SHELLS, SUBSHELLS, AND ORBITALS

4.5. SHELLS, SUBSHELLS, AND ORBITALS

Electrons having the same value of n in an atom are said to be in the same shell. Electrons having the same value of n and the same value of l in an atom are said to be in the same subshell. (Electrons having the same values of n, l and m l in an atom are said to be in the same orbital.) Thus, the first two electrons of aluminum

56 ELECTRONIC CONFIGURATION OF THE ATOM

[ CHAP. 4

(Table 4-3) are in the first shell and in the same subshell. The third and fourth electrons are in the same shell and subshell with each other. They are also in the same shell with the next six electrons (all have n = 2) but a different subshell (l = 0 rather than 1). With the letter designations of Sec. 4.3, the first two electrons of aluminum are in the 1s subshell, the next two electrons are in the 2s subshell, and the next six electrons are in the 2 p subshell. The following two electrons occupy the 3s subshell, and the last electron is in the 3 p subshell.

Since the possible numerical values of l depend on the value of n, the number of subshells within a given shell is determined by the value of n. The number of subshells within a given shell is merely the value of n, the shell number. Thus, the first shell has one subshell, the second shell has two subshells, and so forth. These facts are summarized in Table 4-4. Even the atoms with the most electrons do not have enough electrons to completely fill the highest shells shown. The subshells that hold electrons in the ground states of the biggest atoms are in boldface.

Table 4-4 Arrangement of Subshells in Electron Shells Energy level n

Type of Subshell

Number of Subshells

EXAMPLE 4.8. What are the values of n and l in each of the following subshells? (a) 2 p, (b) 3s, (c) 5d, and (d) 4 f . Ans.

(a) n = 2, l = 1 (b) n = 3, l = 0 (c) n = 5, l = 2 (d ) n = 4, l = 3 EXAMPLE 4.9. Show that there can be only two electrons in any s subshell.

Ans. For any given value of n, there can be a value of l = 0, corresponding to an s subshell. For l = 0 there can be only one possible m l value: m l =

0. Hence, n, l, and m l are all specified for a given s subshell. Electrons can then have spin values of m s =+ 1 2 or m s =− 1 2 . Thus, every possible set of four quantum numbers is used, and there are no other possibilities in that subshell. Each of the two electrons has the first three quantum numbers in common and has a different value of m s . The two electrons are said to be paired.

Depending on the permitted values of the magnetic quantum number m l , each subshell is further broken down into units called orbitals. The number of orbitals per subshell depends on the type of subshell but not on the value of n. Each consists of a maximum of two electrons; hence, the maximum number of electrons that can occupy a given subshell is determined by the number of orbitals available. These relationships are presented in Table 4-5. The maximum number of electrons in any given energy level is thus determined by the subshells it contains. The first shell can contain 2 electrons; the second, 8 electrons; the third, 18 electrons; the fourth, 32 electrons; and so on.

Table 4-5 Occupancy of Subshells

Maximum Number Type of Subshell

Allowed Values

Number of

of m l

Orbitals

of Electrons

CHAP. 4]

ELECTRONIC CONFIGURATION OF THE ATOM

Suppose we want to write the electronic configuration of titanium (atomic number 22). We can rewrite the first 13 electrons that we wrote above for aluminum and then just keep going. As we added electrons, we filled the first shell of electrons first, then the second shell. When we are filling the third shell, we have to ask if the electrons with n = 3 and l = 2 will enter before the n = 4 and l = 0 electrons. Since n + l for the former is 5 and that for the latter is 4, we must add the two electrons with n = 4 and l = 0 before the last 10 electrons with n= 3 and l = 2. In this discussion, the values of m l and m s tell us how many electrons can have the same set of n and l values, but do not matter as to which come first.

n+l

Thus, an important development has occurred because of the n + l rule. The fourth shell has started filling before the third shell has been completed. This is the origin of the transition series elements. Thus, titanium, atomic number 22, has two electrons in its 1s subshell, two electrons in its 2s subshell, six electrons in its 2 p subshell, two electrons in its 3s subshell, six electrons in its 3 p subshell, two electrons in its 4s subshell, and its last two electrons in the 3d subshell.

We note in the electronic configuration for electrons 13 through 20 for titanium that when the (n + l) sum was 4 we added the 3 p electrons before the 4s electrons. Since each of these groups has an (n + l) sum of 4 [the (n + l) values are the same] we add electrons having the lower n value first.

We conventionally use a more condensed notation for electronic configurations, with the subshell notation and a superscript to denote the number of electrons in that subshell. To write the detailed electronic configuration of any atom, showing how many electrons occupy each of the various subshells, one needs to know only the order of increasing energy of the subshells, given above, and the maximum number of electrons that will fit into each, given in Table 4-5. A convenient way to designate such a configuration is to write the shell and subshell designation, and add a superscript to denote the number of electrons occupying that subshell. For example, the electronic configuration of the titanium atom is written as follows:

Number of electrons occupying each subshell

Ti

Shell numbers

Subshell designations

The shell number is represented by 1, 2, 3, . . . , and the letters designate the subshells. The superscript numbers tell how many electrons occupy each subshell. Thus, in this example, there are two electrons in the 1s subshell, two in the 2s subshell, six in the 2 p subshell, two in the 3s subshell, six in the 3 p subshell, two in the 4s subshell, and two in the 3d subshell. (The 3d subshell can hold a maximum of 10 electrons, but in this atom this sub- shell is not filled.) The total number of electrons in the atom can easily be determined by adding the numbers in all the subshells, that is, by adding all the superscripts. For titanium, this sum is 22, equal to its atomic number.

EXAMPLE 4.10. Write the electronic configuration of aluminum. Ans.

1s 2 2s 2 2p 6 3s 2 3p 1 .

58 ELECTRONIC CONFIGURATION OF THE ATOM

[ CHAP. 4