Further, when the gas has no progressive motion as a whole we shall have u2 = v2 = w2. The value of these three equal means is easily found; for if w is the actual velocity of a particle that is, the pressure in a gas which is in equilibrium and at rest as a whole is the same in all directions, and acts always normally to the surface on which it acts. If the gas possesses a progressive motion in which its whole mass takes part, the magnitudes of the mean values are just as easily found. Let a be the velocity with which the gas as a whole moves in the direction of the component u; then, if we put u = u1 + α, u, is the component of the molecular motion which is perceptible not as causing change of position, but as producing heat and pressure in the gas, and, by § 33, the relations hold good, if we now represent the pure molecular velocity freed from that of the flow by w1, where w12 = u12 + v2 + w2. We have in this case also, just as before, expressions of the form for the components of pressure at right angles to the direction of flow. On the contrary, for the pressure in the direction of flow we have X2 = p (u1 + a)2 = p u ̧2 + pa2, since = 0; this is greater than the other components by pa2, u1 for we also have This new magnitude pa2 shows itself as the pressure with which the streaming gas strikes against an opposing surface, or, if we consider the reverse direction, the reacting pressure which the issuing stream exerts on the containing vessel ($$ 34, 35). The pressure which the stream of gas exerts normally to its direction of flow, viz. έρωτα, is smaller than what it would be in the case of rest, viz. όρων; for in coming to rest the gas retains all its energy, so that the mean value of this energy remains unaltered, or The formula found for the pressure in the case of a gas at rest, p = {} p w2, does not differ from that deduced before (§§ 11-13), viz. p = 3 pG2, since the magnitude G herein contained denotes the mean value which is determined by the equation G2 = w2. Since this formula gives the pressure p as proportional to the density p, it expresses Boyle's law. This deduction of the law is preferable to the former in being as free as possible from unproved hypotheses. We have assumed only that the molecules move in rectilinear paths, and have employed no other hypothesis. The single assumption, therefore, of rectilinear molecular motion suffices by itself for the proof of Boyle's law. Since the sum of the kinetic energy in the unit of volume is K = Nm = 3p w2, we obtain by this method also the known relation (§ 16) If we had taken the molecules to be not all of the same kind, but had assumed that two or more kinds of molecules m, m2, ... were mixed together, then we should have obtained, both for the transmitted momentum and for the kinetic energy, formulæ of the same form as those just found, but of greater generality. The equations would then have contained summations with respect to all the different molecules m1, m2, ... Instead of the last formula, therefore, we should have obtained, as is obvious without calculation, the equation which has the same meaning as that in § 17, and thus proves Dalton's law for the pressure of mixed gases. The hypothesis of rectilinear molecular motion therefore is also sufficient by itself for the theoretical proof of Dalton's law. 8*. Kinetic Pressure of Liquefied Substances It is, perhaps, not even absolutely necessary for the molecular motions to be rectilinear. For Boltzmann1 has attempted with good results to extend the foregoing considerations to bodies which are in the liquid state, and therefore to substances whose molecules move, not in straight, but in curved paths. The case is that of substances which are mixed in very great dilution with a liquid; we can imagine, therefore, either a very dilute solution of a solid body, or a liquid which has absorbed small quantities of a gas or contains a small quantity of another liquid. The molecules of the alien body that has been added spread themselves throughout the liquid, and therefore become so widely separated from each other that the forces of cohesion no longer come into play. If we further assume that the liquid also exerts no force on the alien molecules, or that the forces it exerts mutually annul each other, the molecules of the added substance then appear to be quite free from all external forces but that of gravity. They would then move in straight paths just as molecules of gas if they were not hindered by the molecules of the liquid and forced 1 Zeitschr. für phys. Chem. vi. 1890, p. 474; vii. 1891, p. 88. to continually change their direction of motion. Yet they move, and this motion, even if it should happen to be in curved paths, produces a kinetic pressure just as in gases. This pressure may be calculated in the same manner as gaseous pressure, and the calculation leads to exactly the same result. If no attractive or repulsive forces act between the molecules of the liquid and of the alien body the particles of the latter move about in the liquid, not in continuously curved paths, but in straight paths like the particles of gas in vacuum. The only difference is this, that the alien particles collide very much oftener in the liquid, and that therefore the free path traversed between successive collisions is very much shorter. But this difference has no influence on the validity of the calculation. There is only a change in signification of the function f (t, u, v, w), which denotes the probability of the path being straight for the interval t, to this extent that it has values differing from 0 only for very small values of t, and that it vanishes for greater arguments; but thereby no alteration in the final result of the calculation is entailed. The result is similar in the other case which better corresponds to actuality, viz. when forces do indeed act between the liquid and the alien particles, but when the forces to which a particle is subjected from the molecules of the liquid which surround it mutually balance each other on the average. The path of a particle is then continuously curved, as it is continuously under the action of molecular forces; yet we may look on the path as rectilinear which is traversed during an infinitely small interval of time. To this straight bit of path and to the short time needed for it we have to apply the foregoing calculation, which results in the same value as before for the pressure due to the motion of the added alien particles, and gives the same relation between this kinetic pressure p and the kinetic energy K of the particles contained in unit of volume, viz. p = K. This formula remains therefore at least approximately correct when the stretches of which the paths of the molecules are made up are not of finite length. It would therefore be mathematically stricter so to express the condition of its validity that the particles whose motion causes the pressure move under the laws of inertia and collision only, without being subject to external force. On account of this result Boltzmann's theory of osmotic pressure in liquids is important for the kinetic theory of gases. Except for this, however, I should not here have mentioned it, since osmotic pressure is not one of the phenomena which the kinetic theory of gases has to explain. I will also not conceal that I do not think van't Hoff's views of the kinetic nature of osmotic pressure to be correct. For osmose does not arise from the kinetic pressure of the dissolved substance, but from quite different forces which cannot be neglected. At all events, if the formula is to be applied to osmose, it first needs a correction, which G. Elias Müller' has pointed out; viz. from the kinetic pressure of the dissolved molecules there must be subtracted the pressure which the displaced particles of liquid would have exerted by their motion. Not till then does it become intelligible that osmose is able to cause a motion of the liquid towards the side of the greater pressure. Theorie der Muskelcontraction, Leipzig 1891, I. (App.) p. 321. |