CHAPTER XV

THE GROUPING OF THE ELEMENTS AND THE PERIODIC LAW

It is seen from the examples given in the preceding chapters that the sum of the data concerning the chemical transformations proper to the elements (for instance, with respect to the formation of acids, salts, and other compounds having definite properties) is insufficient for accurately determining the relationship of the elements, inasmuch as this may be many-sided. Thus, lithium and barium are in some respects analogous to sodium and potassium, and in others to magnesium and calcium. It is evident, therefore, that for a complete judgment it is necessary to have, not only qualitative, but also quantitative, exact and measurable, data. When a property can be measured it ceases to be vague, and becomes quantitative instead of merely qualitative.

Among these measurable properties of the elements, or of their corresponding compounds, are: (_a_) isomorphism, or the analogy of crystalline forms; and, connected with it, the power to form crystalline mixtures which are isomorphous; (_b_) the relation of the volumes of analogous compounds of the elements; (_c_) the composition of their saline compounds; and (_d_) the relation of the atomic weights of the elements. In this chapter we shall briefly consider these four aspects of the matter, which are exceedingly important for a natural and fruitful grouping of the elements, facilitating, not only a general acquaintance with them, but also their detailed study.

Historically the first, and an important and convincing, method for finding a relationship between the compounds of two different elements is by _isomorphism_. This conception was introduced into chemistry by Mitscherlich (in 1820), who demonstrated that the corresponding salts of arsenic acid, H_{3}AsO_{4}, and phosphoric acid, H_{3}PO_{4}, crystallise with an equal quantity of water, show an exceedingly close resemblance in crystalline form (as regards the angles of their faces and axes), and are able to crystallise together from solutions, forming crystals containing a mixture of the isomorphous compounds. Isomorphous substances are those which, with an equal number of atoms in their molecules, present an analogy in their chemical reactions, a close resemblance in their properties, and a similar or very nearly similar crystalline form: they often contain certain elements in common, from which it is to be concluded that the remaining elements (as in the preceding example of As and P) are analogous to each other. And inasmuch as crystalline forms are capable of exact measurement, the external form, or the relation of the molecules which causes their grouping into a crystalline form, is evidently as great a help in judging of the internal forces acting between the atoms as a comparison of reactions, vapour densities, and other like relations. We have already seen examples of this in the preceding pages.[1] It will be sufficient to call to mind that the compounds of the alkali metals with the halogens RX, in a crystalline form, all belong to the cubic system and crystallise in octahedra or cubes--for example, sodium chloride, potassium chloride, potassium iodide, rubidium chloride, &c. The nitrates of rubidium and cæsium appear in anhydrous crystals of the same form as potassium nitrate. The carbonates of the metals of the alkaline earths are isomorphous with calcium carbonate--that is, they either appear in forms like calc spar or in the rhombic system in crystals analogous to aragonite.[1 bis] Furthermore, sodium nitrate crystallises in rhombohedra, closely resembling the rhombohedra of calc spar (calcium carbonate), CaCO_{3}, whilst potassium nitrate appears in the same form as aragonite, CaCO_{3}, and the number of atoms in both kinds of salts is the same: they all contain one atom of a metal (K, Na, Ca), one atom of a non-metal (C, N), and three atoms of oxygen. The analogy of form evidently coincides with an analogy of atomic composition. But, as we have learnt from the previous description of these salts, there is not any close resemblance in their properties. It is evident that calcium carbonate approaches more nearly to magnesium carbonate than to sodium nitrate, although their crystalline forms are all equally alike. Isomorphous substances which are perfectly analogous to each other are not only characterised by a close resemblance of form (homeomorphism), but also by the faculty of entering into analogous reactions, which is not the case with RNO_{3} and RCO_{3}. The most important and direct method of recognising perfect isomorphism--that is, the absolute analogy of two compounds--is given by that property of analogous compounds of separating from solutions _in homogeneous crystals, containing the most varied proportions_ of the analogous substances which enter into their composition. These quantities do not seem to be in dependence on the molecular or atomic weights, and if they are governed by any laws they must be analogous to those which apply to indefinite chemical compounds.[2] This will be clear from the following examples. Potassium chloride and potassium nitrate are not isomorphous with each other, and are in an atomic sense composed in a different manner. If these salts be mixed in a solution and the solution be evaporated, independent crystals of the two salts will separate, each in that crystalline form which is proper to it. The crystals will not contain a mixture of the two salts. But if we mix the solutions of two isomorphous salts together, then, under certain circumstances, crystals will be obtained which contain both these substances. However, this cannot be taken as an absolute rule, for if we take a solution saturated at a high temperature with a mixture of potassium and sodium chlorides, then on evaporation sodium chloride only will separate, and on cooling only potassium chloride. The first will contain very little potassium chloride, and the latter very little sodium chloride.[3] But if we take, for example, a mixture of solutions of magnesium sulphate and zinc sulphate, they cannot be separated from each other by evaporating the mixture, notwithstanding the rather considerable difference in the solubility of these salts. Again, the isomorphous salts, magnesium carbonate, and calcium carbonate are found together--that is, in one crystal--in nature. The angle of the rhombohedron of these magnesia-lime spars is intermediate between the angles proper to the two spars individually (for calcium carbonate, the angle of the rhombohedron is 105° 8´; magnesium carbonate, 107° 30´; CaMg(CO_{3})_{2}, 106° 10´). Certain of these _isomorphous mixtures_ of calc and magnesia spars appear in well-formed crystals, and in this case there not unfrequently exists a simple molecular proportion of strictly definite chemical combination between the component salts--for instance, CaCO_{3},MgCO_{3}--whilst in other cases, especially in the absence of distinct crystallisation (in dolomites), no such simple molecular proportion is observable: this is also the case in many artificially prepared isomorphous mixtures. The microscopical and crystallo-optical researches of Professor Inostrantzoff and others show that in many cases there is really a mechanical, although microscopically minute, juxtaposition in one whole of the heterogeneous crystals of calcium carbonate (double refracting) and of the compound CaMgC_{2}O_{6}. If we suppose the adjacent parts to be microscopically small (on the basis of the researches of Mallard, Weruboff, and others), we obtain an idea of isomorphous mixtures. A formula of the following kind is given to isomorphous mixtures: for instance, for spars, RCO_{3}, where R = Mg, Ca, and where it may be Fe,Mn ..., &c. This means that the Ca is partially replaced by Mg or another metal. Alums form a common example of the separation of isomorphous mixtures from solutions. They are double sulphates (or seleniates) of alumina (or oxides isomorphous with it) and the alkalis, which crystallise in well-formed crystals. If aluminium sulphate be mixed with potassium sulphate, an alum separates, having the composition KAlS_{2}O_{8},12H_{2}O. If sodium sulphate or ammonium sulphate, or rubidium (or thallium) sulphate be used, we obtain alums having the composition RAlS_{2}O_{8},12H_{2}O. Not only do they all crystallise in the cubic system, but they also contain an equal atomic quantity of water of crystallisation (12H_{2}O). Besides which, if we mix solutions of the potassium and ammonium (NH_{4}AlS_{2}O_{8},12H_{2}O) alums together, then the crystals which separate will contain various proportions of the alkalis taken, and separate crystals of the alums of one or the other kind will not be obtained, but each separate crystal will contain both potassium and ammonium. Nor is this all; if we take a crystal of a potassium alum and immerse it in a solution capable of yielding ammonia alum, the crystal of the potash alum will continue to grow and increase in size in this solution--that is, a layer of the ammonia or other alum will deposit itself upon the planes bounding the crystal of the potash alum. This is very distinctly seen if a colourless crystal of a common alum be immersed in a saturated violet solution of chrome alum, KCrS_{2}O_{8},12H_{2}O, which then deposits itself in a violet layer over the colourless crystal of the alumina alum, as was observed even before Mitscherlich noticed it. If this crystal be then immersed in a solution of an alumina alum, a layer of this salt will form over the layer of chrome alum, so that one alum is able to incite the growth of the other. If the deposition proceed simultaneously, the resultant intermixture may be minute and inseparable, but its nature is understood from the preceding experiments; the attractive force of crystallisation of isomorphous substances is so nearly equal that the attractive power of an isomorphous substance induces a crystalline superstructure exactly the same as would be produced by the attractive force of like crystalline particles. From this it is evident that one isomorphous substance may _induce the crystallisation_[4] of another. Such a phenomenon explains, on the one hand, the aggregation of different isomorphous substances in one crystal, whilst, on the other hand, it serves as a most exact indication of the nearness both of the molecular composition of isomorphous substances and of those forces which are proper to the elements which distinguish the isomorphous substances. Thus, for example, ferrous sulphate or green vitriol crystallises in the monoclinic system and contains seven molecules of water, FeSO_{4},7H_{2}O, whilst copper vitriol crystallises with five molecules of water in the triclinic system, CuSO_{4},5H_{2}O; nevertheless, it may be easily proved that both salts are perfectly isomorphous; that they are able to appear in identically the same forms and with an equal molecular amount of water. For instance, Marignac, by evaporating a mixture of sulphuric acid and ferrous sulphate under the receiver of an air-pump, first obtained crystals of the hepta-hydrated salt, and then of the penta-hydrated salt FeSO_{4},5H_{2}O, which were perfectly similar to the crystals of copper sulphate. Furthermore, Lecoq de Boisbaudran, by immersing crystals of FeSO_{4},7H_{2}O in a supersaturated solution of copper sulphate, caused the latter to deposit in the same form as ferrous sulphate, in crystals of the monoclinic system, CuSO_{4},7H_{2}O.

[1] For instance the analogy of the sulphates of K, Rb, and Cs (Chapter XIII., Note 1).

[1 bis] The crystalline forms of aragonite, strontianite, and witherite belong to the rhombic system; the angle of the prism of CaCO_{3} is 116° 10´, of SrCO_{3} 117° 19´, and of BaCO_{3} 118° 30´. On the other hand the crystalline forms of calc spar, magnesite, and calamine, which resemble each other quite as closely, belong to the rhombohedral system, with the angle of the rhombohedra for CaCO_{3} 105° 8´, MgCO_{3} 107° 10´, and ZnCO_{3} 107° 40´. From this comparison it is at once evident that zinc is more closely allied to magnesium than magnesium to calcium.

[2] Solutions furnish the commonest examples of indefinite chemical compounds. But the isomorphous mixtures which are so common among the crystalline compounds of silica forming the crust of the earth, as well as alloys, which are so important in the application of metals to the arts, are also instances of indefinite compounds. And if in Chapter I., and in many other portions of this work, it has been necessary to admit the presence of definite compounds (in a state of dissociation) in solutions, the same applies with even greater force to isomorphous mixtures and alloys. For this reason in many places in this work I refer to facts which compel us to recognise the existence of definite chemical compounds in all isomorphous mixtures and alloys. This view of mine (which dates from the sixties) upon isomorphous mixtures finds a particularly clear confirmation in B. Roozeboom's researches (1892) upon the solubility and crystallising capacity of mixtures of the chlorates of potassium and thallium, KClO_{3} and TlClO_{3}. He showed that when a solution contains different amounts of these salts, it deposits crystals containing either an excess of the first salt, from 98 p.c. to 100 p.c., or an excess of the second salt, from 63·7 to 100 p.c.; that is, in the crystalline form, either the first salt saturates the second or the second the first, just as in the solution of ether in water (Chapter I.); moreover, the solubility of the mixtures containing 36·3 and 98 p.c. KClO_{3} is similar, just as the vapour tension of a saturated solution of water in ether is equal to that of a saturated solution of ether in water (Chapter I., Note 47). But just as there are solutions miscible in all proportions, so also certain isomorphous bodies can be present in crystals in all possible proportions of their component parts. Van 't Hoff calls such systems 'solid solutions.' These views were subsequently elaborated by Nernst (1892), and Witt (1891) applied them in explaining the phenomena observed in the coloration of tissues.

[3] The cause of the difference which is observed in different compounds of the same type, with respect to their property of forming isomorphous mixtures, must not be looked for in the difference of their volumetric composition, as many investigators, including Kopp, affirm. The molecular volumes (found by dividing the molecular weight by the density) of those isomorphous substances which do give intermixtures are not nearer to each other than the volumes of those which do not give mixtures; for example, for magnesium carbonate the combining weight is 84, density 3·06, and volume therefore 27; for calcium carbonate in the form of calc spar the volume is 37, and in the form of aragonite 33; for strontium carbonate 41, for barium carbonate 46; that is, the volume of these closely allied isomorphous substances increases with the combining weight. The same is observed if we compare sodium chloride (molecular volume = 27) with potassium chloride (volume = 37), or sodium sulphate (volume = 55) with potassium sulphate (volume = 66), or sodium nitrate 39 with potassium nitrate 48, although the latter are less capable of giving isomorphous mixtures than the former. It is evident that the cause of isomorphism cannot be explained by an approximation in molecular volumes. It is more likely that, given a similarity in form and composition, the faculty to give isomorphous mixtures is connected with the laws and degree of solubility.

[4] A phenomenon of a similar kind is shown for magnesium sulphate in Note 27 of the last chapter. In the same example we see what a complication the phenomena of dimorphism may introduce when the forms of analogous compounds are compared.

Hence it is evident that isomorphism--that is, the analogy of forms and the property of inducing crystallisation--may serve as a means for the discovery of analogies in molecular composition. We will take an example in order to render this clear. If, instead of aluminium sulphate, we add magnesium sulphate to potassium sulphate, then, on evaporating the solution, the double salt K_{2}MgS_{2}O_{8},6H_{2}O (Chapter XIV., Note 28) separates instead of an alum, and the ratio of the component parts (in alums one atom of potassium per 2SO_{4}, and here two atoms) and the amount of water of crystallisation (in alums 12, and here 6 equivalents per 2SO_{4}) are quite different; nor is this double salt in any way isomorphous with the alums, nor capable of forming an isomorphous crystalline mixture with them, nor does the one salt provoke the crystallisation of the other. From this we must conclude that although alumina and magnesia, or aluminium and magnesium, resemble each other, they are not isomorphous, and that although they give partially similar double salts, these salts are not analogous to each other. And this is expressed in their chemical formulæ by the fact that the number of atoms in alumina or aluminium oxide, Al_{2}O_{3}, is different from the number in magnesia, MgO. Aluminium is trivalent and magnesium bivalent. Thus, having obtained a double salt from a given metal, it is possible to judge of the analogy of the given metal with aluminium or with magnesium, or of the absence of such an analogy, from the composition and form of this salt. Thus zinc, for example, does not form alums, but forms a double salt with potassium sulphate, which has a composition exactly like that of the corresponding salt of magnesium. It is often possible to distinguish the bivalent metals analogous to magnesium or calcium from the trivalent metals, like aluminium, by such a method. Furthermore, the specific heat and vapour density serve as guides. There are also indirect proofs. Thus iron gives ferrous compounds, FeX_{2}, which are isomorphous with the compounds of magnesium, and ferric compounds, FeX_{3}, which are isomorphous with the compounds of aluminium; in this instance the relative composition is directly determined by analysis, because, for a given amount of iron, FeCl_{2} only contains two-thirds of the amount of chlorine which occurs in FeCl_{3}, and the composition of the corresponding oxygen compounds, _i.e._ of ferrous oxide, FeO, and ferric oxide, Fe_{2}O_{3}, clearly indicates the analogy of the ferrous oxide with MgO and of the ferric oxide with Al_{2}O_{3}.

Thus in the building up of similar molecules in crystalline forms we see one of the numerous means for judging of the internal world of molecules and atoms, and one of the weapons for conquests in the invisible world of molecular mechanics which forms the main object of physico-chemical knowledge. This method[5] has more than once been employed for discovering the analogy of elements and of their compounds; and as crystals are measurable, and the capacity to form crystalline mixtures can be experimentally verified, this method is a numerical and measurable one, and in no sense arbitrary.

[5] The property of solids of occurring in regular crystalline forms--the occurrence of many substances in the earth's crust in these forms--and those geometrical and simple laws which govern the formation of crystals long ago attracted the attention of the naturalist to crystals. The crystalline form is, without doubt, the expression of the relation in which the atoms occur in the molecules, and in which the molecules occur in the mass, of a substance. Crystallisation is determined by the distribution of the molecules along the direction of greatest cohesion, and therefore those forces must take part in the crystalline distribution of matter which act between the molecules; and, as they depend on the forces binding the atoms together in the molecules, a very close connection must exist between the atomic composition and the distribution of the atoms in the molecule on the one hand, and the crystalline form of a substance on the other hand; and hence an insight into the composition may be arrived at from the crystalline form. Such is the elementary and _a priori_ idea which lies at the base of all researches into _the connection between composition and crystalline form_. Haüy in 1811 established the following fundamental law, which has been worked out by later investigators: That the fundamental crystalline form for a given chemical compound is constant (only the combinations vary), and that with a change of composition the crystalline form also changes, naturally with the exception of such limiting forms as the cube, regular octahedron, &c., which may belong to various substances of the regular system. The fundamental form is determined by the angles of certain fundamental geometric forms (prisms, pyramids, rhombohedra), or the ratio of the crystalline axes, and is connected with the optical and many other properties of crystals. Since the establishment of this law the description of definite compounds in a solid state is accompanied by a description (measurement) of its crystals, which forms an invariable, definite, and measurable character. The most important epochs in the further history of this question were made by the following discoveries:--Klaproth, Vauquelin, and others showed that aragonite has the same composition as calc spar, whilst the former belongs to the rhombic and the latter to the hexagonal system. Haüy at first considered that the composition, and after that the arrangement, of the atoms in the molecules was different. This is dimorphism (_see_ Chapter XIV., Note 46). Beudant, Frankenheim, Laurent, and others found that the forms of the two nitres, KNO_{3} and NaNO_{3}, exactly correspond with the forms of aragonite and calc spar; that they are able, moreover, to pass from one form into another; and that the difference of the forms is accompanied by a small alteration of the angles, for the angle of the prisms of potassium nitrate and aragonite is 119°, and of sodium nitrate and calc spar, 120°; and therefore dimorphism, or the crystallisation of one substance in different forms, does not necessarily imply a great difference in the distribution of the molecules, although some difference clearly exists. The researches of Mitscherlich (1822) on the dimorphism of sulphur confirmed this conclusion, although it cannot yet be affirmed that in dimorphism the arrangement of the atoms remains unaltered, and that only the molecules are distributed differently. Leblanc, Berthier, Wollaston, and others already knew that many substances of different composition appear in the same forms, and crystallise together in one crystal. Gay-Lussac (1816) showed that crystals of potash alum continue to grow in a solution of ammonia alum. Beudant (1817) explained this phenomenon as the _assimilation_ of a foreign substance by a substance having a great force of crystallisation, which he illustrated by many natural and artificial examples. But Mitscherlich, and afterwards Berzelius and Henry Rose and others, showed that such an assimilation only exists with a similarity or approximate similarity of the forms of the individual substances and with a certain degree of chemical analogy. Thus was established the idea of _isomorphism_ as an analogy of forms by reason of a resemblance of atomic composition, and by it was explained the variability of the composition of a number of minerals as isomorphous mixtures. Thus all the garnets are expressed by the general formula: (RO)_{3}M_{2}O_{3}(SiO_{2})_{3}, where R = Ca, Mg, Fe, Mn, and M = Fe, Al, and where we may have either R and M separately, or their equivalent compounds, or their mixtures in all possible proportions.

But other facts, which render the correlation of form and composition still more complex, have accumulated side by side with a mass of data which may be accounted for by admitting the conceptions of isomorphism and dimorphism. Foremost among the former stand the phenomena of _homeomorphism_--that is, a nearness of forms with a difference of composition--and then the cases of polymorphism and hemimorphism--that is, a nearness of the fundamental forms or only of certain angles for substances which are near or analogous in their composition. Instances of homeomorphism are very numerous. Many of these, however, may be reduced to a resemblance of atomic composition, although they do not correspond to an isomorphism of the component elements; for example, CdS (greenockite) and AgI, CaCO_{3} (aragonite) and KNO_{3}, CaCO_{3} (calc spar) and NaNO_{3}, BaSO_{4} (heavy spar), KMnO_{4} (potassium permanganate), and KClO_{4} (potassium perchlorate), Al_{2}O_{3} (corundum) and FeTiO_{3} (titanic iron ore), FeS_{2} (marcasite, rhombic system) and FeSAs (arsenical pyrites), NiS and NiAs, &c. But besides these instances there are homeomorphous substances with an absolute dissimilarity of composition. Many such instances were pointed out by Dana. Cinnabar, HgS, and susannite, PbSO_{4}3PbCO_{3} appear in very analogous crystalline forms; the acid potassium sulphate crystallises in the monoclinic system in crystals analogous to felspar, KAlSi_{3}O_{8}; glauberite, Na_{2}Ca(SO_{4})_{2}, augite, RSiO_{3} (R = Ca, Mg), sodium carbonate, Na_{2}CO_{3},10H_{2}O, Glauber's salt, Na_{2}SO_{4},10H_{2}O, and borax, Na_{2}BrO_{7},10H_{2}O, not only belong to the same system (monoclinic), but exhibit an analogy of combinations and a nearness of corresponding angles. These and many other similar cases might appear to be perfectly arbitrary (especially as a _nearness_ of angles and fundamental forms is a relative idea) were there not other cases where a resemblance of properties and a distinct relation in the variation of composition is connected with a resemblance of form. Thus, for example, alumina, Al_{2}O_{3}, and water, H_{2}O, are frequently found in many pyroxenes and amphiboles which only contain silica and magnesia (MgO, CaO, FeO, MnO). Scheerer and Hermann, and many others, endeavoured to explain such instances by _polymetric isomorphism_, stating that MgO may be replaced by 3H_{2}O (for example, olivine and serpentine), SiO_{2} by Al_{2}O_{3} (in the amphiboles, talcs), and so on. A certain number of the instances of this order are subject to doubt, because many of the natural minerals which served as the basis for the establishment of polymeric isomorphism in all probability no longer present their original composition, but one which has been altered under the influence of solutions which have come into contact with them; they therefore belong to the class of _pseudomorphs_, or false crystals. There is, however, no doubt of the existence of a whole series of natural and artificial homeomorphs, which differ from each other by atomic amounts of water, silica, and some other component parts. Thus, Thomsen (1874) showed a very striking instance. The metallic chlorides, RCl_{2}, often crystallise with water, and they do not then contain less than one molecule of water per atom of chlorine. The most familiar representative of the order RCl_{2},2H_{2}O is BaCl_{2},2H_{2}O, which crystallises in the rhombic system. Barium bromide, BaBr_{2},2H_{2}O, and copper chloride, CuCl_{2},2H_{2}O, have nearly the same forms: potassium iodate, KIO_{4}; potassium chlorate, KClO_{4}; potassium permanganate, KMnO_{4}; barium sulphate, BaSO_{4}; calcium sulphate, CaSO_{4}; sodium sulphate, Na_{2}SO_{4}; barium formate, BaC_{2}H_{2}O_{4}, and others have almost the same crystalline form (of the rhombic system). Parallel with this series is that of the metallic chlorides containing RCl_{2},4H_{2}O, of the sulphates of the composition RSO_{4},2H_{2}O, and the formates RC_{2}H_{2}O_{4},2H_{2}O. These compounds belong to the monoclinic system, have a close resemblance of form, and differ from the first series by containing two more molecules of water. The addition of two more molecules of water in all the above series also gives forms of the monoclinic system closely resembling each other; for example, NiCl_{2},6H_{2}O and MnSO_{4},4H_{2}O. Hence we see that not only is RCl_{2},2H_{2}O analogous in form to RSO_{4} and RC_{2}H_{2}O_{4}, but that their compounds with 2H_{2}O and with 4H_{2}O also exhibit closely analogous forms. From these examples it is evident that the conditions which determine a given form may be repeated not only in the presence of an isomorphous exchange--that is, with an equal number of atoms in the molecule--but also in the presence of an unequal number when there are peculiar and as yet ungeneralised relations in composition. Thus ZnO and Al_{2}O_{3} exhibit a close analogy of form. Both oxides belong to the rhombohedral system, and the angle between the pyramid and the terminal plane of the first is 118° 7´, and of the second 118° 49´. Alumina, Al_{2}O_{3}, is also analogous in form to SiO_{2}, and we shall see that these analogies of form are conjoined with a certain analogy in properties. It is not surprising, therefore, that in the complex molecule of a siliceous compound it is sometimes possible to replace SiO_{2} by means of Al_{2}O_{3}, as Scheerer admits. The oxides Cu_{2}O, MgO, NiO, Fe_{3}O_{4}, CeO_{2}, crystallise in the regular system, although they are of very different atomic structure. Marignac demonstrated the perfect analogy of the forms of K_{2}ZrF_{6} and CaCO_{3}, and the former is even dimorphous, like the calcium carbonate. The same salt is isomorphous with R_{2}NbOF_{5} and R_{2}WO_{2}F_{4}, where R is an alkali metal. There is an equivalency between CaCO_{3} and K_{2}ZrF_{6}, because K_{2} is equivalent to Ca, C to Zr, and F_{6} to O_{3}, and with the isomorphism of the other two salts we find besides an equal contents of the alkali metal--an equal number of atoms on the one hand and an analogy to the properties of K_{2}ZrF_{6} on the other. The long-known isomorphism of the corresponding compounds of potassium and ammonium, KX and NH_{4}X, may be taken as the simplest example of the fact that an analogy of form shows itself with an analogy of chemical reaction even without an equality in atomic composition. Therefore the ultimate progress of the entire doctrine of the correlation of composition and crystalline forms will only be arrived at with the accumulation of a sufficient number of facts collected on a plan corresponding with the problems which here present themselves. The first steps have already been made. The researches of the Geneva _savant_, Marignac, on the crystalline form and composition of many of the double fluorides, and the work of Wyruboff on the ferricyanides and other compounds, are particularly important in this respect. It is already evident that, with a definite change of composition, certain angles remain constant, notwithstanding that others are subject to alteration. Such an instance of the relation of forms was observed by Laurent, and named by him _hemimorphism_ (an anomalous term) when the analogy is limited to certain angles, and _paramorphism_ when the forms in general approach each other, but belong to different systems. So, for example, the angle of the planes of a rhombohedron may be greater or less than 90°, and therefore such acute and obtuse rhombohedra may closely approximate to the cube. Hausmannite, Mn_{3}O_{4}, belongs to the tetragonal system, and the planes of its pyramid are inclined at an angle of about 118°, whilst magnetic iron ore, Fe_{3}O_{4}, which resembles hausmannite in many respects, appears in regular octahedra--that is, the pyramidal planes are inclined at an angle of 109° 28´. This is an example of paramorphism; the systems are different, the compositions are analogous, and there is a certain resemblance in form. Hemimorphism has been found in many instances of saline and other substitutions. Thus, Laurent demonstrated, and Hintze confirmed (1873), that naphthalene derivatives of analogous composition are hemimorphous. Nicklès (1849) showed that in ethylene sulphate the angle of the prism is 125° 26´, and in the nitrate of the same radicle 126° 95´. The angle of the prism of methylamine oxalate is 131° 20´, and of fluoride, which is very different in composition from the former, the angle is 132°. Groth (1870) endeavoured to indicate in general what kinds of change of form proceed with the substitution of hydrogen by various other elements and groups, and he observed a regularity which he termed _morphotropy_. The following examples show that morphotropy recalls the hemimorphism of Laurent. Benzene, C_{6}H_{6}, rhombic system, ratio of the axes 0·891 : 1 : 0·799. Phenol, C_{6}H_{5}(OH), and resorcinol, C_{6}H_{4}(OH)_{2}, also rhombic system, but the ratio of one axis is changed--thus, in resorcinol, 0·910 : 1 : 0·540; that is, a portion of the crystalline structure in one direction is the same, but in the other direction it is changed, whilst in the rhombic system dinitrophenol, C_{6}H_{3}(NO_{2})_{2}(OH) = O·833 : 1 : 0·753; trinitrophenol (picric acid), C_{6}H_{2}(NO)_{3}(OH) = 0·937 : 1 : 0·974; and the potassium salt = 0·942 : 1 : 1·354. Here the ratio of the first axis is preserved--that is, certain angles remain constant, and the chemical proximity of the composition of these bodies is undoubted. Laurent compares hemimorphism with architectural style. Thus, Gothic cathedrals differ in many respects, but there is an analogy expressed both in the sum total of their common relations and in certain details--for example, in the windows. It is evident that we may expect many fruitful results for molecular mechanics (which forms a problem common to many provinces of natural science) from the further elaboration of the data concerning those variations which take place in crystalline form when the composition of a substance is subjected to a known change, and therefore I consider it useful to point out to the student of science seeking for matter for independent scientific research this vast field for work which is presented by the correlation of form and composition. The geometrical regularity and varied beauty of crystalline forms offer no small attraction to research of this kind.

The regularity and simplicity expressed by the exact laws of crystalline form repeat themselves in the aggregation of the atoms to form molecules. Here, as there, there are but few forms which are essentially different, and their apparent diversity reduces itself to a few fundamental differences of type. There the molecules aggregate themselves into crystalline forms; here, the atoms aggregate themselves into molecular forms or into _the types of compounds_. In both cases the fundamental crystalline or molecular forms are liable to variations, conjunctions, and combinations. If we know that potassium gives compounds of the fundamental type KX, where X is a univalent element (which combines with one atom of hydrogen, and is, according to the law of substitution, able to replace it), then we know the composition of its compounds: K_{2}O, KHO, KCl, NH_{2}K, KNO_{3}, K_{2}SO_{4}, KHSO_{4}, K_{2}Mg(SO_{4})_{2},6H_{2}O, &c. All the possible derivative crystalline forms are not known. So also all the atomic combinations are not known for every element. Thus in the case of potassium, KCH_{3}, K_{3}P, K_{2}Pt, and other like compounds which exist for hydrogen or chlorine, are unknown.

Only a few fundamental types exist for the building up of atoms into molecules, and the majority of them are already known to us. If X stand for a univalent element, and R for an element combined with it, then eight atomic types may be observed:--

RX, RX_{2}, RX_{3}, RX_{4}, RX_{5}, RX_{6}, RX_{7}, RX_{8}.

Let X be chlorine or hydrogen. Then as examples of the first type we have: H_{2}, Cl_{2}, HCl, KCl, NaCl, &c. The compounds of oxygen or calcium may serve as examples of the type RX_{2}: OH_{2}, OCl_{2}, OHCl, CaO, Ca(OH)_{2}, CaCl_{2}, &c. For the third type RX_{3} we know the representative NH_{3} and the corresponding compounds N_{2}O_{3}, NO(OH), NO(OK), PCl_{3}, P_{2}O_{3}, PH_{3}, SbH_{3}, Sb_{2}O_{3}, B_{2}O_{3}, BCl_{3}, Al_{2}O_{3}, &c. The type RX_{4} is known among the hydrogen compounds. Marsh gas, CH_{4}, and its corresponding saturated hydrocarbons, C_{_n_}H_{2_n_ + 2}, are the best representatives. Also CH_{3}Cl, CCl_{4}, SiCl_{4}, SnCl_{4}, SnO_{2}, CO_{2}, SiO_{2}, and a whole series of other compounds come under this class. The type RX_{5} is also already familiar to us, but there are no purely hydrogen compounds among its representatives. Sal-ammoniac, NH_{4}Cl, and the corresponding NH_{4}(OH), NO_{2}(OH), ClO_{2}(OK), as well as PCl_{5}, POCl_{3}, &c., are representatives of this type. In the higher types also there are no hydrogen compounds, but in the type RX_{6} there is the chlorine compound WCl_{6}. However, there are many oxygen compounds, and among them SO_{3} is the best known representative. To this class also belong SO_{2}(OH)_{2}, SO_{2}Cl_{2}, SO_{2}(OH)Cl, CrO_{3}, &c., all of an acid character. Of the higher types there are in general only oxygen and acid representatives. The type RX_{7} we know in perchloric acid, ClO_{3}(OH), and potassium permanganate, MnO_{3}(OK), is also a member. The type RX_{8} in a free state is very rare; osmic anhydride, OsO_{4}, is the best known representative of it.[6]

[6] The still more complex combinations--which are so clearly expressed in the crystallo-hydrates, double salts, and similar compounds--although they may be regarded as independent, are, however, most easily understood with our present knowledge as aggregations of whole molecules to which there are no corresponding double compounds, containing one atom of an element R and many atoms of other elements RX_{_n_}. The above types embrace all cases of direct combinations of atoms, and the formula MgSO_{4},7H_{2}O cannot, without violating known facts, be directly deduced from the types MgX_{_n_} or SX_{_n_}, whilst the formula MgSO_{4} corresponds both with the type of the magnesium compounds MgX_{2} and with the type of the sulphur compounds SO_{2}X_{2}, or in general SX_{6}, where X_{2} is replaced by (OH)_{2}, with the substitution in this case of H_{2} by the atom Mg, which always replaces H_{2}. However, it must be remarked that the sodium crystallo-hydrates often contain 10H_{2}O, the magnesium crystallo-hydrates 6 and 7H_{2}O, and that the type PtM_{2}X_{6} is proper to the double salts of platinum, &c. With the further development of our knowledge concerning crystallo-hydrates, double salts, alloys, solutions, &c., in the _chemical sense_ of feeble compounds (that is, such as are easily destroyed by feeble chemical influences) it will probably be possible to arrive at a perfect generalisation for them. For a long time these subjects were only studied by the way or by chance; our knowledge of them is accidental and destitute of system, and therefore it is impossible to expect as yet any generalisation as to their nature. The days of Gerhardt are not long past when only three types were recognised: RX, RX_{2}, and RX_{3}; the type RX_{4} was afterwards added (by Cooper, Kekulé, Butleroff, and others), mainly for the purpose of generalising the data respecting the carbon compounds. And indeed many are still satisfied with these types, and derive the higher types from them; for instance, RX_{5} from RX_{3}--as, for example, POCl_{3} from PCl_{3}, considering the oxygen to be bound both to the chlorine (as in HClO) and to the phosphorus. But the time has now arrived when it is clearly seen that the forms RX, RX_{2}, RX_{3}, and RX_{4} do not exhaust the whole variety of phenomena. The revolution became evident when Würtz showed that PCl_{5} is not a compound of PCl_{3} + Cl_{2} (although it may decompose into them), but a whole molecule capable of passing into vapour, PCl_{5} like PF_{5} and SiF_{4}. The time for the recognition of types even higher than RX_{8} is in my opinion in the future; that it will come, we can already see in the fact that oxalic acid, C_{2}H_{2}O_{4}, gives a crystallo-hydrate with 2H_{2}O; but it may be referred to the type CH_{4}, or rather to the type of ethane, C_{2}H_{6}, in which all the atoms of hydrogen are replaced by hydroxyl, C_{2}H_{2}O_{4}2H_{2}O = C_{2}(OH)_{6} (_see_ Chapter XXII., Note 35).

The four lower types RX, RX_{2}, RX_{3}, and RX_{4} are met with in compounds of the elements R with chlorine and oxygen, and also in their compounds with hydrogen, whilst the four higher types only appear for such acid compounds as are formed by chlorine, oxygen, and similar elements.

Among the oxygen compounds the _saline oxides_ which are capable of forming salts either through the function of a base or through the function of an acid anhydride attract the greatest interest in every respect. Certain elements, like calcium and magnesium, only give one saline oxide--for example, MgO, corresponding with the type MgX_{2}. But the majority of the elements appear in several such forms. Thus copper gives CuX and CuX_{2}, or Cu_{2}O and CuO. If an element R gives a higher type RX_{_n_}, then there often also exist, as if by symmetry, lower types, RX_{_n_-2}, RX_{_n_-4}, and in general such types as differ from RX_{_n_} by an even number of X. Thus in the case of sulphur the types SX_{2}, SX_{4}, and SX_{6} are known--for example SH_{2}, SO_{2}, and SO_{3}. The last type is the highest, SX_{6}. The types SX_{5} and SX_{3} do not exist. But even and uneven types sometimes appear for one and the same element. Thus the types RX and RX_{2} are known for copper and mercury.

Among the _saline_ oxides only the _eight types_ enumerated below are known to exist. They determine the possible formulæ of the compounds of the elements, if it be taken into consideration that an element which gives a certain type of combination may also give lower types. For this reason the rare type of the _suboxides_ or quaternary oxides R_{4}O (for instance, Ag_{4}O, Ag_{2}Cl) is not characteristic; it is always accompanied by one of the higher grades of oxidation, and the compounds of this type are distinguished by their great chemical instability, and split up into an element and the higher compound (for instance, Ag_{4}O = 2Ag + Ag_{2}O). Many elements, moreover, form transition oxides whose composition is intermediate, which are able, like N_{2}O_{4}, to split up into the lower and higher oxides. Thus iron gives magnetic oxide, Fe_{3}O_{4}, which is in all respects (by its reactions) a compound of the suboxide FeO with the oxide Fe_{2}O_{3}. The independent and more or less stable saline compounds correspond with the following eight types:--

R_{2}O; salts RX, hydroxides ROH. Generally basic like K_{2}O, Na_{2}O, Hg_{2}O, Ag_{2}O, Cu_{2}O; if there are acid oxides of this composition they are very rare, are only formed by distinctly acid elements, and even then have only feeble acid properties; for example, Cl_{2}O and N_{2}O.

R_{2}O_{2} or RO; salts RX_{2}, hydroxides R(OH)_{2}. The most simple basic salts R_{2}OX_{2} or R(OH)X; for instance, the chloride Zn_{2}OCl_{2}; also an almost exclusively basic type; but the basic properties are more feebly developed than in the preceding type. For example, CaO, MgO, BaO, PbO, FeO, MnO, &c.

R_{2}O_{3}; salts RX_{3}, hydroxides R(OH)_{3}, RO(OH), the most simple basic salts ROX, R(OH)X_{3}. The bases are feeble, like Al_{2}O_{3}, Fe_{2}O_{3}, Tl_{2}O_{3}, Sb_{2}O_{3}. The acid properties are also feebly developed; for instance, in B_{2}O_{3}; but with the non-metals the properties of acids are already clear; for instance, P_{2}O_{3}, P(OH)_{3}.

R_{2}O_{4} or RO_{2}; salts RX_{4} or ROX_{2}, hydroxides R(OH)_{4}, RO(OH)_{2}. Rarely bases (feeble), like ZrO_{2}, PtO_{2}; more often acid oxides; but the acid properties are in general feeble, as in CO_{2}, SO_{2}, SnO_{2}. Many intermediate oxides appear in this and the preceding and following types.

R_{2}O_{5}; salts principally of the types ROX_{3}, RO_{2}X, RO(OH)_{3}, RO_{2}(OH), rarely RX_{5}. The basic character (X, a halogen, simple or complex; for instance, NO_{3}, Cl, &c.) is feeble; the acid character predominates, as is seen in N_{2}O_{5}, P_{2}O_{5}, Cl_{2}O_{5}; then X = OH, OK, &c., for example NO_{2}(OK).

R_{2}O_{6} or RO_{3}; salts and hydroxides generally of the type RO_{2}X_{2}, RO_{2}(OH)_{2}. Oxides of an acid character, as SO_{3}, CrO_{3}, MnO_{3}. Basic properties rare and feebly developed as in UO_{3}.

R_{2}O_{7}; salts of the form RO_{3}X, RO_{3}(OH), acid oxides; for instance, Cl_{2}O_{7}, Mn_{2}O_{7}. Basic properties as feebly developed as the acid properties in the oxides R_{2}O.

R_{2}O_{8} or RO_{4}. A very rare type, and only known in OsO_{4} and RuO_{4}.

It is evident from the circumstance that in all the higher types the _acid hydroxides_ (for example, HClO_{4}, H_{2}SO_{4}, H_{3}PO_{4}) and salts with a single atom of one element contain, like the higher saline type RO_{4}, _not more than four atoms of oxygen_; that the formation of the saline oxides is governed by a certain common principle which is best looked for in the fundamental properties of oxygen, and in general of the most simple compounds. The hydrate of the oxide RO_{2} is of the higher type RO_{2}2H_{2}O = RH_{4}O_{4} = R(HO)_{4}. Such, for example, is the hydrate of silica and the salts (orthosilicates) corresponding with it, Si(MO)_{4}. The oxide R_{2}O_{5}, corresponds with the hydrate R_{2}O_{5}3H_{2}O = 2RH_{3}O_{4} = 2RO(OH)_{3}. Such is orthophosphoric acid, PH_{3}O_{3}. The hydrate of the oxide RO_{3} is RO_{3}H_{2}O = RH_{2}O_{4} = RO_{2}(OH)_{2}--for instance, sulphuric acid. The hydrate corresponding to R_{2}O_{7} is evidently RHO = RO_{3}(OH)--for example, perchloric acid. Here, besides containing O_{4}, it must further be remarked that _the amount of hydrogen in the hydrate is equal to the amount of hydrogen in the hydrogen compound_. Thus silicon gives SiH_{4} and SiH_{4}O_{4}, phosphorus PH_{3} and PH_{3}O_{4}, sulphur SH_{2} and SH_{2}O_{4}, chlorine ClH and ClHO_{4}. This, if it does not explain, at least connects in a harmonious and general system the fact that _the elements are capable of combining with a greater amount of oxygen, the less the amount of hydrogen which they are able to retain_. In this the key to the comprehension of all further deductions must be looked for, and we will therefore formulate this rule in general terms. An element R gives a hydrogen compound RH_{_n_}, the hydrate of its higher oxide will be RH_{_n_}O_{4}, and therefore the higher oxide will contain 2RH_{_n_}O_{4} - _n_H_{2}O = R_{2}O_{8 - _n_}. For example, chlorine gives ClH, hydrate ClHO_{4}, and the higher oxide Cl_{2}O_{7}. Carbon gives CH_{4} and CO_{2}. So also, SiO_{2} and SiH_{4} are the higher compounds of silicon with hydrogen and oxygen, like CO_{2} and CH_{4}. Here the amounts of oxygen and hydrogen are equivalent. Nitrogen combines with a large amount of oxygen, forming N_{2}O_{5}, but, on the other hand, with a small quantity of hydrogen in NH_{3}. _The sum of the equivalents of hydrogen and oxygen_, occurring in combination with an atom of nitrogen, is, as always in the higher types, equal to _eight_. It is the same with the other elements which combine with hydrogen and oxygen. Thus sulphur gives SO_{3}; consequently, six equivalents of oxygen fall to an atom of sulphur, and in SH_{2} two equivalents of hydrogen. The sum is again equal to eight. The relation between Cl_{2}O_{7} and ClH is the same. This shows that the property of elements of combining with such different elements as oxygen and hydrogen is subject to one common law, which is also formulated in the system of the elements presently to be described.[7]

[7] The hydrogen compounds, R_{2}H, in equivalency correspond with the type of the suboxides, R_{4}O. Palladium, sodium, and potassium give such hydrogen compounds, and it is worthy of remark that according to the periodic system these elements stand near to each other, and that in those groups where the hydrogen compounds R_{2}H appear, the quaternary oxides R_{4}O are also present.

Not wishing to complicate the explanation, I here only touch on the general features of the relation between the hydrates and oxides and of the oxides among themselves. Thus, for instance, the conception of the ortho-acids and of the normal acids will be considered in speaking of phosphoric and phosphorous acids.

As in the further explanation of the periodic law only those oxides which give salts will be considered, I think it will not be superfluous to mention here the following facts relative to the peroxides. Of the _peroxides_ corresponding with hydrogen peroxide, the following are at present known: H_{2}O_{2}, Na_{2}O_{2}, S_{2}O_{7} (as HSO_{4}?), K_{2}O_{4}, K_{2}O_{2}, CaO_{2}, TiO_{3}, Cr_{2}O_{7}, CuO_{2}(?), ZnO_{2}, Rb_{2}O_{2}, SrO_{2}, Ag_{2}O_{2}, CdO_{2}, CsO_{2}, Cs_{2}O_{2}, BaO_{2}, Mo_{2}O_{7}, SnO_{3}, W_{2}O_{7}, UO_{4}. It is probable that the number of peroxides will increase with further investigation. A periodicity is seen in those now known, for the elements (excepting Li) of the first group, which give R_{2}O, form peroxides, and then the elements of the sixth group seem also to be particularly inclined to form peroxides, R_{2}O_{7}; but at present it is too early, in my opinion, to enter upon a generalisation of this subject, not only because it is a new and but little studied matter (not investigated for all the elements), but also, and more especially, because in many instances only the hydrates are known--for instance, Mo_{2}H_{2}O_{8}--and they perhaps are only compounds of peroxide of hydrogen--for example, Mo_{2}H_{2}O_{8} = 2MoO_{3} + H_{2}O_{2}--since Prof. Schöne has shown that H_{2}O_{2} and BaO_{2} possess the property of combining together and with other oxides. Nevertheless, I have, in the general table expressing the periodic properties of the elements, endeavoured to sum up the data respecting all the known peroxide compounds whose characteristic property is seen in their capability to form peroxide of hydrogen under many circumstances.

In the preceding we see not only the regularity and simplicity which govern the formation and properties of the oxides and of all the compounds of the elements, but also a fresh and exact means for recognising the analogy of elements. Analogous elements give compounds of analogous types, both higher and lower. If CO_{2} and SO_{2} are two gases which closely resemble each other both in their physical and chemical properties, the reason of this must be looked for not in an analogy of sulphur and carbon, but in that identity of the type of combination, RX_{4}, which both oxides assume, and in that influence which a large mass of oxygen always exerts on the properties of its compounds. In fact, there is little resemblance between carbon and sulphur, as is seen not only from the fact that CO_{2} is the _higher form_ of oxidation, whilst SO_{2} is able to further oxidise into SO_{3}, but also from the fact that all the other compounds--for example, SH_{2} and CH_{4}, SCl_{2} and CCl_{4}, &c.--are entirely unlike both in type and in chemical properties. This absence of analogy in carbon and sulphur is especially clearly seen in the fact that the highest saline oxides are of different composition, CO_{2} for carbon, and SO_{3} for sulphur. In