Question

\(l=3\), then the values of magnetic quantum numbers are? (a) \(\pm 1, \pm 2, \pm 3\) (b) \(0, \pm 1, \pm 2, \pm 3\) (c) \(-1,-2,-3\) (d) \(0,+1,+2,+3\)

Short answer

The correct answer is (b) \(0, \pm 1, \pm 2, \pm 3\).

Step-by-step solution

Understanding the Magnetic Quantum Number

The magnetic quantum number, denoted as \(m\), defines the orientation of the orbital in space relative to the other orbitals and is related to the azimuthal quantum number \(l\). The possible values of \(m\) range from \(-l\) to \(+l\).

Determine the Range for \(m\) When \(l=3\)

Since the azimuthal quantum number \(l=3\), the possible values for the magnetic quantum number \(m\) are given by the integer values from \(-l\) to \(+l\). Thus, \(m = -3, -2, -1, 0, +1, +2, +3\).

Select the Correct Answer

Compare the list of possible \(m\) values \((-3, -2, -1, 0, +1, +2, +3)\) with the provided options. Option (b) \(0, \pm 1, \pm 2, \pm 3\) matches this range.

Key concepts

Azimuthal Quantum Number

In the realm of quantum chemistry, the azimuthal quantum number, often symbolized as \(l\), plays a vital role.
It is also referred to as the angular momentum quantum number. This number indicates the shape of the electron cloud, which dictates the shape of the atomic orbital.
The azimuthal quantum number can take any integer value from 0 to \(n-1\), where \(n\) is the principal quantum number. These values associate with different types of orbitals:

• \(l = 0\) refers to s-orbitals,
• \(l = 1\) pertains to p-orbitals,
• \(l = 2\) is related to d-orbitals,
• \(l = 3\) corresponds to f-orbitals, and so on.

Thus, if you have \(l = 3\), your electron is in an f-orbital. This designation helps chemists understand differences in electron cloud shapes and their possible orientations in space.

Quantum Mechanics in Chemistry

Quantum mechanics provides the foundation for understanding atomic and molecular structures in chemistry.
In the quantum model, electrons are not depicted as particles in fixed paths but as a probability distribution or a cloud around the nucleus.
This probabilistic nature of electrons is captured through quantum numbers:

• The principal quantum number \(n\), which represents the energy level,
• The azimuthal quantum number \(l\) for orbital shape,
• The magnetic quantum number \(m\) for orientation,
• And the spin quantum number \(s\) for direction of the spin.

Each electron in an atom is described by a unique set of these quantum numbers.
This specificity helps explain the chemical behavior of atoms as well as the periodic table's structure.

Atomic Orbitals

Atomic orbitals are regions within an atom where there is a high probability of finding electrons.
Each orbital can hold a maximum of two electrons, each with opposite spins.
The shape and orientation of these orbitals are defined by the quantum numbers:

• The s-orbitals are spherical,
• p-orbitals resemble dumbbells,
• d-orbitals have more complex clover shapes,
• f-orbitals are even more intricate, usually described by a combination of lobes.

The specific arrangement of electrons within these orbitals influences chemical bonding and molecular geometry.
Thus, understanding the concept of atomic orbitals is fundamental in predicting and explaining the chemical properties and reactivity of elements.