8.4 The exclusion principle and the periodic table (Page 2/11)

University physics volume 3 Page 2 / 11

The symbol

(↑)

indicates an unpaired electron in the outer shell, whereas the symbol

(↑ ↓)

indicates a pair of spin-up and -down electrons in an outer shell.

Electron configurations of electrons in an atom
Element	Electron Configuration	Spin Alignment
H	$1 s^{1}$	$(↑)$
He	$1 s^{2}$	$(↑ ↓)$
Li	$1 s^{2} 2 s^{1}$	$(↑)$
Be	$1 s^{2} 2 s^{2}$	$(↑ ↓)$
B	$1 s^{2} 2 s^{2} 2 p^{1}$	$(↑ ↓) (↑)$
C	$1 s^{2} 2 s^{2} 2 p^{2}$	$(↑ ↓) (↑) (↑)$
N	$1 s^{2} 2 s^{2} 2 p^{3}$	$(↑ ↓) (↑) (↑) (↑)$
O	$1 s^{2} 2 s^{2} 2 p^{4}$	$(↑ ↓) (↑ ↓) (↑) (↑)$
F	$1 s^{2} 2 s^{2} 2 p^{5}$	$(↑ ↓) (↑ ↓) (↑ ↓) (↑)$
Ne	$1 s^{2} 2 s^{2} 2 p^{6}$	$(↑ ↓) (↑ ↓) (↑ ↓) (↑ ↓)$
Na	$1 s^{2} 2 s^{2} 2 p^{6} 3 s^{1}$	$(↑)$
Mg	$1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2}$	$(↑ ↓)$
Al	$1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{1}$	$(↑ ↓) (↑)$

The maximum number of electrons in a subshell depends on the value of the angular momentum quantum number, l . For a given a value l , there are $2 l + 1$ orbital angular momentum states. However, each of these states can be filled by two electrons (spin up and down, $↑ ↓$ ). Thus, the maximum number of electrons in a subshell is

N = 2 (2 l + 1) = 4 l + 2 .

In the 2 s $(l = 0)$ subshell, the maximum number of electrons is 2. In the 2 p ( $l = 1$ ) subshell, the maximum number of electrons is 6. Therefore, the total maximum number of electrons in the $n = 2$ shell (including both the $l = 0$ and 1 subshells) is $2 + 6$ or 8. In general, the maximum number of electrons in the n th shell is $2 n^{2} .$

Subshells and totals for $n = 3$

How many subshells are in the

n = 3

shell? Identify each subshell and calculate the maximum number of electrons that will fill each. Show that the maximum number of electrons that fill an atom is

2 n^{2}

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

Because

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 two electrons can be in an s state, and six 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{matrix} 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{matrix}

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 .

Significance

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.

Got questions? Get instant answers now!

The structure of the periodic table ( [link] ) can be understood in terms of shells and subshells, and, ultimately, the total energy, orbital angular momentum, and spin of the electrons in the atom. A detailed discussion of the periodic table is left to a chemistry course—we sketch only its basic features here. In this discussion, we assume that the atoms are electrically neutral; that is, they have the same number of electrons and protons. (Recall that the total number of protons in an atomic nucleus is called the atomic number, Z .)

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Source: OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4

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