Part 35

Footnote 19:

We will suppose, in what follows, the reader to be au courant with the later progress of modern Physics in regard to gaseous substances and heat.

Footnote 20:

M. Poisson, to whom this figure is due, has shown that it accords very well with the result of an experiment of MM. Clement and Desormes on the return of air into a vacuum, or rather, into air slightly rarefied. It also accords very nearly with results found by MM. Gay-Lussac and Welter. (See note, p. 87.)

Footnote 21:

The law of Mariotte, which is here made the foundation upon which to establish our demonstration, is one of the best authenticated physical laws. It has served as a basis to many theories verified by experience, and which in turn verify all the laws on which they are founded. We can cite also, as a valuable verification of Mariotte’s law and also of that of MM. Gay-Lussac and Dalton, for a great difference of temperature, the experiments of MM. Dulong and Petit. (See Annales de Chimie et de Physique, Feb. 1818, t. vii. p. 122.)

The more recent experiments of Davy and Faraday can also be cited.

The theories that we deduce here would not perhaps be exact if applied outside of certain limits either of density or temperature. They should be regarded as true only within the limits in which the laws of Mariotte and of MM. Gay-Lussac and Dalton are themselves proven.

Footnote 22:

When the volume is reduced ¹⁄₁₁₆, that is, when it becomes ¹¹⁵⁄₁₁₆ of what it was at first, the temperature rises one degree. Another reduction of ¹⁄₁₁₆ carries the volume to (¹¹⁵⁄₁₁₆)^2, and the temperature should rise another degree. After x similar reductions the volume becomes (¹¹⁵⁄₁₁₆)^{x}, and the temperature should be raised x degrees. If we suppose (¹¹⁵⁄₁₁₆)^{x} = ¹⁄₁₄, and if we take the logarithms of both, we find

x = about 300°.

If we suppose (¹¹⁵⁄₁₁₆)^{x} = ½, we find

x = 80°;

which shows that air compressed one half rises 80°.

All this is subject to the hypothesis that the specific heat of air does not change, although the volume diminishes. But if, for the reasons hereafter given (pp. 86, 89), we regard the specific heat of air compressed one half as reduced in the relation of 700 to 616, the number 80° must be multiplied by ⁷⁰⁰⁄₆₁₆, which raises it to 90°.

Footnote 23:

MM. Gay-Lussac and Welter have found by direct experiments, cited in the Mécanique Céleste and in the Annales de Chimie et de Physique, July, 1822, p. 267, that the ratio between the specific heat at constant pressure and the specific heat at constant volume varies very little with the density of the gas. According to what we have just seen, the difference should remain constant, and not the ratio. As, further, the specific heat of gases for a given weight varies very little with the density, it is evident that the ratio itself experiences but slight changes.

The ratio between the specific heat of atmospheric air at constant pressure and at constant volume is, according to MM. Gay-Lussac and Welter, 1.3748, a number almost constant for all pressures, and even for all temperatures. We have come, through other considerations, to the number (267 + 116)/(267) = 1.44, which differs from the former (1)/(20), and we have used this number to prepare a table of the specific heats of gases at constant volume. So we need not regard this table as very exact, any more than the table given on p. 89. These tables are mainly intended to demonstrate the laws governing specific heats of aeriform fluids.

Footnote 24:

Note C, Appendix B.

Footnote 25:

Note D, Appendix B.

Footnote 26:

Note E, Appendix B.

Footnote 27:

We find (Annales de Chimie et de Physique, July, 1818, p. 294) in a memoir of M. Petit an estimate of the motive power of heat applied to air and to vapor of water. This estimate leads us to attribute a great advantage to atmospheric air, but it is derived by a method of considering the action of heat which is quite imperfect.

Footnote 28:

Note F, Appendix B.