0.5a: Spin Quantum Number

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May 9, 2021
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- 0.5: Electronic Configurations
- 0.5b: Hund's Rules

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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.
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)?

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).

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Exercise $\PageIndex{1}$ : Tungsten

Exercise $\PageIndex{2}$ : Gold

Exercise $\PageIndex{3}$ : Sulfur

Combinations of Quantum Numbers

Exercise 0.5a.1\PageIndex{1}: Tungsten

Exercise 0.5a.2\PageIndex{2}: Gold

Exercise 0.5a.3\PageIndex{3}: Sulfur

References

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Exercise $\PageIndex{1}$ : Tungsten

Exercise $\PageIndex{2}$ : Gold

Exercise $\PageIndex{3}$ : Sulfur