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ON THE VELOCITIES OF GASES.
BY ALBERT J. MOTT, F.G.S.
My object is to suggest a physical method by which the maximum molecular velocities of gases may be determined within certain limits.
Assuming the truth of the Kinetic theory, a gas is composed of molecules not resting in contact, but in constant and rapid motion in straight lines in all directions. There are other motions also, rotatory and vibratory, but it is the motion of direct translation in straight lines till something is encountered that is here considered.
The cause of this motion is heat communicated to the gas, or more correctly this motion itself is the sensible heat of the gas. In solids the molecules are held together by forces which prevent their heat motion from separating them. In gases these forces are absent or have been overcome, and the molecules, no longer held together by them, move independently under the influence of heat. Every gas has received this motion from some definite source, and the motion would be lost if it were not continually renewed. In a glass jar, for example, full of air, the gaseous molecules constantly beating against the sides of the jar would soon exhaust their motion; that is, the air would become colder and colder till it fell to the temperature at which it would cease to be gaseous, and would take a liquid or solid form. But the jar renews the motion as fast as it is wasted; the glass itself is heated, that is, also in motion, and though the motion in this case is vibratory and the glass molecules do not leave each other, they give back to the gaseous molecules
as great an impulse as they receive from them, the motion of the glass itself being maintained by other sources of heat, which we may finally trace back to the rays of the sun or the temperature of the earth.
But if a gas is not inclosed in anything, so that its molecules meet with no resistance except from their mutual encounters, the effect of their motion is to spread them farther and farther apart, the gas thus expanding without limit unless some force acts upon it from without. And as it expands, and the distance between its molecules increases, their encounters become less and less frequent, till at last some of them, and finally all of them, tend to fly off along independent paths through space.
We never meet with empty space, however, and gases as we know them are always inclosed. Any given portion of air, for example, has the earth beneath it and other portions of air above and around it, and its moving molecules meet in every direction with something from which they receive fresh impulse and by which they are driven back.
In the atmosphere, however, while the earth below it makes a solid wall, the air above any portion of it makes a wall that can be more or less penetrated. This wall also becomes thinner and less substantial as we go higher, and at some upper limit there is no longer anything to meet and confine such of the molecules as are moving upwards. What then will happen to them? They will ascend to a certain height above the earth, and then fall back; the height to which they are carried being determined by the velocity with which they start and its relation to the force of gravity.
The actual velocity with which gaseous molecules move is determined in this way:
The normal pressure of the atmosphere downwards, on a square inch, at the earth's surface, is equal to the pressure of
a dead weight of not quite 15 lbs. The air itself is not, in fact, a dead weight, but its pressure is equal to this, and we measure it by this standard.
Take a tube one square inch in section, closed at the bottom and fitted with a piston at the top, and filled with air. The piston does not move down under the pressure of the atmosphere. Remove that pressure by creating a vacuum above the piston, and put upon it an iron weight of about 15lbs. instead. The piston remains as before. The column of air under it resists the pressure of this weight, and the resisting force is of course equal to that which causes the pressure.
A weight of 15 lbs. is a certain mass acted on by gravity. Gravity at this latitude is a force which, if it acts unresisted for one second on any mass which is free to move, will cause it to move at the end of that time with a velocity of 32.2 feet in a second downwards.
This is the force resisted by the air under the piston. The resistance is caused by the impact of the moving molecules, of which the air consists, striking upon the under side of the piston. What, then, must the velocity be with which they move?
If a weight of 15 lbs. moved upwards with a velocity of 32.2 feet in a second, it would exactly balance the weight of 15lbs. moving downwards with the same speed. And a smaller weight will balance it also if its speed is increased sufficiently. The force exerted by a moving body, if it moves with different velocities, is in proportion to the squares of those velocities. Ten times the speed gives a hundred times the force. A hundred times the speed gives ten thousand times the force.
Suppose the tube of air, one square inch in section, to be 32 feet long. The weight of the air in it is 120 grains. The weight upon the piston is in round numbers 100,000 grains.
The motion of the 120 grains resists the pressure of the 100,000. But the molecules of air are moving in all directions, and not only in an upward direction against the piston, and the final result of their combined motions is that the air acts as if one-third of its molecules moved directly upwards, the others not affecting the piston.* One-third of the molecules weigh 40 grains. We have to learn, therefore, with what velocity 40 grains of matter must move to exert a force equal to that of 100,000 grains moving at the rate of 32 feet per second, at the end of a second. 100,000 is to 40 as 2,500 is to 1. We therefore take the weight upon the piston as 2,500, and the weight of the effective molecules under it as one.
Now, the column of air is 32 feet long. If it moved at the rate of 32 feet in a second, the whole of it would have struck the piston at the end of a second, and if the weights were equal, the force in that time exerted against the under side of the piston would have been exactly equal to the force then acquired by the weight above it. But the weight above
*This is a deduction from Clerk Maxwell's reasoning. (Theory of Heat, p. 294.) The whole pressure exerted in all directions by any portion of gas is the resultant of all the forces due to the velocities of all its molecules. These forces may be resolved in any three directions at right angles to each other. As the pressure of the gas is known to be equal in all directions, these three components are equal, and the force exerted in any one direction is therefore one-third of the total force. And in a straight tube closed by a piston, all the forces exerted by all the molecules within the tube in one direction parallel with the sides of the tube are exerted upon the piston. Or, suppose a cubical box, with six equal sides therefore, with three equal balls in it, moving independently at the same speed along three lines at right angles to each other; one up and down; one backwards and forwards; one from side to side. Each side of the box will be struck at equal intervals by one of the balls. If we can measure the effect of the blows, and if we know the weight of the three balls together, we can tell at once the weight of each, and the velocity with which they move. The three balls represent the whole of the molecules of a gas taken together, moving in such a manner that the force exerted by them on each of the sides of the box is constantly the same. And to make it so, each side must be continually struck by one of the balls when the whole number is three.