QUANTUM NUMBERS AND ENERGIES OF ELECTRONS

4.4. QUANTUM NUMBERS AND ENERGIES OF ELECTRONS

The energy of the electrons in an atom is of paramount importance. The n and l quantum numbers determine the energy of each electron (apart from the effects of external electric and magnetic fields, which are most often not of interest in general chemistry courses). The energies of the electrons increase as the sum n + l increases; the lower the value of n + l for an electron in an atom, the lower is its energy. For two electrons with equal values of n + l, the one with the lower n value has lower energy. Thus, we can fill an atom with electrons starting with its lowest-energy electrons by starting with the electrons with the lowest sum n + l.

CHAP. 4]

ELECTRONIC CONFIGURATION OF THE ATOM

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

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

is 0. With l = 0, the value of m s must be 0 (− 0 · · · + 0). The value of m s can be either − 1 2 or + 1 2 . Thus, the first electron can have either

m l = 0, m s =+ 1 2 The second electron also can have n = 1, l = 0, and m l =

0. Its value of m s can be either + 1 2 or − 1 2 , but not the 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

same values for the first three quantum numbers, we would be stuck when we came to assign the m s value. Both + 1 2 and − 1 2 have already been used, and we would have a duplicate set of quantum numbers for two electrons, which is 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

value for the third electron. Again the value of m l must be 0 since l = 0, and m s can have a value − 1 2 (or + 1 2 ). For the fourth electron, n = 2, l = 0, m l =

0, and m s =+ 1 2 (or − 1 2 if the third were + 1 2 ). The fifth electron can have 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, and m s =− 1 2 are assigned. The rest of the electrons in the aluminum atom are assigned quantum numbers somewhat 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