Example 30.5 from College Physics, 30.9 The Pauli Exclusion Principle

Subshells and Totals for n=3

How many subshells are in the $n = 3$ shell? Identify each subshell, calculate the maximum number of electrons that will fit into each, and verify that the total is $2 n^{2}$ .

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
Subshells are determined by the value of $l$ ; thus, we first determine which values of $l$ are allowed, and then we apply the equation “maximum number of electrons that can be in a subshell $= 2 (2 l + 1)$ ” to find the number of electrons in each subshell.
Solution
Since $n = 3$ , we know that $l$ can be $0, 1$ , or $2$ ; thus, there are three possible subshells. In standard notation, they are labeled the $3 s$ , $3 p$ , and $3 d$ subshells. We have already seen that 2 electrons can be in an $s$ state, and 6 in a $p$ state, but let us use the equation “maximum number of electrons that can be in a subshell = $2 (2 l + 1)$ ” to calculate the maximum number in each:
$\begin{array}{l} 3 s has l = 0; thus, 2 (2 l + 1) = 2 (0 + 1) = 2 \\ 3 p has l = 1; thus, 2 (2 l + 1) = 2 (2 + 1) = 6 \\ 3 d has l = 2; thus, 2 (2 l + 1) = 2 (4 + 1) = 10 \\ Total = 18 \\ (in the n = 3 shell) \end{array}$
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The equation “maximum number of electrons that can be in a shell = $2 n^{2}$ ” gives the maximum number in the $n = 3$ shell to be
$Maximum number of electrons = 2 n^{2} = 2 {(3)}^{2} = 2 (9) = 18.$
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Discussion
The total number of electrons in the three possible subshells is thus the same as the formula $2 n^{2}$ . In standard (spectroscopic) notation, a filled $n = 3$ shell is denoted as $3 s^{2} 3 p^{6} 3 d^{10}$ . Shells do not fill in a simple manner. Before the $n = 3$ shell is completely filled, for example, we begin to find electrons in the $n = 4$ shell.

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