According to quantum mechanics, each electron in an atom is described by four different quantum numbers. There of the quantum numbers (n, 1, and ml) specify the wave function that gives the probability of finding the electron at various points in space. A wave function for an electron in an atom is called an atomic orbital. An orbital should not be confused with a bohr orbit. An atomic orbital is pictured qualitatively by describing the region of space where there is high probability of finding the electrons. The atomic orbital so pictured has a definite shape. A fourth quantum number (ms) refers to a magnetic property of electrons called spin. In this sections we will first look at quantum numbers, then at atomic orbitals.
Quantum Numbers
The allowed values and general meaning of each of the four different quantum numbers of an electron in an atom are as follows:
1. Principal quantum number (n) This quantum number is the modern equivalent of the n quantum number in Bohr’s theory. It can have any positive whole number value, 1, 2, 3, and so on.
The energy of an electron in an atom depends principally on n. The snaller n is, the lower the energy. In the case of the hydrogen atom or any other single-electron atom (such as Li +2 or He + ), it is the only quantum number determining the energy (which is given by Bohr’s formula, discussed. For other atoms, the energy also depends to a slight extent on the l quantum number.
The size of an orbital also depends on n. The larger the value of n is, the larger the orbital. Orbitals of the same quantum state n are said to belong to the same shell. Shells are sometimes designated by the following letters:
Letter K L M N…
n 1 2 3 4…
2. Angular momentum quantum number (l) (also called azimuthal quantum number) Within each shell of quantum number n, there are n different kinds of orbitals, each with a distinctive shape, denoted by the l quantum number. The allowed values of the l quantum number are the integers from 0 to n-1(Note that the value 0 is allowed for l.). For example, if anelectron has a principal quantum number of 3, the possible values for l are 0, 1, and 2. Thus, within the M shell (n=3), there are three kinds of orbitals, each of which has a different shape for the region where the electron is most likely found.
Orbitals of the same n, but different l, are said to belong to different subshells of a given shell. The different subshells are usually denoted by letters.
Letter s p d f g…
l 0 1 2 3 4…
To denote a subshell within a particular shell, we write the value of the n quantum number for the shell followed by the letter designation for the subshell. For example, 2p denotes a subshell with quantum number n=2 and l=1.
3. Magnetic quantum number (ml) Except for an s subshell, there is more than one orbital for each subshell. For a p subshell, for example, there are three different orbitals l. There are always 2l + 1 orbitals in each subshell of quantum number l. Thus, for l=2 (the d subshell), there are (2*2) + 1 = 5 orbitals. One way of designating these orbitals within a subshell is with the magnetic quantum number, ml. The magnetic quantum number can have any whole number value from –l to +l . For l = 2, the possible values of ml are -2, -1, 0, 1, and 2.
Essentionally each of these orbitals of a subshell has the same shape, but a different orientation, or direction, in space. Each orbital of a particular subshell (no matter how it is oriented in space) has the same energy.
4. Spin quantum number (ms) An electron acts as though it were spinning on its axis like the earth. Such an electron spin would give rise to a circulatin electrical charge that would generate a magnetic field. Thus, an electron behaves like a small bar magnet, with a north and south pole. The spin quantum number , ms, refers to the two possible orientations that are allowed im quantum mechanics for this magnet or spin. Values for the spin quantum number are +1/2 and
-1/2.
Table 5.2 lists the permissible quantum numbers for all orbitals through the n = 4 shell. These velues follow from the rules just given. Note that all orbitals with the same principal quantum number, n, have the same energy. For atoms with more than one electron, however, only orbitals in the same subshell (denoted by a given n and l) have the same energy
