616 MOLECULE and therefore the ratio of S Q to SQ, or of the two specific heats at constant pressure and constant volume respectively, becomes a + 3 ^ + 2 dr dr -^ is unknown in all respects except that it must be positive ; also we know that <r must be integral and not less than 3 ; if we denote dy 3 ^ by e we have for the ratio dr J or 1 + + e which, with the necessary limitations of <r and e, cannot be greater than 1 or 1 6, and in this limiting case the gas must be mon- atomic. If, therefore, any value above 1 - 6 of the ratio for mercury vapour be insisted upon, the theory must be abandoned so far as present investigations are concerned. If, however, the difference between 1 6 and any higher value given by the observations be regarded as within reasonable limits of experimental error, this value for mercury vapour, a gas which on chemical grounds is regarded as monatomic, may be viewed as confirming the theory, at least pro tanto. If two spherical atoms were united by a rigid rod to form a mole cule, such a molecule would have five degrees of freedom and the specific heat ratio would in this case be If, for e would then be zero. This value has a plausible approximation to the observed value 1 "408 of the ratio in a great number of two-atom gases, such as hydrogen, nitrogen, oxygen, and others, but all observations agree so completely in the ratio 1 408, or from 1 405 to 1 408, that it hardly seems reasonable to regard the difference 008 as within the limits of experimental error, unless, indeed, we had grounds for suspecting a tendency to excess in all the methods employed for the determination of the ratio. But there are other difficulties more formidable still, arising from the spectroscopic properties of heated gases. The light emitted by such gases, so long as they are of no great density, never presents a continuous spectrum, but a spectrum consisting of bright lines with intervening dark spaces. Thus the spectrum of hydrogen gives thirty-two bright lines, that of mercury vapour six lines, that of nitrogen eighteen, and so on. So long as light is regarded as an energy intercommunicable with heat, and light of definite refrangibility is refeired to vibrations of given period, we must regard these discontinuous spectra as con nected with, and arising from, vibrations of determinate periods in the molecule of the heated gas. And if a gas such as hydrogen or nitrogen be constituted, as we are supposing, of an indefinite repe tition of similar molecules, it must follow that such molecules must be capable, at any rate when not too closely packed together, of as many independent vibrations as there are bright lines in the spec trum ; that is to say, in addition to the three degrees of freedom arising from motion of translation in solid space, each molecule must possess as many additional degrees of freedom or possible relative motion of its parts as are indicated by the number of spec trum lines. The degrees of freedom corresponding to motion of translation cannot well contribute anything to these luminous vibrations owing to their assumed irregularity and independence of any law ; but it is otherwise with the internal or relative degrees of freedom of each molecule, for, unless the gas be very dense, we may easily conceive a sufficient interval of time between one encounter and the next of any molecule with another for very many vibra tions, each according to its own law, to take place in the relative positions of different parts of the molecule. At each encounter the whole molecule would be roughly shaken, and when the encounters increased in frequency the vibrations would become irregular and the spectrum would degenerate into a general diffused light of no definite refrangibility, just as music degenerates into ordinary noise. And this is exactly what occurs in the spectra of dense gases. To bring the theory, therefore, into agreement with observed phenomena, we require very many more degrees of freedom in each molecule than could possibly be assigned to it in accordance with the observed value of the ratios of the specific heats, mercury vapour, for example, admitting with difficulty the minimum number of three such degrees, as we have just now seen, while its spectrum would require at least nine. And the difficulty increases as we pass to hydrogen and other gases. We might perhaps conceive, with the view of possibly explaining this difficulty, that there were in all gases a number of composite molecules with many degrees of freedom mixed up with the other molecules with three or five such degrees, but in so small a propor tion to these molecules that their presence produces no appreciable effect upon the specific heats ; or, since we have no experimental de termination of the specific heats of gases at light-giving temperature, we might, at least until such experimental determination has been arrived at, conceive that our atoms may change their constitution under increased temperature, and become themselves capable of vibration. There is nothing in the conception of an atom as we are considering it which is really inconsistent with such an hypothesis. Certain observed phenomena accompanying dissociation and com bination give rise to other difficulties in the way of the acceptance of the kinetic theory in addition to those arising from the equal distribution of mean kinetic energy just now discussed. For when nitrogen and hydrogen, for example, are mixed in proportion to form ammonia it is observed (1) that at ordinary temperatures they do not exhibit the slightest tendency to combine directly with each other, while, on the other hand, (2) ammonia at ordinary temperatures does not exhibit the slightest tendency to decompose into nitrogen or hydrogen. But ammonia when subjected to certain very high temperatures becomes partially decomposed that is, becomes a mixture of so many parts of ammonia and of so many other parts of nitrogen and hydrogen in the proportions to form ammonia ; and if the temperature be high enough the decomposition may be com plete. But, in accordance with the kinetic theory, the conditions, whatever they may be, which at high temperature cause the ammonia to decompose, must sometimes occur to individual molecules at ordinary temperature, because temperature, as we understand it, merely indicates a certain quantity of kinetic energy, and therefore in a gas, however cold, there will be always some molecules in a state for dissociation ; and this dissociation having taken place can by (1) never be compensated by recombination ; therefore dissocia tion should be going on in ammonia at all temperatures, and this result is contrary to the observed phenomena (2). It might possibly be conceived, as a way of meeting this last-mentioned difficulty, that the dissociation attendant upon high temperature that is, upon an average large molecular velocity of translation requires that there should be a fairly rapid repetition of encounters among molecules moving with dissociation velocity to ensure the production of dis sociation, and that in the case of a gas at low temperature, or small average velocity, the chance of two molecules encountering one another at high velocities is small, and the chance of any molecule meeting with any rapid succession of such encounters is practically insensible, and therefore that the dissociation spoken of really never takes place. As above stated, we conceive that in any gas at ordinary pressure and temperature the intermolecular forces are very small in the aggregate that is, in Clausius s language, have a very small virial, by which is understood, not that the forces themselves, where acting, are small, but that, considering the whole aggregate of molecules at any instant, there are very few pairs near enough to each other to exert any appreciable force on each other. Or, if we could watch any individual molecule for any time, we should find that during by far the greater portion of the time it was sensibly free from any action by surrounding molecules. The distance traversed by the type molecule between the instant when it passes out of the sphere of action of one molecule and the instant when it passes into the sphere of action of the next that is, from one encounter to another is called its free path. We may find the chance that a molecule starting from any point with velocity w in a uniform gas shall have free path between x and x + dx from that point. If a be the chance for such a molecule of free path at least unity, then a- is the chance of a free path at least 2. Hence the chance of free path at least x must be of the form a x . Following the method employed by 0. E. Meyer, 1 let us write this in the form where therefore 1= - I > 1 then the chance of free path x + dx is x + dx e I . The chance that such a molecule shall have its first encounter between x and x + dx is the difference of these two expressions ; that is, __ . T This is the chance of a free path between x and x + dx. The mean free path for such a molecule must then be I This is the meaning of the constant I in e l . But if we denote by B the number of encounters which a molecule moving through space with velocity w experiences on the average per unit of time,
Kinetische Theorie der Case, Breslau, 1877.