Table of Contents
HISTORY OF QUANTUM NUMBERS
The concept of quantum numbers developed in the beginning of the 20th century, alongside the development of quantum mechanics. In 1913, Niels Bohr proposed quantised energy levels for electrons in atoms, which resulted in the first quantum number. Wolfgang Pauli built on this with his exclusion principle in 1925, which introduced more quantum numbers. These numbers describe the properties of particles such as electrons, which transformed atomic theory and laid the groundwork for modern physics.
QUANTUM NUMBERS
The numbers used for completely characterising each electron of an atom are known as Quantum Numbers. For such quantum numbers are found to be necessary for describing completely and electron. They are:
- Principal Quantum Number,
- Azimuthal Quantum Number,
- Magnetic Quantum Number, and
- Spin Quantum Number.
PRINCIPAL QUANTUM NUMBER (n)
It was proposed by Bohr. This number gives an average distance of the electron from the nucleus and corresponds to the principal energy level to which the electron belongs.
Thus, it gives an idea about the position of the electron around the nucleus and also refers to the size of the orbital.
Higher the principal quantum number, greater is its distance from the nucleus, greater is its size and also higher is its energy.
Although, theoretically its value may be from 1 to α, only values from 1 to 7 have so far been established for atoms for the known elements. These are designated either as 1, 2, 3, 4, 5, 6, 7 or K, L, M, N, O, P, Q respectively. The maximum number of electrons in principle quantum number is given by 2n².
SUBSIDIARY OR AZIMUTHAL QUANTUM NUMBER (l)
It was proposed by Sommerfeld. The main energy levels are divided into sub-levels each being denoted by a subsidiary azimuthal quantum number (l). Electrons do not really travel in circular orbits. The volume in space where there is high probability of finding electron is called an orbital.
The subsidiary quantum number (l) describes the shape of the orbital occupied by the electron.
For a given value of the principal quantum number (n), the azimuthal quantum number (l) may have all integral values from 0 to (n-1), each of which represents a different sub-energy level (sub-shell or, sub-orbit) and they are usually denoted by the letters s, p, d, f [s, p, d, f are spectroscopic terms, s = sharp, p = principal, d = diffuse and f = fundamental].
When l = 0, the orbital is spherical and is called s orbital.
When l = 1, the orbital is dumb-bell shaped and is called p orbital.
When l = 2, the orbital is double dumb-bell shaped and is called d orbital.
When l = 3, a more complicated f orbital is formed.
The energy of the various sub-shell in the same shell are in the order s < p < d < f.
The maximum number of electrons which can be held by these sub-shells is given by 2(2l + 1). Thus, s sub-shell (l = 0) can have 2 electrons, p sub-shell (l = 1) 6 electrons, d sub-shell (l = 2) 10 electrons and f sub-shell (l = 3) can accommodate 14 electrons.
MAGNETIC QUANTUM NUMBER (m)
It was proposed by Lande. Each sub-shell is further sub-divided, the sub-divisions being denoted by magnetic quantum number. Magnetic quantum number is determined by the way, in which the lines in atomic spectra split under the influence of magnetic field.
Magnetic quantum number determines the preferred orientations of orbitals in space.
m may have values -l, ………. -3, -2, -1, 0, +1, +2, +3, ………..+l where, l = azimuthal quantum number. There are, therefore, (2l + 1) possible values for magnetic quantum number.
For l = 0 (s sub-shell), m = 1, i.e., m has only one possible value which is zero (0). Hence, there is only one orientation s orbital. For l = 1 (p sub-shell), m = 3 i.e., -1, 0, +1. Hence 3 orientations (px, py, pz) are possible for p sub-shell. For l = 2 (d sub-shell), m = 5 i.e., -2, -1, 0, +1, +2. Hence, d sub-shell can have 5 different orientations (dxy, dyz, dzx, dx²-y², dz²). For l = 3 (f sub-shell), m = 7 i.e., -3, -2, -1, 0, +1, +2, +3, i.e., f sub-shell has 7 different orientations.
SPIN QUANTUM NUMBER (s)
When spectral lines of H, Li, Na, K, etc. were observed by means of the instruments of high resolving power, each line of the spectral series was found consisting of a pair of lines known as doublets or, double-line structure. Hence it should be understood clearly that this doublet if different from fine structure which consists of closely spaced fine lines.
To account for this doublets, Whlenbeck and Goud-Smith in 1925 suggested that the electron, while moving round the nucleus in an orbit, also rotates or, spins about its own axis either in a clockwise direction or, in an anticlockwise direction. Spin quantum number (s) can have two values viz., +1/2 (corresponding to the spinning of the electron in the clockwise direction) and -1/2 (corresponding to the spinning of the electron in anticlockwise direction). Clockwise and anticlockwise spinning of the electron are represented as ↑ and ↓ respectively. An orbital at most can accommodate 2 electrons provided they have opposite spin. Spin quantum number is independent from other quantum numbers.
DIFFERENCES BETWEEN ORBIT AND AN ORBITAL
- Orbit: it is the path around the nucleus in which the electrons revolve with definite amount of energy. Orbit is two dimensional.
- Orbital: Orbital is the volume in space around the nucleus in which the probability of finding electron is maximum. Orbital is three dimensional.
Let’s look at the major differences between an orbit and an orbital.
| ORBIT | ORBITAL |
|---|---|
1. It is the well-defined circular path around the nucleus in which the electron revolves. | 1. It is the region in space around the nucleus, where the probability of finding electron is maximum. |
2. Represents movement of electron in one plane. | 2. Orbital represents the movement of electron in three dimensional space, around the nucleus. |
3. The shape of orbit is circular or, elliptical. | 3. The shape of orbital is either circular or, dumb-bell. |
4. It is non-directional in character and cannot explain the shape of molecules. | 4. It is directional in character and can explain the shape of molecules. |
5. The maximum number of electrons in an orbit is given by 2n² where n is the number of orbits. | 5. An orbital can accommodate maximum 2 electrons. |
DISTRIBUTION OF ELECTRONS OR, ARRANGEMENT OF ELECTRONS IN DIFFERENT ATOMS
As to describe an electron 4 Quantum numbers are necessary, to describe the arrangement and distribution of electrons in shell, the following 3 selective principles and rules are required:
- Pauli’s Exclusion Principle,
- Hund’s Rule of Maximum Multiplicity and,
- Aufbau Principle
Pauli's Exclusion Principle
No two electrons in an atom can have all four quantum numbers the same. In other words, an orbital cannot have more than 2 electrons and moreover, if an orbital has 2 electrons, they must have opposite spin. Therefore, the capacity of s, p, d, f subshells to accommodate electrons is 2, 6, 10 and 14 respectively.
Hund's Rule of Maximum Multiplicity
According to this rule, the maximum number of unpaired electrons in a given energy level is maximum. In a sub-shell, all the available degenerate orbitals (i.e., orbitals of same energy) are occupied singly first and then pairing of electrons in each orbital occurs. Thus, the 3 degenerate orbitals of p sub-shell (px, py, pz), no one will have 2 electrons as long as any other is vacant. This is Hund’s rule of maximum multiplicity.
Aufbau Principle
According to this principle, “electrons are added progressively to the various orbitals in the order of increasing energy starting with the orbital of lowest energy”.
The order of increasing energies are as follows: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p.
The sequence in which the energy levels are filled is shown in the image.
IMPORTANT NOTES
- Orbitals in the same d-sub-shell tend to become completely filled or, exactly half-filled of electrons because the half-filled and full-filled d-orbitals have less energy than any other arrangement and are more stable.
- The total number of electrons that can be accommodated in a shell is equal to 2n² where n refers to principal quantum number. Thus, the 1st shell can accommodate 2 electrons, 2nd shell can accommodate 8 electrons, 3rd shell 18 and 4th shell 32 electrons.
- The total number of electrons that can be accommodate in a sub-shell is equal to twice the number of orbitals it contains. Thus, an s shell can have 2 electrons, p sub-shell 6 electrons due to the presence of 3 orbitals, d sub-shell can have 10 electrons due to 5 orbitals and f sub-shell can have 14 electrons due to seven orbitals.