QUANTUM NUMBERS

4.3. QUANTUM NUMBERS

The modern theory of the electronic structure of the atom stems from a complex mathematical equation (called the Schr¨odinger equation), which is beyond the mathematical requirements for the general chemistry course. We therefore take the results of the solution of this equation as postulates. Solution of the equation yields three quantum numbers, with a fourth and final quantum number obtained from experimental results. The quantum numbers are named and have limitations as shown in Table 4-1. Each electron is specified in terms of its four quantum numbers that govern its energy, its orientation in space, and its possible interactions with other electrons. Thus, listing the values of the four quantum numbers describes the probable location of the electron, somewhat analogously to listing the section, row, seat, and date on a ticket to a rock concert. To learn to express the electronic structure of an atom, it is necessary to learn (1) the names, symbols, and permitted values of the quantum numbers and (2) the order of increasing energy of electrons as a function of their sets of quantum numbers.

Table 4-1 Quantum Numbers

Principal quantum number

Any positive integer

Angular momentum quantum number or Azimuthal quantum number

0, . . . , n − 1 in integer steps Magnetic quantum number

0, . . . , +l in integer steps Spin quantum number

− l, . . . ,

The principal quantum number of an electron is denoted n. It is the most important quantum number in determining the energy of the electron. In general, the higher the principal quantum number, the higher the energy of the electron. Electrons with higher principal quantum numbers are also apt to be farther away from the nucleus than electrons with lower principal quantum numbers. The values of n can be any positive integer: 1, 2, 3, 4, 5, 6,

7, . . . . The first seven principal quantum numbers are the only ones used for electrons in ground states of atoms. The angular momentum quantum number is denoted l. It also affects the energy of the electron, but in general not as much as the principal quantum number does. In the absence of an electric or magnetic field around the atom, only n and l have any effect on the energy of the electron. The value of l can be 0 or any positive integer up to, but not including, the value of n for that electron.

The magnetic quantum number, denoted m l , determines the orientation in space of the electron, but does not ordinarily affect the energy of the electron. Its values depend on the value of l for that electron, ranging

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 does not affect the energy of the electron. Its possible values are − 1 2 and + 1 2 . The value of m s does not depend 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

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?

Ans. − 1 and + 1

2 2 . (The values that the other quantum numbers have do not matter; the m s value must be either − 1 2 or + 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