Electronic configuration and quantum numbers
The quantum numbers are nothing but the details that are required to locate an electron in an atom. In an atom a large number of electron orbitals are permissible. An orbital of smaller size means there is more chance of finding the electron near the nucleus. These orbitals are designated by a set of numbers known as quantum numbers. In order to specify energy, size, shape and orientation of the electron orbital, three quantum numbers are required these are discussed below.
1. The principal quantum number (n)
The electrons inside an atom are arranged in different energy levels called electron shells or orbits. Each shell is characterized by a quantum
number called principal quantum number. This is represented by the letter 'n' and 'n' can have values, 1,2,3,4 etc. The first level is also known as K level. Second as L level, third as M level, fourth as N level and so on. The first or K level is the orbit nearest to the nucleus and next one is second or L level and so on.
2. The subsidiary or azimuthal quantum number (l)
According to Sommerfield, the electron in any particular energy level could have circular path or a variety of elliptical paths about the nucleus resulting in slight differences in orbital shapes with slightly differing energies due to the differences in the attraction exerted by the nucleus on the electron. This concept gave rise to the idea of the existence of sub-energy levels in each of the principal energy levels of the atom. This is denoted by the letter 'l' and have values from 0 to n-1.
Thus, if n=1, l=0 only one value (one level only) s level.
n=2, l=0 and 1 ( 2 values or 2 sub- levels) s and p level.
n=3, l=0, 1 and 2 (3 values or 3 sub-levels) s, p and d level.
n=4, l=0, 1, 2 and 3 (4 values or 4 sub-levels) s, p ,d and f level.
3. Magnetic quantum number (m)
In a strong magnetic field a sub-shell is resolved into different orientations in space. These orientations called orbitals have slight differences in energy. This explains the appearance of additional lines in atomic spectra produced when atoms emit light in magnetic field. Each orbitals is designated by a magnetic quantum number m and its values depends on the value of 'l' . The values are -' l' through zero to +' l' and thus there are (2l+1) values.
Thus when l=0, m= 0 (only one value or one orbital)
l=1, m= -1, 0, +1 (3 values or 3 orbitals)
l=2, m= -2, -1, 0, +1, +2 (5 values or 5 orbitals)
l=3, m= -3,-2, -1, 0, +1, +2, +3 (7 values or 7 orbitals).
The three quantum numbers labeling an atomic orbital can be used equally well to label electron in the orbital. However, a fourth quantum
number, the spin quantum number, (s) is necessary to describe an electron completely.
4. Spin quantum number (s)
The electron in the atom rotates not only around the nucleus but also around its own axis and two opposite directions of rotation are possible (clock wise and anticlock wise). Therefore the spin quantum number can have only two values +1/2 or -1/2. For each values of m including zero, there will be two values for s.
To sum up, the four quantum numbers provide the following informations:
1. n identifies the shell, determines the size of the orbital and also to a large extent the energy of the orbit.
2. There are n subshells in the nth shell. l identifies the subshell and determines the shape of the orbital. There are (2l+1) orbitals of each type in a subshell i.e., one s orbital (l=0), three p orbitals (l=1), and five d orbitals (l=2) per subshell. To some extent l also determines the energy of the orbital in a multi-electron atom.
3. ml designates the orientation of the orbital. For a given value of l, ml has (2l+1) values, the same as the number of orbitals per subshell. It means that the number of orbitals is equal to the number of ways in which they are oriented.
4. ms refers to orientation of the spin of the electron.
What is the total number of orbitals associated with the principal quantum number n=3 ?
For n = 3, the possible values of l are 0,1 and 2. Thus, there is one 3s orbital (n = 3, l = 0 and ml = 0); there are three p orbitals (n = 3, l = 1 and ml = -1, 0, 1) there are five 3d orbitals (n = 3, l = 2, ml = -2, -1, 0, 1, 2).
Therefore, the total number of orbitals is 1+3+5 = 9.
Using s, p, d, f notations, describe the orbital with the following quantum numbers
(a) n=2, l = 1 (b) n = 4, l = 0 (c) n = 5, l = 3 (d) n = 3, l = 2.
n l orbital
(a) 2 1 2p
(b) 4 0 4s
(c) 5 3 5f
(d) 3 2 3d
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