SHELLS, SUBSHELLS, AND ORBITALS

Electrons having the same value ofn in an atom are said to be in the sameshell. Electrons having the same value ofnand the same value oflin an atom are said to be in the samesubshell. (Electrons having the same values ofn,l andml in an atom are said to be in the sameorbital.) Thus, the first two electrons of aluminum

(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 haven =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 1ssubshell, the next two electrons are in the 2ssubshell, and the next six electrons are in the 2p subshell. The following two electrons occupy the 3ssubshell, and the last electron is in the 3psubshell.

Since the possible numerical values ofl depend on the value ofn, the number of subshells within a given shell is determined by the value ofn. The number of subshells within a given shell is merely the value ofn, 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 leveln Type of Subshell Number of Subshells

1 s 1 2 s,p 2 3 s,p,d 3 4 s,p,d, f 4 5 s,p,d, f,g 5 6 s,p,d, f,g,h 6 7 s,p,d, f,g,h,i 7

EXAMPLE 4.8. _{What are the values of}nandlin each of the following subshells? (a) 2p, (b) 3s, (c) 5d, and (d) 4f.

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 anyssubshell.}

Ans. For any given value ofn, there can be a value ofl=_{0, corresponding to an}ssubshell. Forl=_{0 there can be only}

one possibleml value:ml =0. Hence,n,l, andml are all specified for a givenssubshell. Electrons can then have

spin values ofms = +12 orms = −12. 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 ofms. The two electrons are said to bepaired.

Depending on the permitted values of the magnetic quantum numberml, each subshell is further broken

down into units calledorbitals. The number of orbitals per subshell depends on the type of subshell but not on the value ofn. 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

Allowed Values Number of Maximum Number Type of Subshell ofml Orbitals of Electrons

s 0 1 2

p −1,0,1 3 6

d −2,−1,0,1,2 5 10

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 withn =3 andl =2 will enter before then=4 andl =0 electrons. Sincen+lfor the former is 5 and that for the latter is 4, we must add the two electrons withn=4 andl=0 before the last 10 electrons with n=3 andl=2. In this discussion, the values ofmlandmstell us how many electrons can have the same set of nandlvalues, but do not matter as to which come first.

14 15 16 17 18 19 20 21 22 n 3 3 3 3 3 4 4 3 3 l 1 1 1 1 1 0 0 2 2 ml 0 +1 −1 0 +1 0 0 −2 −1 ms −1₂ −1₂ +1₂ +1₂ +1₂ −1₂ +1₂ −1₂ −1₂ n+l 4 4 4 4 4 4 4 5 5

Thus, an important development has occurred because of then+lrule. 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 1ssubshell, two electrons in its 2ssubshell, six electrons in its 2psubshell, two electrons in its 3ssubshell, six electrons in its 3psubshell, two electrons in its 4ssubshell, 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 3pelectrons before the 4selectrons. Since each of these groups has an(n+l)sum of 4 [the(n+l)values are the same] we add electrons having the lowernvalue 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

Shell

numbers Subshell designations

1s2 _2s2 _2p6 _3s2 _3p6 _4s2 _3d2

Ti

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 1ssubshell, two in the 2ssubshell, six in the 2psubshell, two in the 3ssubshell, six in the 3psubshell, two in the 4ssubshell, and two in the 3d subshell. (The 3d subshell can hold a maximumof 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.}