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Spin Quantum Number

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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)?


    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)?


    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.


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

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