admissible. But we can with more ease and exactness determine the other specific heat, that at constant pressure, and infer from the behaviour of the one that of the other. For this the ratio of the two magnitudes need not be known if the gas in question obeys with exactness the laws of Boyle and Gay-Lussac. For the equation

which is the mathematical expression of these laws when, as before, p represents the pressure, p the density, and the absolute temperature, in connection with the formula that we have already used several times, viz.

in which J is the mechanical equivalent of heat, immediately gives the law that the difference Cc of the two specific heats is a constant independent of both pressure and temperature. This law, which was first given by S. Carnot, leads at once to the conclusion that, for those gases whose specific heat C at constant pressure does not alter with the temperature and pressure, the specific heat c at constant volume has also a constant magnitude.

Now Regnault' has experimentally shown for air and hydrogen, and Eilhard Wiedemann2 for carbon monoxide, that the specific heat C at constant pressure does not depend on the temperature. From this we may probably assume that all gases whose molecules contain two atoms will exhibit the same behaviour if they obey Boyle's and Gay-Lussac's laws exactly. In this case,

therefore, there would be no doubt as to the ratio of the atomic energy to the molecular energy E having a constant value.

For other gases, on the contrary, whose molecules are composed of more than two atoms, it has been observed that C is by no means constant. Regnault found with carbonic acid, and E. Wiedemann with carbonic acid, ethylene, nitrous oxide, and ammonia, an unmistakable

Mém. de l'Acad. de Paris, xxvi.

• Habilitationsschrift, Leipzig 1875; Pogg. Ann. clvii. 1876, p. 1.

dependence of the specific heat C on temperature. Lussana1 found that for a series of gases it varies also with the pressure-increasing, in fact, with the pressure. For these gases, therefore, the specific heat c at constant volume cannot be looked upon as invariable; we shall consequently have also to expect that the ratio of the atomic energy & to the molecular energy E depends on the temperature, and perhaps also on the pressure.

This expectation is confirmed by some of the numbers given in § 55. Wüllner, as was there indicated, has found the following values for the ratio of the specific heats at 0° and 100° :

For these gases, therefore, there appears a diminution of the ratio as the temperature rises. On the contrary, the value of the ratio of the energy of an atom e to the molecular energy E, as calculated from these numbers, increases as the temperature rises, thus:

For carbon monoxide, the molecule of which contains only two atoms, the variation is slight; it is of the same order of magnitude as the deviation from the laws of Boyle and Gay-Lussac. But with the other gases the variation

1 Nuovo Cim. [3] 1894, xxxvi. ; [4] 1895, i.; 1896, iii.; 1897, vi.; 1898, vii. Atti del Ist. Veneto [7] viii.

is much greater, and the variation seems the greater the more atoms there are in the molecule.

This remark leads to a simple explanation on the basis of the previous discussions upon the nature of the molecular and atomic energy. A possible cause was there indicated in the circumstance that the atomic energy is not, like the molecular energy, simply kinetic, but partly consists of the potential energy of chemical affinity, and that the latter is perhaps subject to different laws. If this is true, the variation must prove the greatest where the greatest forces of affinity come into play, and, therefore, will be the larger the more atoms are chemically combined. And this is in fact the case.

It still remains for us to explain why the ratio of the atomic energy e to the molecular energy E increases, and does not diminish, as the temperature rises. This fact is indeed to be explained only thus, that the so-called disgregation or loosening of the chemical bonds of the atoms combined in the molecule requires an expenditure of energy which is the greater the further it has already proceeded in consequence of increase of the temperature.

This assumption contains nothing improbable, provided that we suppose that the force of chemical affinity does not bring the atoms into direct contact, but endeavours to hold them at a certain distance from each other; they can then oscillate about their assigned positions of equilibrium, and rotate or move about in any other way. By addition of heat these motions will be accelerated, and the amplitude of the oscillations and, above all, the lengths of the paths will be increased without at first causing the atom to escape out of the range of action of the forces of affinity. For this proportionally little energy is needed. Far more energy, however, is required when the atom begins by its increased speed to break loose from the bonds of the molecule, because now the opposing chemical forces are to be

overcome.

This explanation of the behaviour observed by Wüllner seems to contradict the fact that, according to the observations of P. A. Müller, some substances act exactly

oppositely to the gases investigated by Wüllner. Müller found, as in the table of § 55,

For these substances, therefore, the ratio of the specific heats does not decrease, but increases, as the temperature rises; and their atomic energy e increases with the temperature not more, but less, rapidly than the molecular energy E.

It is, however, easy to see that we have here to do with quite different circumstances; for the two methyl compounds which Müller examined have not the same right as the five bodies examined by Wüllner to be considered gases. We can hardly assume that they obey the laws of Boyle, Gay-Lussac, and Avogadro, at least, not at the temperatures at which Müller made his observations. Deviations from these laws are in most cases to be ascribed to the heat not being sufficient to overcome the cohesion so far that a breaking up into simple molecules of the same kind is attained. In such vapours an addition of heat augments the number of molecules, and the value of the molecular energy thereby increases, and increases, indeed, more rapidly than the atomic energy. Müller's observations are therefore just as easy to interpret as those made by Wüllner.

At the present stage of our knowledge we shall have therefore to assume that the law discovered by Clausius, according to which the molecular and atomic energies should bear to each other a ratio that is always constant, holds good in its full strictness only for diatomic gaseous molecules; for other molecules, however, that ratio is variable with the temperature, and may, indeed, according to circumstances, either increase or decrease as the temperature rises.

But it can also happen that for diatomic molecules this law does not hold good. For among the gases whose

molecules consist of two atoms there are several which do not obey the laws of Boyle and Avogadro with exactness; for these, then, Clausius' law cannot be exactly true. To this class of gases belongs, for instance, chlorine and many gaseous chlorine compounds.

Possibly we may also refer to this behaviour of chlorine the striking circumstance that, of the possibly eight substances for which the ratio of the atomic energy to the molecular energy E was found greater than 1, seven are chlorine compounds. For these substances the temperature at which the ratio of the specific heats C and c has been determined will perhaps not have been high enough for a complete breaking up of the vapour into single molecules to have been attained.

The deviation exhibited by ethyl ether will be explainable in the same way. If the measurements had been made at higher temperatures, there would doubtless have been found a greater value of the ratio of C to c, as in the case of methyl ether; and the calculation would then have given a smaller value for the ratio borne by the atomic energy e to the molecular energy E.

The law that in real gases the share of energy possessed by each atom of a molecule is always less than the translatory kinetic energy of the molecule would not therefore be confuted by the eight exceptions.

59. Dependence of the Specific Heat on the Number of Atoms

From the foregoing remarks on the nature of atomic energy it at once follows that the total amount of atomic energy depends on the number of atoms contained in the moleculé, and, indeed, must increase with this number. That such is the case is at once seen by a glance at the values of the ratio E to E in the sixth column of the table in § 55.

On closer examination further regularities come to view. Among them the fact is especially striking that the ratios for the gases O2, N2, H2, CO, NO that head the table are very nearly identical; for them the whole atomic energy & is