By taking appropriate differences the following facts will be observed: (1) the replacement of potassium by rubidium occasions an increase in the equivalent volumes by about eight units, and of rubidium by caesium by about eleven units; (2) replacement in the same order is attended by a general increase in the three topic parameters, a greater increase being met with in the replacement of rubidium by caesium; (3) the parameters χ and ψ are about equally increased, while the increase in ω is always the greatest. Now consider the effect of replacing sulphur by selenium. It will be seen that (1) the increase in equivalent volume is about 6.6; (2) all the topic parameters are increased; (3) the greatest increase is effected in the parameters χ and ψ, which are equally lengthened.
These observations admit of ready explanation in the following manner. The ordinary structural formula of potassium sulphate is
If the crystal structure be regarded as composed of three interpenetrating point systems, one consisting of sulphur atoms, the second of four times as many oxygen atoms, and the third of twice as many potassium atoms, the systems being so arranged that the sulphur system is always centrally situated with respect to the other two, and the potassium system so that it would affect the vertical axis, then it is obvious that the replacement of potassium by an element of greater atomic weight would specially increase the length of ω (corresponding to the vertical axis), and cause a smaller increase in the horizontal parameters χ and ψ; moreover, the increments would advance with the atomic weight of the replacing metal. If, on the other hand, the sulphur system be replaced by a corresponding selenium system, an element of higher atomic weight, it would be expected that a slight increase would be observed in the vertical parameter, and a greater increase recorded equally in the horizontal parameters.
Muthmann (Zeit. f. Kryst., 1894), in his researches on the tetragonal potassium and ammonium dihydrogen phosphates and arsenates, found that the replacement of potassium by ammonium was attended by an increase of about six units in the molecular volume, and of phosphorus by arsenic by about 4.6 units. In the topic parameters the following changes were recorded: replacement of potassium by ammonium was attended by a considerable increase in ω, χ and ψ being equally, but only slightly, increased; replacement of phosphorus by arsenic was attended by a considerable increase, equally in χ and ψ, while ω suffered a smaller, but not inconsiderable, increase. It is thus seen that the ordinary plane representation of the structure of compounds possesses a higher significance than could have been suggested prior to crystallographical researches.
Identity, or approximate identity, of crystal form is not in itself sufficient to establish true isomorphism. If a substance deposits itself on the faces of a crystal of another substance of similar crystal form, the substances are probably isomorphous. Such parallel overgrowths, termed episomorphs, are very common among the potassium and sodium felspars; and K. von Hauer has investigated a number of cases in which salts exhibiting episomorphism have different colours, thereby clearly demonstrating this property of isomorphism. For example, episomorphs of white potash alum and violet chrome alum, of white magnesium sulphate and green nickel sulphate, and of many other pairs of salts, have been obtained. More useful is the property of isomorphous substances of forming mixed crystals, which are strictly isomorphous with their constituents, for all variations in composition. In such crystals each component plays its own part in determining the physical properties; in other words, any physical constant of a mixed crystal can be calculated as additively composed of the constants of the two components.
Fig. 7 represents the specific volumes of mixtures of ammonium and potassium sulphates; the ordinates representing specific volumes, and the abscissae the percentage composition of the mixture. Fig. 8 shows the variation of refractive index of mixed crystals of potash alum and thallium alum with variation in composition.
In these two instances the component crystals are miscible in all proportions; but this is by no means always the case. It may happen that the crystals do not form double salts, and are only miscible in certain proportions. Two cases then arise: (1) the properties may be expressed as linear functions of the composition, the terminal values being identical with those obtained for the individual components, and there being a break in the curve corresponding to the absence of mixed crystals; or (2) similar to (1) except that different values must be assigned to the terminal values in order to preserve collinearity. Fig. 9 illustrates the first case: the ordinates represent specific volumes, and the abscissae denote the composition of isomorphous mixtures of ammonium and potassium dihydrogen phosphates, which mutually take one another up to the extent of 20% to form homogeneous crystals. The second case is illustrated in fig. 10. Magnesium sulphate (orthorhombic) takes up ferrous sulphate (monoclinic) to the extent of 19%, forming isomorphous orthorhombic crystals; ferrous sulphate, on the other hand, takes up magnesium sulphate to the extent of 54% to form monoclinic crystals. By plotting the specific volumes of these mixed crystals as ordinates, it is found that they fall on two lines, the upper corresponding to the orthorhombic crystals, the lower to the monoclinic. From this we may conclude that these salts are isodimorphous: the upper line represents isomorphous crystals of stable orthorhombic magnesium sulphate and unstable orthorhombic ferrous sulphate, the lower line isomorphous crystals of stable monoclinic ferrous sulphate and unstable monoclinic magnesium sulphate.
