Question

Give the 4 quantum numbers (n,l,m­_l,m_s) for an electron, what they describe and their values.

Answer 1

There are four quantum numbers which can describe an electron in an atom completely.

1) Principal quantum number (n)

2) Azimuthal quantum number (ℓ)

3) Magnetic quantum number (m)

and

4) Spin quantum number (s)

1) The principal quantum number (n) : Describes the electron shell, or energy level, of an atom. The value of n ranges from 1 to the shell containing the outermost electron of that atom.

n = 1, 2, ... .

For example, in caesium (Cs), the outermost valence electron is in the shell with energy level 6, so an electron in caesium can have an n value from 1 to 6.

This number therefore has a dependence only on the distance between the electron and the nucleus.The average distance increases with n, and hence quantum states with different principal quantum numbers (different n values) are said to belong to different shells.

2) The azimuthal quantum number (ℓ) : It is also known as the angular quantum number or orbital quantum number, Describes the subshell, and gives the magnitude of the orbital angular momentum through the relation.

L² = ħ²ℓ (ℓ + 1).

In chemistry and spectroscopy the vaiues are sa fallows

"ℓ = 0" is called an s orbital"

ℓ = 1" a p orbital,

"ℓ = 2" a d orbital,

and

"ℓ = 3" an f orbital.

The value of ℓ ranges from 0 to n − 1, because the first p orbital (ℓ = 1) appears in the second electron shell (n = 2), the first d orbital (ℓ = 2) appears in the third shell (n = 3), and so on.

ℓ = 0, 1, 2,..., n − 1.

A quantum number beginning in 3, 0, … describes an electron in the s orbital of the third electron shell of an atom. In chemistry, this quantum number is very important, since it specifies the shape of an atomic orbital and strongly influences chemical bonds and bond angles.

3) The magnetic quantum number (m) : Describes the specific orbital within that subshell, and yields the projection of the orbital angular momentum along a specified axis:

L_z = m_ℓ ħ.

The values of m_ℓ range from −ℓ to ℓ, with integer steps between them

The s subshell (ℓ = 0) contains only one orbital, and therefore the m_ℓ of an electron in an s orbital will always be 0.

The p subshell (ℓ = 1) contains three orbitals, So the m_ℓ of an electron in a p orbital will be −1, 0, and +1. The d subshell (ℓ = 2) contains five orbitals, with m_ℓ values of −2, −1, 0, +1, and +2.

4) The spin projection quantum number (S) : Describes the spin (intrinsic angular momentum) of the electron within that orbital, and gives the projection of the spin angular momentum S along the specified axis.

An electron has spin number s = ½, consequently s will be ±½, referring to "spin up" and "spin down" states.

Each electron in any individual orbital must have different quantum numbers because of the Pauli exclusion principle, therefore an orbital never contains more than two electrons.