Dalton and others had endeavoured to measure directly the rise of temperature produced by the compression of a gas. Dalton had observed a rise of 50° F. in a gas when suddenly compressed to half its volume, but no thermometers at that time were sufficiently sensitive to indicate more than a fraction of the change of temperature. Laplace was the first to see in this phenomenon the probable explanation of the discrepancy between provided that there was no solution or chemical action. He | compression of a gas was due to the diminution of its thermal showed that the condensation or saturation-pressure of a vapour capacity. depended only on the temperature, and increased by nearly the same fraction of itself per degree rise of temperature, and that the pressures of different vapours were nearly the same at equal distances from their boiling points. The increase of pressure per degree C. at the boiling point was about th of 760 mm. or 27.2 mm., but increased in geometrical progression with rise of temperature. These results of Dalton's were confirmed, and in part corrected, as regards increase of vapour-pressure, by Gay-Newton's calculation of the velocity of sound and the observed Lussac, Dulong, Regnault and other investigators, but were found to be as close an approximation to the truth as could be obtained with such simple expressions. More accurate, empirical expressions for the increase of vapour-pressure of a liquid with temperature were soon obtained by Thomas Young, J. P. L. A. Roche and others, but the explanation of the relation was not arrived at until a much later date (see VAPORIZATION). 11. Specific Heats of Gases.-In order to estimate the quantities of heat concerned in experiments with gases, it was necessary in the first instance to measure their specific heats, which presented formidable difficulties. The earlier attempts by Lavoisier and others, employing the ordinary methods of calorimetry, gave very uncertain and discordant results, which were not regarded with any confidence even by the experimentalists themselves. Gay-Lussac (Mémoires d'Arcueil, 1807) devised an ingenious experiment, which, though misinterpreted at the time, is very interesting and instructive. With the object of comparing the specific heats of different gases, he took two equal globes A and B connected by a tube with a stop-cock. The globe B was exhausted, the other A being filled with gas. On opening the tap between the vessels, the gas flowed from A to B and the pressure was rapidly equalized. He observed that the fall of temperature in A was nearly equal to the rise of temperature in B, and that for the same initial pressure the change of temperature was very nearly the same for all the gases he tried, except hydrogen, which showed greater changes of temperature than other gases. He concluded from this experiment that equal volumes of gases had the same capacity for heat, except hydrogen, which he supposed to have a larger capacity, because it showed a greater effect. The method does not in reality afford any direct information with regard to the specific heats, and the conclusion with regard to hydrogen is evidently wrong. At a later date (Ann. de Chim., 1812, 81, p. 98) Gay-Lussac adopted A. Crawford's method of mixture, allowing two equal streams of different gases, one heated and the other cooled about 20° C., to mix in a tube containing a thermometer. The resulting temperature was in all cases nearly the mean of the two, from which he concluded that equal volumes of all the gases tried, namely, hydrogen, carbon dioxide, air, oxygen and nitrogen, had the same thermal capacity. This was correct, except as regards carbon dioxide, but did not give any information as to the actual specific heats referred to water or any known substance. About the same time, F. Delaroche and J. E. Bérard (Ann. de chim., 1813, 85, p. 72) made direct determinations of the specific heats of air, oxygen, hydrogen, carbon monoxide, carbon dioxide, nitrous oxide and ethylene, by passing a stream of gas heated to nearly 100° C. through a spiral tube in a calorimeter containing water. Their work was a great advance on previous attempts, and gave the first trustworthy results. With the exception of hydrogen, which presents peculiar difficulties, they found that equal volumes of the permanent gases, air, oxygen and carbon monoxide, had nearly the same thermal capacity, but that the compound condensible gases, carbon dioxide, nitrous oxide and ethylene, had larger thermal capacities in the order given. They were unable to state whether the specific heats of the gases increased or diminished with temperature, but from experiments on air at pressures of 740 mm. and 1000 mm., they found the specific heats to be 269 and 245 respectively, and concluded that the specific heat diminished with increase of pressure. The difference they observed was really due to errors of experiment, but they regarded it as proving beyond doubt the truth of the calorists' contention that the heat disengaged on the value. The increase of pressure due to a sudden compression, in which no heat was allowed to escape, or as we now call it an "adiabatic" compression, would necessarily be greater than the increase of pressure in a slow isothermal compression, on account of the rise of temperature. As the rapid compressions and rarefactions occurring in the propagation of a sound wave were perfectly adiabatic, it was necessary to take account of the rise of temperature due to compression in calculating the velocity. To reconcile the observed and calculated values of the velocity, the increase of pressure in adiabatic compression must be 1.410 times greater than in isothermal compression. This is the ratio of the adiabatic elasticity of air to the isothermal elasticity. It was a long time, however, before Laplace saw his way to any direct experimental verification of the value of this ratio. At a later date (Ann. de chim., 1816, 3, p. 238) he stated that he had succeeded in proving that the ratio in question must be the same as the ratio of the specific heat of air at constant pressure to the specific heat at constant volume. In the method of measuring the specific heat adopted by Delaroche and Bérard, the gas under experiment, while passing through a tube at practically constant pressure, contracts in cooling, as it gives up its heat to the calorimeter. Part of the heat surrendered to the calorimeter is due to the contraction of volume. If a gramme of gas at pressure p, volume v and temperature T abs. is heated 1° C. at constant pressure p, it absorbs a quantity of heat S=238 calorie (according to Regnault) the specific heat at constant pressure. At the same time the gas expands by a fraction 1/T of v, which is the same as 1/273 of its volume at o° C. If now the air is suddenly volume, and its temperature will be raised by the liberation of a compressed by an amount v/T, it will be restored to its original quantity of heat R', the latent heat of expansion for an increase of volume /T. If no heat has been allowed to escape, the air will now be in the same state as if a quantity of heat S had been communicated to it at its original volume v without expansion. The rise of temperature above the original temperature T will be S/s degrees, where s is the specific heat at constant volume, which is obviously equal to S-R'. Since p/T is the increase of pressure for 1°C. rise of tempera. ture at constant volume, the increase of pressure for a rise of S/s of pressure produced by a sudden compression v/T, and is seen to be degrees will be yp/T, where y is the ratio S/s. But this is the rise times the rise of pressure p/T produced by the same compression at constant temperature. The ratio of the adiabatic to the isothermal elasticity, required for calculating the velocity of sound, is therefore the same as the ratio of the specific heat at constant pressure to that at constant volume. 12. Experimental Verification of the Ratio of Specific Heats.-This was a most interesting and important theoretical relation to discover, but unfortunately it did not help much in the determination time to measure the specific heat of air at constant volume in a of the ratio required, because it was not practically possible at that closed vessel. Attempts had been made to do this, but they had signally failed, on account of the small heat capacity of the gas as compared with the containing vessel. Laplace endeavoured to Delaroche and Bérard for the specific heat of air at 1000 and 740 extract some confirmation of his views from the values given by mm. pressure. On the assumption that the quantities of heat contained in a given mass of air increased in direct proportion to its volume when heated at constant pressure, he deduced, by some rather obscure reasoning, that the ratio of the specific heats Sand s should be about 1.5 to 1, which he regarded as a fairly satisfactory agree. ment with the value y 1.41 deduced from the velocity of sound. The ratio of the specific heats could not be directly measured, but a few years later, Clément and Désormes (Journ. de Phys., Nov. 1819) succeeded in making a direct measurement of the ratio of the elasticities in a very simple manner. They took a large globe containing air at atmospheric pressure and temperature, and removed a small quantity of air. They then observed the defect of pressure po when the air had regained its original temperature. By suddenly opening the globe, and immediately closing it, the pressure was restored almost instantaneously to the atmospheric, the rise of pressure o corresponding to the sudden compression produced. The air, having been heated by the compression, was to steam for developing motive power from heat? In discussing this problem, we cannot do better than follow Carnot's reasoning which, in its main features, could hardly be improved at the present day. allowed to regain its original temperature, the tap remaining closed, | sion or "cycle" of operations continuously? Is the efficiency and the final defect of pressure pl was noted. The change of pressure limited, and, if so, how is it limited? Are other agents preferable for the same compression performed isothermally is then p.-p'; The ratio po/(Po-p) is the ratio of the adiabatic and isothermal elasticities, provided that p. is small compared with the whole atmospheric pressure. In this way they found the ratio 1.354, which is not much smaller than the value 1.410 required to reconcile the observed and calculated values of the velocity of sound. GayLussac and J. J. Welter (Ann. de chim., 1822) repeated the experiment with slight improvements, using expansion instead of compression, and found the ratio 1.375. The experiment has often been repeated since that time, and there is no doubt that the value of the ratio deduced from the velocity of sound is correct, the defect of the value obtained by direct experiment being due to the fact that the compression or expansion is not perfectly adiabatic. Gay-Lussac and Welter found the ratio practically constant for a range of pressure 144 to 1460 mm., and for a range of temperature from -20° to +40° C. The velocity of sound at Quito, at a pressure of 544 mm. was found to be the same as at Paris at 760 mm. at the same temperature. Assuming on this evidence the constancy of the ratio of the specific heats of air, Laplace (Mécanique céleste, v. 143) showed that, if the specific heat at constant pressure was independent of the temperature, the specific heat per unit volume at a pressure must vary as py, according to the caloric theory. The specific which he found agreed heat per unit mass must then vary as p precisely with the experiment of Delaroche and Bérard already cited. This was undoubtedly a strong confirmation of the caloric theory; Poisson by the same assumptions (Ann. de chim., 1823, 23, p. 337) obtained the same results, and also showed that the relation between the pressure and the volume of a gas in adiabatic compression or expansion must be of the form pv-constant. P. L. Dulong (Ann. de chim., 18-9, 41, p. 156), adopting a method due to E. F. F. Chladni, compared the velocities of sound in different gases by observing the pitch of the note given by the same tube when filled with the gases in question. He thus obtained the values of the ratios of the clasticities or of the specific heats for the gases employed. For oxygen, hydrogen and carbonic oxide, these ratios were the same as for air. But for carbonic acid, nitrous oxide and olefiant gas, the values were much smaller, showing that these gases experienced a smaller change of temperature in compression. On comparing his results with the values of the specific heats for the same gases found by Delaroche and Bérard, Dulong observed that the changes of temperature for the same compression were in the inverse ratio of the specific heats at constant volume, and deduced the important conclusion that " Equal volumes of all gases under the same conditions evolve on compression the same quantity of heat." This is equivalent to the statement that the difference of the specific heats, or the latent heat of expansion R' per 1°, is the same for all gases if equal volumes are taken. Assuming the ratio y=1.410, and taking Delaroche and Bérard's value for the specific heat of air at constant pressure S=267, we have s=S/1.41=189, and the difference of the specific heats per unit mass of air S-s=R' = .078. Adopting Regnault's value of the specific heat of air, namely, S=238, we should have S-s-069. This quantity represents the heat absorbed by unit mass of air in expanding at constant temperature T by a fraction 1/T of its volume v, or by rd of its volume o° C. If, instead of taking unit mass, we take a volume v, 22.30 litres at o° C. and 760 mm. being the volume of the molecular weight of the gas in grammes, the quantity of heat evolved by a compression equal to /T will be approximately 2 calories, and is the same for all gases. The work done in this compression is pv/TR, and is also the same for all gases, namely, 8.3 joules. Dulong's experimental result, therefore, shows that the heat evolved in the compression of a gas is proportional to the work done. This result had previously been deduced theoretically by Carnot (1824). At a later date it was assumed by Mayer, Clausius and others, on the evidence of these experiments, that the heat evolved was not merely proportional to the work done, but was equivalent to it. The further experimental evidence required to justify this assumption was first supplied by Joule. Latent heat of expansion R'=069 calorie per gramme of air, per 1° C. =2.0 calories per gramme-molecule of any gas. Work done in expansion R=287 joule per gramme of air per 1° C. =8.3 joules per gramme-molecule of any gas. 13. Carnot: On the Motive Power of Heat -A practical and theoretical question of the greatest importance was first answered by Sadi Carnot about this time in his Reflections on the Motive Power of Heat (1824). How much motive power (defined by Carnot as weight lifted through a certain height) can be obtained from heat alone by means of an engine repeating a regular succes Carnot points out that in order to obtain an answer to this question, it is necessary to consider the essential conditions of the process, apart from the mechanism of the engine and the working substance or agent employed. Work cannot be said to be produced from heat alone unless nothing but heat is supplied, and the working substance and all parts of the engine are at the end of the process in precisely the same state as at the beginning.1 Carnot's Axiom.-Carnot here, and throughout his reasoning, makes a fundamental assumption, which he states as follows: "When a body has undergone any changes and after a certain number of transformations is brought back identically to its original state, considered relatively to density, temperature and mode of aggregation, it must contain the same quantity of heat as it contained originally." Heat, according to Carnot, in the type of engine we are considering, can evidently be a cause of motive power only by virtue of changes of volume or form produced by alternate heating and cooling. This involves the existence of cold and hot bodies to act as boiler and condenser, or source and sink of heat, respectively. Wherever there exists a difference of temperature, it is possible to have the production of motive power from heat; and conversely, production of motive power, from heat alone, is impossible without difference of temperature. In other words the production of motive power from heat is not merely a question of the consumption of heat, but always requires transference of heat from hot to cold. What then are the conditions which enable the difference of temperature to be most advantageously employed in the production of motive power, and how much motive power can be obtained with a given difference of temperature from a given quantity of heat? Carnot's Rule for Maximum Effect.-In order to realize the maximum effect, it is necessary that, in the process employed, there should not be any direct interchange of heat between bodies at different temperatures. Direct transference of heat by conduction or radiation between bodies at different temperatures is equivalent to wasting a difference of temperature which might have been utilized to produce motive power. The working substance must throughout every stage of the process be in equilibrium with itself (i.e. at uniform temperature and pressure) and also with external bodies, such as the boiler and condenser, at such times as it is put in communication with them. In the actual engine there is always some interchange of heat between the steam and the cylinder, and some loss of heat to external bodies. There may also be some difference of temperature between the boiler steam and the cylinder on admission, or between the waste steam and the condenser at release. These differences represent losses of efficiency which may be reduced indefinitely, at least in imagination, by suitable means, and designers had even at that date been very successful in reducing For instance a mass of compressed air, if allowed to expand in a cylinder at the ordinary temperature, will do work, and will at the same time absorb a quantity of heat which, as we now know, is the thermal equivalent of the work done. But this work cannot be said to have been produced solely from the heat absorbed in the process, because the air at the end of the process is in a changed condition, and could not be restored to its original state at the same temperature without having work done upon it precisely equal to that obtained by its expansion. The process could not be repeated indefinitely without a continual supply of compressed air. The source of the work in this case is work previously done in compressing the air, and no part of the work is really generated at the expense of heat alone, unless the compression is effected at a lower temperature than the expansion Clausius (Pogg Ann. 79, p. 369) and others have misinterpreted this assumption, and have taken it to mean that the quantity of heat required to produce any given change of state is independent of the manner in which the change is effected, which Carnot does not here assume. them. All such losses are supposed to be absent in deducing the ideal limit of efficiency, beyond which it would be impossible to go. 14. Carnot's Description of his Ideal Cycle.-Carnot first gives a rough illustration of an incomplete cycle, using steam much in the same way as it is employed in an ordinary steam-engine. After expansion down to condenser pressure the steam is completely condensed to water, and is then returned as cold water to the hot boiler. He points out that the last step does not conform exactly to the condition he laid down, because although the water is restored to its initial state, there is direct passage of heat from a hot body to a cold body in the last process. He points out that this difficulty might be overcome by supposing the difference of temperature small, and by employing a series of engines, each working through a small range, to cover a finite interval of temperature. Having established the general notions of a perfect cycle, he proceeds to give a more exact illustration, employing a gas as the working substance. He takes as the basis of his demonstration the well-established experimental fact that a gas is heated by rapid compression and cooled by rapid expansion, and that if compressed or expanded slowly in contact with conducting bodies, the gas will give out heat in compression or absorb heat in expansion while its temperature remains constant. He then goes on to say:- "This preliminary notion being settled, let us imagine an elastic fluid, atmospheric air for example, enclosed in a cylinder abcd, fig. 4, fitted with a movable diaphragm or piston cd. Let there also be two bodies A, B, each maintained at a constant temperature, that of A being more elevated than that of B. Let us now A B suppose the following series of operations to be performed: "1. Contact of the body A with the air contained in the space abcd, or with the bottom of the cylinder, which we will suppose to transmit heat easily. The air is now at the temperature of the body A, and cd is the actual position of the piston. 2. The piston is gradually raised, and takes the position ef. The air remains in contact with the body A, and is thereby maintained at a constant temperature during the expansion. The body A furnishes the heat necessary to maintain the constancy of temperature. " 3. The body A is removed, and the air no longer being in contact with any body capable of giving it heat, the piston continues nevertheless to rise, and passes from the position ef to gh. The air expands without receiving heat and its temperature falls. Let us imagine that it falls until it is just equal to that of the body B. At this moment the piston is stopped and occupies the position gh. FIG. 4. Carnot's Cylinder. 4. The air is placed in contact with the body B; it is compressed by the return of the piston, which is brought from the position gh to the position cd. The air remains meanwhile at a constant temperature, because of its contact with the body B to which it gives up its heat. 5. The body B is removed, and the compression of the air is continued. The air being now isolated, rises in temperature. The compression is continued until the air has acquired the temperature of the body A. The piston passes meanwhile from the position cd to the position ik. 6. The air is replaced in contact with the body A, and the piston returns from the position ik to the position ef, the temperature remaining invariable. 7. The period described under (3) is repeated, then successively the periods (4), (5), (6); (3), (4), (5), (6); (3), (4), (5), (6); and so on. During these operations the air enclosed in the cylinder exerts an effort more or less great on the piston. The pressure of the air varies both on account of changes of volume and on account of changes of temperature; but it should be observed that for equal volumes, that is to say, for like positions of the piston, the temperature is higher during the dilatation than during the compression. Since the pressure is greater during the expansion, the quantity of motive power produced by the dilatation is greater than that consumed by the compression. We shall thus obtain a balance of motive power, which may be employed for any purpose. The air has served as working substance in a heat-engine; it has also been employed in the most advantageous manner possible, since no useless re-establishment of the equilibrium of heat has been allowed to occur. All the operations above described may be executed in the reverse order and direction. Let us imagine that after the sixth period, that is to say, when the piston has reached the position ef, we make it return to the position ik, and that at the same time we keep the air in contact with the hot body A; the heat furnished by this body during the sixth period will return to its source, that is, to the body A, and everything will be as it was at the end of the fifth period. If now we remove the body A, and if we make the piston move from ik to cd, the temperature of the air will decrease by just as many degrees as it increased during the fifth period, and will become that of the body B. We can evidently continue in this way described; it suffices to place oneself in the same circumstances and a series of operations the exact reverse of those which were previously to execute for each period a movement of expansion in place of a movement of compression, and vice versa. "The result of the first series of operations was the production of a certain quantity of motive power, and the transport of heat from the body A to the body B; the result of the reverse operations is the consumption of the motive power produced in the first case, and the return of heat from the body B to the body A, in such sort that these two series of operations annul and neutralize each other. quantity of motive power greater than that which we have obtained "The impossibility of producing by the agency of heat alone a in our first series of operations is now easy to prove. It is demonstrated by reasoning exactly similar to that which we have already given. The reasoning will have in this case a greater degree of exactitude; the air of which we made use to develop the motive power is brought back at the end of each cycle of operations precisely to its initial state, whereas this was not quite exactly the case for the vapour of water, as we have already remarked." 15. Proof of Carnot's Principle.-Carnot considered the proof too obvious to be worth repeating, but, unfortunately, his previous demonstration, referring to an incomplete cycle, is not so exactly worded that exception cannot be taken to it. We will therefore repeat his proof in a slightly more definite and exact form. Suppose that a reversible engine R, working in the cycle above described, takes a quantity of heat H from the source in each cycle, and performs a quantity of useful work W.. If it were possible for any other engine S, working with the same two bodies A and B as source and refrigerator, to perform a greater amount of useful work W. per cycle for the same quantity of heat H taken from the source, it would suffice to take a portion W, of this motive power (since W, is by hypothesis greater than W,) to drive the engine R backwards, and return a quantity of heat H to the source in each cycle. The process might be repeated indefinitely, and we should obtain at each repetition a balance of useful work W.-W., without taking any heat from the source, which is contrary to experience. Whether the quantity of heat taken from the condenser by R is equal to that given to the condenser by S is immaterial. The hot body A might be a comparatively small boiler, since no heat is taken from it. The cold body B might be the ocean, or the whole earth. We might thus obtain without any consumption of fuel a practically unlimited supply of motive power. Which is absurd. Carnot's Statement of his Principle.'-If the above reasoning be admitted, we must conclude with Carnot that the motive power obtainable from heat is independent of the agents employed to realize it. The efficiency is fixed solely by the temperatures of the bodies between which, in the last resort, the transfer of heat is effected. "We must understand here that each of the methods of developing motive power attains the perfection of which it is susceptible. This condition is fulfilled if, according to our rule, there is produced in the body no change of temperature that is not due to change of volume, or in other words, if there is no direct interchange of heat between bodies of sensibly different temperatures." It is characteristic of a state of frictionless mechanical equili brium that an indefinitely small difference of pressure suffices to upset the equilibrium and reverse the motion. Similarly in thermal equilibrium between bodies at the same temperature, an indefinitely small difference of temperature suffices to reverse the transfer of heat. Carnot's rule is therefore the criterion of the reversibility of a cycle of operations as regards transfer of heat. It is assumed that the ideal engine is mechanically 1 Carnot's description of his cycle and statement of his principle have been given as nearly as possible in his own words, because some injustice has been done him by erroneous descriptions and statements. of cycle. Carnot in making the same calculation did not obtain quite reversible, that there is not, for instance, any communication | steam with the same range of temperature, but a very different kind between reservoirs of gas or vapour at sensibly different pressures, and that there is no waste of power in friction. If there is equilibrium both mechanical and thermal at every stage of the cycle, the ideal engine will be perfectly reversible. That is to say, all its operations will be exactly reversed as regards transfer of heat and work, when the operations are performed in the reverse order and direction. On this understanding Carnot's principle may be put in a different way, which is often adopted, but is really only the same thing put in different words: The efficiency of a perfectly reversible engine is the maximum possible, and is a function solely of the limits of temperature between which it works. This result depends essentially on the existence of a state of thermal equilibrium defined by equality of temperature, and independent, in the majority of cases, of the state of a body in other respects. In order to apply the principle to the calculation and prediction of results, it is sufficient to determine the manner in which the efficiency depends on the temperature for one particular case, since the efficiency must be the same for all reversible engines. to satisfy him that his theor etical deductions were cor- AXIS OF PRESSURE AXIS OF VOLUME FIG. 5.-Elementary Carnot Cycle for Gas. 17. Carnot's Function. Variation of Efficiency with Temperature.-By means of calculations, similar to those given above, Carnot endeavoured to find the amount of motive power obtainable from one unit of heat per degree fall at various temperatures with various substances. The value found above, namely 1-13 kilogrammetre per kilo-calorie per 1° fall, is the value of the efficiency per 1° fall at 100° C. He was able to show that the efficiency per degree fall probably diminished with rise of temperature, but the experimental data at that time were too inconsistent to suggest the true relation. He took as the analytical expression of his principle that the efficiency W/H of a perfect engine taking in heat H at a temperature 1° C.. and rejecting heat at the temperature o° C., must be some function Ft of the temperature 1, which would be the same for all substances. The efficiency per degree fall at a temperature he represented by F't, the derived function of Ft. The function F't would be the same for all substances at the same temperature, but would have different values at different temperatures. In terms of this function, which is generally known as Carnot's function, the results obtained in the previous section might be expressed as follows:--. 16. Experimental Verification of Carnot's Principle.-Carnot endeavoured to test his result by the following simple calculations. Suppose that we have a cylinder fitted with a frictionless piston, containing 1 gram of water at 100° C., and that the pressure of the steam, namely 760 mm., is in equilibrium with the external pressure on the piston at this temperature. Place the cylinder in connexion with a boiler or hot body at 101° C. The water will then acquire the temperature of 101° C., and will absorb 1 gram-calorie of heat. Some waste of motive power occurs here because heat is allowed to pass from ore body to another at a different temperature, but the waste in this case is so small as to be immaterial. Keep the cylinder in contact with the hot body at 101° C. and allow the piston to rise. It may be made to perform useful work as the pressure is now 27.7 mm. (or 37:7 grams per sq. cm.) in excess of the external pressure. Continue the process till all the water is converted into steam. The heat absorbed from the hot body will be nearly 540 gramcalories, the latent heat of steam at this temperature. The increase of volume will be approximately 1620 c.c., the volume of 1 gram of steam at this pressure and temperature. The work done by the excess pressure will be 37.7X1620=61,000 gram-centimetres or 0-61 of a kilogrammetre. Remove the hot body, and allow the steam to expand further till its pressure is 760 mm. and its tempera-F't by the latent heat of vaporization. ture has fallen to 100° C. The work which might be done in this expansion is less than roboth part of a kilogrammetre, and may be neglected for the present purpose. Place the cylinder in contact with the cold body at 100° C., and allow the steam to condense at this temperature. No work is done on the piston, because there is Since the last two coefficients are the same for all gases if equal equilibrium of pressure, but a quantity of heat equal to the latent volumes are taken, Carnot concluded that: "The difference of the heat of steam at 100° C. is given to the cold body. The water is specific heats at constant pressure and volume is the same for equal now in its initial condition, and the result of the process has been to volumes of all gases at the same temperature and pressure.' gain 0-61 of a kilogrammetre of work by allowing 540 gram-calories Taking the expression WRT log for the whole work done by a of heat to pass from a body at 101° C. to a body at 100° C. by means gas obeying the gaseous laws po= RT in expanding at a temperature of an ideally simple steam-engine. The work obtainable in this T from a volume 1 (unity) to a volume r, or for a ratio of expansion way from 1000 gram-calories of heat, or 1 kilo-calorie, would evidently, and putting W'R log for the work done in a cycle of range 10. be 1.13 kilogrammetre (=0·61X1000). Carnot obtained the expression for the heat absorbed by a gas in isothermal expansion "The increase of volume of a mixture of liquid and vapour per unit-mass vaporized at any temperature, multiplied by the increase of vapour-pressure per degree, is equal to the product of the function "The difference of the specific heats, or the latent heat of expansion for any substance multiplied by the function F't, is equal to the product of the expansion per degree at constant pressure by the increase of pressure per degree at constant volume." H =R log7/F'! F't=R/C(+b). (2) Taking the same range of temperature, namely 101° to 100° C., we may perform a similar series of operations with air in the cylinder, instead of water and steam. Suppose the cylinder to contain i gramme of air at 100° C. and 760 mm, pressure instead of water. He gives several important deductions which follow from this formula, Compress it without loss of heat (adiabatically), so as to raise its which is the analytical expression of the experimental result already temperature to 101° C. Place it in contact with the hot body at quoted as having been discovered subsequently by Dulong. Employ101 C., and allow it to expand at this temperature, absorbing heating the above expression for the latent heat of expansion, Carnot from the hot body, until its volume is increased by 4th part (the deduced a general expression for the specific heat of a gas at constant expansion per degree at constant pressure). The quantity of heat volume on the basis of the caloric theory. He showed that if the absorbed in this expansion, as explained in § 14, will be the difference specific heat was independent of the temperature (the hypothesis of the specific heats or the latent heat of expansion R'069 calorie. already adopted by Laplace and Poisson) the function F't must be Remove the hot body, and allow the gas to expand further without of the form gain of heat till its temperature falls to 100° C. Compress it at (3) 100° C. to its original volume, abstracting the heat of compression by where C and to are unknown constants. A similar result follows contact with the cold body at 100° C. The air is now in its original from his expression for the difference of the specific heats. If this is state, and the process has been carried out in strict accordance with Carnot's rule. The quantity of external work done in the cycle R/CT, which is the same as the above if fo=273. Assuming the assumed to be constant and equal to C, the expression for F't becomes is easily obtained by the aid of the indicator diagram ABCD (fig. 5). specific heat to be also independent of the volume, he shows that the which is approximately a parallelogram in this instance. The area function F't should be constant. But this assumption is inconsistent of the diagram is equal to that of the rectangie BEHG, being the with the caloric theory of latent heat of expansion, which requires product of the vertical height BE, namely, the increase of pressure the specific heat to be a function of the volume. It appears in fact per 1° at constant volume, by the increase of volume BG, which is impossible to reconcile Carnot's principle with the caloric theory ard of the volume at o° C. and 760 mm., or 2.83 c.c. The increase on any simple assumptions. As Carnot remarks: "The main prinof pressure BE is 18, or 2-03 mm., which is equivalent to 2.76 ciples on which the theory of heat rests require most careful examina gm. per sq. cm. The work done in the cycle is 2-76X2·83=7.82 tion. Many experimental facts appear almost inexplicable in the gm. cm., or 0782 gram-metre. The heat absorbed at 101° C. was present state of this theory." 069 gram-calorie, so that the work obtained is .0782/-069 or 1.13 gram-metre per gram-calorie, or 113 kilogrammetre per kilogramcalorie. This result is precisely the same as that obtained by using Carnot's work was subsequently put in a more complete analytical form by B. P. E. Clapeyron (Journ, de l'éc. polytechn Paris, 1832, 14, p. 153), who also made use of Watt's indicator | definitely, deducing it from the old principle, causa aequat diagram for the first time in discussing physical problems. effectum. Assuming that the sinking of a mercury column by Clapeyron gave the general expressions for the latent heat of a which a gas was compressed was equivalent to the heat set free vapour, and for the latent heat of isothermal expansion of any by the compression, he deduced that the warming of a kilosubstance, in terms of Carnot's function, employing the notation gramme of water 1° C. would correspond to the fall of a weight of the calculus. The expressions he gave are the same in form as of one kilogramme from a height of about 365 metres. But those in use at the present day. He also gave the general Mayer did not adduce any fresh experimental evidence, and expression for Carnot's function, and endeavoured to find its made no attempt to apply his theory to the fundamental variation with temperature; but having no better data, he equations of thermodynamics. It has since been urged that the succeeded no better than Carnot. Unfortunately, in describing experiment of Gay-Lussac (1807), on the expansion of gas from Carnot's cycle, he assumed the caloric theory of heat, and made one globe to another (see above, § 11), was sufficient justification some unnecessary mistakes, which Carnot (who, we now know, for the assumption tacitly involved in Mayer's calculation. was a believer in the mechanical theory) had been very careful But Joule was the first to supply the correct interpretation of to avoid. Clapeyron directs one to compress the gas at the lower this experiment, and to repeat it on an adequate scale with suittemperature in contact with the body B until the heat disengaged able precautions. Joule was also the first to measure directly is equal to that which has been absorbed at the higher temperature. the amount of heat liberated by the compression of a gas, and to He assumes that the gas at this point contains the same quantity | prove that heat was not merely rendered latent, but disappeared of heat as it contained in its original state at the higher tempera- altogether as heat, when a gas did work in expansion. ture, and that, when the body B is removed, the gas will be restored to its original temperature, when compressed to its initial volume. This mistake is still attributed to Carnot, and regarded as a fatal objection to his reasoning by nearly all writers at the present day. 18. Mechanical Theory of Heat.-Accordingto the caloric theory, the heat absorbed in the expansion of a gas became latent, like the latent heat of vaporization of a liquid, but remained in the gas and was again evolved on compressing the gas. This theory gave no explanation of the source of the motive power produced by expansion. The mechanical theory had explained the production of heat by friction as being due to transformation of visible motion into a brisk agitation of the ultimate molecules, but it had not so far given any definite explanation of the converse production of motive power at the expense of heat. The theory could not be regarded as complete until it had been shown that in the production of work from heat, a certain quantity of heat disappeared, and ceased to exist as heat; and that this quantity was the same as that which could be generated by the expenditure of the work produced. The earliest complete statement of the mechanical theory from this point of view is contained in some notes written by Carnot, about 1830, but published by his brother (Life of Sadi Carnot, Paris, 1878). Taking the difference of the specific heats to be 078, he estimated the mechanical equivalent at 370 kilogrammetres. But he fully recognized that there were no experimental data at that time available for a quantitative test of the theory, although it appeared to afford a good qualitative explanation of the phenomena. He therefore planned a number of crucial experiments such as the porous plug" experiment, to test the equivalence of heat and motive power. His early death in 1836 put a stop to these experiments, but many of them have since been independently carried out by other observers. 66 The most obvious case of the production of work from heat is in the expansion of a gas or vapour, which served in the first instance as a means of calculating the ratio of equivalence, on the assumption that all the heat which disappeared had been transformed into work and had not merely become latent. Marc Séguin, in his De l'influence des chemins de fer (Paris, 1839), made a rough estimate in this manner of the mechanical equivalent of heat, assuming that the loss of heat represented by the fall of temperature of steam on expanding was equivalent to the mechanical effect produced by the expansion. He also remarks (loc. cit. p. 382) that it was absurd to suppose that "a finite quantity of heat could produce an indefinite quantity of mechanical action, and that it was more natural to assume that a certain quantity of heat disappeared in the very act of producing motive power." J. R. Mayer (Liebig's Annalen, 1842, 42, p. 233) stated the equivalence of heat and work more It was for this reason that Professor W. Thomson (Lord Kelvin) stated (Phil. Mag., 1852, 4) that "Carnot's original demonstration utterly fails," and that he introduced the "corrections" attributed to James Thomson and Clerk Maxwell respectively. In reality Carnot's original demonstration requires no correction. 19. Joule's Determinations of the Mechanical Equivalent.-The honour of placing the mechanical theory of heat on a sound experimental basis belongs almost exclusively to J. P. Joule, who showed by direct experiment that in all the most important cases in which heat was generated by the expenditure of mechanical work, or mechanical work was produced at the expense of heat, there was a constant ratio of equivalence between the heat generated and the work expended and vice versa. His first experiments were on the relation of the chemical and electric energy expended to the heat produced in metallic conductors and voltaic and electrolytic cells; these experiments were described in a series of papers published in the Phil. Mag., 1840-1843. He first proved the relation, known as Joule's law, that the heat produced in a conductor of resistance R by a current C is proportional to CR per second. He went on to show that the total heat produced in any voltaic circuit was proportional to the electromotive force E of the battery and to the number of equivalents electrolysed in it. Faraday had shown that electromotive force depends on chemical affinity. Joule measured the corresponding heats of combustion, and showed that the electromotive force corresponding to a chemical reaction is proportional to the heat of combustion of the electrochemical equivalent. He also measured the E.M.F. required to decompose water, and showed that when part of the electric energy EC is thus expended in a voltameter, the heat generated is less than the heat of combustion corresponding to EC by a quantity representing the heat of combustion of the decomposed gases. His papers so far had been concerned with the relations between electrical energy, chemical energy and heat which he showed to be mutually equivalent. The first paper in which he discussed the relation of heat to mechanical power was entitled "On the Calorific Effects of Magneto-Electricity, and on the Mechanical Value of Heat" (Brit. Assoc., 1843; Phil. Mag., 23, p. 263). In this paper he showed that the heat produced by currents generated by magneto-electric induction followed the same law as voltaic currents. By a simple and ingenious arrangement he succeeded in measuring the mechanical power expended in producing the currents, and deduced the mechanical equivalent of heat and of electrical energy. The amount of mechanical work required to raise 1 lb of water 1° F. (1 B.Th.U.), as found by this method, was 838 foot-pounds. In a note added to the paper he states that he found the value 770 foot-pounds by the more direct method of forcing water through fine tubes. In a paper "On the Changes of Temperature produced by the Rarefaction and Condensation of Air" (Phil. Mag., May 1845), he made the first direct measurements of the quantity of heat disengaged by compressing air, and also of the heat absorbed when the air was allowed to expand against atmospheric pressure; as the result he deduced the value 798 foot-pounds for the mechanical equivalent of 1 B.Th. U. He also showed that there was no appreciable absorption of heat when air was allowed to expand in such a manner as not to develop mechanical power, and he pointed out that the mechanical equivalent of heat could not be satisfactorily deduced from |