§ 4. Deductions as to the two Specific Heats, and transformation of the foregoing equations. In the same way as we see from equation (10) that the quantity C, which appears as factor of dT, denotes the specific heat at constant temperature, we may see from equation (12) that the factor of dT in that equation, viz. C+R, expresses the Specific Heat at constant pressure. If therefore we denote this Specific Heat by C, we may put ...(14), which equation gives the relation between the two Specific Heats. Since R is a constant, and C,, as shewn above, is a function of temperature only, it follows from equation (14) that C, also can only be a function of temperature. When the author first drew in this manner from the Mechanical Theory of Heat the conclusion that the two Specific Heats of a permanent gas must be independent of its density, or in other words of the pressure to which it is subjected, and could depend only on its temperature; and when he added the further remark that they were thus in all probability constant; he found himself in opposition to the then prevailing views on the subject. At that time it was considered to be established from the experiments of Suermann, and from those of de la Roche and Bérard, that the specific heat of a gas depended on the pressure; and the circumstance that the new theory led to an opposite conclusion produced mistrust of the theory itself, and was used by A. von Holtzmann as a weapon of attack against it. Some years later, however, followed the first publication of the splendid experiments of Regnault on the specific heat of gases*, in which the influence of pressure and temperature on the specific heat was made a subject of special * Vol. II. Comptes Rendus, Vol. xxxvI., 1853; also Relation des expériences, investigation. Regnault tested atmospheric air at pressures from 1 to 12 atmospheres, and hydrogen at from 1 to 9 atmospheres, but could detect no difference in their specific heats. He tested them also at different temperatures, viz. between -30° and +10°, between 0° and 100°, and between 0' and 200°; and here also he found the specific heat always the same The result of his experiments may thus be expressed by saying that, within the limits of pressure and temperature to which his observations extended, the specific heat of permanent gases was found to be constant. It is true that these direct explanatory researches were confined to the specific heat at constant pressure; but there will be little scruple raised as to assuming the same to be correct for the other specific heat, which by equation (14) differs from the former only by the constant R. Accordingly in what follows we shall treat the two specific heats, at least for perfect gases, as being constant quantities. By help of equation (14) we may transform the three equations (11), (12) and (13), which express the first main principle of the Mechanical Theory of Heat as applied to gases, in such a way that they may contain, instead of the Specific Heat at constant volume, the Specific Heat at constant density; which may perhaps appear more suitable, since the latter, as being determined by direct observation, ought to be used more frequently than the former. The resulting equations are: Lastly, we may introduce both Specific Heats into the equations, and eliminate R, by which means the resulting The numbers obtained for atmospheric air (Rel. des Exp. Vol. 11., p. 108) are as follows in ordinary heat units: between -30° and + 10° 0.23771, 99 which may be taken as practically the same. equations become symmetrical as to p and v, as follows: In the above equations the specific heats are expressed in mechanical units. If we wish to express them in ordinary heat units, we have only to divide these values by the Mechanical Equivalent of Heat. Thus if we denote the specific heats, as expressed in ordinary heat units, by c, and c,, we may put Applying these equations to equation (14), and dividing by E, we have § 5. Relation between the two Specific Heats, and its application to calculate the Mechanical Equivalent of Heat. If a system of Sound-waves spreads itself through any gas, e.g. atmospheric air, the gas becomes in turn condensed and rarefied; and the velocity with which the sound spreads depends, as was seen by Newton, on the nature of the changes of pressure produced by these changes of density. For very small changes of density and pressure the relation between. the two is expressed by the differential coefficient of the pressure with respect to the density, or (if the density, i. e. the weight of a unit of volume, is denoted by p) by the differential dp coefficient Applying this principle we obtain for the dp velocity of sound, which we will call u, the following equa tion dp u = 9 dp .(19), in which g represents the accelerating force of gravity. Now in order to determine the value of the differential dp coefficient Newton used the law of Mariotte*, according to dp which pressure and density are proportional to each other. He therefore put p constant, whence by differentiation: and therefore P whence (19) becomes The velocity calculated by this formula did not however agree with experiment, and the reason of this divergence, after it had been long sought for in vain, was at last discovered by Laplace. The law of Mariotte in fact holds only if the change of density takes place at constant temperature. But in sound vibrations this is not the case, since in every condensation a heating of the air takes place, and in every rarefaction a cooling. Accordingly at each condensation the pressure is increased, and at each rarefaction diminished, to a greater extent than accords with Mariotte's law. The question now dp arises how, under these circumstances, can the value of be determined. dp Since the condensations and rarefactions follow each other with great rapidity, the exchange of heat that can take place during each short period between the condensed and rarefied parts of the gas must be very small. Neglecting this, we have to do with a change of density, in which the quantity of gas under consideration receives no heat and gives forth none; and we may thus, in applying to this case the differential equa This law is commonly known in England as 'Boyle's law,' as being originally due to Boyle. (Translator.) tions of the last section, put dQ=0. Hence, e. g. from the last of equations (16), we obtain: Now, since the volume v of one unit of weight is the re 1 ciprocal of the density, we may put v=- and therefore dv == 2 Ρ ρ , dp dp This value of the Differential Coefficient differs from that deduced from Mariotte's law, and given in (20), by containing as factor the ratio of the two Specific Heats. If for simplicity we put C From this equation the velocity of sound u can be calculated if k is known; or, on the other hand, if the velocity of sound is known by experiment, we can apply the equation to calculate k, changing it first into the form |