Question

(a) When \(n=4,\) what are the possible values of \(\ell ?\) (b) When \(\ell\) is \(2,\) what are the possible values of \(m_{\ell} ?\) (c) For a \(4 s\) orbital, what are the possible values of \(n, \ell,\) and \(m_{\ell} ?\) (d) For a 4 forbital, what are the possible values of \(n, \ell,\) and \(m_{\ell} ?\)

Short answer

(a) 0, 1, 2, 3; (b) -2, -1, 0, 1, 2; (c) \(n=4, \ell=0, m_\ell=0\); (d) \(n=4, \ell=3, m_\ell=-3 \text{ to } 3\).

Step-by-step solution

Determine possible values of \( \ell \) for \( n=4 \)

The quantum number \( \ell \), the azimuthal quantum number, can have integer values ranging from 0 to \( n-1 \). Here, \( n = 4 \), so the possible values of \( \ell \) are 0, 1, 2, and 3.

Determine possible values of \( m_\ell \) for \( \ell=2 \)

The magnetic quantum number \( m_\ell \) can have integer values ranging from \(-\ell\) to \(\ell\). For \( \ell = 2 \), the possible values of \( m_\ell \) are -2, -1, 0, 1, and 2.

Determine possible quantum numbers for 4s orbital

For a 4s orbital, we assign \( n = 4 \). An s orbital corresponds to \( \ell = 0 \). Therefore, for this orbital, \( n = 4 \), \( \ell = 0 \), and since \( m_\ell \) ranges from \(-\ell\) to \(\ell\), \( m_\ell = 0 \).

Determine possible quantum numbers for 4f orbital

For a 4f orbital, we assign \( n = 4 \). An f orbital corresponds to \( \ell = 3 \). Therefore, \( n = 4 \), \( \ell = 3 \), and \( m_\ell \) ranges from -3 to 3. Thus, the possible values of \( m_\ell \) are -3, -2, -1, 0, 1, 2, and 3.

Key concepts

Azimuthal Quantum Number

The azimuthal quantum number, often denoted as \( \ell \), plays a crucial role in understanding the shape and angular momentum of an electron's orbital. It's more commonly referred to as the angular momentum quantum number or the subshell quantum number. This quantum number provides insight into the shape of the electron cloud surrounding the nucleus of an atom. The azimuthal quantum number can have integer values ranging from \( 0 \) to \( n - 1 \), where \( n \) is the principal quantum number representing the principal energy level.

• For \( \ell = 0 \), the subshell is an s orbital, which is spherical.
• For \( \ell = 1 \), the subshell is a p orbital, often depicted as a dumbbell shape.
• For \( \ell = 2 \), it forms a d orbital with a more complex shape.
• For \( \ell = 3 \), it constitutes an f orbital, characterized by even more intricacies in shape.

The different shapes influence the electron’s distribution in space and also the energy levels of an atom. Understanding these shapes helps us predict how electrons occupy different energy levels, which is key in the study of chemical bonds and reactions.

Magnetic Quantum Number

The magnetic quantum number, represented as \( m_\ell \), specifies the orientation of an orbital in space. Given a value of \( \ell \), the magnetic quantum number can assume values that range from \( -\ell \) to \( \ell \), including zero. This results in a total of \( 2\ell + 1 \) possible values for \( m_\ell \). These values pertain to the orientation of the shape of the orbital in three-dimensional space.

• For instance, if \( \ell = 2 \), \( m_\ell \) can take the values -2, -1, 0, 1, and 2.
• Each different value corresponds to a specific orientation of the d orbital.

Magnetic quantum numbers are essential for understanding how electrons behave in magnetic fields, which plays a crucial part in the electron's magnetic sub-properties. These properties directly impact the chemical behavior and interaction of atoms, manifested through spectral lines seen in spectroscopy.

Orbitals

In atomic theory, orbitals represent regions in an atom where there is a high probability of locating an electron. They are defined by the quantum numbers \( n \), \( \ell \), and \( m_\ell \), each contributing to the properties and shapes of these orbitals.

• The principal quantum number \( n \) determines the orbital's size and energy level.
• The azimuthal quantum number \( \ell \) dictates the orbital's shape.
• The magnetic quantum number \( m_\ell \) provides the orbital's orientation in space.

Orbitals can be thought of as a three-dimensional map of where electrons are likely to be found. For each combination of quantum numbers \( n \), \( \ell \), and \( m_\ell \), a distinct orbital is defined, representing a specific electron configuration and associated energy states. As such, orbitals form the foundation of atomic interactions, determining how atoms bond and react with each other to form molecules.