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# 107

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7. Some More Mathematics!

There is logic in everything. Even the most unaccountable phenomenon has its logic It may not be perceptible at first, and then inconsistencies appear. Inconsistencies are very unpleasant things for any theory or hypothesis. They either disclose the wrongness of the theory or make one think hard. And it often so happens that this hard thinking helps to penetrate deeper into the uncomprehensible.

Here is an example of such an inconsistency. Equality reigns only in the first two periods of the Periodic Table. There are exactly as many elements in each of these periods as the corresponding outer shell can hold electrons. Thus, in the atoms of the elements of the first period, hydrogen and helium, the K-shell is filled. It cannot contain more than two electrons, and therefore there are only two elements in the first period. An eight-electron (octet) shell fills up completely through the atoms of the elements of the second period, from lithium to neon, and that is why the second period contains eight elements. After this things get more involved.

Count up the number of elements in the subsequent periods. There are 8 in the third, 18 in the fourth, 18 in the fifth, 32 in the sixth, and there should also be 32 in the seventh (which is incomplete as yet). But what about the corresponding shells? Here the figures are different: 18, 32, 50, and 72.

Now, were we not too hasty in concluding that in undertaking to explain the structure of the Periodic Table the physicists found no defects in its construction? It would be a good thing if a definite electron shell were filled regularly in the inhabitants of each floor of the Big House, and if each floor started with an alkali metal and ended in an inert gas The capacity of each period would then be equal to the capacity of the electron shell.

Alas, we are obliged to speak of this in the subjunctive mood: if this, if that. Actually the balance does not tally. The third period of Mendeleyev�s Table accommodates less inhabitants than there are electrons in the third shell, the M-shell. And so on.

A sad incongruity. But this incongruity holds the clue to the essence of the Periodic System. Though the third period ends in argon, the third M-shell of the latter�s atom is not completed. The completed shell should contain 18 electrons, but so far there are only 8 in it. Argon is followed by potassium which belongs to the fourth period, being the first inhabitant of the fourth floor. But instead of placing its latest electron in its third shell, the potassium atom prefers having it in its fourth, N-shell. This is no accident, but again a strict regularity established by the physicists. It is simply that no atom can have more than 8 electrons in its outside shell. The combination of 8 outer electrons is a very stable arrangement.

In calcium, potassium�s next-door neighbour, the newest electron also finds it �more advantageous� to occupy the outermost shell, because then the energy supply of the calcium atom is smaller than with any other electron distribution. But in scandium, which follows calcium, the tendency to continue filling the outer shell of the atom vanishes. Its new electron �dives� into the incomplete second-last M-shell. And since this shell has ten vacancies (we already know that the maximum capacity of the M -shell is 18 electrons), the atoms of the next ten elements, from scandium to zinc, gradually fill up their M-shells. Finally, in zinc, all the electrons of the M-shell are in place. After this the N-shell again begins to accept electrons. As soon as it contains a total of 8 electrons we get the inert gas krypton. In rubidium the old story repeats itself: the fifth shell appears before the fourth is complete.

Such a step-by-step filling of electron shells is a �standard of behaviour� for the inhabitants of the Periodic Table from the fourth period on. This is the basic strict regulation of the Big House of Chemical Elements. For this reason main and secondary subgroups are distinguished in it. The elements in which the outermost electron shells are being filled comprise the main subgroups. Those in which inner shells are being completed make up the secondary subgroups. But the fourth, N-shell, does not fill up in one step. Its completion is drawn out into three whole storeys of the Big House. The first electron of this shell appears in potassium which occupies flat 19. But the 32nd electron appears in it only in lutetium, a representative of the sixth period. Its atomic number is 71.

So you see, the incongruity has a silver lining. In trying to account for it we and the physicists got a better insight into the structure of the Periodic System.

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by Ian Ellis