| C. | O. | H. | Cl. | Br. | I. | S. |
| 11 | 12.2 | 5.5 | 22.8 | 27.8 | 37.5 | 22.6. |
These values hold fairly well when compared with the experimental values determined from other compounds, and also with the molecular volumes of the elements themselves. Thus the actually observed densities of liquid chlorine and bromine at the boiling-points are 1.56 and 2.96, leading to atomic volumes 22.7 and 26.9, which closely correspond to Kopp’s values deduced from organic compounds.
These values, however, require modification in certain cases, for discrepancies occur which can be reconciled in some cases by assuming that the atomic value of a polyvalent element varies according to the distribution of its valencies. Thus a double bond of oxygen, as in the carbonyl group CO, requires a larger volume than a single bond, as in the hydroxyl group -OH, being about 12.2 in the first case and 7.8 in the second. Similarly, an increase of volume is associated with doubly and trebly linked carbon atoms.
Recent researches have shown that the law originally proposed by Kopp—“That the specific volume of a liquid compound (molecular volume) at its boiling-point is equal to the sum of the specific volumes of its constituents (atomic volumes), and that every element has a definite atomic value in its compounds”—is by no means exact, for isomers have different specific volumes, and the volume for an increment of CH2 in different homologous series is by no means constant; for example, the difference among the esters of the fatty acids is about 57, whereas for the aliphatic aldehydes it is 49. We may therefore conclude that the molecular volume depends more upon the internal structure of the molecule than its empirical content. W. Ostwald (Lehr. der allg. Chem.), after an exhaustive review of the material at hand, concluded that simple additive relations did exist but with considerable deviations, which he ascribed to differences in structure. In this connexion we may notice W. Städel’s determinations:
| CH3CCl3 | 108 | CHClBr·CH3 | 96·5 |
| CH2Cl·CHCl2 | 102.8 | CH2Br·CH2Cl | 88 |
These differences do not disappear at the critical point, and hence the critical volumes are not strictly additive.
Theoretical considerations as to how far Kopp was justified in choosing the boiling-points under atmospheric pressure as being comparable states for different substances now claim our attention. Van der Waal’s equation (p+a/v²)(v-b) = RT contains two constants a and b determined by each particular substance. If we express the pressure, volume and temperature as fractions of the critical constants, then, calling these fractions the “reduced” pressure, volume and temperature, and denoting them by π, φ and θ respectively, the characteristic equation becomes (π+3/φ²)(3φ-1) = 8θ; which has the same form for all substances. Obviously, therefore, liquids are comparable when the pressures, volumes and temperatures are equal fractions of the critical constants. In view of the extremely slight compressibility of liquids, atmospheric pressure may be regarded as a coincident condition; also C.M. Guldberg pointed out that for the most diverse substances the absolute boiling-point is about two-thirds of the critical temperature. Hence within narrow limits Kopp’s determinations were carried out under coincident conditions, and therefore any regularities presented by the critical volumes should be revealed in the specific volumes at the boiling-point.
The connexion between the density and chemical composition of solids has not been investigated with the same completeness as in the case of gases and liquids. The relation between the atomic Volume relations of solids. volumes and the atomic weights of the solid elements exhibits the periodicity which generally characterizes the elements. The molecular volume is additive in certain cases, in particular of analogous compounds of simple constitution. For instance, constant differences are found between the chlorides, bromides and iodides of sodium and potassium:—
| I. | Diff. | II. | Diff. | Diff. I. & II. |
| KCl = 37.4 | 6.9 | NaCl = 27.1 | 6.7 | 10.3 |
| KBr = 44.3 | 9.7 | NaBr = 33.8 | 9.7 | 10.5 |
| KI = 54.0 | NaI = 43.5 | 10.5 |