178
REINOHAKALA
Vol. 56
THE PERIODIC LAW IN MATHEMATICAL FORM BY REINOHAKAW Depurfment of Chemistry, Syracuse University, Syracuse, N . Y . Received November 5 , 1060
Desiderata of the most useful form of eriodic table are given. Attention is focused on Janet’s table’ which best meets these requirements. A compact array tEat describes the order in which the subshells of atoms are filled is iven. It most clearly brings out the systematics involved and Janet’s table can be derived directly from it. The “general ofSservation that alternate members of a valency group in the periodic table show the greatest chemical resemblance”* is provided with a firm mathematical foundation by an analysis of this array. Equations relating the atomic numbers of the alkaline-earth elements and helium (because of its similar electronic configuration) to the principal and orbital quantum numbers of their subshells of highest ener y are derived on the basis of an extrapolation of the order in which the subshells of atoms are filled. An equation that fescribes the author’s version of Janet’s table is derived. It relates the atomic numbers of the elements to their positions in the table. Various forms of it are given. The periodic law is given mathematical ex ression. This relates the atomic numbers of the elements to their positions in any periodic table that is based upon the eyectron configurations of the elements. Fundamentally, then, it relates atomic numbers to electron configurations. (The e uation neglects changes in configuration due to the exceptional stability of completed, half-completed, and empty shells. %owever, these changes occur in only 10% of the known elements and do not always occur when expected, e.g., W has the configuration 6s25d4instead of 6s5d5.) An introduction is given to the determination of general mathematical expressions for both periodic and nonperiodic properties of the elements and their compounds. A method is outlined by which expressions may be found that relate several properties to one another independent of electron configurations and positions in periodic tables. A simple way of evaluating the arbitrary constants of functions +(y) = uo u J ( z ) . . . an[j(?)I” is outlined. A known formula of finite differences is generalized to give other formulas which find application in evaluatmg these arbitrary constants.
+
The periodic law is without doubt the most important generalization in chemistry. (“The periodic series is a brilliant and adequate means of producing an easily surveyed system of facts which by gradually becoming complete will take the place of an assemblage of the known facts.”-Ernst Mach) It is therefore not surprising that so many modifications of the periodic table have been devised, and that so many attempts have been made to obtain a mathematical expression for the periodic law. Only a few of these tables, however, have found widespread use; and, unfortunately, the many attempts to express the periodic law mathematically were none too successful-mainly because they were based on the old idea that the properties of the elements are periodic functions of their atomic weights. Since the periodicity in chemical and physical properties arises from a periodicity in the electronic structures of the elements, the mathematical representation of the periodic law must be based on the mathematical analysis of a periodic table founded on the electronic structures of the elements. Many such tables have been published, in various shapes, both in two dimensions and three. Some preserve the atomic number continuity while others do not. The simplest and most logical of these would be expected to have the simplest shape, t o occur in two dimensions only, and t o preserve the atomic number continuity. Charles Janet’ was the first to publish a table that mests these requirements.3 Slight modifications of i t have been published recently by*Simmons* and by Gibson.6 (1) C. Janet, “La Classification Hdlicoldale des Ihments Chimiques,” Fascicle No. 4, Beauvais, Imprimerie Ddpartmentale de I’Oise, Nov. 1928. (2) J. D. Main Smith, “Chemistry and Atomio Structure,” Ernest Benn, Ltd., London, 1924. p. 126. (3) Janet’s work was brought to the author’s attention last year b y Clifton P. Idyll, of Adelphi College, Garden City, Long Island, New York, to whom the author is very grateful. (4) L. M. Simmons, J . Chem. Education, 24, 588 (1947). (5) D. T. Gibson, Chem. and Ind., 12 (1948).
+
+
The author’s version, brought up to date, appears in Table I. This is now to be analyzed.6 Two Series of Elements.-When the atomic number sequence of the alkaline-earth family (including helium because it has an electronic structure characteristic of the alkaline-earth elements) is subjected to differencing, the familiar sequence 2, 8, 8, 18, 18, 32 results as the first order difference. Besides indicating the number of elements contained in each period of the table, this sequence, because of its twofold nature, indicates that the atomic number sequence consists of the terms of two separate sequences added alternately together. These separate sequences are 2, 12, 38,88 and 4, 20, 56. Differencing these separate sequences should give a clue as to the nature of the function that relates the terms. However, when a table of differences is constructed, it is seen that there are not enough terms present to arrive a t a column of constants. Hence additional terms are needed to continue the analysis. These can be obtained by extrapolating the sequence in which the subshells of atoms are filled 18
2s
2
4f 5f 6f 7f
3s
3d
2p 3p 4p
4d
513
4s 5s 6a
5d 6d 7d 8d
7p 8p Bp
8s 9s 10s
7s
(This array is read from top to bottom and from left to right, i.e., Is, 2s, 2p, 3s, 3p, 4s, 3d, . . . .) The resulting atomic number sequences are 2, 12,38, 88, 170, and 4, 20, 56, 120, 220 (the g shell can hold no more than 18 electrons), and the difference tables are (6) Part of the author’s seminar on “The Periodic Law,” Department of Chemistry, Columbia University, Oct. 31, 1946.
PERIODIC LAWIN MATHEMATICAL FORM
Feb., 1952 Z 4
A2
AtZ
A32
A2
2
16
56
36 64
120 220
AS2
2
10
_.
20
ASZ
~~
20
8
28
12
and
8
36
100
38
88 170
16
26
24
50
32
8 8
+
+ b(n + I ) + c(n + I ) * + d ( n + Z)a
where Z is the atomic number of the element whose subshell containing the last-added electron? has the principal and orbital quantum numbers n and 1, and a, b, c and d are constants to be evaluated. The last of these, d, can be found directly by inspection.
= a.
+ alx + . . .
+ anxn, for equi-spaced values of the argument an = (l/n!)[A"y/(Az)"]s d = (1/3!)(8/2a) = 1/6
To determine the three remaining unknowns, a,
b and c, three simultaneous equations are needed.
The values necessary in setting up these equations for the sequence 2, 12, 38,88 are (7) I.e., the subshell of highest energy. Henceforth, for aimplicity's sake, it will be called the "last-filled" subshell (although it need not be cloeed). (8) This expression is readily verified by taking successive derivatives of 2": (d/dz)z" = n~"-';(d/dX)5" = n(n 1)~"-*; (d/dz)%" = n(n l)(n 2)x"+; * * ; (d/dz)"z" = n(n l)(n-2) * 3.2.1 = n! And (d/dz)%p
