allowed to regain its original temperature, the tap remaining closed, and the final defect of pressure p1 was noted. The change of pressure for the same compression performed isothermally is then p0 − p1. The ratio p0/(p0 − p1) is the ratio of the adiabatic and isothermal elasticities, provided that p0 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. Gay-Lussac 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 p must vary as p1γ, according to the caloric theory. The specific heat per unit mass must then vary as p1γ−1 which he found agreed 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 pvγ = constant.
P. L. Dulong (Ann. de chim., 1829, 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 elasticities 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 γ = 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 1273rd of its volume 0° C.
If, instead of taking unit mass, we take a volume v0 = 22.30 litres at 0° 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 v/T will be approximately 2 calories, and is the same for all gases. The work done in this compression is pv/T = R, 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 succession or “cycle” of operations continuously? Is the efficiency limited, and, if so, how is it limited? Are other agents preferable 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.
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.”[2]
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.
