Question 42.8: The Maximum Number of Electrons in Each Subshell Argue that ......
The Maximum Number of Electrons in Each Subshell
Argue that Pauli’s exclusion principle means that the maximum number of electrons allowed in each subshell is
N = 2(2\ell + 1) \quad \quad (42.27)Check this for the 2p subshell.
Interpret and Anticipate
A subshell is characterized by a particular value of n and \ell. Table 42.4 will help us to determine the possible quantum numbers for a subshell, and Pauli’s exclusion principle says there can only be one electron with each unique set of quantum numbers.
Solve
Because m=0, \pm 1, \pm 2, \ldots \pm \ell, for each value of \ell , there are 2\ell + 1 possible values of m.
The range in m allows for 2\ell + 1 electrons in a subshell.
For each value of m, the electron can be either spin up (m_s = 1/2) or spin down (m_s = -1/2). The spin magnetic quantum number doubles the number of electrons (found above) that can possibly be in the subshell.
N=2(2 \ell+1) \quad \checkmark\quad \quad (42.27)Check and Think
The 2p subshell has \ell= 1, so we expect N = 2(2 · 1 + 1) = 6 electrons. Check this by listing all the unique combinations of quantum numbers. (Since n = 2 and \ell= 1 for the 2p subshell, we don’t bother to list them separately here.)
6 combinations
| Table 42.4 Allowed values for quantum numbers. | ||
| Name | Symbol | Allowed Values |
| Principal | n | 1, 2, 3,… |
| Orbital | \ell | 0, 1, 2, … (n – 1) |
| Orbital magnetic | m | 0, \pm 1, \pm 2, \ldots \pm \ell |
| Spin | s | 1 / 2 |
| Spin magnetic | m_s | \pm 1 / 2 |