The chief novelties of Carnot’s work are the introduction of the idea of a cycle of operations, and the invaluable discovery of the special property of a reversible cycle. It is not too much to say that, without these wonderful novelties, thermodynamics as a theoretical science could not have been developed.

Carnot’s work seems to have excited no attention at the time of its publication. Ten years later (1834) Clapeyron gave some of its main features in an anally tical form, and he also employed Watt’s diagram for the exhibition of others. Even this, however, failed to call attention properly to the extremely novel processes of Carnot, and it was reserved for Sir W. Thomson (in 1848, and more at length in 1849) to point out to scientific men their full value. His papers on Carnot’s treatise, following closely after the splendid experimental researches of Colding and Joule, secured for the dynamical theory of heat its position as a recognized branch of science. James Thomson, by Carnot’s methods, predicted in 1849 the lowering of the freezing point of water by pressure, which was verified experimentally in the same year by his brother. Von Helmholtz had published, two years before, a strikingly original and comprehensive pamphlet on the conservation of energy. The start once given, Rankine, Clausius, and W. Thomson rapidly developed, though from very different standpoints, the theory of thermodynamics. The methods adopted by Thomson differed in one special characteristic from those of his concurrents,—they were based entirely on the experimental facts and on necessary principles; and, when hypothesis was absolutely required, attention was carefully directed to its nature and to the reasons which appeared to justify it.

Three specially important additions to pure science followed almost directly from Carnot’s methods :—(1) the absolute definition of temperature; (2) the thermodynamic function or entropy; (3) the dissipation of energy. The first (in 1848) and the third (in 1852) were given by W. Thomson. The second, though introduced by Rankine, was also specially treated by Clausius.

In giving a brief sketch of the science, we will not adhere strictly to any of the separate paths pursued by its founders, but will employ for each step what appears to be most easily intelligible to the general reader. And we will arrange the steps in such an order that the necessity for each may be distinctly visible before we take it.

1. General Motions. — The conversion of mechanical work into heat can always be effected completely. In fact, friction, without which even statical results would be all but unrealizable in practical life, interferes to a marked extent in almost every problem of kinetics,—and work done against friction is (as a rule) converted into heat. But the conversion of heat into work can be effected only in part, usually in very small part. Thus heat is regarded as the lower or less useful of these forms of energy, and when part of it is elevated in rank by conversion into work the remainder sinks still lower in the scale of usefulness than before.

There are but two processes known to us for the conversion of heat into work, viz., that adopted in heat-engines, where the changes of volume of the “working substance” are employed, and that of electromagnetic engines driven by thermoelectric currents (see Electricity, vol. viii. p. 96). To the latter we will not again refer. And for simplicity we will suppose the working substance to be fluid, so as to have the same pressure throughout, or, if it be solid, to be isotropic, and to be subject only to hydrostatic pressure, or to tension uniform in all directions and the same from point to point.

The state of unit mass of such a substance is known by experiment to be fully determined when its volume and pressure are given, even if (as in the case of ice in presence of water, or of water in presence of steam) part of it is in one molecular state and part in another. But, the state being determinate, so must be the temperature, and also the amount of energy which the substance contains. This consideration is insisted on by Carnot as the foundation of his investigations. In other words, before we are entitled to reason upon the relation between the heat supplied to and the work done by the working substance, Carnot says we must bring that substance, by means of a cycle of operations, back to precisely its primitive state as regards volume, temperature, and. molecular condition.

2. Watt's Diagram.— Watt’s indicator-diagram (see Steam-Engine) enables us to represent our operations graphically. For if OM (fig. 1) represent the volume, at any instant, of the unit mass of working substance, MP its pressure, the point P is determinate and corresponds to a definite temperature, definite energy, &c. If the points of any curve, as PP', in the diagram repre sent the successive states through which the working substance is made to pass, the work done is (loc. cit.) represented by the area MPP'M'. Hence, a cycle of operations, whose essential nature is to bring the working substance back to its primitive state, is necessarily represented by a closed boundary, such as PP'Q'Q, in the diagram. The area enclosed is the excess of the work done by the working substance over that spent on it during the cycle. [This is positive if the closed path be described clockwise, as indicated by the arrow-heads.]

3. Carnot’s Cycle.— For a reason which will immediately appear, Carnot limited the operations in his cycle to two kinds, employed alternately during the expansion and during the compression of the working substance. The first of these involves change of volume at constant temperature ; the second, change of volume without direct loss or gain of heat. [In his hypothetical engine the substance was supposed to be in contact with a body kept at constant temperature, or to be entirely surrounded by non-conducting materials.] The corresponding curves in the diagram are called isothermals, or lines of equal temperature, and adiabatic lines respectively. We may consider these as having been found, for any particular working substance, by the direct use of Watt’s indicator. It is easy to see that one, and only one, of each of these kinds of lines can be found for an assigned initial state of the working substance; also that, because in expansion at constant temperature heat must be constantly supplied, the pressure will fall off less rapidly than it does in adiabatic expansion. Thus in the diagram the adiabatic lines PQ, P'Q' cut the lines of equal temperature PP', QQ' downwards and to the right. Thus the boundary of the area PP'Q'Q does not cross itself. To determine the behaviour of the engine we have therefore only to find how much heat is taken in along PP' and how much is given out in Q'Q. Their difference is equivalent to the work expressed by the area PP'Q'Q.

4. Carnot's Principle of Reversibility.— It will be observed that each operation of this cycle is strictly reversible; for instance, to take the working substance along the path P'P we should have to spend on it step by step as much work as it gave out in passing along PP', and we should thus restore to the source of heat exactly the amount of heat which the working substance took from it during the expansion. In the case of the adiabatics the work spent during compression is the same as that done during the corresponding expansion, and there is no question of loss or gain of heat directly.

If, however, a transfer of heat between the working substance and its surroundings have taken place on account of a finite difference of temperature, it is clear that such an operation is not reversible. Strictly speaking, isothermal expansion or contraction is unattainable in practice, but it is (without limit) more closely approximated to as the operation is more slowly performed. The adiabatic condition, on the other hand, is more closely approximated to in practice the more swiftly the operation is performed. We have an excellent instance of this in the compression and dilatation of air caused by the propagation of a soundwave.

And now we have Carnot’s invaluable proposition, a reversible heat-engine is a perfect engine,— perfect, that is, in the sense that no other heat-engine can be superior to it. Before giving the proof, let us see the immense consequences of this proposition. Reversibility is the sole test of perfection; so that all heat-engines, whatever be the working substance, provided only they be reversible, convert into work (under given circumstances) the same fraction of the heat supplied to them. The only circumstances involved are the temperatures of the source and condenser. Thus we are furnished with a general principle on which to reason about transformation of heat, altogether independently of the properties of any particular substance.

The proof, as Carnot gave it on the hypothesis of the materiality of heat, is ex absurdo. It is as follows. Suppose a heat-engine A to be capable of giving more work from a given amount of heat than is a reversible engine B, the temperatures of source and condenser being the same for each. Use the two as a compound engine, A working direct and B reversed. By hypothesis B requires to be furnished with part only of the work given by A to be able to restore to the source the heat abstracted by A, and thus at every complete stroke of the compound engine the source has its heat restored to it, while a certain amount of external work has been done. This wou1d be the Perpetual Motion (q.ν.). 5. The Basis of the Second Law of Thermodynamics.— Carnot’s reasoning, just given, is based on the hypothesis that heat (or caloric) is indestructible, and that (under certain conditions) it does work in being let down from a higher to a lower temperature, just as does water when falling to a lower level. It is clear from several expressions in his work that Carnot was not at all satisfied with this view, even in 1824, and we have seen that he soon afterwards reached the true theory. But it is also clear that such an assumption somewhat simplifies the reasoning, for in his hypothetical heat-engine all the heat which leaves the boiler goes to the condenser, and vice versa in the reversed working. The precise point of Carnot’s investigation where the supposed indestructibility of heat introduces error is when, after virtually saying compress from Q' to a state Q determined by the condition that the heat given out shall be exactly equal to that taken in during the expansion from P to P', he assumes that, on farther compressing adiabatically to the original volume, the point P will be reached and the cycle completed. J. Thomson, in 1849, rectified this by putting it in the true form :— compress from Q' to a state Q, such that subsequent adiabatic compression will ultimately lead to the state P.

We have now to consider that, if an engine (whether simple or compound) does work at all by means of heat, less heat necessarily reaches the condenser than left the boiler. Hence, if there be two engines A and B as before, and the joint system be worked in such a way that B constantly restores to the source the heat taken from it by A, we can account for the excess of work done by A over that spent on B solely by supposing that B takes more heat from the condenser than A gives to it. Such a compound engine would transform into work heat taken solely from the condenser. And the work so obtained might be employed on B, so as to make it convey heat to the source while farther cooling the condenser.

Clausius, in 1850, sought to complete the proof by the simple statement that “this contradicts the usual behaviour of heat, which always tends to pass from warmer bodies to colder.” Some years later he employed the axiom, “it is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.” W. Thomson, in 1851, employed the axiom, “it is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.” But he was careful to supplement this by further statements of an extremely guarded character. And rightly so, for Clerk-Maxwell has pointed out that such axioms are, as it were, only accidentally correct, and that the true basis of the second law of thermodynamics lies in the extreme smallness and enormous number of the particles of matter, and in consequence the steadiness of their average behaviour. Had we the means of dealing with the particles individually, we could develop on the large scale what takes place continually on a very minute scale in every mass of gas, —the occasional, but ephemeral, aggregation of warmer particles in one small region and of colder in another.

6. The Laws of Thermodynamics.— I. "When equal quantities of mechanical effect are produced by any means whatever from purely thermal sources, or lost in purely thermal effects, equal quantities of heat are put out of existence, or are generated. [To this we may add, after Joule, that in the latitude of Manchester 772 foot-pounds of work are capable of raising the temperature of a pound of water from 50° F. to 51° F. This corresponds to 1390 foot-pounds per centigrade degree, and in metrical units to 425 kilogramme-metres per calorie (see Heat).]

II. If an engine be such that, when it is worked backwards, the physical and mechanical agencies in every part of its motions are all reversed, it produces as much mechanical effect as can be produced by any thermodynamic engine, with the same temperatures of source and refrigerator, from a given quantity of heat.

7. Absolute Temperature.— We have seen that the fraction of the heat supplied to it which a reversible engine can convert into work depends only on the temperatures of the boiler and of the condenser. On this result of Carnot’s Sir W. Thomson based his absolute definition of temperature. It is clear that a certain freedom of choice is left, and Thomson endeavoured to preserve as close an agreement as possible between the new scale and that of the air thermometer. Thus the definition ultimately fixed on, after exhaustive experiments, runs :—“The temperatures of two bodies are proportional to the quantities of heat respectively taken in and given out in localities at one temperature and at the other respectively, by a material system subjected to a complete cycle of perfectly reversible thermodynamic operations, and not allowed to part with or take in heat at any other temperature; or, the absolute values of two temperatures are to one another in the proportion of the heat taken in to the heat rejected in a perfect thermodynamic engine, working with a source and refrigerator at the higher and lower of the temperatures respectively.” If we now refer again to fig. 1, we see that, t and f being the absolute temperatures corresponding to PP' and QQ', and H, H' the amounts of heat taken in during the operation PP' and given out during the operation Q'Q respectively, we have

H∕i = H'∕i', whatever be the values of t aud t'. Also, if heat be measured in terms of work, we have

H-H' = area PP'Q'Q.

Thus with a reversible engine working between temperatures t and f the fraction of the heat supplied which is converted into work is (í - i')∕i.

It is now evident that we can construct Watt’s diagram in such a way that the lines of equal temperature and the adiabatics may together intercept a series of equal areas. Thus let PP' (fig. 2) be the isothermal t, and on it so take points P', P', P''', <fcc., that, as the working substance passes from P to P', P' to P", <tc., t units of heat (the unit being of any assigned value) shall in each case be taken in. Let QQ', RR', <fcc., be other isothermals, so drawn that the successive areas PQ', QR', <fcc., between any two selected adiabatics, may be equal. Then, as it is clear that all the successive areas between each one pair of isothermals are equal (each representing the area t-t'), it follows that all the quadrilateral areas in the figure are equal.

It is now clear that the area included between PP and the two adiabatics PQR, P'Q'R' is essentially finite, being numerically equal to t. Thus the temperature for each isothermal is represented by the corresponding area. This is indicated in the cut by the introduction of an arbitrary line SS', supposed to be the isothermal of absolute zero. The lower parts of the adiabatics also are unknown, so that we may draw them as we please, subject to the condition that the entire areas PS', PS", P"S'", <fcc., shall all be equal. To find, on the absolute scale, the numerical values of two definite temperatures, such as the usually employed freezing and boiling points of water, we must therefore find their ratio (that of the heat taken and the heat rejected by a reversible engine working between these temperatures), and assign the number of degrees in the interval.

Thomson and Joule experimentally showed that this ratio is about L365. Hence, if we assume (as in the centigrade scale) 100 degrees as the range, the temperatures in question are 274 and 374 nearly. A full discussion of this most important matter will be found under Heat.

8. Entropy.— Just as the lines PP, QQ', <tc., are characterized by constant temperature along each, so we figure to ourselves a quantity which is characteristic of each adiabatic line,—being constant along it. The equation of last section at once points out such a quantity. If we write √> for its value along PQ, <∕>' for P'Q', we may define thus ≠'-≠=H∕i.

From the statements as to the equality of the areas in fig. 2 the reader will see at once that the area bounded by t, f, φ, φ' is (tt')(φ' - φ). We are concerned only with the changes of φ, not with its actual magnitude, so that any one adiabatic may be chosen as that for which φ = 0.

9. The Dissipation of Energy.— In the before-cited article Energy (vol. viii. p. 210) this part of the subject has already been treated. Since that article was written Sir William Thomson has introduced the term thermodynamic motivity to signify “the possession the waste of which is called dissipation.” We speak of a distribution of heat in a body or system of bodies as having motivity, and we may regard it from without or from within the system.

In the first case it expresses the amount of work which can be obtained by means of perfect engines employed to reduce the whole system to some definite temperature, that, say, of the surrounding medium. In the second case the system is regarded as self-contained, its hotter parts acting as sources, and...