It is apparent that the law of Dulong and Petit is not rigorously true, and that deviations are observed which invalidate the law as originally framed. Since the atomic heat of the same element varies with its state of aggregation, it must be concluded that some factor taking this into account must be introduced; moreover, the variation of specific heat with temperature introduces another factor.
We now proceed to discuss molecular heats of compounds, that is, the product of the molecular weight into the specific heat. The earliest generalization in this direction is associated with F.E. Neumann, who, in 1831, deduced from observations on many carbonates (calcium, magnesium, ferrous, zinc, barium and lead) that stoichiometric quantities (equimolecular weights) of compounds possess the same heat capacity. This is spoken of as “Neumann’s law.” Regnault confirmed Neumann’s observations, and showed that the molecular heat depended on the number of atoms present, equiatomic compounds having the same molecular heat. Kopp systematized the earlier observations, and, having made many others, he was able to show that the molecular heat was an additive property, i.e. each element retains the same heat capacity when in combination as in the free state. This has received confirmation by the researches of W.A. Tilden (Phil. Trans., 1904, 203 A, p.139) for those elements whose atomic heats vary considerably with temperature.
The specific heat of a compound may, in general, be calculated from the specific heats of its constituent elements. Conversely, if the specific heats of a compound and its constituent elements, except one, be known, then the unknown atomic heat is readily deducible. Similarly, by taking the difference of the molecular heats of compounds differing by one constituent, the molecular (or atomic) heat of this constituent is directly obtained. By this method it is shown that water, when present as “water of crystallization,” behaves as if it were ice.
Deductions from Dulong and Petit’s Law.—Denoting the atomic weight by W and the specific heat by s, Dulong and Petit’s law states that 6.4 = Ws. Thus if s be known, an approximate value of W is determinate. In the determination of the atomic weight of an element two factors must be considered: (1) its equivalent weight, i.e. the amount which is equivalent to one part of hydrogen; and (2) a factor which denotes the number of atoms of hydrogen which combines with or is equivalent to one atom of the particular element. This factor is termed the valency. The equivalent weight is capable of fairly ready determination, but the settlement of the second factor is somewhat more complex, and in this direction the law of atomic heats is of service. To take an example: 38 parts of indium combine with 35.4 parts of chlorine; hence, if the formula of the chloride be InCl, InCl2 or InCl3, indium has the atomic weights 38, 76 or 114. The specific heat of indium is 0.057; and the atomic heats corresponding to the atomic weights 38, 76 and 114 are 3.2, 4.3, 6.5. Dulong and Petit’s law thus points to the value 114, which is also supported by the position occupied by this element in the periodic classification. C. Winkler decided the atomic weight of germanium by similar reasoning.
Boiling-Point and Composition.—From the relation between the critical constants PkVk/Tk = (1/3.7)R or Tk/Pk = 3.7Vk/R, and since Vk is proportional to the volume at absolute zero, the ratio Tk/Pk should exhibit additive relations. This ratio, termed by Guye the critical coefficient, has the following approximate values:—
| C. | H. | Cl. | -O-. | =O. | N. | N=. | P. | Double linkage. | Triple linkage. |
| 1.35 | 0.57 | 2.66 | 0.87 | 1.27 | 1.6 | 1.86 | 3.01 | 0.88 | 1.03 |
Since at the boiling-point under atmospheric pressure liquids are in corresponding states, the additive nature of the critical coefficient should also be presented by boiling-points. It may be shown theoretically that the absolute boiling-point is proportional to the molecular volume, and, since this property is additive, the boiling-point should also be additive.
These relations have been more thoroughly tested in the case of organic compounds, and the results obtained agree in some measure with the deductions from molecular volumes. In general, isomers boil at about the same temperature, as is shown by the isomeric esters C9H18O2:—
| Methyl octoate | 192.9° | Amyl butyrate | 184.8° |
| Ethyl heptoate | 187.1° | Heptyl acetate | 191.3° |
| Propyl hexoate | 185.5° | Octyl formate | 198.1° |
| Butyl pentoate | 185.8 |
Equal increments in the molecule are associated with an equal rise in the boiling-point, but this increment varies in different homologous series. Thus in the normal fatty alcohols, acids, esters, nitriles and ketones, the increment per CH2 is 19°-21°; in the aldehydes it is 26°-27°. In the aromatic compounds there is no regularity between the increments due to the introduction of methyl groups into the benzene nucleus or side chains; the normal value of 20°-21° is exhibited, however, by pyridine and its derivatives. The substitution of a hydrogen atom by the hydroxyl group generally occasions a rise in boiling-point at about 100°. The same increase accompanies the introduction of the amino group into aromatic nuclei.