# Spin Quantum Number

- Page ID
- 2581

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 (
*n*,*l*, 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.

- If
- 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.

- If
- 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

- s:

Exercise \(\PageIndex{1}\): Tungsten

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

**Answer**-
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)?

**Answer**-
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)?

**Answer**-
Sulfur has 4 electrons in the 3p orbitals. The 4th electron in this orbital will be the first one to pair up with another electron, therefore giving it a -1/2 spin.

## References

- Housecroft, Catherine E., and Alan G. Sharpe.
*Inorganic Chemistry*. 3rd ed. Harlow: Pearson Education, 2008. Print. (pg 15). - Nostrand, Van. Encyclopedia of Chemistry. 5th ed. John Wiley and Sons, Inc., 2005. Print. (pg 1396).