The Gas-Laws

THE GAS-LAWS One of the earliest triumphs of the kinetic theory was its explanation of the laws which had been found to connect the density-pressure and temperature of a gas. The kinetic theory regards a mass of gas as a collection of independently moving molecules ; if these are shut up in a closed vessel, molecules must continually collide with the walls of the vessel, exerting pressure at the moment of collision, and the kinetic theory interprets the pressure of the gas as the aggregate of the pressure exerted by the various molecules when they collide with the boundary.

Boyle had found that as a gas was compressed its pressure in creased directly as its density. On the kinetic theory the necessity for this law is obvious, since doubling the density of a gas doubles the number of molecules in any small volume and so doubles the number of collisions with any small area of its boundary. Charles and Gay-Lussac had found that as a gas was heated its pressure increased directly as its temperature, and the kinetic theory provides a simple explanation of this also. For heating a gas increases the mean velocity of the molecules; this has a twofold effect on the pressure. If the mean velocity is increased n-fold, the force of each collision is increased n-fold, and col lisions also occur n times as frequently, so that the pressure is increased to times its original value. The pressure is accord ingly proportional to the square of the molecular velocity, and this, we have seen, is proportional to the absolute temperature. This is the law of Charles and Gay-Lussac.

Calculation of the Pressure in a Gas.

The actual amount of the pressure is readily calculated. The molecules can be divided up into a system of showers, such that all the molecules in any one shower have all approximately the same velocity and are all moving in the same direction. Let us fix our attention on any one shower, which we shall call shower A, this being specified by the condition that the three velocity components u, v, w of its molecules lie within the small range du, dv, dw already mentioned. Let us consider the impact of the molecules of this shower on an area A of the boundary of the containing vessel. For convenience we may suppose this to be parallel to the plane of yz. Then each molecule of shower A which strikes this area gives up momentum mu to the boundary before it is brought to rest, and as it rebounds with equal velocity it also acquires momentum mu in the opposite direction from the boundary, so that the total transfer of momentum is 2mu. The number of molecules belonging to shower A which strike the area S in a small interval of time dt is equal to the number which lay within a distance u dt of the area S, at the beginning of this interval, and so occupied a small disk of volume Sudt of the gas. If the gas contained v molecules in all

per unit volume, the total number of molecules inside this small disk was Suvdt, and hence, from formula (I), the number belong ing to shower A was Each of these molecules transfers momentum of amount 2 mu to the area S of the boundary, so that the total impact of mole cules of shower A on this area in time dt is equal to 2MU times the above expression, or \ / The total impact of the molecules of all showers is obtained by integrating this expression over all possible values of a, v, w; it is found to be Since this must be equal to p S dt, where p is the total pressure per unit area on the boundary, the value of p is clearly Brownian Movements.—From the value R =1.37 X it is readily calculated that a molecule, or aggregation of molecules, of mass grammes, ought to have a mean velocity of about 2MM. a second at o° Centigrade. Such a velocity ought accord ingly to be set up in a particle of grammes mass immersed in air or liquid at o° C., by the continual jostling of the surrounding molecules or particles. A particle of this mass is easily visible microscopically, and a velocity of 2MM. per second would of course be visible if continued for a sufficient length of time. Each bombardment will, however, change the motion of the particle, so that changes are too frequent for the separate motions to be individually visible. But it can be shown that from the aggrega tion of these separate short motions the particle ought to have a resultant motion, described with an average velocity which, although much smaller than 2MM. a second, ought still to be microscopically visible. R. von S. Smoluchowski and Einstein have shown that this theoretically predicted motion is simply that seen in the "Brownian movements" first observed by the botanist Robert Brown in 1827. Thus the "Brownian movements" provide visual demonstration of the reality of the heat-motion postulated by the kinetic theory. (See Brownian Movement.) Pressure in a Mixture of Gases.—Imagine that a gas con sists of a mixture of gases of different kinds, so that unit volume contains v of one kind, V of another, V' of a third, and so on. The pressure on the boundary now is the sum of the pressures exerted at all collisions by molecules of all kinds. The pressure exerted by the first kind of molecules is of course v RT, that exerted by the second kind is V RT, and so on, so that the total pressure p is given by p = (11-Ev' d-v" .)RT Dalton's Law.—Since the pressure as given by this formula can be written as the sum of a number of separate terms, one for each gas in the mixture, we have Dalton's law : The pressure of a mixture of gases is the sum of the pressures which would be exerted separately by the several constituents if each alone were present.