinder closed by a piston so that the heat may be furnished the working gas by compressicn of the reservoir gas, then, during any infinitesimal step of the path the change in entropy of the working gas is 6q,/T, and that of the reservoir gas is -6q,/T,. Thus the total change of entropy at each infinitesimal step is zero and the same must be true of the total suctession of such steps from A to B. Therefore in any arbitrary change of state the total change in entropy is zero, provided the change of state be reversible. Change in S during Irreversible Changes of State I t is instructive to examine the mathematical significance of reversibility and irreversibility. From the physical point of view reversibility is characterized by uniformity of parameter values throughout the system. For example, temperature, pressure, and density are nowhere more than infinitesimally different from their equilibrium or macroscopic values. This means that, durinq a reversible change of state, the system passes through a succession of states of equilibrium. If a t any moment the reversible passage of the system from state A to state B 1s brought to a complete stop, no further observable alteration of the determining variables, say $, T, will occur during a finite period of time, and the state of the system may be represented by a point in the $-T diagram. Similarly, if the process were stopped a t any other moment it would be discovered that at that instant also the system was in a state completely determined by the same variables used to define the state of the system when a t rest. In this
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series the states merge imperceptibly into one another and the representative points form a mathematically continuous sequence, a curve. Mathematically, then, reversibility is characterized by the fact that the state of the system is at all times a function of the n variables representing the n degrees of freedom of the system when it is in static equilibrium. Irreversibility may be identically defined as the aggregate of properties characteristic of changes of state which are not reversible. In particular the values of the "static determining variables" are at any instant different for different points in the system. If the passage of the system from state A to state B is arrested at a particular instant the finite variations in temperature, pressure, and density throughout the system require a finite time to become ironed out into uniform equilibrium values. Thus, although the arrested system finally attains a state of equilibrium capable of definition in terms of, say, two variables, yet at the initial instant at which the irreversible process was arrested the two variables were multiplevalued. In order at that moment to define the state completely it would be necessary to know the values of T and p not only a t any one point but at all points: the number of defining variables is very Zurge. The state could not be represented by a point in a two-dimensional diagram; it is no longer a function of two variables; and the whole process cannot be represented by a continuous curve as it was in the case of reversibility. Nor can the process be represented, as might be supposed, by the series of equilibrium states which could be obtained by taking a very large number of identically similar systems simultanegusly under identical irreversible conditions from state A toward state B and arresting them one at a time at the end of successive short time intervals. It is true that in this way, from any given irreversible process, a corresponding reversible process could be uniquely derived, but the converse is not true; for, to any given reversible process, a very large number of irreversible processes might conceivably correspond. Hence an analysis of the reversible process arrived at in this way would give only ambiguous information concerning the irreversible process. Mathematically, the state of the system considered as a function of two variables does not exist. We conclude that the mathematical distinction between reversibility and irreversibility is that, in the case of the former, the state of the system is a (continuous) function of two variables, whereas, in that of the latter, the state of the system is not definable in terms of two variables. Applying this idea to the defining equation for a change in entropy dS = 6p/T,
it is evident that this relationship which is universally valid for reversible changes in a system of perfect gases is true for only the first infinitesimal step of an irreversible process. During the first step the temperature will
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change only infinitesimally and so there will still be a value of T applicable to the whole system. But it is essentially characteristic of irreversibility that succeeding steps occur so rapidly that the inequalities of temperature (and pressure), initially very small, have no time to be ironed out before new ones arise, so that in a short time tinite variations in temperature (and pressure) exist. In such a case it is obviously impossible to assign a value of T (or p) to the whole system. The symbol 6q/T becomes meaningless. Of course S is still a point function; its value when the system has finally reached state B is the same regardless of what sort of changes the system has suffered in its passage from A to B. The change in entropy Sg - SA is likewise independent of the process. I t is only the right-hand member of the above equation, 6q/T, which has altered its significance. We seek an expression which shall take the place of 6q/T for irreversible as well as reversible changes of state. In order to make the problem more amenable to treatment, consider all the heat gained or lost by the system to be exchanged with a (perfect gas) beat reservoir of h e d temperature To. Further, choose one particular spot on the surface of the system, a spot so small that T may be considered constant all over it, and let the heat be transferred to the system from the reservoir by means of a perfect gas "carrier" performing reversible Camot cycles. Suppose the lower temperature of each cycle to be the momentary temperature of the selected element of the system. By this arrangement the heat exchange with the surroundings is made reversible; all irreversibility of heat exchange occurs within the system dnring the equalization of temperature which is constantly taking place. This does not really limit the generality *of the proceeding, for we may use as the locality of heat transfer not one spot only but as many as we like, taking care to have one heat carrier for each small region over which T may be considered constant. In this way the total surface might be utilized. In such a case let 6q:) represent the heat taken in by the rth dt (the time region on the surface during the short time interval t to 1 interval must be small to secure the sensible constancy of T dnring the heat transfer). Then the total heat absorbed by the system during that small where the summation extends over the whole time interval will be surface Similarly, the total heat taken from the reservoir during the same small time interval will be given by:
+
x6&',
&go = T"Z@("/T.
where T, is the temperature of the rth region of the surface at the time t, and the summation extends, as before, over the entire surface. I t is evidently 6p0/T0which corresponds to 8p/T for the reversible case. Disregarding for the moment any changes in entropy due to equalization of temperature within the system the total change in entropy is given by AS = SB - Sr - Q / T o
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where @ / T o = f Gpo/To, the integration extending from the moment the systemleft the state A to the moment when it finally attained equilibrium a t the state B. The problem resolves itself into a determination of the sign of AS. For reversible changes of state the quantity is evidently zero. For irreversible changes it is conceivable that the sign might be always positive, or always negative, or sometimes the one and sometimes the other. An interesting sidelight on the last possibility is presented in an article by C. Raveau (1) in which the whole significanceof the Second Law is made to depend on the permanence of irrewersibility.' For our purposes we rely on the mechanical conditions governing the quantity of work which can be performed by the system in passing from a given initial to a given final state. The usual mechanical formula, 6w = @a, accepted in thermodynamics as characteristic of all systems, is evidently true only for reversible changes of state, as has been pointed out frequently in thermodynamic treatises. It is only in the reversible case that the symbol p has a definite significance. In all other cases p is multiple-valued. However, certain qualitative conclusions as to the value of p may be drawn. It is mechanically evident that the seat of the pressure or force acting on the piston is a t the interface between the gas and the piston and that the actual pressure which does work on the piston is the pressure in the interface. At the first instant of an irreversible expansion p has its equilibrium value; thereafter, in calculating the work done by the gas, p shoujd be evaluated a t the interface. We kuow, for example, that in the extreme case of irreversibility, expansion Q into a vacuum, zero work is performed, corresponding to the zero pressure acting on the "piston." The initial step of an irreversible process lowers the pressure in the immediate vicinityof the piston. A wave of dilatation spreads through the gas, tending in the end to be reduced to heat by viscosity. When such a reduction has been accomplished the pressure is again the same throughout. But the process of transmission and transformation of the wave takes a finite length of time and if the piston moves farther during the time required-and it does so in the case of irreversibility-the pressure is not only not equalized but becomes even more widely different in different regions. In particular, it will be even less at the piston-gas interface. Evidently, for a given volume change, the irreversible work is less than the reversible work due to the fact that the actual pressure is usually less and certainly never greater than the equilibrium pressure. But, for a given change of state, the change in internal energy is constant. Therefore, if the work done be less for the irreversible case than that for the reversible case, so also is the heat absorbed. That is, 6~7.~~. > Q.,
Rut the change in entropy of the system is ASerz. = QrSv,/TQ > Q;r.dT',
VOL.9, NO.6 PRESENTATION OF SECOND LAW THEORY. I1
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where Qr., and Qi,. represent the total heat taken from the reservoir a t constant temperature To in the reversible and irreversible cases, respectively; and the change in entropy of the reservoir is in any case AS"
=
Consequently, AS = ASz,
-Q/T0.
+ AS" = 0 or is > 0
according as the process is or is not reversible. When compression is carried on irreversibly the work done on the gas (-6w) is greater than in the case where the opposing pressures are equal. This is true because, due to the fact that density variations are equalized with finite velocity, the initial increase of density of the layer of material next to the piston, which, in the irreversible case, is not only permanent, but increasing with time offers a greater resistance (pressure) than would be offered by a layer a t equilibrium density. Since -6 2 ~< ~ ~- . 8wirrsv.,
thereiore Sq-".
>
Spi".".
and the conclusion arrived a t for irreversible expansion is also valid here. Irreversibility of heat transfer occurs whenever two media at different temperatures are placed in contact. Heat flows from the medium at high to that at low temperature.* In point of fact there must exist an intermediate region in which the temperature varies from point to point in a physically continuous manner, but this 'need not concern us. Only the initial and final states are of practical valuerin calculation because the change in entropy may be determined entirely from them. Let there be inequalities of temperature within the system under consideration. Suppose a small region p, to possess a momentary temperature T I and a neighboring small region pz to possess a temperature Te< TI. A flow of heat from p1 to 0%will take place. Suppose 6g calories to have been transported. Then the changes in entropy of the two small regions are, respectively; dSL = -8q/Tx; dS2 = Sp/T*;
and the total change in entropy is the sum of these, viz.: dS
=
d.7,
+ dS2 = 8q(TL- T 2 ) / T , T 2> 0,
that is, positive. Consequently, all internal equalization of temperature in a system of perfect gases is attended by a net increase of entropy. The case of irreversible transfer of heat with the surroundings remains, but it is evident that any such regions of irreversibility may legitimately be included in the "system" if desired. The surroundings are thus arbitrarily
* Statistically heat may also flow from low to high temperature, but it is evident that in such a case temperature has not the same significance as in thermodynsmics where it is defined as the parameter determining the direction of flow of heat.
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selected so that all contacts between it and the system are made the seats of only reversible energy transformations or transfers.
The case of diffusion is diierent from those treated above because it necessitates the presence of at least two molecular species. For simplicity consider a system composed of one mole of A and one mole of B confined in a container of volume v at a total pressure p. First suppose the volume u divided into two equal parts and suppose all of A to be in the right half and all of B in the left. (Figure 2.) The total volume is v and the pressure throughout is p. Now suppose the partition removed. That the gases will interpenetrate is an experimental fact giving occasion for the name diffusion. Further, since both portions were at pressure p before the partition was removed, there will be no sudden mass motion of gas in either direction. Each gas acts as though it were expanding against a sensibly equal pressure and it continues to "expand" until it fills the volume uniformly. The work done by each gas is R T v In - = R T ln 2. But (v/2) the work is done by one constituent of the mixture on the other and zero work is produced externally. I t is possible to devise
",r"
hypothetical mechanisms by means of which the work here expended on internal reaction might be utilized. Suppose the partition to consist of two pistons, @ and a, of which @ is permeable to B but not to A and a to A but not to B. By moving a very slowly from its original position (Figure 2 ) to the right wall of the container B is allowed to expand reversibly into A. The expansion may be taken to be isothermal, the requisite heat being reversibly supplied from a reservoir at the temperature T of the system itself. In this expansion FIGURE 2
AE
=
0, JSq
= JSw = R T l n Z
Similarly the piston @ may be moved very slowly from the center of the container to the left wall, producing a quantity of work R T in 2 and absorbing an equal quantity of heat. (The two expansions may, of course, be carried out simultaneously.) In the present reversible case of diffusion the change in S for the system is J G ~ / T= 2RT in 2, and for the reservoir -f Gq/T = -2RT In 2, so that the total change in entropy is zero. For the irreversible case previously described the change in entropy must be the same for the system since the initial and final states are identical,
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whereas the change in entropy of the reservoir is zero. Since the A S for the system is positive it is evident that here too the total change in entropy is zero or positive according as the changes of state occurring are or are not reversible. Discussion The value of the foregoing as a contribntion to human knowledge may well be questioned on the grounds of a helpful criticism received by the author as follows. "Broadly speaking, it may be taken to illustrate one way in which a considerable part of a known result can be obtained by using less hypotheses than are necessary t o establish the whole of the known result. It seems evident that an infinity of attempts of this sort are possible." However, the pedagogical problem of presenting Second Law theory t o a class with little thermodynamic background has led me to seek for ways oE avoiding until the last possible moment the appeal to the "principles" on one or the other of which the standard development of the law depends. A treatment such as the above yields many of the pertinent results of the law in a f o m more readily grasped because based on no "selfevident" statements which seem to most students the reverse of plausible. It is hoped that there will shortly be developed a new postulate on which t o found the classical second law-a postulate just as general but more plausible than the usual ones. In July of last year G. N. Lewis (2) pub,lished a highly important article in which he stated that he had found a new and really valid statement of the Second Law. This claim is really less thian the fact, since the new postulate accounts not only for the phenomena treated by the classical law but also the fluctuations about equilibrium, which are quite foreign t o it. I should like to propose the title "The Lewis Postulate" for the cardinal principle set forth by Prof. Lewis, a principle which appears to bear to the classical Second Law the same relation that Relativistic Mechanics does to Newtonian. Summary In a system composed of perfect gases or of substances for which i t can be shown that 6p/T = dS, a perfect differential, i t has been shown that the total change in the point function S is zero for reversihle and positive for irreversible processes. In the proof no use has been made of any "principles" such as are usually made a basis for the Second Law of Thermodynamics. The only assumptions are the First Law, the Perfect Gas, and those auxiliary assumptions as t o the mechanical properties of all matter which are always used in thermodynamics t o define "work done" and "reversibility." The pfobable pedagogical value of the presentation is pointed out. The suggestion is made that the new postulate recently
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discovered by Prof. G. N. Lewis to replace the Second Law be called the Lewis Postulate. Literature Cited ( I ) KAveAu. C o m l . rend., 188,313 (1929). ( 2 ) G.N. LEWIS,J. Am. Chem. Soc., 53, 2578 (1931)
