Maxwell’s mathematical methods are, in their generality and elegance, far in advance of anything previously attempted in the subject.
So far it has been assumed that the particles in the vessel are all alike. Maxwell next takes the case of a mixture of two kinds of particles, and inquires what relation must exist between the average velocities of these different particles, in order that the state may be steady.
Now, it can be shown that when two elastic spheres impinge the effect of the impact is always such as to reduce the difference between their kinetic energies.
Hence, after a very large number of impacts the kinetic energies of the two balls must be the same; the steady state, then, will be reached when each ball has the same kinetic energy.
Thus if m₁, m₂ be the masses of the particles in the two sets respectively, v₁, v₂ their mean velocities we must have finally—
½ m₁ v₁² = ½ m₂ v₂²
This is the second of the two great laws enunciated by Waterston in 1845 and 1851, but which, as we have seen, had remained unknown until 1859, when it was again given by Maxwell.
Now, when gases are mixed their temperatures become equal. Hence we conclude, in Maxwell’s words, “that the physical condition which determines that the temperature of two gases shall be the same, is that the mean kinetic energy of agitation of the individual molecules of the two gases are equal.”
Thus, as the result of Maxwell’s more exact researches on the motion of a system of spherical particles, we find that we again can obtain the equations—
T = ½ mv² p = ⅓ N mv² = ⅔ NT = ⅔ ρ T/m