30.9 The pauli exclusion principle  (Page 3/11)

The number of electrons that can be in a subshell depends entirely on the value of $l$ . Once $l$ is known, there are a fixed number of values of $m_l$ , each of which can have two values for $m_s$ First, since $m_l$ goes from $-l$ to l in steps of 1, there are $2l+1$ possibilities. This number is multiplied by 2, since each electron can be spin up or spin down. Thus the maximum number of electrons that can be in a subshell is $2\left(2l+1\right)$ .

For example, the $2s$ subshell in [link] has a maximum of 2 electrons in it, since $2\left(2l+1\right)=2\left(0+1\right)=2$ for this subshell. Similarly, the $2p$ subshell has a maximum of 6 electrons, since $2\left(2l+1\right)=2\left(2+1\right)=6$ . For a shell, the maximum number is the sum of what can fit in the subshells. Some algebra shows that the maximum number of electrons that can be in a shell is $2n^2$ .

For example, for the first shell $n=1$ , and so $2n^2=2$ . We have already seen that only two electrons can be in the $n=1$ shell. Similarly, for the second shell, $n=2$ , and so $2n^2=8$ . As found in [link] , the total number of electrons in the $n=2$ shell is 8.

Shell filling and the periodic table

[link] shows electron configurations for the first 20 elements in the periodic table, starting with hydrogen and its single electron and ending with calcium. The Pauli exclusion principle determines the maximum number of electrons allowed in each shell and subshell. But the order in which the shells and subshells are filled is complicated because of the large numbers of interactions between electrons.