J. M. Honig Purdue University West Lofoyette, Indiana 47907
On a Heuristic Formulation of Caratheodory's Version of the Second Law
Discussions concerning the second law of thermodynamics tend to fall into one of two categories. Those in the first grouping refer to the operation of cyclically operating engines; considerable effort must then be spent in generalizing such statements so that they may become useful in areas of thermodynamics of interest to chemists. Moreover, in the discussion of Carnot engines the working substance is usually chosen to be the perfect gas whose very existence is subsequently denied as a consequence of the third law. These esthetically displeasing features are absent from the second group of assertions, which are based on CarathBodory's Theorem1 and therefore possess the sweep and generality necessary to handle all fields of thermodynamics. Unfortunately, these versions tend to he abstruse and to be so preoccupied with mathematical rigor and detail as to become virtually inaccessible to nonspecialists. This is deplorable because the power and generality of the approach deserve to be much more widely known and appreciated. In this paper an attempt is made to review the essential features as well as some of the principal consequences of Carathmdory's theorem, without going into excessive mathematical detail. While some background material concernine essentials in analvtical eeometrv" is orovided. . most of tKe arguments are prkentez in heuristic fashion: This tvpe of oresentation has been reasonahlv successful in clas&orn discussions and is hopefully of use to a wider audience. For a proper exposition of the subject matter interested readers are referred to two excellent monographs by H . A. Buchdah12 and by H . R e i ~ s . ~ Representation of Equations; a Specific Example In preparation for later sections we consider a specific example which illustrates the diagrammatic representation of equations. As is well known, the kinetic energy Efp,,p,;m,c) of a particle moving in a plane is given by
t E0
Figure 1. Sketch of eqn. ( 1 ) in momentum space. The upper plane represents the intersection of the plane E = EO with the paraboloid of revolution. Vf represents the gradient according to eqn. (8).
guise: One may introduce a new function f which includes
E amongp, and p,, to write where p,,p, are the momentum along the x and y direction, m is the mass of the particle, and e is an arbitrary reference value. Equation (1) may be represented by a surface which is a paraboloid of revolution in a space of three dimensions spanned by the p,, p , and E axes, as shown in Figure 1. The shape and location of the paraboloid which is the image of eqn. (1) is determined by the values of the parameters e and m. Note that the differential of the energy has the form
+
(aEldp,Mp, dE = (aEldp,Mp, = (p,lm)dp, + ( P J ~ M P , and that the above may be rewritten symbolically as
This latter relation may be used in place of eqn. (2) if one imposes the constraint
in which - c has been replaced by fo. Then
For, with this condition eqn. (5) may be rewritten as (2) Note that eqn. (7) is indeed equivalent to eqn. (2); i.e.
Equations (1) and (2) may he restated under a different 'Carathi.odory, C., Moth Ann., 67, 355 (1909);Sitzb.d. preuss. Akad. Wiss, 39,(1925). =Buchdahl, H. A,, "The Concepts of Classical Thermodynamics," Cambridge University Press, 1966. %Reiss, H., "Methods of Thermodynamics," Blaisdell, New York, 1965. 418 / Journal of Chemical Education
and similarly for (aE1ap.J. The point of introducing eqn. (4) is that the equations f = fo, f = fi, f = f2, f = fa, . . . (or in general, f = constant) have as their images a set of nested paraboloids which are surfaces of revolution, of which the one drawn in Figure 1 is representative. This illustrates the more general feature
that any equation of the type f(x,y,z) = fi is represented by a surface in three dimensions or by a hyper-surface in a hyperspace (as is required when f involves more than three independent variables). Equation (5) may be rewritten in more compact notation: The gradient of eqn. (4) is given by
-
vf
= i(af/ap.)
-.f=-
+ j(af/ap,) + k ( a f / a ~ )
--
(8)
where i = p,, j - p,, k = E are unit vectors along the three coordinate axes. Furthermore, the general displacement vector is given by the relation
Usingeqns. (8) and (9) one may reexpress eqn. (5) as df
=
Vf.ds
-t
(11)
is an equation for a circie. Note further that the dimensions of the space required t o depict the circle are now reduced from three (E, p,, p,) to two ( p , p,). The surface representing conditions of constant energy Eo relative to a reference value c for a particle of mass m is given by eqn. (11). Also the differential expression now reads
-
=
(pZZ/2m)+ (p,Z/Zm) + (pZ2/2m)+
(14)
requires a four dimensional hyperspace to depict the dependence of E on p,, p,, p,. However, imposition of the constraint E = const enables us to sketch in p,, p , p, (momentum) space a set of surfaces representing the eonditions of constant energy. According to eqn. (14) these are the set of concentric spheres centered about p, = p, = p, = 0, corresponding to E = 6 . A given sphere represents those values of p,, p,, p, which can be realized by eqn. (14) under the constraint E = El = const. Any point on this spherical surface represents an energy state of the system consistent with this constrgnt. If th&systemis altered infinitesimally-according20 I dp, +j- dp, + k dp, while also requiring i p,dp, + j p,dp, k p,dp, = 0 the
+
Here, the various partial derivatives must first he specified to determine dE explicitly; for example, a E p V = p must be known in its dependence on V, A, B, D, . . . 8. Then the calculation of d E is straightforward. More gen) obtains the differential erally, from R = R(x1,. . . , x ~ one expression
where fii = fii(xl,. . .,xN). O b s e ~ ethe correspondence to eqn. (3) and note that every one of the independent variahles in the argument of R appears as a term in eqn. (16); the latter is said to be of the standard Pfaffian form. A situation that is frequently encountered and is of considerable interest arises through the requirement
where Cis a constant. When this conditionholds
...
in which the line element is given by ds = idp, + jdp,, where dp, and dp,.are interrelated as shown by eqn. (12). Both the gradient X -%I + 3 2 as well as the line element dx now lie on the two-dim_ensional plane E = Eo; also, dx is everywhere normal to X, i.e. is everywhere tangent to the circle. Note that if the intersections of the various planes E = Ei with the ~araboloidof revolution were = 0 one would obtain a projected onto the basal pla-E series of concentric circles (1, j) space, each representing pairs of allowed (p,, p,) values which maintain a constant energy E = Ei according to eqn. (1). Generalization of the latter to three dimensions E
In thermodynamics one frequently encounters expressions of the form R = R(x= . . . ,xN), where XI, . . . ,XN-I are deformation coordinates, X N is a constitution coordinate, and R a dependent variable of interest. For example, R may represent the energy E, the set (xi/ (i # N) may represent the volume V, surface area A, magnetic induction B, electric displacement D, . . . , and X N may represent an empirical temperature 0. Once E(V,A,B,D, . . . ,0) is specified one can ascertain how E is changed by infinitesimal alterations of all coordinates
(10)
which involves the dot product of the gradient with the infinitesimal displacement vector. Notice that constraint (eqn. (6)) does not require the gradient to vanish; according to eqn. (10) one need demand only that d s be ortbogonal to vf. Now a gradient has the property that a t any point it is normal to the surface characterized by f = fi; accordingly, to satisfy eqn. (6) d s must be on the surface (more precisely, must be a tangent to the surface) a t that point. To make contact with the later analysis consider next the special case d E = 0 or E = Eo, a constant. Geometrically this is depicted as the intersection of the paraboloid of revolution f with the horizontal plane E = Eo, as shown in Figure 1. The resulting intersecting "surface" is now the circle on the horizontal plane E = Eo. Mathematically this may be verified by noting that pX2/2m+ pv2/2m = E,
initial point is displaced infinitesimally to another location on the same spherical surface. Points interconnected in this manner are said to form "solution curves" to eqn. (14). Generalizations to Thermodynamics
which corresponds with eqn. (13). I t is convenient to be able to visualize (17) and (18) pictorially: Construct a "state space" in which the various xi form mutually orthogonal axes. Assignment of specific values xi0 to all N variables xi locates a particular point P= (214 . . . ,IN") in N-space.. Note that any such point represents a specific state of the system, namely that for which every variable xi has the value xio. When the state of the system is altered the position of the representative point likewise changes. Since eqn. (17) specifies a relation between the {xiJ(see eqn. (1) as a very special case) it represents a hypersurface in this space. Returning to the earlier example, this means that when the energy of the system is held constant, changes in 0 cannot be made independently of changes in V,A,B,D, . . . ;rather, once dV,dA,dB,dD, . . . are prescribed dB. is found from the requirement d E = 0, (eqn. (15)). dx = 0; As before, Xi Xr_dxl may be abbreviated as here the "vector".X has the components (aR/al-i) (where i = 1,. . . ,N), making it the gradient of R, and dx is an infinitesimal curve elementwith components ( d x ~ . . .,&N). Again, the requirement X d x =O may be met by requiring dx to be perpendicular to X. Since VR is a surface normal. dx must lie on the surface R = C. We conclude that to satisfy eqn. (17), the changes in XI, . . . ,XNmust be made in such a wav that the initial point (~10,. . . ,xwO)and the final point (XI; + dxl, . . . ,xW0+ &N). which are con-
.
Volume 52, Number 7. July 1975 / 419
nected by the line element dx, both lie on the surface R =
C.
The above has ah obvious implication: When condition (18) holds there are vast numbers of points not accessible from XI", . . . ,xN" by solution curves to (la), namely all those points not on the surface R = C. Physically, this means that there is a multitude of states that cannot be reached by forcing a system to undergo changes subject to the restrictions of eqn. (17) or (18)-which is physically a very sensible conclusion. Note finally that when eqn. (16) is integrated the result depends only on the initial and final states and is independent of the path: Since
the integration yields a t once the result R121 - RCII,regardless of the path selected to reach x,lZ' from xt'l' for all i; dR is an exact differential. Unfortunately, matters are frequently more complicated. In thermodynamics one often encounters the Pfaffian forms
which do not possess the property of being the differential of a quantity L. This manifests itself in several ways: For example, even though the dx, are infinitesimal, the c o m sponding bL need not be small. For example, in a phase transition a very small change in temperature may lead to a very sizeable latent heat transfer. Also, the value of AL depends on the path. Finally, bL = 0 does not imply that there exists an L = C; tbxs CK is nothing more than a shorthand notation for Z,,,Y,dx,. I t is clearly awkward to have to deal with quantities of this type. One is then led to ask whether it is possible to find a function q(xl, . . . ,XN)such that the ratio ZL/q does become an exact differential dR. If this is the case then -
-
Furthermore, ZI1,Y,dx, = 0 implies ZltlqXidxl = 0 or 'i,iX,dx, = 0. That is, one can replace aL by qdR, where the latter now involves the function of state R. Clearly, this sets the stage for the crucial equality ZQ,,, = TdS, to be established at a later stage. It is of great import to be able to ascertain under what conditions, if any, the Pfaffian form (eqn. (19)) possesses an integrating denominator q, satisfying eqn. (20) and converting bL into an exact differential. Here, one can again apply the earlier geometric imagery: The solution curves to the Pfaffian form bL = 0, for which an integrating denominator may be found, must all lie on a hypersurface R = C. (Translated into a more physical example this reads as follows: Assume the relation bQ,, = TdS has been established as correct. Then, a system altered under the adiabatic constraint dare" = 0 must change its properties in such a way as to maintain constant entropy. In phase space the image of the constant entropy requirement is conveyed by a surface Sfxl,. . ..IN) = C.) Invoking the requirement that one must remain on the surface R = C one can again make the very important assertion: Solutions to the Pfaffian form iEL = 0 possessing integrating denominators must have the property that in an arbitrary neighborhood of a point P i n state space there exists others not accessible from P through solution curves of bL = 0. An understanding of this statement goes a long way toward unravelling the mystery of Carathbodory's version of the second law. The above should be reasonably transparent; the question is whether the reverse statement holds. It is not tw difficult to prove the affirmative statement: If a Pfaffian expression 420 / Journal of ChernIcalEd~~~Non
has the property that in its state space every arbitrary neighborhood of a point P contains other points which are inaccessible from P along paths corresponding to bL = 0, then bL does oossess an inteeratine denominator. Thus the nonaccessil;ility criterion is botcnecessary and sufficient to guarantee the integrability of ZL = 0. Mathematical Tests for Integrability
Given a particular Pfaffian differential expression it is of importance to establish by mathematical tests whether the same is an exact differential or holonomic, i.e., whether it possesses an integrating denominator by means of which it can be converted into an exact differential. To develop the necessary tests would lead us too far afield; hence, the required results are simply cited below. The sum Z~i)Xidxtmay be shown to be integrable as i t stands if and only if
for all sets of distinct indices i and j appearing in said sum. It turns out that a Pfaffian form containing only two terms is always integrable. This no longer is the case when there are three or more terms in the sums. If eqn. (21) fails to hold then said Pfaffian form is holonomic under the following condition which is both necessary and sufficient
x,(%
-2)+ x,($
-z)+ -
aX
ax
T,($ - -)=o ax,
where i, j, k runs over all distinct triplets of indices occurring in the Pfaffian differential form. Caratheodory's Theorem and the Second Law of Thermodynamics
The Theorem of Carathkodory asserts that if every neinhborhood of anv arbitran, uoint P in the uhase space spanned by the deformation coordinates xi contains points P' igaccessible from P by solution curves of the equation ZiiiXidxi .. . = 0. then the Pfaffian form is holonomic. The discussion of the section Generalizations to Thermodvnamics shows that the above theorem. considered as a sufficient condition is indeed correct: Given the integrability of Z,,iX,dxl = 0, one cannot reach any arbitrary point P' in phase space from an initial point P along the solution curves of the Pfaffian differential expression. More precisely, the multitude of points P' not lying on the surface R(XL . . . ,x,) = C to which P belongs is inaccessible. The proof of the converse as given by the statement of the theorem, is much more elaborate and will not be given, but the assertion is perfectly plausible in light of the discussion of the earlier section. Based on the above one can provide the following general statement of the Second Law of Thermodynamics as an experience of mankind: In every neighborhood of any state of an adiabatically isolated system there exist states not accessible from it. The implications of this statement follow at once from the recognition that the state of an adiabatically isolated system may be represented as a point P in state space, suhiect to the reauirement it9 = Z,.) Y,dx, = 0.The above implies immediately that & muit have an integrating denominator X (XI. . . . .XN)such that dQIX = ds, where ds is an exact differential, and where s, termed the empirical entropy, is a function of state of the x,. It remains to provide an interpretation for h and for s. Consequences of the Second Law
We come now to one of the most amsting consequences of the second law, which is arrived a t solely by the process
of combining two distinct systems into a composite. Let two closed systems A and B, characterized by the deformation coordinates XI, . . . .x,-1, t and yl, . . . .y,-I, t be in diathermal contact, so that in equilibrium they are a t a common empirical temperature t and let the totality of these coordinates characterize the compound system. Because any heat exchange involving the two parts of the system is additive we find that dQc =
+ dQs
~ Q A
(23)
Further, from the holonomic character of bQ i t follows that
Since SA and SB are functions of state they must depend onxl, . . . , x , _ ~ , tand ony,, . . . ,ym_l,t, respectively. In principle we can invert these relations to solve for X,_I in ~ for ym-1 in terms of terms of XI,. . . , x ~ - z , s A ,and Yl.. . . .Ym-m%,t. The following argument which identifies A and s is, in its simplicity, sweep, and compelling nature, among the most elegant in science. It proceeds according to the following steps: (1) sc must be independent of the deformation coordinates XI, . . . ,xn-2;ylr . . . .ym s t : As discussed in conjunction with eqn. (16), if this were not so, the differentials of these coordinates would have to appear on the right hand side of eqn. (24). (2) The ratios AA/Ac and Ag/Ac are likewise independent of these coordinates: If this were not so, sc would necessarily depend on the quantities, in contradiction to step (1). (3) Ac cannot depend on yl, . . . , y,z; for if it did, then AA would have to depend on these deformation coordinates in the same manner in order for these quantities to cancel out from the ratio AA/Ac as is necessary for consistency with steps (2) and (1). But AA cannot possibly depend on the deformation coordinates of system B, thus verifying the initial assertion. By similar reasoning A, cannot depend on the deformation coordinates y ~. ,. . , ym-2. (4) AA cannot depend on XI, . . . , x,_z and AB cannot depend on y ~ .,. . , y,_,; for according to (3) these quantities are not functionally involved in Ac, and must therefore he missing from An and AB in order to satisfy step (2). However, this chain of arguments does not apply to the quantity t which is common to A, B, and C. (5) The three functions An, AB, and Ac are a t most functions of (SA, t), (SB,t) and ( s ~ , s ~ , t )re, spectively; this is an elementary consequence of assertion (4). ( 6 ) The functional dependences referred to in step (5) must have the form @n(sn)T(t), +~(sg)T(t), and &(sA,sB)T(~). respectively, in which T(t) is a common function of the empirical temperature t. If this were not so, the ratios AnlAc and A B / ~would necessarily depend on t, in contravention to step (2). It finally follows that A , = rn,(s,)'Rt) A, = r n u ( s a ' 0 ) A, = C(s,s,)T(t)
(25)
and dQ*
T(t)$,(sdds~ dQn = T(t) %SBMSS dQr = T ( t ) & ( a , s ~ ) d s ~ =
regardless of the specific properties of the system under study, it is called the absolute temperature (function) in thermodynamics. Note finally that the substitution of eqn. (27) into (23) leads to the additivity relation I t follows that Sc = SA+ SB, where we have set the arbitrary constant of integration to zero. There is a further consequence of the second law that must he taken up a t this point, namely that for any adiabatic process which is not reversihle the entropy must either always increase or always decrease. Consider the composite system used earlier and let it he characterized by the deformation coordinates XA,XB and by S = Sc = SA+ SB. (If more deformation coordinates are required for the characterization of A and B the arguments given below may be readily generalized.) Let the initial state of the compound system be given by xnO,XBO,and Soand let the final state be reached in two stages: (1) The deformation coordinates are changed reversibly from xnO and r e o to XA and XB. respectively, in an adiabatic process during which the entropy is kept constant. In the composite system of our choice XA and X B can he varied continuously and independently; therefore, all XA and X B values consisO can he reached. (2) The entropy tent with the entropy S now is altered hy initiating a second process which is to he camed out irreversibly and adiabatically, while keeping XA and XB fixed. The final state of the system is then characterized by the quantities XA, XB, and S. Now if under some conditions of the process (xnO, xgO, So) (XA, XB, S ) one were to find S > So, and under others one were to find S < So, every neighboring state to xaO, XBO,and So would have been covered, in contravention to the second law. We see that the only way of avoiding this impasse is to reauire either that S > SO or that S 5 SO under all conceivable adiabatic processes. The choice S > So renders the temperature T positive and is adopted hv convention. lmplicitin the above is the well-known pointthat one can encounter the case S 5 SO for processes carried out under nonadiabatic conditions. However, somewhere else in the universe one must then encounter a compensating process such that the total entropy change of the system and of the surroundings, where the compensating process occurs, is always nonnegative.
-
Relation of Caratheodory's Theorem to Other Versions of the Second Law
It is instructive to show how Carath6odory9sversion of the second law is correlated with the more conventional statements encountered in the elementary treatments of thermodynamics. This will he arcomplirhed in the con. text of processes involving two adjacent points in state space as represented in Figure 2. Choose a particular quasistatic process such that the transition when carried out reversibly involves a heat flow bQ,. Select a second, irreversible path which effects the same change and involves a heat flow bQ,. This process is indicated by a dotted line to show that the irreversible process cannot be represented as a succession of equilihri-
(26)
It turns out convenient to define the metrical entropy S as that function for which d S = +(s)ds; then
where ~SC(A,SB) =~ASA,SB)~SC(SA,SB). Inasmuch as the same function T of an empirical temperature scale serves as an integrating denominator of bQ,
Figure 2. Two paths connecting states 1 and 2 in configuration space. The irreversible path is dotted to indicate that it cannot be represented by a succession of equilibrium states in configuration space.
Volume 52, Number 7. July 1975 / 421
um states in the chosen state space. Then, by the first law, bQ, = dE + bW, and bQi = d E + itW, (Note that d E is the same in both cases). Hence, dQi - dQ, = dW; - dW, (29) The above quantity cannot vanish. If it did, the reversihle and irreversible paths would have to coincide, since dQ, and bQ, depend on the chosen paths. This obviously would lead to a contradiction of terns. Note also that -ZQr and -dWr are the elements of heat and work involved in going from state 2 to state 1along the reversible path. The algebraic sums bQi - dQr and bW, dW, thus also represent the total heat transfer and work involved in completing a cycle, going irreversibly from 1to 2 and reversibly back from 2 to 1. We observe that TdSi TdS, = 0 for the cyclic process; the question which now arises is whether dQi - dQ, = bWi - bW, are positive or negative quantities. Suppose first bQ; - dQ, > 0. Then in completing a cycle by going from 1to 2 via path i and returning to 1via path r the work accomplished is bWi - bW, > 0;work is done by the system when it undergoes a cyclic process, a t the expense of an equivalent amount of heat dQi - bQ, > 0 which flows into the system from suitable reservoirs. After comnletine the cvcle no other chanees have occurred in the universe. S u n ~ o s einstead that bQi - 3% < 0. A similar line of reasoning now shows t h a t - i n one cycle this quantity of heat is extracted from the system and transferred into the surroundings a t the expense of an equivalent amount of work bWi - dW, < 0 which is performed on the system. No other changes have occurred in the universe. Which of these alternatives do we choose? If we set bQi - ZQ, < 0 then TdS, = TdS; > bQi. Suppose we specialize to the case where the irreversihle path is an adiabatic one for which dQi = 0. We conclude that then TdSi > 0. With T > 0, dSi > 0 and no contradictions are uncovered. On the other hand, had we selected bQ; bQ, > 0 then the possibility dSi < 0 would have emerged, in contravention to the second law. It follows that bQ, > 88;; since the opposite is led out it follows that one cannot convert heat completely into work in cyclic processes without also incurring other changes in the universe. Further, in any reversible process TdS exceeds the heat transfer accompanying an irreversible process. We thus find dQ, > dQ, dW, > dW, (30) TdS > dQ, dS > dQ;/T (31)
-
-
Notice that for all adiabatic irreversihle processes where bQ; = 0, d S > 0. All closed systems are adiabatic by definition; for this entire class undergoing irreversible processes it necessarily follows that d S > 0. For a system undergoing a cyclic process for which & = Sz, (S being a function of state) eqn. (32) leads to the Clausius inequality
Note that in any reversihle cyclic process LQ, since Sz = S1 Hence $ ~ Q , / T= o Both statements may he combined to read
= TdS = 0 (34)
We have so far passed over one most important point: In order to arrive a t the conclusion that TdS > 0 i t was necessary to invoke both irreversibility and adiabaticity. But i t is in the nature of an irreversihle process that one 422 / Journal of Chemical Educathn
can never return to the starting point without incurring other changes in the universe. Hence if in an adiabatic system there occurs an irreversihle process, there is no possibility of ever returning to the starting point. Moreover, there is no way of forcing a process to occur in an isolated system; if an irreversihle process proceeds a t all on its own i t must go in a spontaneous direction. We conclude that for an adiabatically isolated system in which spontaneous processes occur d S > 0. However, the inequality does not necessarily hold in nonadiabatic, nonisolated systems. For an isolated system where reversihle changes occur we have d S = 0. Thus the state function S serves as a means of monitoring whether a given process in an adiabatically isolated system is indeed possible. No process for which S decreases can occur in an adiabatically isolated system; conversely, any process for which S increases in such a system will he spontaneous. For a system which exchanges heat with surroundings we consider an enlarged system in which the original system and its surroundings form a composite system. Then dStm =,dS1 + dSz > 0. Cons~dernow the operation of a heat engine. To be useful it must he capable of being operated in cycles, so that a t the end of every cycle the engine gets back to the same state in which it started a t the beginning. All changes accompanying the operation of the engine thus occur in the universe. We operate the engine between two reservoirs kept a t temperatures Th and T,, with TI, > T,. The reservoirs are assumed to be large enough that heat transfers do not appreciably alter the temperatures of each. Let Q he the heat transfer per cycle across the boundary between reservoir h and the engine (Qh > 0 or Qh < 0 according as heat flows into or out of the engine) and let Q, be the heat transfer per cycle across the boundary separating the engine and the reservoir c(Q, > 0 or Q, < 0 according as heat flows into or out of the engine). If now Qh > Qc or Qn - Qc > 0, then in the course of one cycle we must have W > 0 in order that the first law as applied to the cyclic process (Qh - Q,) - W = 0 may hold. That is, by arranging matters so that heat is transferred into the engine from the hot reservoir and heat is rejected from the engine to the cold reservoir, the difference Qh - Qe is transformed into work W > 0 performed by the engine. We now ask how efficiently this energy conversion process can be carried out. The efficiency is measured by a quantity it defined as q = Work out/Heat in (36) To get a t this quantity recall the Clausius inequality (eqn. (35)). In the present case the integration in eqn. (35) degenerates into a sum
(where again Qh is heat into and -Q, is heat out of the engine operating in one cycle). Hence (38) Qh/Th 5 QI/Tc where Qh,Q, are both positive quantities, inasmuch as a negative sign precedes Q, whenever we consider the heat removed from the engine. Hence eqn. (38) may be rewritten as Q,/Qh 2 T,/Th (39) or -Q,/Qh 5 -T,/Tb C40) Thus
Note the following points: (1) The complete generality of our approach. We have not restricted ourselves as to types of engines, reversibility, or types of processes other than the restriction to cyclic events. (2) The efficiency relates to the heat transfer across the hot junction. (3) The equality sign holds only for reversible processes, so that for any irreversible Drocess the efficiency is inevitably less than that for a r&rsible one. (4) ~ v k na reversible process is never 100% efficient, except in the inaccessible limits T, = 0 or Th m ; i t becomes the more efficient the larger Th/T, is. (5) rt depends only on the temperatures of the "boiler" and "condenser." Statements (5) and (3) are sometimes called the first and second theorems of Carnot. On the basis of the ahove we can now restate the second law in a variety of ways. Statement (4) above provides the basis for Kelvin's formulation of the second law: I t is impossible to devise an engine which, working in cycles, shall ~ r o d u c eno effect other than to extract heat from a reservoir and perform an equal amount of work. Related to the above is the Clausius statement: I t is impossible to devise a machine which, operating in cycles, transfers heat from a colder to a hotter body without producing any other effects in the universe.
-
This may be demonstrated by noting what would happen if the Clausius statement were incorrect. If no other changes are to occur in the universe then the heat extracted from the cold reservoir must be transferred without loss to the hot reservoir. Equation (37) would then read -Qh/Th + Qc/Tc5 0, with the requirement that Q = Q, be the heat flow into the engine and that Qh = Qc. In these circumstances one would have Th/Tc 5 1, which is a nonsensical result because by definition Th > T,. Epilogue
An attempt has been made in the foregoing to make Carathbodory's version of the second law of thermodynamics more generally accessible, and thereby, to demonstrate again the advantages of his approach over the more conventiorial discussions of the second law. This writer contends that it is highly desirable to expose students to a statement of the second law that deals with all conceivable processes rather than to start with a set of statements ~ e r t a i n i n eto cvclic en~ines.The resent a o ~ r o a c h contains many statements which are bacied only by plausibility arguments. However, once the basic principles are grasped, a student interested in mathematical rigor may then he referred to several excellent treatises where the required mathematical proofs are provided
Volume 52. Number 7. July 1975 / 423
