# How to Determine the Number of Electrons With Quantum Numbers ••• knowlesgallery/iStock/GettyImages

Describing the states of electrons in atoms can be a complicated business. Like if the English language had no words to describe orientations like "horizontal" or "vertical," or "round" or "square," a lack of terminology would lead to many misunderstandings. Physicists also need terms to describe the size, shape and orientation of the electron orbitals in an atom. But instead of using words, they use numerals called quantum numbers. Each of these numbers corresponds to a different attribute of the orbital, which allows physicists to identify the exact orbital they want to discuss. They are also related to the total number of electrons an atom can hold if this orbital is its outer, or valence, shell.

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Determine the number of electrons using quantum numbers by first counting the number of electrons in each full orbital (based on the last fully-occupied value of the principle quantum number), then adding the electrons for the full subshells of the given value of the principle quantum number, and then adding two electrons for each possible magnetic quantum number for the last subshell.

Subtract 1 from the first, or principle, quantum number. Since the orbitals must fill in order, this tells you number of orbitals that must already be full. For example, an atom with the quantum numbers 4,1,0 has a principal quantum number of 4. This means that 3 orbitals are already full.

Add the maximum number of electrons that each full orbital can hold. Record this number for later use. For example, the first orbital can hold two electrons; the second, eight; and the third, 18. Therefore the three orbitals combined can hold 28 electrons.

Identify the subshell represented by the second, or angular, quantum number. The numbers 0 through 3 represent the "s", "p," "d" and "f" subshells, respectively. For example, 1 identifies a "p" subshell.

Add the maximum number of electrons that each previous subshell can hold. For example, if the quantum number indicates a "p" subshell (as in the example), add the electrons in the "s" subshell (2). However, if your angular quantum number was "d," you'd need to add the electrons contained in both the "s" and "p" subshells.

Add this number to the electrons contained in the lower orbitals. For example, 28 + 2 = 30.

Determine how many orientations of the final subshell are possible by determining the range of legitimate values for the third, or magnetic, quantum number. If the angular quantum number equals "l," the magnetic quantum number can be any number between "l" and " −l," inclusive. For example, when the angular quantum number is 1, the magnetic quantum number may be 1, 0 or −1.

Count the number of possible subshell orientations up to and including the one that is indicated by the magnetic quantum number. Begin with the lowest number. For example, 0 represents the second possible orientation for the sublevel.

Add two electrons for each of the orientations to the previous electron sum. This is the total number of electrons an atom can contain up through this orbital. For example, since 30 + 2 + 2 = 34, an atom with a valence shell described by the numbers 4,1,0 holds a maximum of 34 electrons.

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