If the given gas contains molecules of sev eral different kinds, then the kinetic theory teaches us (1) that the foregoing statements are true for each species of molecule, separately and independently of all the others; and (2) that the average kinetic energy of translation is identically the same in each of the several species of molecules that may be presenti and furthermore, (3) that the average kinetic energy of translation, in any one set of mole cules, is proportional to the absolute tempera ture of the gas as a whole.
The foregoing conclusions are based upon the researches of numerous mathematicians and physicists, beginning with Maxwell; but the principle of the equable partition of kinetic energy, as here described, was established, in its full generality, by Boltzmann, and is often called °Boltzmann's H (See, in addi tion, Watson, 'Kinetic Theory of Gases'). There has been considerable controversy as to the soundness of the proofs that have been given of it, yet the principle itself has been exceedingly fruitful, and it has played an im portant part in the development of the molec ular theory in general. The consensus of opinion among physicists appear, to he, in fact, that the Boltzmann theorem, whether the proofs of it that we have are sound or not, constitutes a good general approximation to the real facts —provided we can regard molecules as behav ing, in their excursions and encounters, sub stantially like solid and perfectly elastic bodies, which obey the same laws of mechanics that are followed by bodies large enough to be vis ible—and provided, furthermore, the mole cules are not thrown into a state of internal vibration by reason of the shocks that they experience upon encountering one another.
It should be carefully noted, however, that even if we admit that Boltzmann's law of the partition of kinetic energy holds true in the case of a gas, it by no means follows that it is applicable to liquids and solids. In fact, cer tain of Boltzmann's fundamental assumptions are not fulfilled, even approximately, in liquids and solids; and hence his reasoning will not apply to such bodies, even though the conclu sions at which he arrived may possibly still hold true. At first thought it appears highly improbable that molecular motions in solids and liquids will be found to conform with the principles stated above, except that for any given substance the kinetic energy due to the translatory motions parallel to the three fixed axes will probably be equally divided, as before —though it is not obvious that even this would be true, in general, in crystals. Nevertheless, there is indirect evidence that tends to show that a distribution of kinetic energy similar in a general way to that established by Boltzmann for gases prevails also in liquids and solids. The numerous useful analogies that have been observed between gases and dilute solutions are strongly suggestive of a close relation in ulti mate molecular behavior between liquids and gases, and the law of Dulong and Petit is similarly suggestive in connection with the solid state.
In physical chemistry it is often convenient to deal with the so-called (or agramme-atom')), a gramme-molecule (or gramme-atom) of any element or compound being defined as that quantity of it that has a weight in grammes equal to the molecular weight (or atomic weight, as the case may be) of the element or compound under consider ation. The convenience of this unit depends mainly upon the fact that when we measure out any two definite chemical substances in quantities proportional to their respective mo lecular weights, we know that each sample then contains the same number of molecules. The actual determination of this number is a sep arate and special problem; but even if it proved to be forever impossible to enumerate the mole cules, we should nevertheless find it useful, for many purposes, to know that the number of them is the same in two given cases. Assuming the atomic weight of magnesium to be 24.32 and that of platinum to be 195.2, to obtain a gramme-atom of each we weigh out 24.32 grammes of magnesimti and 195.2 of platinum. Then we know, at all events, that we have taken the same number of atoms in both cases (ex cept for slight errors in weighing and in deter mining the atomic weights). Now the specific heat of magnesium is 0.250 and that of plati num is 0.032. Hence 24.32X calo ries of heat will be required to raise the tem perature of the gramme-atom of magnesium by one Centigrade degree. Similarly, 195.2 X 0.032=6.25 calories will be required to raise the temperature of the gramme-atom of platinum by the same amount. The point to be specially noted is that approximately the same amount of heat is required, per atom, to raise the tem perature of either metal by one degree; and this appears to be the ultimate significance of the law of Dulong and Petit — which states that the product of the specific heat and atomic weight of a solid chemical element has an ap proximately constant value. It is true that there are apparent exceptions to this rule, but it is often found that there is a marked tend ency to conform to the law when the tempera ture of the exceptional substance is raised. The law of Dulong and Petit has been the subject of many investigations, and it has been shown to be applicable to numerous compounds as well as to the elements. °This is the case even at the ordinary temperature,') says Perrin, °for the fluorides, chlorides, bromides, iodides and sulphides of various metals, hut not for oxygen ated compounds. A specimen of quartz weigh ing 60 grammes, composed of one gramme atom of silicon and two of oxygen (or three in all), absorbs only 10 calories per degree of rise of temperature. But above 400° C. this same fragment uniformly absorbs 18 calories per degree, or precisely 6 for each gramme atom,* which is substantially the same result as was obtained above, for magnesium and platinum at ordinary temperature.