Page 66 - Theory and Problems of BEGINNING CHEMISTRY

CHAP. 4] ELECTRONIC CONFIGURATION OF THE ATOM 55

EXAMPLE 4.6. Arrange the electrons in the following list in order of increasing energy, lowest first:

n l m l m s n l m l m s

(a) 4 1 1 − 1 (c) 3 2 −1 1
2 2
(b) 2 1 −1 1 (d) 4 0 0 − 1
2 2
Ans. Electron (b) has the lowest value of n+l (2+1 = 3), and so it is lowest in energy of the four electrons. Electron (d)
has the next-lowest sum of n+l (4+0 = 4) and is next in energy (despite the fact that it does not have the next-lowest
n value). Electrons (a) and (c) both have the same sum of n + l (4 + 1 = 3 + 2 = 5). Therefore, in this case,
electron (c), the one with the lower n value, is lower in energy. Electron (a) is highest in energy.

The Pauli exclusion principle states that no two electrons in the same atom can have the same set of four
quantum numbers. Along with the order of increasing energy, we can use this principle to deduce the order of
filling of electron shells in atoms.

EXAMPLE 4.7. Use the Pauli principle and the n + l rule to predict the sets of quantum numbers for the 13 electrons in
the ground state of an aluminum atom.
Ans. We want all the electrons to have the lowest energy possible. The lowest value of n +l will have the lowest n possible
and the lowest l possible. The lowest n permitted is 1 (Table 4-1). With that value of n, the only value of l permitted
1
1
is 0. With l = 0, the value of m s must be 0 (−0 ··· + 0). The value of m s can be either − or + . Thus, the first
2 2
electron can have either
n = 1, l = 0, m l = 0, m s =− 1 or n = 1, l = 0, m l = 0, m s =+ 1
2 2
1
1
The second electron also can have n = 1,l = 0, and m l = 0. Its value of m s can be either + or − , but not the
2 2
same as that for the first electron. If it were, this second electron would have the same set of four quantum numbers
that the first electron has, which is not permitted by the Pauli principle. If we were to try to give the third electron the
1
same values for the first three quantum numbers, we would be stuck when we came to assign the m s value. Both + 2
1
and − have already been used, and we would have a duplicate set of quantum numbers for two electrons, which is
2
not permitted. We cannot use any other values for l or m l with the value of n = 1, and so the third electron must have
the next-higher n value, n = 2. The l values could be 0 or 1, and since 0 will give a lower n + l sum, we choose that
1
1
value for the third electron. Again the value of m l must be 0 since l = 0, and m s can have a value − (or + ). For
2
2
1
1
1
the fourth electron, n = 2,l = 0, m l = 0, and m s =+ (or − if the third were + ). The fifth electron can have
2 2 2
n = 2 but not l = 0, since all combinations of n = 2 and l = 0 have been used. Therefore, n = 2,l = 1, m l =−1,
1
and m s =− are assigned. The rest of the electrons in the aluminum atom are assigned quantum numbers somewhat
2
arbitrarily as shown in Table 4-3.
Table 4-3 Quantum Numbers of the Electrons of Aluminum
Electron 1 2 3 4 5 6 7 8 9 10 11 12 13
n 1 1 2 2 2 2 2 2 2 2 3 3 3
l 0 0 0 0 1 1 1 1 1 1 0 0 1
m l 0 0 0 0 −1 0 1 −1 0 1 0 0 −1
1 1 1 1 1 1 1 1 1 1 1 1 1
m s − + − + − − − + + + − + −
2 2 2 2 2 2 2 2 2 2 2 2 2
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