Question

What atomic orbitals are related to a d sublevel?

Short answer

The atomic orbitals related to a d sublevel are dxy (\(m=-2\)), dxz (\(m=-1\)), dyz (\(m=0\)), dx²-y² (\(m=1\)), and d3z²-r² (\(m=2\)). These five orbitals have unique shapes and orientations in 3-dimensional space around the nucleus of the atom.

Step-by-step solution

1. Understanding orbital notation

Atomic orbitals are defined by three quantum numbers: the principal quantum number (n), the azimuthal quantum number (l), and the magnetic quantum number (m). The principal quantum number (n) defines the energy level and overall size of the orbital, while the azimuthal quantum number (l) corresponds to the sublevel and shape of the orbital, and the magnetic quantum number (m) distinguishes between different orbitals within the same sublevel.

2. Identifying the d sublevel

In the context of atomic orbitals, the azimuthal quantum number (l) determines the sublevel. The values of l are 0, 1, 2, 3, ... and correspond to the s, p, d, and f sublevels (and so on) respectively. Since we're dealing with the d sublevel, we know that the azimuthal quantum number (l) is 2.

3. Determining possible magnetic quantum numbers (m)

The magnetic quantum number (m) has values ranging from -l to +l, including zero. Since we know the value of l (which is 2), we can determine the possible values of m for the d sublevel. In this case, m can take the values -2, -1, 0, 1, or 2.

4. Listing the atomic orbitals related to the d sublevel

With the possible values of m defined, we can now list the atomic orbitals related to the d sublevel. Since there are five possible values for the magnetic quantum number (m) in the d sublevel, we have five different d atomic orbitals: - dxy (\(m = -2\)) - dxz (\(m = -1\)) - dyz (\(m = 0\)) - dx²-y² (\(m = 1\)) - d3z²-r² (\(m = 2\)) These five atomic orbitals are associated with the d sublevel, and each one has a unique shape and orientation in 3-dimensional space around the nucleus of the atom.

Key concepts

Quantum Numbers

Quantum numbers are essential in understanding atomic orbitals. They are like an address system for electrons in an atom, helping us locate and describe them. There are four quantum numbers:

• Principal quantum number (n): This indicates the main energy level or shell of the electron. It's a positive integer (1, 2, 3, ...), and the larger the number, the higher the energy level and larger the orbital.
• Azimuthal quantum number (l): This defines the shape of the orbital and the sublevel. It ranges from 0 to (n-1). Each value of l corresponds to a different sublevel (s, p, d, f, etc.).
• Magnetic quantum number (m): This describes the orientation of the orbital in space and ranges from -l to +l.
• Spin quantum number (ms): Indicates the spin direction of the electron, with possible values of +1/2 or -1/2.

Each of these numbers provides unique information about the electron's position and behavior within an atom.

d Sublevel

The d sublevel refers to the set of orbitals within an atom where the azimuthal quantum number l is equal to 2. This sublevel can exist in any principal quantum level starting from n = 3.
The d sublevel is composed of five distinct orbitals due to the range of possible magnetic quantum numbers (m), which also contribute to its varied geometric shapes and orientations.
In atoms, the d sublevel becomes more important as you move to transition metals, where it plays a crucial role in defining electron configurations and chemical properties.
The understanding of d orbitals is critical, especially in fields like chemistry and material science, as they greatly influence an atom's bonding and magnetic characteristics.

Magnetic Quantum Number

The magnetic quantum number, often represented by the symbol m, is pivotal in detailing the orientation of an orbital.
Given a specific azimuthal quantum number, the magnetic quantum number can take on multiple values between -l and +l, including zero. For the d sublevel, where l is 2, m can have five possible values: -2, -1, 0, 1, and 2.
These values correspond to the different d orbitals:

• dxy
• dxz
• dyz
• dx²-y²
• d3z²-r²

Each of these orbitals has a unique spatial orientation, affecting how they bond with other orbitals in chemical reactions. Understanding these orientations is vital for predicting the behavior of molecules, especially in complex transition metal chemistry.

Azimuthal Quantum Number

The azimuthal quantum number, symbolized by l, is key to defining an electron's sublevel in an atom. It not only determines the sublevel's shape but also the energy distribution among orbitals.
This number can take values from 0 up to n-1 for each principal quantum number (n). Each value corresponds to a specific type of sublevel:

• l = 0: s sublevel
• l = 1: p sublevel
• l = 2: d sublevel (as seen in our example)
• l = 3: f sublevel

In the context of electron configurations, the azimuthal quantum number is indispensable for predicting chemical properties and electronic structure, as it affects how different atoms interact and bond in various molecular forms.