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29.9 The pauli exclusion principle  (Page 3/11)

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The number of electrons that can be in a subshell depends entirely on the value of l size 12{l} {} . Once l size 12{l} {} is known, there are a fixed number of values of m l size 12{m rSub { size 8{l} } } {} , each of which can have two values for m s size 12{m rSub { size 8{s} } } {} First, since m l size 12{m rSub { size 8{l} } } {} goes from − l size 12{ - l} {} to l in steps of 1, there are 2 l + 1 size 12{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 2 l + 1 size 12{2 left (2l+1 right )} {} .

For example, the 2 s size 12{2s} {} subshell in [link] has a maximum of 2 electrons in it, since 2 2 l + 1 = 2 0 + 1 = 2 size 12{2 left (2l+1 right )=2 left (0+1 right )=2} {} for this subshell. Similarly, the 2 p size 12{2p} {} subshell has a maximum of 6 electrons, since 2 2 l + 1 = 2 2 + 1 = 6 size 12{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 2 n 2 size 12{2n rSup { size 8{2} } } {} .

For example, for the first shell n = 1 size 12{n=1} {} , and so 2 n 2 = 2 size 12{2n rSup { size 8{2} } =2} {} . We have already seen that only two electrons can be in the n = 1 size 12{n=1} {} shell. Similarly, for the second shell, n = 2 size 12{n=2} {} , and so 2 n 2 = 8 size 12{2n rSup { size 8{2} } =8} {} . As found in [link] , the total number of electrons in the n = 2 size 12{n=2} {} shell is 8.

Subshells and totals for n = 3 size 12{n=3} {}

How many subshells are in the n = 3 size 12{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 size 12{2n rSup { size 8{2} } } {} .

Strategy

Subshells are determined by the value of l size 12{l} {} ; thus, we first determine which values of l size 12{ ital "ls"} {} are allowed, and then we apply the equation “maximum number of electrons that can be in a subshell = 2 2 l + 1 size 12{2 left (2l+1 right )} {} ” to find the number of electrons in each subshell.

Solution

Since n = 3 size 12{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 size 12{3d} {} subshells. We have already seen that 2 electrons can be in an s state, and 6 in a p size 12{p} {} state, but let us use the equation “maximum number of electrons that can be in a subshell = 2 2 l + 1 size 12{2 left (2l+1 right )} {} ” to calculate the maximum number in each:

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 )

The equation “maximum number of electrons that can be in a shell = 2 n 2 size 12{2n rSup { size 8{2} } } {} ” gives the maximum number in the n = 3 size 12{n=3} {} shell to be

Maximum number of electrons = 2 n 2 = 2 3 2 = 2 9 = 18.

Discussion

The total number of electrons in the three possible subshells is thus the same as the formula 2 n 2 size 12{2n rSup { size 8{2} } } {} . In standard (spectroscopic) notation, a filled n = 3 size 12{n=3} {} shell is denoted as 3 s 2 3 p 6 3 d 10 size 12{3s rSup { size 8{2} } 3p rSup { size 8{6} } 3d rSup { size 8{"10"} } } {} . Shells do not fill in a simple manner. Before the n = 3 size 12{n=3} {} shell is completely filled, for example, we begin to find electrons in the n = 4 size 12{n=4} {} shell.

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.

Electron configurations of elements hydrogen through calcium
Element Number of electrons (Z) Ground state configuration
H 1 1 s 1 size 12{1s rSup { size 8{1} } } {}
He 2 1 s 2 size 12{1s rSup { size 8{2} } } {}
Li 3 1 s 2 size 12{1s rSup { size 8{2} } } {} 2 s 1 size 12{2s rSup { size 8{1} } } {}
Be 4 " 2 s 2 size 12{2s rSup { size 8{2} } } {}
B 5 " 2 s 2 size 12{2s rSup { size 8{2} } } {} 2 p 1 size 12{2p rSup { size 8{1} } } {}
C 6 " 2 s 2 size 12{2s rSup { size 8{2} } } {} 2 p 2 size 12{2p rSup { size 8{2} } } {}
N 7 " 2 s 2 size 12{2s rSup { size 8{2} } } {} 2 p 3 size 12{2p rSup { size 8{3} } } {}
O 8 " 2 s 2 size 12{2s rSup { size 8{2} } } {} 2 p 4 size 12{2p rSup { size 8{4} } } {}
F 9 " 2 s 2 size 12{2s rSup { size 8{2} } } {} 2 p 5 size 12{2p rSup { size 8{5} } } {}
Ne 10 " 2 s 2 size 12{2s rSup { size 8{2} } } {} 2 p 6 size 12{2p rSup { size 8{6} } } {}
Na 11 " 2 s 2 size 12{2s rSup { size 8{2} } } {} 2 p 6 size 12{2p rSup { size 8{6} } } {} 3 s 1 size 12{3s rSup { size 8{1} } } {}
Mg 12 " " " 3 s 2 size 12{3s rSup { size 8{2} } } {}
Al 13 " " " 3 s 2 size 12{3s rSup { size 8{2} } } {} 3 p 1 size 12{3p rSup { size 8{1} } } {}
Si 14 " " " 3 s 2 size 12{3s rSup { size 8{2} } } {} 3 p 2 size 12{3p rSup { size 8{2} } } {}
P 15 " " " 3 s 2 size 12{3s rSup { size 8{2} } } {} 3 p 3 size 12{3p rSup { size 8{3} } } {}
S 16 " " " 3 s 2 size 12{3s rSup { size 8{2} } } {} 3 p 4 size 12{3p rSup { size 8{4} } } {}
Cl 17 " " " 3 s 2 size 12{3s rSup { size 8{2} } } {} 3 p 5 size 12{3p rSup { size 8{5} } } {}
Ar 18 " " " 3 s 2 size 12{3s rSup { size 8{2} } } {} 3 p 6 size 12{3p rSup { size 8{6} } } {}
K 19 " " " 3 s 2 size 12{3s rSup { size 8{2} } } {} 3 p 6 size 12{3p rSup { size 8{6} } } {} 4 s 1 size 12{4s rSup { size 8{1} } } {}
Ca 20 " " " " " 4 s 2 size 12{4s rSup { size 8{2} } } {}

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Source:  OpenStax, College physics -- hlca 1104. OpenStax CNX. May 18, 2013 Download for free at http://legacy.cnx.org/content/col11525/1.1
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