“There exists a positive function belonging to a group of molecules which, as they settle themselves into a steady state—on the average derived from a great number of configurations—maintains a steady downward trend. The Maxwell-Boltzmann steady state is the one in which this function has finally attained its minimum value, and is thus a unique steady state, it still being borne in mind that this is only a proposition of averages derived from a great number of instances in which nothing is conserved in encounters, except the energy, and that exceptional circumstances may exist, comparatively very few in number, in which the trend is, at any rate, temporarily the other way.”
This theorem, when applied to cases of motion, such as that of a gas at constant temperature enclosed in a rigid envelope impermeable to heat, appears to be proved. For such a case, therefore, the Maxwell-Boltzmann law is the only one possible.
But whether this be so or not, the law first introduced by Maxwell is one of those possible, and the advance in molecular science due to its introduction is enormous.
We come now to the second result, the equal partition of the energy among all the degrees of freedom of each molecule. Lord Kelvin has pointed out a flaw in Maxwell’s proof, but Boltzmann showed (Philosophical Magazine, March, 1893) how this flaw can easily be corrected, and it may be said that in all cases in which the Boltzmann-Maxwell law of the distribution of velocities holds, Maxwell’s law of the equal partition of energy holds also.
Three cases are considered by Mr. Bryan, in which the law of distribution fails for rigid molecules: the first is when the molecules have all, in addition to their velocities of agitation, a common velocity of translation in a fixed direction; the second is when the gas has a motion of uniform rotation about a fixed axis; while the third is when each molecule has an axis of symmetry. In this last case the forces acting during a collision necessarily pass through the axis of symmetry, the angular velocity, therefore, of any molecule about this axis remains constant, the number of molecules having a given angular velocity will remain the same throughout the motion, and the part of the kinetic energy which depends on this component of the motion will remain fixed, and will not come into consideration when dealing with the equal partition of the energy among the various degrees of freedom.
Such a molecule has five, and not six, degrees of freedom; three quantities are needed to determine the position of its centre of gravity, and two to fix the position of the axis of symmetry.
In this case, then, as Boltzmann points out, in the expression for the ratio of the specific heats, we must have n equal to 5, and hence
γ = 1 + 2/n = 1 + 2/5 = 1·4
agreeing fairly with the value found for air and various other permanent gases.
For cases, then, in which we consider each atom as a single rigid body, the Boltzmann-Maxwell theorem appears to give a unique solution, and the Maxwell law of the distribution of the energy to be in fair accordance with the results of observation.[55]