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# Thermal Properties of Gases

THERMAL PROPERTIES OF GASES 13. The most characteristic property of a gaseous or elastic fluid, namely, the elasticity, or resistance to compression, was first investigated scientifically by Robert Boyle (1662), who showed that the pressure p of a given mass of gas varied inversely as the volume v, provided that the temperature remained constant. This is generally expressed by the f or mula pv=C, where C is a constant for any given tempera ture, and v is taken to represent the specific volume, or the volume of unit mass, of the gas at the given pressure and temperature. Boyle was well aware of the effect of heat in expanding a gas, but he was unable to investigate this properly as no thermometric scale had been defined at that date. According to Boyle's law, when a mass of gas is compressed by a small amount at constant temperature, the percentage increase of pressure is equal to the percentage diminution of volume (if the compression is v/too, the increase of pressure is very nearly p/ioo). Adopting this law, Newton showed, by a most ingenious piece of reasoning (Principia, ii., sect. 8), that the velocity of sound in air should be equal to the velocity acquired by a body falling under gravity through a distance equal to half the height of the atmosphere, considered as being of uniform density equal to that at the surface of the earth. This gave the result 918ft. per sec. (28o metres per sec.) for the velocity at the freezing point. Newton was aware that the actual velocity of sound was somewhat greater than this, but supposed that the difference might be due in some way to the size of the air particles, of which no account could be taken in the calcula tion. The first accurate measurement of the velocity of sound by the French Academie des Sciences in 1738 gave the value 332 metres per sec. as the velocity at o° C. The true explanation of the discrepancy was not discovered till nearly loo years later.

The law of expansion of gases with change of temperature was investigated by Dalton and later by Gay-Lussac (1802), who found that the volume of a gas under constant pressure increased by -s 7 of its volume at o° C for each I° C rise in temperature. This value was generally assumed in all calculations for nearly 5o years. More exact researches, especially those of Regnault, at a later date, showed that the law was very nearly correct for all permanent gases, but that the value of the coefficient should be According to this law the volume of a gas at any temperature t° C should be proportional to 273+t, i.e., to the temperature reckoned from a zero 273° below that of the Centi grade scale, which was called the _absolute zero of the gas ther mometer. If T= 2 73+t, denotes the temperature measured from this zero, the law of expansion of a gas may be combined with Boyle's law in the simple formula pv=RT (I) which is generally taken as the expression of the gaseous laws. If equal volumes of different gases are taken at the same tempera ture and pressure, it follows that the constant R is the same for all gases. If equal masses are taken, the value of the constant R for different gases varies inversely as the molecular weight or as the density relative to hydrogen.

Dalton also investigated the laws of vapours, and of mixtures of gases and vapours. He found that condensible vapours approxi mately followed Boyle's law when compressed, until the condensa tion pressure was reached, at which the vapour liquefied without further increase of pressure. He found that when a liquid was introduced into a closed space, and allowed to evaporate until the space was saturated with the vapour and evaporation ceased, the increase of pressure in the space was equal to the condensation pressure cf the vapour, and did not depend on the volume of the space or the presence of any other gas or vapour provided that there was no solution or chemical action. He showed that the condensation or saturation-pressure of a vapour 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 of 76omm. or 27.2mm., 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-Lussac, Dulong, Regnault and other investigators, but were found to be as close an approxi mation to the truth as could be obtained with such simple expres sions. 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 explana tion of the relation was not arrived at until a much later date (see VAPORIZATION).

## 14.

Specific Heats of Gases.-In order to estimate the quanti ties 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 (Mernoires d'Arcueil, 1807) devised an ingenious experi ment, which, though misinterpreted at the time, is very interest ing and instructive. With the object of comparing the specific heats of different gases, he took two equal globes A and B con nected 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) Gay-Lussac adopted A. Crawford's method of mixture, allowing two equal streams of different gases, one heated and the other cooled about 2o° C, to mix in a tube containing a thermometer. The resulting tem perature 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. Berard (Ann. de Chim., 1813) made direct determinations of the specific heats of air, oxygen, hydro gen, carbon monoxide, carbon dioxide, nitrous oxide and ethy lene, by passing a stream of gas heated to nearly roo° C through a spiral tube in a calorimeter containing water. Their work was a great advance on previous attempts, and gave the first trust worthy results. With the exception of hydrogen, which presents peculiar difficulties, they found that equal volumes of the perma nent gases, air, oxygen and carbon monoxide, had nearly the same thermal capacity, but that the compound condensible gases, car bon dioxide, nitrous oxide and ethylene, had larger thermal capaci ties in the order given. They were unable to state whether the specific heats of the gases increased or diminished with tempera ture, but from experiments on air at pressures of 74omm. and r,000mm., they found the specific heats to be •269 and 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 dis engaged on the compression of a gas was due to the diminution of its thermal capacity.

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 5o° F in a gas when suddenly compressed to half its volume, but no thermometers at that time were suffi ciently 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 Newton's calcu lation of the velocity of sound and the observed value. The in crease of pressure due to a sudden compression, in which no heat was allowed to escape, or as we now call it an adiabatic compres sion, would necessarily be greater than the increase of pressure in a slow isothermal_ compression, on account of the rise of tem perature. 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 1.41 o times greater than in iso thermal 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 verifica tion of the value of this ratio. At a later date (Ann. de chins., 1816) 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.

15. 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 of the ratio required, because it was not practically possible at that time to measure the specific heat of air at con stant volume in a 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. La place endeavoured to extract some confirmation of his views from the values given by Delaroche and Berard for the specific heat of air at 1,00o and 74omm. 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 S and s should be about 1.5 to I, which he re garded as a fairly satisfactory agreement with the value y= I.41 deduced from the velocity of sound.

The ratio of the specific heats could not be directly measured, but a few years later, N. Clement-Desormes (Journ. de Phys., 1819) succeeded in making a direct measurement of the ratio of the elasticities in a very simple manner. He took a large globe containing air at atmospheric pressure and temperature, and re moved a small quantity of air. He 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 corresponding to the sudden compression produced. The air, having been heated by the compression, was allowed to regain its original temperature, the tap remaining closed, and the final defect of pressure p, was noted. The change of pressure for the same compression performed isothermally is then The ratio is the ratio of the adiabatic and isothermal elasticities, provided that is small compared with the whole atmospheric pressure. In this way he found the ratio 1.354, which is not much smaller than the value 1.410 re quired to reconcile the observed and calculated values of the ve locity of sound. Gay-Lussac and J. J. Welter (Ann. de chins., 18 2 2) 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 experi ment 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 1,46omm., and for a range of temperature from —2o° to +4o° C. The velocity of sound at Quito, at a pressure of 544mm. was found to be the same as at Paris at 76omm. at the same temperature. Assuming on this evidence the constancy of the ratio of the specific heats of air, Laplace (Mecanique celeste, v.) 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 p1/7 according to the caloric theory. The specific heat per unit mass must then vary as pl/y-1 which he found agreed precisely with the experiment of Delaroche and Berard already cited. This was undoubtedly a strong confirmation of the caloric theory. Poisson by the same assumptions (Ann. de chim., 1823) 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 7 = constant.

P. L. Dulong (Ann. de chim., 1829), 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 val ues 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 Berard, Dulong observed that the changes of temperature for the same compression were in the inverse ratio of the specific heats at con stant volume, and deduced the important conclusion that "Equal volumes of all gases under the same conditions evolve on com pression 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 r °, is the same for all gases if equal volumes are taken. Assuming the ratio -y =1 •4 i o, and taking Delaroche and Berard's value for the specific heat of air at con stant pressure S= • 267, we have s=S/i •4 i = • 189, and the differ ence of the specific heats per unit mass of air S—s=R'=•o78. 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 i/T of its volume v, or by of its volume at o° C.

If, instead of taking unit mass, we take a volume 22.30 litres at o° C and 76omm. 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, Claus ius 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.