Since each of these molecules, being of mass m and moving with speed G from left to right, carries over into the right half the momentum mG, the molecular momentum of this half from left to right will have been increased by the passage of molecules over this unit area in the unit of time by the amount NG × MG= NmG2; while simultaneously the same number of molecules cross the area from right to left, diminishing thereby the oppositely directed from right to left-momentum of the right half by the same amount; and therefore there is produced in the right half an excess of the left-to-right molecular momentum over that from right to left of twice this amount, or of NmG2. This excess acts continuously during the given time-viz. the time-unit-as a force from left to right on the right half of the medium, and it is nothing else than the pressure p = NmG2, which is balanced by the oppositely directed pressure of the other half. This formula is the same as that found before, and thus proves that its validity is not bound up with the assumption. before made, which assimilated the problem to that of elastic collision. 13. Absolute Value of the Molecular Speed The product Nm in the last formula, of the mass of a molecule m into the number N of the molecules contained in unit volume, has a simple meaning, for it obviously represents the mass of gas in the unit of volume; but this may be shortly called the density of the gas, the density of water-of which the mass-unit, one kilogram, occupies the volume-unit, one litre-being the unit density. This definition of the density p gives Nm = p. Consequently the formula may be written p = pG2, in which form its agreement with Boyle's law, viz. that the pressure of a gas is proportional to its density, is more directly seen. In this new form, however, it teaches us much more; it empowers us to draw a remarkable and very important conclusion. Since two of the magnitudes occurring in the formula, viz. the pressure p and the density p, are directly amenable to observation and measurement, the formula allows us to deduce from them the third, viz. the value of G, the mean speed of the molecules, in absolute measure. It was Joule who by this conception opened up to investigation a field which one would have been tempted to think was closed to human knowledge; and Clausius 2 followed him along the path thus trodden to explore an unseen world. Though measured by the height h of a column of mercury, the pressure p is not identical with this height, but with the action of gravity on the column when taken of unit area. If, then, q denotes the density of mercury and g the acceleration of gravity, we have p = gqh, and therefore G is given by = G2 = 3ggh/p. Let us make this calculation for the temperature 0° C. and the pressure of one atmosphere, i.e. of a column of mercury 0.76 metre high. We will take Regnault's 3 value, q 13.596, and his values for the density of the various gases; we must therefore take the value of gravity for Paris, where Regnault made his observations, and put g = 9.80896 metres per sec. per sec. The density p of the gas is, like the density q of mercury, to be referred to water as unity; but if instead it is referred to air, which under the 1 Mem. of the Manch. Lit. and Phil. Soc. [2] ix. 1851, p. 107; Phil. Mag. [4] xiv. 1857, p. 211. 2 Pogg. Ann. c. 1857, p. 375; Abhandl. über d. Wärmetheorie, pt. ii. 1867, p. 254; transl. Phil. Mag. [4] xiv. 1857, p. 108. 3 Mér. de l'Acad. de Paris, xxi. 1847, p. 162. given circumstances is 773-3 times lighter than water at 4° C.,' we must put p = s/773·3, where s is the specific gravity of the gas. We obtain in this manner, according to Clausius's calculation, a general formula for the value of the mean molecular speed of a gas at 0° C., which we will denote by 6, viz. 6485/s metres per second,2 which holds good for all pressures and places, though the special circumstances of Regnault's observations were employed in its calculation. From this formula Clausius3 has deduced the following values for the mean molecular speeds of atmospheric air and other gases at 0° C. in metres per second: The surprising magnitude of these numbers may serve as new evidence of the degree in which heat, the cause of these rapid motions, is superior to the mechanical forces which are at our disposal in capability for doing work; and they further justify the assertion in § 4, which is there not proved, that the speeds produced by gravity in short periods are too small in comparison with these speeds to cause any sensible parabolic curvature in the paths of the molecules. But, on the other hand, these molecular speeds are not so great that in comparison with them gravity can be absolutely neglected. If this were so, the continuance of an atmosphere about the earth would be impossible, as all the 1/773-270, and Broch (Trav. et Mém. 0.00129305 = 1/773-365. 'Regnault found p = 0·00129321 = du Bureau Int. des Poids et Mes. 1881, pt. i. p. 49) p 2 = [Regnault's value of p gives 484-898, and Broch's 484-928, and the number 773-3 for the value of s/p gives 484.907.-TR.] 3 Pogg. Ann. c. 1857, p. 377; Abh. i. Wärmetheorie, pt. ii. 1867, p. 256; Mechanische Wärmetheorie, 1889-91, 2nd ed. iii. p. 35, edited by Planck and Pulfrich. molecules of the air would disperse into space in consequence of their speed. A body thrown vertically upward with a speed of 485 metres per second would rise to a height of 12,000 metres and then fall back again. A molecule of air, therefore, which moves at the earth's surface with a calculated mean speed of 485 metres per second, cannot in consequence of this rise higher than 12,000 metres, and remains, therefore, within the much higher atmosphere. Just as little can molecules in higher layers leave the atmosphere, as these layers are colder, and the molecular speed is therefore smaller. To entirely escape from the earth without returning, a molecule of air must have at the earth's surface a speed of at least 11,000 metres per second, which is twenty times larger than the mean speed at 0°; but we are in a position to assume that such a speed can never occur―or, at most, only very exceptionally. On the moon, whose diameter is four times less than that of the earth, the acceleration of gravity is nearly six times less than on the earth; consequently a molecule of air with the speed 485 metres per second could rise to a height of 74 kilometres, and to escape entirely from the moon it would require a speed of only 2,400 metres per second. From this we may conclude that if the moon possesses an atmosphere at its general low temperature it can in any case have only a very thin one." 14. Temperature The values found for the mean molecular speed are those at 0° C., and it has already been pointed out in § 9 that this speed increases with the temperature. To determine in what ratio this increase takes place we have to compare the formula which theoretically expresses the pressure with the laws regarding the pressure that have been deduced from Compare §§ 24, 26, 28. 2 Bull, Liveing, and Bryan, Meteorologische Zeitschrift, xi. 1894, p. 76. G. Johnstone Stoney, Astrophys. Journ. 1898, vii. p. 25 ; Journ. de Phys. [3] vii. p. 528. experiment. This comparison leads to a definition of the nature of temperature in reference to the conceptions underlying our theory. The empirical law, discovered nearly simultaneously by Gay-Lussac1 and Dalton,2 which expresses the dependence of the pressure of a gas on its temperature, is contained in this amended form of Boyle's law, viz. : p = kp(1 + ad), where p and p denote the pressure and density as before, 9 is the temperature C., k a constant, and a the thermal coefficient of expansion, or, more correctly, the coefficient of increase of pressure.3 From this and the formula proved before, viz. p = 3pG2, we obtain the value of k by taking the temperature 9 = 0, thus finding where denotes the mean molecular speed at the temperature 90; and it further necessarily follows that the square of the molecular speed G increases in linear proportion with the temperature 9, the relation between them being G2 = ®2(1 + ad). We thus find that the square of the molecular speed of a gas, and therefore the kinetic energy of its molecular motion, increases proportionally with the temperature. The speed itself is given by G = (1 + ad). This law is in complete agreement with the conclusion obtained in § 9 from Bernoulli's theory, viz. that the kinetic energy of the molecular motion is the mechanical measure of heat and temperature. 1Annales de Chimie et de Physique, xliii. 1802, p. 137; Gilb. Ann. xii. 1802, p. 257. 2 Mem. of the Manch. Lit. and Phil. Soc. v. 1802, p. 595; Gilb. Ann. xii. 1802, p. 310. * See § 46. |