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The number of electrons that can be in a subshell depends entirely on the value of . Once is known, there are a fixed number of values of , each of which can have two values for First, since goes from to l in steps of 1, there are 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 .
For example, the subshell in [link] has a maximum of 2 electrons in it, since for this subshell. Similarly, the subshell has a maximum of 6 electrons, since . 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 .
For example, for the first shell , and so . We have already seen that only two electrons can be in the shell. Similarly, for the second shell, , and so . As found in [link] , the total number of electrons in the shell is 8.
How many subshells are in the shell? Identify each subshell, calculate the maximum number of electrons that will fit into each, and verify that the total is .
Strategy
Subshells are determined by the value of ; thus, we first determine which values of are allowed, and then we apply the equation “maximum number of electrons that can be in a subshell ” to find the number of electrons in each subshell.
Solution
Since , we know that can be , or ; thus, there are three possible subshells. In standard notation, they are labeled the , , and subshells. We have already seen that 2 electrons can be in an state, and 6 in a state, but let us use the equation “maximum number of electrons that can be in a subshell = ” to calculate the maximum number in each:
The equation “maximum number of electrons that can be in a shell = ” gives the maximum number in the shell to be
Discussion
The total number of electrons in the three possible subshells is thus the same as the formula . In standard (spectroscopic) notation, a filled shell is denoted as . Shells do not fill in a simple manner. Before the shell is completely filled, for example, we begin to find electrons in the shell.
[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.
Element | Number of electrons (Z) | Ground state configuration | |||||
---|---|---|---|---|---|---|---|
H | 1 | ||||||
He | 2 | ||||||
Li | 3 | ||||||
Be | 4 | " | |||||
B | 5 | " | |||||
C | 6 | " | |||||
N | 7 | " | |||||
O | 8 | " | |||||
F | 9 | " | |||||
Ne | 10 | " | |||||
Na | 11 | " | |||||
Mg | 12 | " | " | " | |||
Al | 13 | " | " | " | |||
Si | 14 | " | " | " | |||
P | 15 | " | " | " | |||
S | 16 | " | " | " | |||
Cl | 17 | " | " | " | |||
Ar | 18 | " | " | " | |||
K | 19 | " | " | " | |||
Ca | 20 | " | " | " | " | " |
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