Page 65 - Theory and Problems of BEGINNING CHEMISTRY

54 ELECTRONIC CONFIGURATION OF THE ATOM [CHAP. 4

from −l through 0 to +l in integral steps. Thus, for an electron with an l value of 3, the possible m l values are
−3, −2, −1, 0, 1, 2, and 3.
The spin quantum number, denoted m s , is related to the “spin” of the electron on its “axis.” It ordinarily
1
1
does not affect the energy of the electron. Its possible values are − and + . The value of m s does not depend
2 2
on the value of any other quantum number.
The permitted values for the other quantum numbers when n = 2 are shown in Table 4-2. The following
examples will illustrate the limitations on the values of the quantum numbers (Table 4-1).
Table 4-2 Permitted Values of Quantum Numbers When n = 2

n 2 2 2 2 2 2 2 2
l 0 0 1 1 1 1 1 1
m l 0 0 −1 0 +1 −1 0 +1
1 1 1 1 1 1 1 1
m s − + − − − + + +
2 2 2 2 2 2 2 2
Note: Each vertical set of four quantum numbers represents values for one electron.

EXAMPLE 4.2. What are the first seven permitted values for n?
Ans. 1, 2, 3, 4, 5, 6, and 7.

EXAMPLE 4.3. What values of l are permitted for an electron with principal quantum number n = 3?
Ans. 0, 1, and 2. (l can have integer values from 0 up to n − 1.)

EXAMPLE 4.4. What values are permitted for m l for an electron in which the l value is 2?
Ans. −2, −1, 0, 1, and 2. (m l can have integer values from −l through 0 up to +l.)

EXAMPLE 4.5. What values are permitted for m s for an electron in which n = 2,l = 1, and m l = 0?
1
1
1
1
Ans. − and + . (The values that the other quantum numbers have do not matter; the m s value must be either − or + .)
2 2 2 2
The values of the angular momentum quantum number are often given letter designations, so that when they
are stated along with principal quantum numbers, less confusion results. The letter designations of importance
in the ground states of atoms are the following:
l Value Letter Designation
0 s
1 p
2 d
3 f

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.