8 theory forms the true explanation of the constancy of the chemical equivalents, since a molecule can contain only an integral and not a fractional number of atoms. The relations considered above may be made useful in theoretical chemistry in two ways. We may either, with Gay-Lussac, calculate by means of the given proportion the unknown density of a gas or vapour from its chemical equivalent which has been determined from its chemical action; or, inversely, from its observed density we may deduce its chemical equivalent. For this purpose Avogadro's law, which is discussed in the next paragraph, is of service. or the theorem that the densities P1, P2 of two gases at the same temperature and under the same pressure are in the ratio of their molecular masses m1, m, in the gaseous state, is capable of a very simple interpretation, which is, therefore, the more important. In the meaning assigned in § 13 to the idea of density, p is nothing else than the mass of all the N molecules contained in unit volume, or where the meaning of the symbols is plain. If we substitute these values of the densities in the above proportion we obtain the equation N1 = N2, 29 which, expressed in words, gives the theorem that two different gases, when they are at the same temperature and under the same pressure, contain equal numbers of molecules in equal volumes. This is called Avogadro's law, after its discoverer. Although the considerations by which Avogadro1 arrived at it are closely bound up with views which were then universally accepted, but are now rejected, viz. with the assumption of a material caloric, yet the experimental results from which he started, and the conclusions he founded on them, agree substantially with those which we have here employed. He relied especially on the law of Gay-Lussac, which was discussed in the last paragraph, and from which, even without this special theory of gases, Avogadro's law can be easily deduced with at least very great probability. On this ground Clausius, who had already pointed out the significance of this law for theoretical chemistry in his first memoir 2 on the kinetic theory of gases, was able to proceed the reverse way. From the laws of Gay-Lussac and Avogadro he inferred the law, first given by him, that two gases have the same temperature when the mean kinetic energy of their molecular motion is the same, a law which, aided by Maxwell's later researches, we have deduced from the mechanics of molecules. Avogadro's law forms one of the most important foundations of theoretical chemistry; for by its aid we are in a position to calculate the molecular mass of a substance from its density in the state of gas. For if in the formula p = N is a number which is the same for all gases and vapours, we can express the values of the molecular mass for all gases in terms of any arbitrary unit, such, for instance, as the molecular mass of hydrogen. The values of the molecular masses so found do not all agree with the atomic masses, but are in cases multiples of them. A molecule, therefore, must in general consist of several atoms, as we have already assumed to be possible. Further inquiry into this interesting subject, which is more concerned with chemistry than with physics, I must 1 Journ. de Phys. par Delametherie, lxxiii. 1811, p. 58; lxxviii. 1814, p. 131; Mem. di Torino, xxvi. 1821, p. 440. 2 Pogg. Ann. c. 1857, p. 353; Abhandl. über Wärmetheorie, 2. Abth. 1867, p. 229; transl. Phil. Mag. [4] xiv. 1857, p. 108. here abandon, and all the more so as it has obtained a thorough treatment in a treatise by my brother.1 32. Coefficient of Expansion From the relation found in § 16 between the temperature of a gas and the mean value of the kinetic energy of its molecules, follows another law which has in like manner been confirmed by experiment. Since two gases have the same temperature 9 when the mean values of the kinetic energy of their molecules are equal, i.e. when the values also of their molecular energies are equal when they are both at the temperature 0°, or, in the notation already used in § 14, But, if a,, a, are the thermal expansibilities of the two gases, G12 = & ̧2(1+a,D), G22 = Œ¿2(1+ a¿D). 2 From these four formulæ we obtain the equation or the law that the thermal expansibilities of all gases are the same. This law, which has been already mentioned in § 15, in the determination of the absolute zero of temperature, was empirically established by Gay-Lussac2 and Dalton,3 and still earlier, as the former tells us, by Charles. If it now appears as a logical deduction from the theory, we must see in this coincidence a weighty and convincing argument for the truth of the theoretical views from which we have started in explaining gaseous pressure. Lothar Meyer, Die modernen Theorien der Chemie, 5. Aufl., Breslau 1884; 6. Aufl. I. 1895. 2 Ann. Chim. Phys. xliii. 1802, p. 137; Gilb. Ann. xii. p. 257. 3 Mem. of the Manchester Lit. and Phil. Soc. v. 1802, p. 595; Gilb. Ann. xii. p. 310. F 33. Maxwell's Law for a Gas in Motion In the simple form in which we have hitherto used it, Maxwell's law of distribution rests on the assumption that the gas is in equilibrium and at rest as a whole. Hitherto, we have always assumed that there is no other motion in the gas than the invisible to-and-fro motion of its particles. Beside this molecular motion, the effect of which we perceive only in the pressure and heat of the gas, there should be no directly perceptible motion, no flow, no rotation, no change of the volume occupied; there should, therefore, be no sort of cause for the centroid of the whole mass of gas to change its position, nor, indeed, for that of any portion of the gas of finite magnitude; only the single atoms were endowed with independent motions, which they executed without disturbing the equilibrium of the gas as a whole. If we discard this assumption Maxwell's law must be modified, and the necessary modification in a special simple case is easy to see. If we impart to the whole mass of gas and its containing vessel a uniform motion of translation, there is no reason at all for any change in the to-and-fro motion of the molecules. Both motions, the molecular and the molar, will exist together without mutually disturbing each other. If we compound them by the known rule of the parallelogram of velocities, we get for each molecule the direction and magnitude of the velocity with which it moves when the gas as a whole is in translatory motion. Herewith, then, the law of distribution for this case is determined. It does not seem necessary to express here in mathematical formulæ this more general law; for the more general law is easily to be deduced from Maxwell's known law of distribution. If we diminish, that is to say, the actual velocity of a molecule by the velocity of translation of the centroid of the gas as a whole, Maxwell's simple law for the probability of a definite speed again comes to view. It is obvious that the subtraction of the velocity of the centroid of the whole gas from that of a molecule amounts to bringing the prin 1 See §§ 16*, 17* of the Mathematical Appendices. ciple of the parallelogram into play, or, what comes to the same thing, to subtracting the components of both velocities from each other. The most general case can be at once deduced from this very simple one. If the gaseous mass so moves that the molar velocity is not everywhere the same, but in different places is different in magnitude and direction, we have at each particular place to subtract the velocity of flow at that place (which is the same thing as the molar velocity), and the remaining molecular motion will satisfy Maxwell's law. 34. Pressure of a Gas in Motion. Resistance If such a distribution of molecular velocities exists, the different directions can no longer be looked upon as having no distinction. The pressure, too, of a streaming gas will, therefore, no longer be equally great in all directions. In the direction of the flow the velocity and pressure will be greater than in any other direction; the pressure is increased by the stress which the gas by its motion exerts on a surface in its way. It is easy to calculate this increase of pressure if we remember that, according to the kinetic theory, the pressure consists in a transference of momentum. In a gas at rest this transference is effected by the to-and-fro motion of the molecules. In a gas in motion there is an additional cause in the velocity of flow by which not only momentum but also mass is transferred. Through a surface F at right angles to the direction of flow there passes in unit time a volume Fa and a mass pFa, if a denotes the velocity. This mass possesses the momentum pFa2. In consequence of the flow, therefore, the momentum transferred in the direction of the flow increases in unit time by pFa2. Since, now, according to our theory, the pressure is |