Atomic Orbitals

What are Atomic Orbitals?

Atomic orbitals are mathematical functions that provide insight into the wave nature of electrons (or pairs of electrons) that exist around the nuclei of atoms. In the fields of quantum mechanics and atomic theory, these mathematical functions are often employed in order to determine the probability of finding an electron (belonging to an atom) in a specific region around the nucleus of the atom. It is important to note that the term ‘atomic orbital’ can also be used to refer to the physical space or physical region around an atom’s nucleus in which the probability of a specific electron being present is maximum. The presence of an electron in such a region is predicted by the mathematical form of the atomic orbital.

It is important to note that the characteristics of each atomic orbital are dependent upon the values of the following quantum numbers:

  • The principal quantum number (denoted by the symbol ‘n’)
  • The azimuthal quantum number, also known as the orbital angular momentum quantum number (denoted by the symbol ‘l’)
  • The magnetic quantum number (denoted by the symbol ‘ml’)

Furthermore, it can be noted that each atomic orbital can hold a maximum of two electrons. In completely occupied atomic orbitals, i.e. the atomic orbitals containing two electrons, each of the electrons has an equal and opposite spin when compared to the other. Insight into the electron spin is provided by the value of the spin quantum number, which is denoted by the symbol ‘ms’. Thus, insight into any electron residing in any atomic orbital in a given atom can be obtained by determining the values of the four quantum numbers that describe it, namely the principal quantum number, the azimuthal quantum number, the magnetic quantum number, and the electron spin quantum number.

Names of Atomic Orbitals and the Relationship Between the Different Quantum Numbers that Describe Them

The name of an atomic orbital is usually expressed in terms of a combination of the principal quantum number (n) and the azimuthal quantum number (l). The simple names of the atomic orbitals and the corresponding value of the azimuthal quantum number are listed below.

  • The s orbital, where the value of the azimuthal quantum number is equal to 0.
  • The p orbital, where the value of the azimuthal quantum number is equal to 1.
  • The d orbital, where the value of the azimuthal quantum number is equal to 2.
  • The f orbital, where the value of the azimuthal quantum number is equal to 3.
  • The g orbital, where the value of the azimuthal quantum number is equal to 4.
  • The h orbital, where the value of the azimuthal quantum number is equal to 5.

It can be noted that the next atomic orbitals can be named alphabetically, omitting the letter ‘j’ (which is done because certain languages do not distinguish between the letters ‘j’ and ‘i’). Therefore, when l=6, the name of the atomic orbital will be ‘i’ and when l=7, the name of the atomic orbital will be ‘k’. It can also be noted that the names of the first four orbitals (s, p, d, and f) are derived from the descriptions that were initially provided by the spectroscopists who studied the spectroscopic lines of the alkali metals and described them as ‘sharp’, ‘principal’, ‘diffuse’, and ‘fundamental’.

When naming a specific atomic orbital, the value of the principal quantum number must be added as a prefix to the alphabetical description of the azimuthal quantum number. It is important to note that the value of the azimuthal quantum number is dependent on the value of the principal quantum number. For any given value of ‘n’, the value of ‘l’ can range from zero to (n-1). For example, if the value of ‘n’ is equal to 3, the possible values of ‘l’, which range from zero to (3-1), are 0, 1, and 2. The names of these atomic orbitals will be 3s (for n=3 and l=0), 3p (for n=3 and l=1), and 3d (for n=3 and l=2). It can also be noted that it is not possible for the 3f orbital to exist because that would require the value of ‘n’ and ‘l’ both to be equal to 3, which is not possible since the value of the azimuthal quantum number must always be lower than that of the principal quantum number.

Table of All Possible Atomic Orbitals where the Value of ‘n’ Ranges from 0 to 5

Value of the Principal Quantum Number (n)

Possible Values of the Azimuthal Quantum Number (l)

Names of all the Possible Atomic Orbitals for the Given Value of ‘n’

1

  • l = 0 (s orbital)
  • The 1s orbital

2

  • l = 0 (s orbital)
  • l = 1 (p orbital)
  • The 2s orbital
  • The 2p orbital

3

  • l = 0 (s orbital)
  • l = 1 (p orbital)
  • l = 2 (d orbital)
  • The 3s orbital
  • The 3p orbital
  • The 3d orbital

4

  • l = 0 (s orbital)
  • l = 1 (p orbital)
  • l = 2 (d orbital)
  • l = 3 (f orbital)
  • The 4s orbital
  • The 4p orbital
  • The 4d orbital
  • The 4f orbital

5

  • l = 0 (s orbital)
  • l = 1 (p orbital)
  • l = 2 (d orbital)
  • l = 3 (f orbital)
  • l = 5 (g orbital)
  • The 5s orbital
  • The 5p orbital
  • The 5d orbital
  • The 5f orbital
  • The 5g orbital

It is important to note that electrons are filled into these orbitals in compliance with several rules such as the Aufbau principle, the Pauli exclusion principle, and Hund’s rule of maximum multiplicity.

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