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
volumes and the atomic weights of the solid elements exhibits the periodicityVolume relations
of solids. 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 |
According to H. Schroeder the silver salts of the fatty acids exhibit additive relations; an increase in the molecule of CH2 causes an increase in the molecular volume of about 15.3.
Thermal Relations.
Specific Heat and Composition.—The nature and experimental determination of specific heats are discussed in the article Calorimetry; here will be discussed the relations existing between the heat capacities of elements and compounds.
In the article Thermodynamics it is shown that the amount of heat required to raise a given weight of a gas through a certain range of temperature is different according as the gas Specific heat of gases. is maintained at constant pressure, the volume increasing, or at constant volume, the pressure increasing. A gas, therefore, has two specific heats, generally denoted by Cp and Cv, when the quantity of gas taken as a unit is one gramme molecular weight, the range of temperature being 1° C. It may be shown that Cp – Cv = R, where R is the gas-constant, i.e. R in the equation PV = RT. From the ratio Cp/Cv conclusions may be drawn as to the molecular condition of the gas. By considerations based on the kinetic theory of gases (see Molecule) it may be shown that when no energy is utilized in separating the atoms of a molecule, this ratio is 5/3 = 1.67. If, however, an amount of energy a is taken up in separating atoms, the ratio is expressible as Cp/Cv = (5+a)/(3+a), which is obviously smaller than 5/3, and decreases with increasing values of a. These relations may be readily tested, for the ratio Cp/Cv is capable of easy experimental determination. It is found that mercury vapour, helium, argon and its associates (neon, krypton, &c.) have the value 1.67; hence we conclude that these gases exist as monatomic molecules. Oxygen, nitrogen, hydrogen and carbon monoxide have the value 1.4; these gases have diatomic molecules, a fact capable of demonstration by other means. Hence it may be inferred that this value is typical for diatomic molecules. Similarly, greater atomic complexity is reflected in a further decrease in the ratio Cp/Cv. The following table gives a comparative view of the specific heats and the ratio for molecules of variable atomic content.
The abnormal specific heats of the halogen elements may be due to a loosening of the atoms, a preliminary to the dissociation into monatomic molecules which occurs at high temperatures. In the more complex gases the specific heat varies considerably with temperature; only in the case of monatomic gases does it remain constant. Le Chatelier (Zeit. f. phys. Chem. i. 456) has given the formula Cp = 6.5 + aT, where a is a constant depending on the complexity of the molecule, as an expression for the molecular heat at constant pressure at any temperature T (reckoned on the absolute scale). For a further discussion of the ratio of the specific heats see Molecule.
| Molecular Content. | Examples. | Cp. | Cv. | Cp/Cv. |
| Monatomic | Hg, Zn, Cd, He, Ar, &c. | 5 | 3 | 1.66 |
| Diatomic | H2, 02, N2 (0°–200°) | 6.83 | 4.83 | 1.41 |
| Cl2, Br2, I2 (0°–200°) | 8.6 | 6.6 | 1.30 | |
| HCl, HBr, HI, NO, CO | . . . | . . . | 1.41 | |
| Triatomic | H2O, H2S, N2O, CO2 | 9.2 | 7.2 | 1.28 |
| Tetratomic | As4, P4 | 13.4 | 11.4 | 1.175 |
| NH3, C2H2 | 11.6 | 9.6 | 1.21 | |
| Pentatomic | CHCl3 | 14 | 12 | 1.17 |
| Hexatomic | C2H4, C2H3Br | 16.4 | 14.4 | 1.14 |
Specific Heats of Solids.—The development of the atomic theory and the subsequent determination of atomic weights in the opening decades of the 19th century inspired A. T. Petit and P. L. Dulong to investigate relations (if any) existing between specific heats and the atomic weight. Their observations on the solid elements led to a remarkable generalization, now known as Dulong and Petit’s law. This states that “the atomic heat (the product of the atomic weight and specific heat) of all elements is a constant quantity.” The value of this constant when H = 1 is about 6.4; Dulong and Petit, using O = 1, gave the value .38, the specific heat of water being unity in both cases. This law—purely empirical in origin—was strengthened by Berzelius, who redetermined many specific heats, and applied the law to determine the true atomic weight from the equivalent weight. At the same time he perceived that specific heats varied with temperature and also with allotropes, e.g. graphite and diamond. The results of Berzelius were greatly extended by Hermann Kopp, who recognized that carbon, boron and silicon were exceptions to the law. He regarded these anomalies as solely due to the chemical nature of the elements, and ignored or regarded as insignificant such factors as the state of aggregation and change of specific heat with temperature.
The specific heats of carbon, boron and silicon subsequently formed the subject of elaborate investigations by H. F. Weber, who showed that with rise of temperature the specific (and atomic) heat increases, finally attaining a fairly constant value; diamond, graphite and the various amorphous forms of carbon having the value about 5.6 at 1000°, and silicon 5.68 at 232°; while he concluded that boron attained a constant value of 5.5. Niison and Pettersson’s observations on beryllium and germanium have shown that the atomic heats of these metals increase with rise of temperature, finally becoming constant with a value 5.6. W. A. Tilden (Phil. Trans., 1900, p. 233) investigated nickel and cobalt over a wide range of temperature (from –182.5° to 100°); his results are:—
| Cobalt. | Nickel. | |
| From –182.5° to –78.4° | 4.1687 | 4.1874 |
| –78.4° to 15° | 5.4978 | 5.6784 |
| 15° to 100° | 6.0324 | 6.3143 |
It is evident that the atomic heats of these intimately associated elements approach nearer and nearer as we descend in temperature, approximating to the value 4. Other metals were tested in order to determine if their atomic heats approximated to this value at low temperatures, but with negative results.
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,
