Shells and Subshells
Because of the Pauli exclusion principle, only hydrogen and helium can have all of their electrons in the state. Lithium—see the periodic table—has three electrons, and so one must be in the level. This leads to the concept of shells and shell filling. As we progress up in the number of electrons, we go from hydrogen to helium, lithium, beryllium, boron, and so on, and we see that there are limits to the number of electrons for each value of Higher values of the shell correspond to higher energies, and they can allow more electrons because of the various combinations of and that are possible. Each value of the principal quantum number thus corresponds to an atomic shell into which a limited number of electrons can go. Shells and the number of electrons in them determine the physical and chemical properties of atoms, since it is the outermost electrons that interact most with anything outside the atom.
The probability clouds of electrons with the lowest value of are closest to the nucleus and, thus, more tightly bound. Thus when shells fill, they start with progress to and so on. Each value of thus corresponds to a subshell.
The table given below lists symbols traditionally used to denote shells and subshells.
Shell |
Subshell |
|
|
Symbol |
1 |
0 |
|
2 |
1 |
|
3 |
2 |
|
4 |
3 |
|
5 |
4 |
|
|
5 |
|
|
6 |
|
Table 13.2 Shell and Subshell Symbols
To denote shells and subshells, we write with a number for and a letter for For example, an electron in the state must have and it is denoted as a electron. Two electrons in the state is denoted as Another example is an electron in the state with written as The case of three electrons with these quantum numbers is written This notation, called spectroscopic notation, is generalized as shown in Figure 13.60.
Counting the number of possible combinations of quantum numbers allowed by the exclusion principle, we can determine how many electrons it takes to fill each subshell and shell.
Example 13.4 How Many Electrons Can Be in This Shell?
List all the possible sets of quantum numbers for the shell, and determine the number of electrons that can be in the shell and each of its subshells.
Strategy
Given for the shell, the rules for quantum numbers limit to be 0 or 1. The shell therefore has two subshells, labeled and Since the lowest subshell fills first, we start with the subshell possibilities and then proceed with the subshell.
Solution
It is convenient to list the possible quantum numbers in a table, as shown below.
Discussion
It is laborious to make a table like this every time we want to know how many electrons can be in a shell or subshell. There exist general rules that are easy to apply, as we shall now see.
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 one, there are possibilities. This number is multiplied by two, 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 Example 13.4 has a maximum of two electrons in it, since for this subshell. Similarly, the subshell has a maximum of six 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 Example 13.4, the total number of electrons in the shell is eight.
Example 13.5 Subshells and Totals for
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 two electrons can be in an state, and six 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:
13.55 The equation maximum number of electrons that can be in a shell = gives the maximum number in the shell to be
13.56
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.