Quantum Numbers and the Periodic Table T H O M A S H . HAZLEHURST Lehigh University, Bethlehem, Pennsylvania
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HIS paper formulates and attempts to solve a problem in the presentation of the Periodic Table to elementary chemistry students. The more modem texts (la-li) usually include a tabulation of the distribution of electrons in shells or subshells for each atom. In the tabulation, the shells are arranged, quite naturally, according to the principal quantum numbers of the electrons. Hence, when the transition elements are reached (at Sc, No. 21), a shell already "complete" in the sense of being chemically inert in argon begins to find places for more electrons and actually acquires ten of them before it is "full." During this process of completion of the third quantum shell, there are two electrons in the fourth shell and the question inevitably arises: "Why do these two not 'fall back' into the third shell, and, if the third shell is not full a t argon, why is argon an inert gas?" The same difficultiesarise for the other transition series commencing a t yttrium and lanthanum. The question is either ignored or sidestepped in most texts,' not, of course, because the authors did not know the answer, but because it probably seemed a bit beyond the comprehension of elementary students. I t is proposed to present the actual explanation in briefest outline and then to suggest a method of presentation suitable for elementary students and plausible without being incorrect. The capacities of shells with a given principal quantum number are fixed by (1) the rules governing the permitted values of the quantum numbers and (2) the Pauli Exclusion Principle. The permitted values of the quantum numbers are:
quantum numbers, because 1 may have the values 0, 1, . . ., n-1, and for each value of 1 there are 21 1 values of m, and for each set of values of 1 and m there are just two choices for s. Hence the capacity of the nth quantum shell is
+
In Table 1 are listed the capacities of quantum shells with given n. For comparison the lengths of the periods in the Periodic Table are also inserted. There is obviously a close relationship between the capacities of the shells and the periods, for the numbers 2, 8, 18, 32 appear in both sets and in the same order, but the numbers 8 and 18 appear twice in the lengtbs of the periods. Why? TABLE 1
Quantum shell number Capacity Period number Capacity
1 2 1 2
2 8 2 8
3 4 5 18 ' 32 50 3 4 5 f 8 18 18
6 72 i 32
The nth quantum shell is actually a group of "subshells," for there are 2n3 electrons there and they do not all have exactly the same energy. Actually, it is a su&ciently good approximation for our purpose to put all the electrons with the same n and 1 (but different m and s) into the same subshell, so that there will be n subshells in the nth shell corresponding to the n possible choices for 1. Frequently the tables showP;incipal quantum number n 1 to m ing electronic configurations of elements are arranged Az~muthalquantum number 1 0 to n-1 (n values) Magnetic quantum number rn -1 to +1 (21 + 1 values) by subshells. s -'/, or +I/, (2 values) Spin quantum number The order in which the shells or subshells are filled The Pauli Exclusion Principle states that no two elec- is fixed by their relative energies, the electrons always trons in the same atom may have the same values of all going into the available space of lowest energy. An four quantum numbers. It follows that, for a given approximate arrangement of subshells by energies is value of n, there are 2n2 different sets of values for the given by Pauling (2) and a modified diagram of the same sort is shown in Figure I t is clear that the 3d Notable exceptions: DEMING( l h ) , FOSTERAND ALYEA * Fosren AND ALYEA(Id) have one of the same general type. (Id).
subshell, although it belongs to the shell for which 7% = 3, lies higher in the energy scale than does the 4s snhshell and is therefore filled later.% The inert gases come just where the largest gaps appear in the energy diagram, that is, after the completion of subshells i s , 2p, 3p, 4p, 5p, and Gp, respectively. The various subshells grouped together. between these positions have, of course, exactly the capacities of the periods of the Periodic Table, and each group might well he termed a "valence shell" in distinction to the grouping of suhshells into "quantum shells."
the fact that there is no need to stress or, perhaps, even to mention the designation of subshells by quantum numbers, a bald statement that the numbering of the suhshells "follows a convention established to facilitate FIGURE 1.-ENERGYLEYELSOP SHELLS AND SUBSHELLS OP ELECTRONS I N ATOMS. EACH CIRCLECANHOLDTWO ELECTRONS Subrhrll
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