# Spin Quantum Number

The Spin Quantum Number ($$m_s$$) describes the angular momentum of an electron. An electron spins around an axis and has both angular momentum and orbital angular momentum. Because angular momentum is a vector, the Spin Quantum Number (s) has both a magnitude (1/2) and direction (+ or -).

Each orbital can only hold two electrons. One electron will have a +1/2 spin and the other will have a -1/2 spin. Electrons like to fill orbitals before they start to pair up. Therefore the first electron in an orbital will have a spin of +1/2. After all the orbitals are half filled, the electrons start to pair up. This second electron in the orbital will have a spin of -1/2. If there are two electrons in the same orbital, it will spin in opposite directions.

### Combinations of Quantum Numbers

• The three quantum numbers (nl, and m) that describe an orbital are integers: 0, 1, 2, 3.
• The principal quantum number (n) cannot be zero. The allowed values of n are therefore 1, 2, 3, 4...
• The angular quantum number (l) can be any integer between 0 and n - 1.
• If n = 3, l can be either 0, 1, or 2.
• The magnetic quantum number (m) can be any integer between -l and +l.
• If l = 2, m can be -2, -1, 0, +1, or +2.
• Orbitals that have same value of principal quantum number form a Shell(n).
• Orbitals within the shells are divided into subshell (l)
• s:l = 0 p:l = 1 d:l = 2 f:l = 3

Exercise $$\PageIndex{1}$$: Tungsten

What is the spin quantum number for Tungsten (symbol W)?

Tungsten has 4 electrons in the 5d orbital. Therefore 1 electron will go into each orbital (no pairing). The 4th electron will have a +1/2 spin.

Exercise $$\PageIndex{2}$$: Gold

What is the spin quantum number for Gold (symbol Au)?

Gold has 9 electrons in the 5d orbital. Therefore the electrons will start to pair up, which means the 9th electron will pair up, giving it a -1/2 spin.

Exercise $$\PageIndex{3}$$: Sulfur

What is the spin quantum number for Sulfur (symbol S)?