By Kylene Arnold – Updated Aug 30, 2022
Describing the states of electrons in atoms can be challenging. Without precise terminology, misunderstandings abound. Physicists solve this by using four quantum numbers that uniquely identify each electron’s orbital. These numbers also reveal how many electrons an atom can hold in its outer, or valence, shell.
To calculate the total electron count for a given set of quantum numbers: 1) Count the fully occupied orbitals below the principal quantum number. 2) Add the electrons in those full orbitals. 3) Add the electrons in all fully occupied subshells up to the angular quantum number. 4) For the final subshell, add two electrons for each allowed magnetic quantum number value up to the specified one. The sum is the maximum number of electrons the atom can contain.
The principal quantum number (n) tells you how many shells are filled before the current one. Subtract one from n to find the number of completely filled lower shells.
Each shell can hold a specific maximum number of electrons: 2 in the first (n=1), 8 in the second (n=2), 18 in the third (n=3), 32 in the fourth (n=4), and so on. Sum these values for all shells that are fully occupied.
The angular quantum number (l) designates the subshell type: 0 = s, 1 = p, 2 = d, 3 = f. For example, l=1 corresponds to a p subshell.
For each subshell below the one indicated by l, add its maximum electron capacity: s = 2, p = 6, d = 10, f = 14. If l=1 (p), add 2 from the s subshell. If l=2 (d), add 2 from s and 6 from p, totaling 8.
Add the numbers from steps 2 and 4. This gives the electron count up to the last fully occupied subshell of the current shell.
The magnetic quantum number (m) specifies the orientation of orbitals within a subshell. It ranges from –l to +l. For l=1, possible m values are –1, 0, +1.
List all allowed m values in ascending order and count how many precede or equal the given m. Each orientation can hold two electrons (spin up and spin down).
Multiply the number of counted orientations by 2 and add this to the total from step 5. The result is the maximum electron count for the atom described by the quantum numbers (n, l, m).
Example: For (n=4, l=1, m=0), there are 3 fully filled shells (28 electrons) + 2 electrons from the s subshell (total 30). With l=1, possible m values are –1, 0, +1; up to m=0, there are two orientations, adding 4 electrons. The atom can hold 34 electrons in total.
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