QUANTUM NUMBER

What are Quantum Numbers?

A quantum number is a value that is used when describing the energy levels available to atoms and molecules. By solving the Schrödinger equation (Hy = Ey), we obtain a set of mathematical equations, called wave functions (y), which describe the probability of finding electrons at certain energy levels within an atom.

A wave function for an electron in an atom is called an atomic orbital; this atomic orbital describes a region of space in which there is a high probability of finding the electron. Energy changes within an atom are the result of an electron changing from a wave pattern with one energy to a wave pattern with a different energy (usually accompanied by the absorption or emission of a photon of light).

Each electron in an atom is described by four different quantum numbers. The first three (n, l, ml) specify the particular orbital of interest, and the fourth (ms) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n):  n = 1, 2, 3, …, ∞

Specifies the energy of an electron and the size of the orbital (the distance from the nucleus of the peak in a radial probability distribution plot). All orbitals that have the same value of n are said to be in the same shell (level). For a hydrogen atom with n=1, the electron is in its ground state; if the electron is in the n=2 orbital, it is in an excited state. The total number of orbitals for a given n value is n2.

Angular Momentum (Secondary, Azimuthal) Quantum Number (l):  l = 0, …, n-1.

Specifies the shape of an orbital with a particular principal quantum number. The secondary quantum number divides the shells into smaller groups of orbitals called subshells (sublevels). Usually, a letter code is used to identify l to avoid confusion with n:

l 0 1 2 3 4 5 …

Letter s p d f g h …

The subshell with n=2 and l=1 is the 2p subshell; if n=3 and l=0, it is the 3s subshell, and so on. The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (ml):  ml = -l, …, 0, …, +l.

Specifies the orientation in space of an orbital of a given energy (n) and shape (l). This number divides the subshell into individual orbitals which hold the electrons; there are 2l+1 orbitals in each subshell. Thus the s subshell has only one orbital, the p subshell has three orbitals, and so on.

Spin Quantum Number (ms):  ms = +½ or -½.

Specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions (sometimes called up and down).

The Pauli Exclusion Principle (Wolfgang Pauli, Nobel Prize 1945) states that no two electrons in the same atom can have identical values for all four of their quantum numbers. What this means is that no more than two electrons can occupy the same orbital, and that two electrons in the same orbital must have opposite spins.

Because an electron spins, it creates a magnetic field, which can be oriented in one of two directions. For two electrons in the same orbital, the spins must be opposite to each other; the spins are said to be paired. These substances are not attracted to magnets and are said to be diamagnetic. Atoms with more electrons that spin in one direction than another contain unpaired electrons. These substances are weakly attracted to magnets and are said to be paramagnetic.

What are Electron Configurations?

The electron configuration of an element describes how electrons are distributed in its atomic orbitals. Electron configurations of atoms follow a standard notation in which all electron-containing atomic subshells (with the number of electrons they hold written in superscript) are placed in a sequence. For example, the electron configuration of sodium is 1s22s22p63s1.

However, the standard notation often yields lengthy electron configurations (especially for elements having a relatively large atomic number). In such cases, an abbreviated or condensed notation may be used instead of the standard notation. In the abbreviated notation, the sequence of completely filled subshells that correspond to the electronic configuration of a noble gas is replaced with the symbol of that noble gas in square brackets. Therefore, the abbreviated electron configuration of sodium is [Ne] 3s1 (the electron configuration of neon is 1s22s22p6, which can be abbreviated to [He] 2s22p6). 

Electron Configurations are useful for:

• Determining the valence of an element.

• Predicting the properties of a group of elements (elements with similar electron configurations tend to exhibit similar properties).

• Interpreting atomic spectra.

This notation for the distribution of electrons in the atomic orbitals of atoms came into practice shortly after the Bohr model of the atom was presented by Ernest Rutherford and Niels Bohr in the year 1913.

Writing Electron Configurations

• Shells

The maximum number of electrons that can be accommodated in a shell is based on the principal quantum number (n). It is represented by the formula 2n2, where ‘n’ is the shell number. The shells, values of n, and the total number of electrons that can be accommodated are tabulated below.

• Subshells

The subshells into which electrons are distributed are based on the azimuthal quantum number (denoted by ‘l’).

This quantum number is dependent on the value of the principal quantum number, n. Therefore, when n has a value of 4, four different subshells are possible.

When n=4. The subshells correspond to l=0, l=1, l=2, and l=3 and are named the s, p, d, and f subshells, respectively.

The maximum number of electrons that can be accommodated by a subshell is given by the formula 2*(2l + 1).

Therefore, the s, p, d, and f subshells can accommodate a maximum of 2, 6, 10, and 14 electrons, respectively.

All the possible subshells for values of n up to 4 are tabulated below.

• Notation

The electron configuration of an atom is written with the help of subshell labels.

These labels contain the shell number (given by the principal quantum number), the subshell name (given by the azimuthal quantum number) and the total number of electrons in the subshell in superscript.

For example, if two electrons are filled in the ‘s’ subshell of the first shell, the resulting notation is ‘1s2’.

With the help of these subshell labels, the electron configuration of magnesium (atomic number 12) can be written as 1s2 2s2 2p6 3s2.

• Filling of Atomic Orbitals

Aufbau Principle

This principle is named after the German word ‘Aufbeen’ which means ‘build up’.

The Aufbau principle dictates that electrons will occupy the orbitals having lower energies before occupying higher energy orbitals.

The energy of an orbital is calculated by the sum of the principal and the azimuthal quantum numbers.

According to this principle, electrons are filled in the following order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p…

The order in which electrons are filled in atomic orbitals as per the Aufbau principle is illustrated below.

Pauli Exclusion Principle

The Pauli Exclusion Principle states that a maximum of two electrons, each having opposite spins, can fit in an orbital.

This principle can also be stated as “no two electrons in the same atom have the same values for all four quantum numbers”.

Therefore, if the principal, azimuthal, and magnetic numbers are the same for two electrons, they must have opposite spins.

Hund’s Rule

This rule describes the order in which electrons are filled in all the orbitals belonging to a subshell.

It states that every orbital in a given subshell is singly occupied by electrons before a second electron is filled in an orbital.

In order to maximize the total spin, the electrons in the orbitals that only contain one electron all have the same spin (or the same values of the spin quantum number).