THE ELEMENTS OF THE HIGHER GROUPS Displacements Element Atomic

Number Displacements Element Atomic Number

3-2-1 Potassium 19 3-3-1 Rubidium 37

3-2-2 Calcium 20 3-3-2 Strontium 38

3-2-3 Scandium 21 3-3-3 Yttrium 39

3-2-4 Titanium 22 3-3-4 Zirconium 40

3-2-5 Vanadium 23 3-3-5 Niobium 41

3-2-6 Chromium 24 3-3-6 Molybdenum 42

3-2-7 Manganese 25 3-3-7 Technetium 43

3-2-8 Iron 26 3-3-8 Ruthenium 44

3-2-9

3-3-(9) Cobalt 27 3-3-9

3-3-(9) Rhodium 45

3-3-(8) Nickel 28 4-3-(8) Palladium 46

3-3-(7) Copper 29 4-3-(7) Silver 47

3-3-(6) Zinc 30 4-3-(6) Cadmium 48

3-3-(5) Gallium 31 4-3-(5) Indium 49

3-3-(4) Germanium 32 4-3-(4) Tin 50

3-3-(3) Arsenic 33 4-3-(3) Antimony 51

3-3-(2) Selenium 34 4-3-(2) Tellurium 52

3-3-(1) Bromine 35 4-3-(1) Iodine 53

3-3-0 Krypton 36 4-3-0 Xenon 54

The final two groups of elements, Table 3, contain 2x4², or 32, members each. The heaviest elements of the last group have not yet been observed, as they are highly radioactive, and consequently unstable in the terrestrial environment. In fact, uranium, element number 92, is the heaviest that exists naturally on earth in any substantial quantities. As we will see later, however, there are other conditions under which the elements are stable all the way up to number 117.

TABLE 3

THE ELEMENTS OF THE HIGHER GROUPS Displacements Element Atomic

Number Displacements Element Atomic Number

4-3-1 Cesium 55 4-4-1 Francium 87

4-3-2 Barium 56 4-4-2 Radium 88

4-3-3 Lanthanum 57 4-4-3 Actinium 89

4-3-4 Cerium 58 4-4-4 Thorium 90

4-3-5 Praseodymium 59 4-4-5 Protactinium 91

4-3-6 Neodymium 60 4-4-6 Uranium 92

4-3-7 Promethium 61 4-4-7 Neptunium 93

4-3-8 Samarium 62 4-4-8 Plutonium 94

4-3-9 Europium 63 4-4-9 Americium 95

4-3-10 Gadolinium 64 4-4-10 Curium 96

4-3-11 Terbium 65 4-4-11 Berkelium 97

4-3-12 Dysprosium 66 4-4-12 Californium 98 4-3- 13 Holmium 67 4-4-13 Einsteinium 99

4-3-14 Erbium 68 4-4-14 Fermium 100

4-3-15 Thulium 69 4-4-15 Mendelevium 101

4-3-16

4-3-(16) Ytterbium 70 4-4-16

5-4-(16) Nobelium 102 4-4-(15) Lutetium 71 5-4-(15) Lawrencium 103 4 4-(14) Hafnium 72 5-4-(14) Rutherfordium 104 4 4-(13) Tantalum 73 5 4-(13) Hafnium 105

4 4-(12) Tungsten 74 5-4-(12) 106

4 4-(11) Rhenium 75 5 4-(11) 107

4 4-(10) Osmium 76 5-4-(10) 108

4-4-(9) Iridium 77 5-4-(9) 109

4-4-(8) Platinum 78 5-4-(8) 110

4-4-(7) Gold 79 5-4-(7) 111

4-4-(6) Mercury 80 5-4-(6) 112

4-4-(5) Thallium 81 5-4-(5) 113

4-4-(4) Lead 82 5-4-(4) 114

4-4-(3) Bismuth 83 5-4-(3) 115

4-4-(2) Polonium 84 5-4-(2) 116

4-4-(1) Astatine 85 5-4-(1) 117

4-4-0 Radon 86

For convenience in the subsequent discussion these groups of elements will be identified by the magnetic n value, with the first and second groups in each pair being designated A and B respectively. Thus the sodium group, which is the second of the 8-element groups (n=2) will be called Group 2B.

At this point it will be appropriate to refer back to this statement that was made in Chapter 9:

The (mathematical) development will begin with nothing more than the series of cardinal numbers and the geometry of three dimensions. By subjecting these to simple mathematical processes, the applicability of which to the physical universe of motion is specified in the fundamental postulates, the combinations of rotational motions that can exist in the theoretical universe will be ascertained, and it will be shown that these rotational

combinations that theoretically can exist can be individually identified with the atoms of the chemical elements and the sub-atomic particles that are observed to exist in the physical universe. A unique group of numbers representing the different rotational components will be derived for each of these combinations.

A review of the manner in which the figures presented in Tables 1 to 3 were derived will show that this commitment, so far as it applies to the elements, has been carried out in full.

This is a very significant accomplishment. Both the existence of a series of theoretical elements identical with the observed series of chemical elements, and the numerical values which theoretically characterize each individual element have been derived from the general

properties of mathematics and geometry, without making any supplementary assumptions or introducing any numerical values specifically applicable to matter. The possibility that the agreement between the series of elements thus derived and the known chemical elements could be accidental is negligible, and this derivation is, in itself, a conclusive proof that the atoms of matter are combinations of motions, as asserted by the Reciprocal System of theory. But this is only the beginning of a vast process of mathematical development. The numerical values at which we have arrived, the atomic numbers and the three displacement values for each element, now provide us with the basis from which we can derive the quantitative relations in the areas that we will examine.

The behavior characteristics, or properties, of the elements are functions of their respective displacements. Some are related to the total net effective displacement (equal to the atomic number in the combinations thus far discussed), some are related to the electric

displacement, others to the magnetic displacement, while still others follow a more complex pattern. For instance, valence, or chemical combining power, is determined by either the electric displacement or one of the magnetic displacements, while the inter-atomic distance is affected by both the electric and magnetic displacements, but in different ways. The manner in which the magnitudes of these properties for specific elements and compounds can be calculated from the displacement values has been determined for many properties and for many classes of substances. These subjects will be considered individually in the

chapters that follow.

One of the most significant advances toward an understanding of the relations between the structures of the different chemical elements and their properties was the development of the periodic table by Mendeleeff in l869. In this diagram the elements are arranged horizontally in periods and vertically in groups, the order within the period being that of the atomic number (approximately defined in the original work by the atomic weights). When the elements are correctly assigned in the periods, those in the vertical groups are quite similar in their properties. On comparing the periodic table with the rotational characteristics of the elements, as tabulated in this chapter, it is evident that the horizontal periods reflect the magnetic rotational displacement, while the vertical groups represent the electric rotational displacement. In revising the table to take advantage of the additional information derived from the Reciprocal System of theory we may therefore replace the usual group and period numbering by the more meaningful displacement values.

When this is done it is apparent that a further revision of the tabular arrangement is required in order to put all of the elements into their proper positions. Mendeleeff's original table included nine vertical groups, beginning with the inert gases, Group O, and ending with a group in which the three elements iron, cobalt, and nickel, and the corresponding elements in the higher periods, were all assigned to a single vertical position. In the more modern

versions of the table the number of vertical groups has been expanded to avoid splitting each of the longer periods into two sub-periods, as was done by Mendeleeff. One of the most popular of these revised versions utilizes 18 vertical groups, and puts 15 elements of each of the last two periods into one of these l8 positions in order to accommodate the full number of elements.

In the light of the new information now available, it can be seen that Mendeleeff based his

arrangement on the relations existing in the 8-element rotational groups, 2A and 2B in the notation used in this work, and forced the elements of the larger groups into conformity with this 8-element pattern. The modern revisers have made a partial correction by setting up their tables on the basis of the l8-element rotational groups, 3A and 3B, leaving blank spaces where the 8-element groups have no counterparts of the l8-element values. But these tables still retain a part of the original distortion, as they force the members of the 32-element groups into the l8-element pattern. To construct a complete and accurate table, it is only necessary to carry the revision procedure one step farther, and set up the table on the basis of the largest of the magnetic groups, the 32-element groups 4A and 4B.

For some purposes a simple extension of the current versions of the table to the full 32 position width necessary to accommodate Grcups 4A and 4B is probably all that is needed.

On the other hand, the useful chemical information displayed by the table is confined mainly to the elements with electric displacements below l0, and separating the central elements of the two upper groups from the main portion of the table, as in the conventional

arrangements, has considerable merit. The particular elements that are thus separated on the basis of the electric displacement are not the same ones that are treated separately in the conventional tables, but the general effect is much the same.

When the table is thus divided into two sections, it also appears that there are some

advantages to be gained by a vertical, rather than a horizontal, arrangement, and the revised table, Fig. 1, has been set up on this basis. The new concept of ―divisions,‖ which is

emphasized in this table, will be explained in Chapter 18. Inasmuch as carbon and silicon play both positive and negative roles rather freely, they have each been assigned to two positions in the table, but hydrogen, which is usually shown in two positions in the conventional tables, is necessarily negative on the basis of the principles that have been developed in this work and is only shown in one position. The aspects of its chemical behavior that have led to its classification with the electropositive elements will also be explained in Chapter 18.

Figure 1

The Revised Periodic Table of the Elements Magnetic Displacement Div. Electric

6

In the original construction of the periodic table the known properties of certain elements were combined with the atomic number sequence to establish the existence of the relations between the elements of the various periods and groups, and thereby to predict previously undetermined properties, and even the existence of some previously unknown elements. The table thus added significantly to the chemical knowledge of the time. In this work, however,

the revised table is not being presented as an addition to the information contained in the preceding pages, but merely as a convenient graphic method of expressing some portions of that information. Everything that can be learned from the table has already been set forth in more detailed form, verbally and mathematically, in this and the earlier chapters. Some of the implications of this information, such as its application to the property of valence, will have further consideration later.

CHAPTER 11