Thus far in the discussion of the inter-atomic distances we have been dealing with aggregates composed of like atoms. The same general principles apply to aggregates of unlike atoms, but the existence of differences between the components of such systems introduces some new factors that we will now want to examine.
The matters to be considered in this chapter have no relevance to direct combinations of electropositive elements (aggregates of which are mixtures or alloys, rather than chemical compounds). As noted in Chapter18, Volume I, the proportions in which such elements can combine may be determined, or limited, by geometrical considerations, but aside from such effects, unlike atoms of this kind can combine on the same basis as like atoms. Here the forces are identical in character and concurrent, the type of combination that we have called the positive orientation. The resultant specific electric rotation, according to the principles previously set forth, is (t1t2)½, the geometric mean of the two constituents. If the two elements have different magnetic rotations, the resultant is also the geometric mean of the individual rotations, as the magnetic rotations always have positive displacements, and these combine in the same manner as the positive electric displacements. The effective electric and magnetic specific rotations thus derived can then be entered in the applicable force and distance equations from Chapter 1.
Combinations of unlike positive atoms may also take place on the basis of the reverse orientation, the alternate type of structure that is available to the elemental aggregates. Where the electric rotations of the components differ, the resultant specific rotation of the two-atom combination will not be the required neutral 5 or 10, but a second pair of atoms inversely oriented to the first results in a four-atom group that has the necessary rotational balance.
As brought out in Volume I, the simplest type of combination in chemical compounds is based on the normal orientation, in which Division I electropositive elements are joined with Division IV electronegative elements on the basis of numerically equal displacements. The resultant effective specific magnetic rotation can be calculated in the same manner as in the all-positive structures, but, as we saw in our consideration of the inter-atomic distances of the elements, where an equilibrium is established between positive and negative electric rotations, the resultant is the sum of the two individual values, rather than the mean.
When this arrangement unites one electropositive atom with each electronegative atom the resulting structure is usually a simple cube with the atoms of each element occupying alternate comers of the cube. This is called the Sodium Chloride structure, after the most familiar member of the family of compounds crystallizing in this form. Table 7 gives the inter-atomic distances of a number of common NaCl type crystals. From this tabulation it can be seen that the special rotational characteristics
which certain of the elements possess in the elemental aggregates carry over into their compounds. The second element in each group shows the same preference for rotation on the basis of vibration two that we encountered in examining the structures of the elements. Here, again, this preference extends to some of the following elements, and in such series of compounds as CaO, ScN, TiC, one component keeps the vibration two status throughout the series, and the resulting effective rotations are 5½, 7, 8½, rather than 6, 8, 10. The elements of the lower groups have inactive force dimensions in the compounds just as in the elemental structures previously examined. If the active dimensions are not the same in both components, the full rotational force of the more active component is effective in its excess dimensions, the effective rotation in an inactive dimension being unity. For example, the value of ln t for magnetic rotation 3 is 1.099 in three dimensions, or 0.7324 in two dimensions. If this two-dimensional rotation is combined with a three-dimensional magnetic rotation x, the resultant value of ln t is (0.7324 x)½, the geometric mean of the individual values, in two dimensions, and x in the third. The average value for all three dimensions is (0.7324 x2)1/3.
This dimensional inactivity in the lower groups plays only a minor role in the structures of the elements, as can be seen from the fact that it did not need any attention until almost the end of Table 8.
The compounds of lithium with valence one negative elements follow the regular pattern, and were included in Table 7, but the compounds with valence two elements are irregular, and they have therefore been omitted from Table 8. As we will see in Chapter 6, the irregularity is due to the fact that the two lithium atoms in a molecule of the CaF2 type act as a radical rather than as independent constituents of the molecule.
These two normal orientation tables, 7 and 8, provide an impressive confirmation of the validity of the theoretical findings. One of the problems in dealing with the inter-atomic distances of the elements is that because of the relatively small total number of elements, the number to which any particular magnetic rotational combination is applicable is quite small, and consequently it is rather difficult to establish a prima facie case for the authenticity of the rotational values. But this is not true of the normal type compounds, as they are more numerous and less variable. There are two elements in these tables, sulfur and chlorine, that have different magnetic rotations under different conditions. These elements have 4-3 rotation in the CaF2 type crystals, and in the NaCl type combinations with elements of group 4A. In the other compounds of the NaCl type they take the 3½-3½ rotations. There are also two more elements, each of which, according to the information now available, deviates from its normal rotations in one of the listed compounds. Otherwise, all of the elements entering into the 60 compounds in the two tables have the same specific magnetic rotations in every compound in which they participate.
Furthermore, when the inherent differences between the elemental and compound aggregates are taken into account, there is also agreement between these rotations in the compounds and the specific rotations of the same elements in the elemental aggregates. The most common difference of this kind is a result of the fact that the Division IV element in a compound has a purely negative role. For this reason it takes the magnetic rotation of the next higher group. In the elemental aggregates half of the atoms are reoriented to act in a positive capacity. Consequently, they tend to retain the normal rotation of the group to which they actually belong. For example, the Division IV elements of Group 3A, germanium, arsenic, selenium, and bromine, have the normal specific rotation of their group, 4-3, in the crystals of the elements, but in the compounds they take the 4-4 specific rotation of Group 3B, acting as negative members of that group.
Another difference between the two classes of structures is that those elements of the higher groups that have the option of extending their rotation to a second vibrational unit are less likely to do so if they are combining with an element which is rotating entirely on the basis of vibration one. Aside from these deviations due to known causes, the values of the specific magnetic rotation determined for the elements in Chapter 2 are also generally applicable to the compounds. This equivalence does not apply to the specific electric rotations, as they are determined by the way in which the rotations of the constituents of each aggregate are oriented relative to each other, a relation that is different in the two classes of structures.
This applicability of the same equations and, in general, the same numerical values, to the calculation of distances in both elements and compounds contrasts sharply with the conventional theory that regards the inter-atomic distance as being determined by the “sizes” of the atoms. The sodium atom, or “ion,” in the NaCl crystal, for example, is asserted to have a radius only about 60 percent as large as the radius of the atom in the elemental aggregate. If this atom takes part in a compound which cannot be included in the “ionic” class, current theory gives it still a different “size”: what is called a “covalent” radius. The need for assuming any extraordinary changeability in the size of what, so far as we can tell, continues to be the same object, is now eliminated by the finding that the variations in the inter-atomic distance have nothing to do with the sizes of the atoms, but merely indicate differences in the location of the equilibrium between the inward and the outward forces to which the atoms are subject.
Another type of orientation that forms a relatively simple binary compound is the rotational combination that we found in the diamond structure. As in the elements, this is an equilibrium between an atom of a Division IV element and one of Division III, the requirement being that t1+ t2 = 8. Obviously, the only elements that can meet this requirement by themselves are those whose negative rotational displacement (valence) is 4, but any Division IV element can establish an equilibrium of this kind with an appropriate Division III element.
Closely associated with this cubic diamond-like Zinc Sulfide class of crystals is a hexagonal structure based on the same orientation, and containing the same equal proportions of the two constituents. Since these controlling factors are identical in the two forms, the crystals of the hexagonal Zinc Oxide class have the same inter-atomic distances as the corresponding Zinc Sulfide structures. In such instances, where the inter-atomic forces are the same, there is little or not probability advantage of one type of crystal over the other, and either may be formed under appropriate conditions. Table 9 lists the inter-atomic distances for some common crystals of these two classes.
The comments that were made about the consistency of the specific rotation values in Tables 7 and 8 are applicable to the values in Table 9 as well. Most of the elements participating in the compounds of this table have the same specific rotations as in the previous tabulations, and where there are exceptions, the deviations are of a regular and predictable nature.
A feature of Table 9 is the appearance of one of the normally electropositive elements of group 2B, Aluminum, in the role of a Division III element. Beryllium and magnesium also form ZnS type compounds, but like the lithium compounds previously mentioned they are irregular, probably for the same reason, and have not been included in the tabulation. The Division III behavior of these normally Division I elements is a result of the small size of the lower groups, which puts their their Division I elements into the same positions with respect to the electronegative zero point as the Division III elements of the larger groups. This relationship is indicated in the following tabulation, where the asterisks identify those elements that are normally in Division I.
None of the orientations thus far considered is applicable to compounds of the Division II elements. The normal orientation does not exist above a specific rotation of 5, as the higher value would put the relative rotation above the limiting value 10. The Zinc Oxide and Zinc Sulfide types of combination are electronegative structures, and the reverse orientation of the Division II elemental structures is not available for compounds with negative elements. The Division II elements therefore form their compounds on the basis of the magnetic orientation. This type of structure is theoretically available for any element, but its use is limited by probability considerations. It is utilized in many of the compounds of Divisions III and IV, especially in the higher rotational groups, but rarely appears in Division I combinations because of the very high probability of the normal orientation in this division.
Since the magnetic rotation is distributed over all three dimensions, its effective component is not altered by a change in position, and has the same value in the magnetic orientations as in the corresponding compounds based on the electric orientations. In order to establish the magnetic type of equilibrium, however, the axis of the negative electric rotation has to be parallel to that of one of the magnetic rotations, and it is therefore perpendicular to the axis of the positive electric rotation. Consequently, the latter takes no part in the normal inter-atomic force equilibrium, and it constitutes an additional orienting influence, the effects of which were discussed in Volume I. In these compounds of the magnetic type the displacement of the negative component (-x) is balanced by a numerically equal positive displacement (x). Thus the magnetic orientation is somewhat similar to the normal orientation. However, the magnetic rotation is opposite in vectorial direction to the electric rotation, and the resultant relative rotation effective in the dimension of combination is therefore one of the neutral values 10, 5, or a combination of these two, rather than the 2x of the normal orientation.
Compounds based on the magnetic orientation occur in a variety of crystal forms, the nature of which depends on the degree of force symmetry and the number of atoms of each kind in the equilibrium system. In some cases there is enough symmetry to make isometric structures of the NaCl, CaF2, and similar types possible. Other crystals are asymmetric. A common arrangement for the binary compounds is the Nickel Arsenide structure, a hexagonal crystal in which the positive atoms occupy the face positions and the negative atoms are in the central positions, spaced alternately ¼ and ¾ along the c axis. Table 10 shows the inter-atomic distances calculated for some NiAs and NaCl, type crystals of binary magnetic orientation compounds of Group 3A.
Almost all of the NiAs type compounds that have been examined in the course of this present work take the vibration one value of the specific electric rotation: 10. The magnetic orientation compounds with the NaCl structure are quite evenly divided between the 10 rotation and the combination 5-10 in the 3A group, but utilize the 5-10 rotation almost exclusively in the higher groups. In order to show as wide a variety of the features of these magnetic type compounds as is possible in the limited amount of space that can be allotted to them, Table 10 has been restricted to Group 3A compounds, and the following Table 11 gives the data for a representative sample of the compounds of the rare earth elements (from Group 4A), together with a selection of compounds from Group 4B, in which the identical values of the inter-atomic distance in the combinations of the elements of this group with the Division IV elements of Group 2A are emphasized.
Thus far the calculation of equilibrium distances has been carried out by crystal types as a matter of convenience in identifying the effect of various atomic characteristics on the crystal form and dimensions. It is apparent from the points brought
|NiAs (Hexagonal) Class—Group 3A|
|NaCl (Cubic) Class-Group 3A|
out in the discussion, however, that identification of the crystal type is not always essential to the determination of the inter-atomic distance. For example, let us consider the series of compounds NaBr, Na2Se, and Na3As. From the relations that have been established in the preceding pages we may conclude that these Division I compounds are formed on the basis of the normal orientation. We therefore apply the known value of the relative specific electric rotation of a normal orientation Sodium compound, 4, and the known values of the normal specific magnetic rotations of Sodium and the Group 3B elements, 3-3½ and 4-4 respectively, to equation 1-10, from which we ascertain that the most probable inter-atomic distance in all three compounds is 2.95, irrespective of the crystal structure. (Measured values are 2.97, 2.95, and 2.94 respectively.)
The possible inter-atomic distances in the more complex compounds can be calculated in a similar manner, without the necessity of analyzing the great variety of geometrical structures in which these compounds crystallize. The usefulness of
this procedure in application to compounds in general is limited, at the present stage of the theoretical development, because we are not normally able to define the specific rotations from theoretical premises as definitely as in the foregoing illustration. It is of considerable value, however, in dealing with the lower electronegative elements, whose specific electric rotations are confined to the neutral values, and whose variability in the magnetic dimensions is only in the number of inactive dimensions (that is, dimensions in which the specific rotation is 2). The elements involved are those of groups 1B and 2A; hydrogen, carbon, nitrogen, oxygen, and fluorine, together with Boron, one of the normally electropositive elements of Group 2A. The other two positive elements of this group, lithium and beryllium, are also two-dimensional under most conditions, but they take the positive orientation, and have much greater inter-atomic distances.
Table 12 gives the theoretically possible inter-atomic distances of these lower group elements, with some examples of the measured values corresponding to the calculated distances
The experimental results are not all in agreement with the theory. On the contrary, they are widely scattered. The measured C-C distances, for example, cover almost the entire range from 1.18, the minimum for this combination, to the maximum 1.54. However, the basic compounds of each class do with the theoretical values. The paraffin hydrocarbons, benzene, ethy and acetylene, have C-C distances approximating the theoretical 1.54, 1.41, 1.30, and 1.18 respectively. All C-H distances are close to the theoretical 0.92 and 1.06, and so on. It can reasonably be concluded, therefore, that the significant deviations from the theoretical values are due to special factors that apply to the less regular structures.
A detailed investigation of the reasons for these deviations is beyond the scope of this present work. However, there are two rather obvious causes that are worth mentioning. One is that forces exerted by adjacent atom may modify the normal result of a two-atom interaction. An interesting point in this connection is that the effect, where it occurs, is inverse; that is, it increases the atomic separation, rather than decreasing it as might be expected. The natural reference system always progresses at unit speed, irrespective of the positions of the structures to which it applies, and consequently the inward force due to this progression always remains the same. Any interaction with a third atom introduces an additional rotational outward) force, and therefore moves the point of equilibrium outward. This is illustrated in the measured distances in the polynuclear derivatives of benzene. The lowest C-C distances in these compounds, 1.38 and 1.39, are found along the outer edges of the molecular structures, while the corresponding distances in the interiors of the compounds, where the influence of adjoining atoms is at a maximum, characteristically range from 1.41 to 1.43.
Another reason for discrepancies is -that in many instances the measurement and the theoretical calculation do not apply to the same quantity. The calculation gives us the distance between structural units, whereas the measurements apply to the distances between specific atoms. Where the atoms are the structural units, as in the compounds of these, or where the inter-group distance is the same as the inter-atomic distance, as in the normal paraffins, there is no problem, but exact agreement cannot be expected otherwise. Again we can use benzene as an example. The C-C distance in benzene is generally reported as 1.39, whereas the corresponding theoretical distance, as indicated in Table 12, is 1.41. But, according to the theory, benzene is not a ring of carbon atoms with hydrogen atoms attached; it is a ring of CH neutral groups, and the 1.41 neutral value applies to the distance between these neutral groups, the structural units of the atom. Since the hydrogen atoms are known to be outside the carbon atoms, if these atoms are coplanar it follows that the distance between the effective centers of the CH groups must be somewhat greater than the distance between the carbon atoms of these groups. The 1.39 measurement between the carbon atoms is therefore entirely consistent with the theoretical distance calculations.
The same kind of a deviation from the results of the (apparent) direct interaction between two individual atoms occurs on a larger scale where there is a group of atoms that is acting structurally as a radical. Many of the properties of molecules composed in part, or entirely, of radicals or neutral groups are not determined directly by the characteristics of the atoms, but by the characteristics of the groups. The NH4 radical, for example, has the same specific rotations, when acting as a group, as the rubidium atom, and it can be substituted in the NaCl type crystals of the rubidium halides without altering the volume. Consequently, the inter-atomic distances have no direct significance in compounds containing these groups. It is theoretically feasible to locate the effective centers of the various groups, and to measure the inter-group distances that correspond to those calculated from theory, but this task has not yet been undertaken, and it will not be possible it this time to present a comparison between theoretical and experimental distances in compounds containing radicals comparable to the comparisons in Tables 1 to12.
Some preliminary results have been made, however, on the relation between the theoretical distances and the density in complex compounds. There are a number of factors, not yet investigated in detail, that have some influence on the density of solid matter, and for that reason the conclusions thus far derived from theory are somewhat tentative, and the correlations between theory and observation are only approximate. Nevertheless, certain aspects of these tentative results are significant, and are of enough interest to justify giving them some attention.
If we divide the molecular mass, in terms of atomic weight units, by the density, we arrive at the molecular volume in terms of the units entering into the density measurement. For present purposes it will be convenient to convert this quantity to natural units of volume. The applicable conversion factor is the cube of the time region unit of distance divided by the mass unit atomic weight. In the cgs system of units it has the numerical value 14.908.
In Table 13 the average volumes per volumetric group of a representative number of inorganic compounds containing radicals (V), as calculated from the measured densities, are compared with the cubes of the inter-group distances (S03), as calculated on the theoretical basis previously described.
The specific electric rotation (c) for the compounds with the normal orientation is 4, as in the valence one binary compounds. Those with the magnetic orientation take the neutral value 5. The applicable specific magnetic rotations for the positive component and the negative radical are shown in the columns headed ab1 and ab2 respectively. Columns 2, 3, and 4 give the molecular mass (m), the density of the solid compound (d), and the number of volumetric units in the molecule (n). Here, again, as in the earlier tables, the calculated and empirical values are not exactly comparable, as the measured values of the densities have been used directly, rather than being projected back to zero temperature, a refinement that would be required for accuracy, but is not justified at this early stage of the investigation.
In this table there are five pairs of compounds, such as Ca(NO3)2 and KNO3 in which the inter-group distances are the same, and the only difference between the pairs, so far as the volumetric factors are concerned, is in the number of structural groups. Because of the uncertainties involved in the measured densities, it is difficult to reach firm conclusions on the basis of each pair considered individually, but the average volume per group, calculated from the density, in the five two-group structures is 1.267, whereas in the five three-group structures the average is 1.261. It is evident from this that the volumetric equality of the group and the independent atom which we noted in the case of the NH4 radical is a general proposition, in this class of compounds at least. This is a point that will have a special significance when we take up consideration of the liquid volume relations.
In closing the discussion in this chapter it is appropriate to reiterate that the values of the inter-atomic and inter-group distance derived from theory apply to the separations as they would exist if the equilibrium were reached at zero temperature and zero pressure. In the next two chapters we will consider how these distances are modified when the solid structure is subjected to finite pressures and temperatures.