THE KINETIC THEORY OF GASES A number of molecules whose motion is determined by dynam ical laws will pass through a series of configurations which can be theoretically determined as soon as the structure of each molecule and the initial position and velocity of every part of it are known. The kinetic theory is not, however, concerned with the problem of determining the sequence of positions of individual molecules which will develop out of given conditions; it aims instead at discovering properties which are true for all systems of molecules, no matter what the initial conditions of the system may have been.
The first and most fundamental theorem of the kinetic theory shows that, for any given system of molecules, there exist a series of "normal states" such that, whatever the initial state of the system may have been, it tends, after the lapse of sufficient time, to pass into one of these normal states and to stay there. The normal states can, of course, only be specified statistically; they have no reference to the motions of individual molecules. Boltzmann has shown that the normal states correspond thermodynamically to states of maximum entropy. Thus the passage of a system to a normal state is accompanied by an increase of its entropy, and when the system has once attained a normal state it cannot leave it, except under the action of some outside agency, because the entropy, being already a maximum, cannot increase any further. The different normal states which are possible for a given system of molecules are specified by giving different values to a single quantity, which may be regarded as measuring either the total energy of the system or its temperature. There is only one normal state for a system of given energy, and all systems having this given amount of energy will move towards this state. Thus— A system starting from an abnormal state tends to assume that normal state which has the same total energy; and A system starting from a normal state remains in the normal state.
The kinetic theory deduces all the ordinary physical properties of gases from the supposition that a gas is a collection of freely and independently moving molecules which have attained to the normal state.
When a gas has attained its normal state, the collisions between its molecules do not remove it from that state, so that the statistical effect of all these collisions must be nil. Let us divide all possible velocities up into distinct ranges, a, b, c, . . . When two molecules moving with velocities a, b, collide, the collision changes the motion of both and they assume new velocities p, q. Thus the collision takes one molecule away from each of the classes a, b, and adds one to each of the classes p, q. In order that the statistical state may remain unchanged, as it must if the gas is in the normal state, there must be a corresponding collision which takes one molecule from each of the classes p, q, and adds one to each of the classes a, b. By the use of this consideration Maxwell, Boltz mann and Lorentz determined the law according to which the velocities are distributed among the different possible ranges of velocity. Confining our attention to one component of velocity u, it is found that the values of u for the different molecules con form to what is called the law of trial and error—they are distributed at random around the value u= o according to the same law as determines the distribution of shots around the middle point of a line-target. To express the result mathematic ally, if u, v, w denote the three components of velocity in three directions at right angles in space, then the number of molecules whose velocity is such that Here N is the total number of molecules and m is the mass of each, while h is a constant, different values of h corresponding to the different normal states possible for the system. This law is commonly known as Maxwell's law of distribution of velocities.