# 30.9 The pauli exclusion principle  (Page 5/11)

 Page 5 / 11

Of course, other groups are also of interest. Carbon, silicon, and germanium, for example, have similar chemistries and are in Group 4 (Group IV). Carbon, in particular, is extraordinary in its ability to form many types of bonds and to be part of long chains, such as inorganic molecules. The large group of what are called transitional elements is characterized by the filling of the $d$ subshells and crossing of energy levels. Heavier groups, such as the lanthanide series, are more complex—their shells do not fill in simple order. But the groups recognized by chemists such as Mendeleev have an explanation in the substructure of atoms.

## Phet explorations: build an atom

Build an atom out of protons, neutrons, and electrons, and see how the element, charge, and mass change. Then play a game to test your ideas!

## Section summary

• The state of a system is completely described by a complete set of quantum numbers. This set is written as $\left(\mathrm{n, l,}\phantom{\rule{0.25em}{0ex}}{m}_{l},\phantom{\rule{0.25em}{0ex}}{m}_{s}\right)$ .
• The Pauli exclusion principle says that no two electrons can have the same set of quantum numbers; that is, no two electrons can be in the same state.
• This exclusion limits the number of electrons in atomic shells and subshells. Each value of $n$ corresponds to a shell, and each value of $l$ corresponds to a subshell.
• The maximum number of electrons that can be in a subshell is $2\left(2l+1\right)$ .
• The maximum number of electrons that can be in a shell is $2{n}^{2}$ .

## Conceptual questions

Identify the shell, subshell, and number of electrons for the following: (a) $2{p}^{3}$ . (b) $4{d}^{9}$ . (c) $3{s}^{1}$ . (d) $5{g}^{\text{16}}$ .

Which of the following are not allowed? State which rule is violated for any that are not allowed. (a) $1{p}^{3}$ (b) $2{p}^{8}$ (c) $3{g}^{\text{11}}$ (d) $4{f}^{2}$

## Problem exercises

(a) How many electrons can be in the $n=4$ shell?

(b) What are its subshells, and how many electrons can be in each?

(a) 32. (b) $2\phantom{\rule{0.25em}{0ex}}\text{in}\phantom{\rule{0.25em}{0ex}}s,\phantom{\rule{0.25em}{0ex}}\text{6 in}\phantom{\rule{0.25em}{0ex}}p,\phantom{\rule{0.25em}{0ex}}\text{10 in}\phantom{\rule{0.25em}{0ex}}d,$ and 14 in $f$ , for a total of 32.

(a) What is the minimum value of 1 for a subshell that has 11 electrons in it?

(b) If this subshell is in the $n=\text{5}$ shell, what is the spectroscopic notation for this atom?

(a) If one subshell of an atom has 9 electrons in it, what is the minimum value of $l$ ? (b) What is the spectroscopic notation for this atom, if this subshell is part of the $n=3$ shell?

(a) 2

(b) $3{d}^{9}$

(a) List all possible sets of quantum numbers $\left(n,\phantom{\rule{0.25em}{0ex}}l,\phantom{\rule{0.25em}{0ex}}{m}_{l},\phantom{\rule{0.25em}{0ex}}{m}_{s}\right)$ for the $n=3$ shell, and determine the number of electrons that can be in the shell and each of its subshells.

(b) Show that the number of electrons in the shell equals $2{n}^{2}$ and that the number in each subshell is $2\left(2l+1\right)$ .

Which of the following spectroscopic notations are not allowed? (a) $5{s}^{1}$ (b) $1{d}^{1}$ (c) $4{s}^{3}$ (d) $3{p}^{7}$ (e) $5{g}^{15}$ . State which rule is violated for each that is not allowed.

(b) $n\ge l$ is violated,

(c) cannot have 3 electrons in $s$ subshell since $3>\left(2l+1\right)=2\phantom{\rule{0.25em}{0ex}}$

(d) cannot have 7 electrons in $p$ subshell since $7>\left(2l+1\right)=2\left(2+1\right)=6$

Which of the following spectroscopic notations are allowed (that is, which violate none of the rules regarding values of quantum numbers)? (a) $1{s}^{1}$ (b) $1{d}^{3}$ (c) $4{s}^{2}$ (d) $3{p}^{7}$ (e) $6{h}^{\text{20}}$

(a) Using the Pauli exclusion principle and the rules relating the allowed values of the quantum numbers $\left(n,\phantom{\rule{0.25em}{0ex}}l,\phantom{\rule{0.25em}{0ex}}{m}_{l},\phantom{\rule{0.25em}{0ex}}{m}_{s}\right)$ , prove that the maximum number of electrons in a subshell is $2{n}^{2}$ .

(b) In a similar manner, prove that the maximum number of electrons in a shell is 2 n 2 .

(a) The number of different values of ${m}_{l}$ is $±l,±\left(l-1\right),\text{...,}0$ for each $l>0$ and one for $l=0⇒\left(2l+1\right)\text{.}$ Also an overall factor of 2 since each ${m}_{l}$ can have ${m}_{s}$ equal to either $+1/2$ or $-1/2⇒2\left(2l+1\right)$ .

(b) for each value of $l$ , you get $2\left(2l+1\right)$

$=\text{0, 1, 2, ...,}\left(n\text{–1}\right)⇒2\left\{\left[\left(2\right)\left(0\right)+1\right]+\left[\left(2\right)\left(1\right)+1\right]+\text{.}\text{.}\text{.}\text{.}+\left[\left(2\right)\left(n-1\right)+1\right]\right\}=\underset{\underset{n\phantom{\rule{0.25em}{0ex}}\text{terms}}{︸}}{2\left[1+3+\text{.}\text{.}\text{.}+\left(2n-3\right)+\left(2n-1\right)\right]}$ to see that the expression in the box is $={n}^{2},$ imagine taking $\left(n-1\right)$ from the last term and adding it to first term $=2\left[1+\left(n\text{–1}\right)+3+\text{.}\text{.}\text{.}+\left(2n-3\right)+\left(2n-1\right)–\left(n-1\right)\right]=2\left[n+3+\text{.}\text{.}\text{.}\text{.}+\left(2n-3\right)+n\right]\text{.}$ Now take $\left(n-3\right)$ from penultimate term and add to the second term $2\underset{\underset{n\phantom{\rule{0.25em}{0ex}}\text{terms}}{︸}}{\left[n+n+\text{.}\text{.}\text{.}+n+n\right]}={2n}^{2}$ .

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