The Electron Configuration: Quantum Numbers: Videos & Practice Problems

Understanding quantum numbers is essential for determining the location and energy levels of electrons within an atom. The principal quantum number (n) indicates the size and energy of an electron shell, with higher numbers representing electrons in outer shells. The angular momentum quantum number (l), also known as the azimuthal quantum number, defines the shape of orbitals (s, p, d, f) within a subshell, with values of 0 for s, 1 for p, 2 for d, and 3 for f. The magnetic quantum number (m_l) specifies the orientation of an orbital within a subshell, with possible values ranging from -l to +l. Lastly, the spin quantum number (m_s) indicates the spin of an electron within an orbital, either +1/2 for up-spin or -1/2 for down-spin. These quantum numbers collectively describe an electron's unique position in an atom, adhering to the Aufbau principle which dictates the order of electron filling in orbitals.

Electron Configuration and Number of Electrons

To understand the relationship between quantum numbers and electron configurations, it's essential to first grasp how to number electrons within an electron orbital diagram. Electrons are numbered according to the Aufbau principle, which dictates the order in which orbitals are filled. This principle begins with the 1s orbital, followed by the 2s, then the 2p, and continues in this sequence through the various subshells.

By following this order, you can accurately assign numbers to the electrons as they fill the orbitals. Each electron configuration reflects the distribution of electrons among the available orbitals, and this distribution is directly linked to the quantum numbers that describe each electron's state. The quantum numbers include the principal quantum number (n), which indicates the energy level, the azimuthal quantum number (l), which defines the subshell, the magnetic quantum number (m_l), which specifies the orbital orientation, and the spin quantum number (m_s), which describes the electron's spin direction.

Understanding the Aufbau principle and the correct numbering of electrons is crucial for linking these quantum numbers to their respective electron configurations, thereby providing a comprehensive view of an atom's electronic structure.

Electron Configuration and Number of Electrons Example

To determine the number of a specific electron within an electron orbital diagram, we utilize the Aufbau principle, which guides the order in which electrons fill atomic orbitals. In this case, the electron configuration is represented as 1s² 2s² 2p⁶ 3s² 3p⁴.

We begin by numbering the electrons sequentially, starting from the lowest energy level. The filling order is as follows:

1. For the 1s orbital, we have two electrons: one spin-up (↑) and one spin-down (↓), numbered as 1 and 2.

2. Moving to the 2s orbital, we again fill with two electrons: 3 (↑) and 4 (↓).

3. The 2p orbitals consist of three degenerate orbitals. According to Hund's rule, we first fill each orbital with one electron before pairing them. Thus, we fill them as follows: 5 (↑), 6 (↑), 7 (↑), and then pair them: 8 (↓), 9 (↓), 10 (↓).

4. Next, we fill the 3s orbital with two electrons: 11 (↑) and 12 (↓).

5. Finally, for the 3p orbitals, we again follow Hund's rule. We fill them as: 13 (↑), 14 (↑), 15 (↑), and then pair them: 16 (↓).

In this configuration, the electron we are focusing on is the 14th electron, which is located in the 3p orbital. Understanding this numbering system is crucial as it lays the groundwork for connecting electrons to their respective quantum numbers, which describe their energy levels, angular momentum, and spin.

Quantum Numbers Review

Quantum numbers are essential values that describe the energy levels and locations of electrons within an atom. There are four primary quantum numbers, each serving a distinct purpose in defining the characteristics of electrons.

The principal quantum number, represented by the variable n, indicates the size and energy of an electron shell. For instance, if we consider the 7th shell, then n equals 7, which signifies that the electrons are located in this energy level.

The angular momentum quantum number, also known as the azimuthal quantum number and denoted by l, describes the shape of an orbital within a subshell. The subshells are categorized by the letters s, p, d, and f, corresponding to specific l values: s is 0, p is 1, d is 2, and f is 3. Understanding the subshell letter allows you to determine the l value.

The magnetic quantum number, represented as m_l, specifies the orientation of electrons within a set of orbitals. For example, in the case of 7 d orbitals, which consist of five orbitals, the m_l values are -2, -1, 0, +1, and +2. These values indicate the numerical location of each orbital within the d subshell.

Lastly, the spin quantum number, denoted by m_s, describes the spin of an electron within an orbital. Each orbital can accommodate a maximum of two electrons, with one spinning up (having a m_s value of +1/2) and the other spinning down (with a m_s value of -1/2).

Understanding these quantum numbers is crucial for grasping the fundamental principles of electron configuration and the behavior of atoms. They provide a framework for predicting how electrons are arranged and how they interact within an atom.

Quantum Numbers and Electron Configuration Example

To determine the set of four quantum numbers for the sixth electron of an oxygen atom, we first need to understand the electron configuration of oxygen. With an atomic number of 6, the electron configuration is represented as 1s² 2s² 2p⁴. This indicates that oxygen has a total of six electrons distributed across its orbitals.

In the electron orbital diagram, we fill the orbitals according to the Aufbau principle, Pauli exclusion principle, and Hund's rule. The filling sequence starts with the 1s orbital, which can hold two electrons (1s²), followed by the 2s orbital (2s²), and then the 2p orbitals. For the 2p orbitals, we first half-fill each of the three available orbitals before pairing up the electrons, resulting in the configuration of 2p⁴ being filled as follows: up, up, up, down, down.

To identify the quantum numbers for the sixth electron, we note its position in the orbital diagram. The sixth electron is the last one added to the 2p orbitals. The relevant quantum numbers are defined as follows:

• Principal quantum number (n): This indicates the energy level of the electron. For the sixth electron in the 2p orbital, n = 2.
• Azimuthal quantum number (l): This describes the subshell of the electron. For a p orbital, l = 1.
• Magnetic quantum number (m_l): This specifies the orientation of the orbital. The sixth electron occupies the m_l = 0 orbital within the 2p subshell.
• Spin quantum number (m_s): This indicates the spin direction of the electron. Since the sixth electron is represented as pointing up, m_s = +1/2.

Thus, the complete set of quantum numbers for the sixth electron in an oxygen atom is:

n = 2, l = 1, m_l = 0, m_s = +1/2.

Understanding these quantum numbers is essential for grasping the behavior of electrons in atoms and their arrangement in various energy levels and subshells.

Give the set of four quantum numbers that represent the indicated electron in each of the following elements. 
Mg (7th electron) 

• A. n = 2; l = 2; ml = 0; ms = +1/2
• B. n = 2; l = 1; ml = 0; ms = -1/2
• C. n = 2; l = 1; ml = +1; ms = +1/2
• D. n = 2; l = 1; ml = +1; ms = -1/2
• E. n = 2; l = 1; ml = -1; ms = +1/2

Answer: (C)

Give the set of four quantum numbers that represent the indicated electron in each of the following elements. 
Cl^– (18th electron)

• A. n = 3; l = 0; ml = +1; ms = -1/2
• B. n = 3; l = 0; ml = -1; ms = -1/2
• C. n = 3; l = 2; ml = +1; ms = -1/2
• D. n = 3; l = 1; ml = +1; ms = +1/2
• E. n = 3; l = 1; ml = +1; ms = -1/2

Answer: (E)

Give the set of four quantum numbers that represent the indicated electron in each of the following elements. 
Cr³⁺ (21^st electron) 

• A. n = 3; l = 1; ml = 0; ms = -1/2
• B. n = 3; l = 2; ml = 1; ms = -1/2
• C. n = 3; l = 1; ml = 1; ms = +1/2
• D. n = 3; l = 2; ml = 0; ms = +1/2
• E. n = 3; l = 1; ml = 0; ms = +1/2

Answer: (D)

Here’s what students ask on this topic:

What are the four quantum numbers and what do they represent?

The four quantum numbers are the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l), and the spin quantum number (m_s). The principal quantum number (n) indicates the size and energy level of an electron shell, with higher values representing electrons in outer shells. The angular momentum quantum number (l) defines the shape of the orbital within a subshell, with values of 0 for s, 1 for p, 2 for d, and 3 for f. The magnetic quantum number (m_l) specifies the orientation of an orbital within a subshell, with possible values ranging from -l to +l. The spin quantum number (m_s) indicates the spin of an electron within an orbital, either +1/2 for up-spin or -1/2 for down-spin.

How do you determine the electron configuration using the Aufbau principle?

The Aufbau principle states that electrons fill orbitals starting from the lowest energy level to the highest. To determine the electron configuration, begin with the 1s orbital and proceed to higher orbitals in the order of increasing energy: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, and so on. Each orbital can hold a specific number of electrons: s orbitals hold 2, p orbitals hold 6, d orbitals hold 10, and f orbitals hold 14. By following this order and filling each orbital according to its capacity, you can determine the electron configuration of an atom.

What is the significance of the magnetic quantum number (m_l)?

The magnetic quantum number (m_l) specifies the orientation of an orbital within a subshell. It determines the number of orbitals and their spatial orientation in a given subshell. The possible values of m_l range from -l to +l, where l is the angular momentum quantum number. For example, if l = 2 (d subshell), m_l can be -2, -1, 0, +1, or +2, corresponding to the five d orbitals. This quantum number is crucial for understanding the distribution of electrons in different orbitals and their magnetic properties.

How do the quantum numbers relate to the electron configuration of an atom?

Quantum numbers provide a unique address for each electron in an atom, describing its energy level, shape, orientation, and spin. The principal quantum number (n) indicates the electron's shell, the angular momentum quantum number (l) specifies the subshell and shape of the orbital, the magnetic quantum number (m_l) defines the orbital's orientation, and the spin quantum number (m_s) indicates the electron's spin direction. By following the Aufbau principle and using these quantum numbers, we can determine the electron configuration, which shows the arrangement of electrons in an atom's orbitals.

What is the role of the spin quantum number (m_s) in electron configuration?

The spin quantum number (m_s) indicates the spin direction of an electron within an orbital. It can have two possible values: +1/2 for an up-spin and -1/2 for a down-spin. Each orbital can hold a maximum of two electrons, and they must have opposite spins to comply with the Pauli exclusion principle. This principle states that no two electrons in an atom can have the same set of four quantum numbers. The spin quantum number helps in determining the specific arrangement of electrons within orbitals and ensures that electrons are paired correctly within each orbital.