bei48482_FM

240 Chapter Seven

Shell and Subshell Capacities

The exclusion principle limits the number of electrons that can occupy a given subshell.

A subshell is characterized by a certain principal quantum number nand orbital quan-

tum number l, where lcan have the values 0, 1, 2,... , (n1). There are 2l 1

different values of the magnetic quantum number mlfor any l, since ml0, 1,

2,... , l. Finally, the spin magnetic quantum number mshas the two possible

values of ^12 and ^12 for any ml. The result is that each subshell can contain a maximum

of 2(2l1) electrons (Table 7.3).

The maximum number of electrons a shell can hold is the sum of the electrons in

its filled subshells. This number is

Nmax

ln 1

l 0

2(2l1)2[1 3  5 .. .20(n1)1]

2[1 3  5 .. . 2 n1]

The quantity in brackets has nterms whose average value is ^12 [1(2n1)]. The num-

ber of electrons in a filled shell is therefore

Nmax(n)(2)(^12 )[1(2n1)] 2 n^2 (7.14)

Thus a closed Kshell holds 2 electrons, a closed Lshell holds 8 electrons, a closed M

shell holds 18 electrons, and so on.

7.6 EXPLAINING THE PERIODIC TABLE

How an atom’s electron structure determines its chemical behavior

The notion of electron shells and subshells fits perfectly into the pattern of the periodic

table, which mirrors the atomic structures of the elements. Let us see how this pattern

arises.

An atomic shell or subshell that contains its full quota of electrons is said to be

closed.A closed ssubshell (l0) holds two electrons, a closed psubshell (l1)

six electrons, a closed dsubshell (l2) ten electrons, and so on.

The total orbital and spin angular momenta of the electrons in a closed subshell

are zero, and their effective charge distributions are perfectly symmetrical (see Ex-

ercise 23 of Chap. 6). The electrons in a closed shell are all very tightly bound,

since the positive nuclear charge is large relative to the negative charge of the inner

shielding electrons (Fig. 7.9). Because an atom with only closed shells has no di-

pole moment, it does not attract other electrons, and its electrons cannot be easily

Table 7.3Subshell Capacities in the M(n3) Shell of an Atom

ml 0 ml 1 ml 1 ml 2 ml 2

l0: ↓↑↑ms^12 

l1: ↓↑↓↑↓↑↓ms^12 

l2: ↓↑↓↑↓↑↓↑↓↑

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