In this explainer, we will learn how to use quantum numbers to describe an electron within an atom.

An electron within an atom can be completely described with values that are known as quantum numbers. There are four quantum numbers (, , , and ), and they determine how electrons successively fill atomic orbitals. The four quantum numbers also explain why elements should be grouped into periodic table blocks and why so many elements have similar chemical properties.

The Pauli exclusion principle states that no two electrons in any one atom can have the same set of four quantum numbers. The highest-energy electron in a potassium atom has one set of four quantum numbers, and the highest-energy electron in a cesium atom has a different set of four quantum numbers.

The principal quantum number determines the size of an atomic orbital. The principal quantum number is always a positive integer, and it can be stated that

The 3s atomic orbital has a principal quantum number of and the 2s and 1s atomic orbitals have principal quantum numbers of and respectively. The 3s atomic orbital is wider than the 2s atomic orbital and the 2s atomic orbital is wider than the 1s atomic orbital. The use of “wider” here is to describe the effective radius of an atomic orbital relative to the central section of an atomic nucleus.

### Definition: Principal Quantum Number (𝑛)

The principal quantum number determines the size of an atomic orbital, and it can have any value that is a positive integer from one to seven.

The principal quantum number is based on the Bohr model of the atom, and it determines which energy level or shell an electron will occupy. The principal quantum number can be squared to determine how many orbitals there are per energy level. It can also be squared and multiplied by two to determine how many electrons there are per electron shell. Chemists sometimes use capital letters to describe particular electron shells like the or electron shell. The letter K is used for the electron shell and the letter L is used for the electron shell. This information is summarized in the following table.

Principal Quantum Number | Electron Shell Notation | Number of Orbitals | Number of Electrons |
---|---|---|---|

1 | K | 1 | 2 |

2 | L | 4 | 8 |

3 | M | 9 | 18 |

4 | N | 16 | 32 |

The formula is generally not applied to principal quantum numbers that are greater than or equal to five because the fifth and higher electron shells contain some subshells that are not occupied by the electrons of any known chemical elements. The fifth electron shell contains the 5f subshell that is not occupied by the electrons of any known chemical element and the sixth electron shell has the 6f and 6g subshells that are not occupied by the electrons of any known chemical elements.

### Example 1: Calculating the Number of Atomic Orbitals from the Principal Quantum Number

What is the relationship between the principal quantum number, , and the total number of orbitals?

### Answer

The principal quantum number determines the size of all the atomic orbitals, and it can be used to determine the total number of orbitals and electrons in any one energy level. The principal quantum number is squared to determine the number of orbitals in any one energy level, and it is squared and multiplied by two to determine the total number of electrons in any one energy level. The formula for calculating the number of orbitals can be expressed as , and the formula for calculating the total number of electrons can be expressed as . We can use these statements to determine that option C is the correct answer for this question.

The second quantum number can be referred to as the subsidiary, azimuthal, or orbital angular momentum quantum number. The subsidiary quantum number determines the shape of an atomic orbital. Subshells with a subsidiary quantum number of 0 have a spherical shape and are termed s-type subshells. The 1s and 2s orbitals both have a subsidiary quantum number of zero.

### Definition: Subsidiary Quantum Number (𝑙)

The subsidiary quantum number describes the shape of an atomic orbital, and it is described by the expression , where is the principal quantum number.

The subsidiary quantum number can have any integer value ranging from 0 to . This statement could also be rephrased to state that where is the principal quantum number.

This means that the first electron shell can only have subshells with a subsidiary quantum number of zero because when , and therefore can only equal 0. The second electron shell can have subsidiary quantum numbers of both zero and one because when and so . The third electron shell can have subshells with subsidiary quantum numbers of zero, one, and two because when and so . Physical chemists classify the first few subshells as follows: zero (s), one (p), two (d), and three (f). The relationship between the values of and the different types of subshells is shown in the table below.

Value of | Subshell |
---|---|

0 | s |

1 | p |

2 | d |

3 | f |

The 1s subshell has a principal quantum number of one and a subsidiary quantum number of zero . Electrons always fill the 1s subshell first and then they fill the 2s and 2p subshells after that. Electrons tend to always fill the lowest-energy subshells first and then they fill the higher-energy subshells after that. The order of subshell energies in any one electron shell can be described by the expression . The s subshell always has the lowest energy in any one electron shell and the p subshell has the next lowest energy value.

The s subshells are spherical and the p subshells are shaped like dumbbells. The d and f orbitals have much more complex geometries and they are not easy to draw or to describe in a single sentence. The following image shows the relatively simple shapes of the s and p orbitals first and the much more complex shapes of the d and f orbitals second.

The magnetic quantum number determines how many orbitals there are per subshell because can have any value that ranges from through to . This statement could also be rephrased to state that where is a subsidiary quantum number.

This means that s subshells can only have one orbital but p subshells can have three orbitals. The d subshells can have five orbitals because can have any value between and . The total number of orbitals per subshell can always be determined with the formula.

### Example 2: Determining How Many Electrons Are in a Subshell with 𝑛 = 2 and 𝑙 = 1

How many electrons in total can have the quantum numbers and ?

### Answer

The question is focusing on the electron subshell that has a principal quantum number of two and a subsidiary quantum number of 1 . The magnetic quantum number is known to determine how many orbitals there are per subshell because can have any value that ranges from through to . This statement could also be rephrased to state that where is a subsidiary quantum number. We can use this statement to determine that the magnetic quantum number can have values of , 0, and here since . This conclusion can be used to determine that the question is focusing on a subshell that contains a total of three orbitals.

We know that each orbital can hold up to two electrons. This means there can be six electrons in the subshell with and because it contains three orbitals. The correct answer for this question is the number six (6).

The magnetic quantum number is commonly related to the orientation in space of each orbital within a subshell.

### Definition: Magnetic Quantum Number (𝑚_{𝑙})

The magnetic quantum number is commonly related to the orientation in space of each orbital within a subshell and it is described by the expression , where is a subsidiary quantum number.

The 2p orbital can be used as a representative example to better understand the magnetic quantum number. The 2p orbital has a principal quantum number of two and a subsidiary quantum number of one . The 2p orbital has three different atomic orbitals that have magnetic quantum numbers of , 0, and . The orbitals correspond to the three separate , , and orbitals. The following table shows the different possible magnetic quantum numbers and orbitals for the first four subsidiary quantum numbers. It is important to stress here that the table is filled for the sake of completeness and to help students understand how the values of the subsidiary and magnetic quantum numbers determine the numbers of orbitals in an electron subshell. Our content will only assess knowledge of the orbitals that make up the s or p subshells and not the orbitals that make up any d or f type subshells.

Value of | Subshell | Value of | Possible Orbitals |
---|---|---|---|

0 | s | 0 | s |

1 | p | , , | |

2 | d | , , , , | |

3 | f | , , , , , , |

### Example 3: Understanding the Relationship between the 𝑙 and 𝑚_{𝑙} Quantum Numbers

If , how many possible values of are there?

### Answer

The subsidiary quantum number determines the shape of an atomic orbital and it can have any integer value that ranges from 0 through to . The magnetic quantum number is commonly related to the orientation in space of each orbital within a subshell, and it can have any integer value that ranges from through to . The to range includes the value of zero.

The magnetic quantum number would only be able to have a value of zero if the subsidiary quantum number has a value of zero because and . Therefore, there is only one possible value of when . The correct answer is the number one (1).

The first three quantum numbers determine the size , shape ,
and orientation of an atomic orbital. The last quantum number has been termed the *spin
quantum number* and it determines the spin state of an electron. It is important to
stress here that spin is considered to be an intrinsic property and electrons should not be thought of as discrete spheres
that rotate about one principal axis like Earth. Electrons can have spin quantum numbers of either
or . Each atomic orbital can hold one spin-up-state
electron and one spin-down-state electron . This explains why each s-type subshell can hold two electrons and why each one of the three
p-type orbitals in any electron shell can hold two electrons.

### Definition: Spin Quantum Number (𝑚_{𝑠})

The spin quantum number determines the spin state of an electron and every atomic orbital can hold one spin-up-state electron and a second spin-down-state electron .

Periodic tables are broken down into blocks that are based on the subsidiary quantum number and rows that are based on the principal quantum number . There is a section of the periodic table that corresponds to the 1s atomic orbital and sections that correspond to the other atomic orbitals such as the 2p and 3d atomic orbitals. The size of each periodic table block is partly determined with the spin quantum number because each atomic orbital can contain one spin-up-state electron and a second spin-down-state electron. We can read a periodic table from the lowest atomic number through to the highest atomic number to determine how atomic orbitals tend to be filled one after another.

### Example 4: Determining the Quantum Numbers That Represent an Electron in an Atom

The quantum numbers for the valence electrons in an atom of lithium are , , , . What are the quantum numbers for the second valence electron in an atom of beryllium?

### Answer

The periodic table can be divided into sections that are associated with different types of atomic orbitals. Periodic tables are broken down into blocks that are based on the subsidiary quantum number and rows that are based on the principal quantum number . There is a section of the periodic table that corresponds to the 1s atomic orbital and sections that correspond to the other atomic orbitals such as the 2s and 3d atomic orbitals.

The electrons of any one atom tend to fill the lowest-energy atomic orbitals before they start to fill other higher-energy atomic orbitals. The lithium atom has three electrons and its first two electrons fill the lowest-energy 1s atomic orbital and its valence electron occupies the slightly higher-energy 2s atomic orbital. The 2s atomic orbital has a principal quantum number of two and a subsidiary quantum number of zero . The single valence electron of a lithium atom must have a principal quantum number of two and a subsidiary quantum number of zero because it is in the 2s atomic orbital. The magnetic quantum number of this electron is zero because , and we have confirmed that . It must also have a spin quantum number of positive one-half because by convention electrons occupy a spin-up state before they occupy a spin-down state .

Beryllium has four electrons and its two lowest-energy electrons fill its 1s atomic orbital. The other two electrons fill the slightly higher-energy 2s atomic orbital. We have already confirmed through comparison that electrons in the 2s atomic orbital have a principal quantum number of two and a subsidiary quantum number of zero , so the first two quantum numbers of these valence electrons are and . We can then determine that the magnetic quantum number of these electrons is zero because , and we have confirmed that . The lowest-energy valence electron has a spin quantum number of positive one-half and this enables us to determine that the slightly higher-energy valence electron must have a spin quantum number of negative one-half . We can put all of this information together to determine that the second valence electron of a beryllium has the following four quantum numbers: , , , and .

### Key Points

- Each electron within an atom can be described with its own set of four quantum numbers.
- The principal quantum number determines the size of an atomic orbital and can have any value that is a positive integer.
- The subsidiary quantum number describes the shape of an atomic orbital and is described by the expression , where is the principal quantum number.
- The magnetic quantum number is commonly related to the orientation in space of each orbital within a subshell and is described by the expression , where is a subsidiary quantum number.
- The spin quantum number determines the spin state of an electron and every atomic orbital can hold one spin-up-state electron and a second spin-down-state electron .