Question

What are the possible values for the quantum numbers \(n, \ell,\) and \(m_{\ell} ?\)

Short answer

The possible values for the quantum numbers \(n\), \(\ell\), and \(m_{\ell}\) are: 1. \(n\) can be any positive integer (e.g. \(n = 1, 2, 3, \dots\)). 2. \(\ell\) ranges from \(0\) to \(n-1\) (e.g. if \(n = 3\), then \(\ell = 0, 1, 2\)). 3. \(m_{\ell}\) ranges from \(-\ell\) to \(+\ell\) (e.g. if \(\ell = 2\), then \(m_{\ell} = -2, -1, 0, 1, 2\)).

Step-by-step solution

Possible values for n (principal quantum number)

The principal quantum number, n, represents the energy level of an electron within an atom and can be any positive integer (n = 1, 2, 3, ...). There is no upper limit for n, as it can have any positive integer value.

Possible values for ℓ (azimuthal quantum number)

The azimuthal quantum number, ℓ, represents the shape of the electron's orbital and is dependent on the principal quantum number (n). The possible values for ℓ range from 0 to n-1. For example, if n = 1, then the possible values for ℓ would be: ℓ = 0. If n = 2, then the possible values for ℓ would be: ℓ = 0, 1. If n = 3, then the possible values for ℓ would be: ℓ = 0, 1, 2. And so on for higher values of n.

Possible values for mₗ (magnetic quantum number)

The magnetic quantum number, mₗ, represents the orientation of the electron's orbital and is dependent on the azimuthal quantum number (ℓ). The possible values for mₗ range from -ℓ to +ℓ. For example, if ℓ = 0, then the possible values for mₗ would be: mₗ = 0. If ℓ = 1, then the possible values for mₗ would be: mₗ = -1, 0, 1. If ℓ = 2, then the possible values for mₗ would be: mₗ = -2, -1, 0, 1, 2. And so on for higher values of ℓ. To summarize, the possible values for the quantum numbers n, ℓ, and mₗ are: 1. n = 1, 2, 3, ... 2. ℓ = 0, 1, 2, ... , n-1 3. mₗ = -ℓ, -(ℓ-1), ... , 0, ... , (ℓ-1), ℓ

Key concepts

Principal Quantum Number

The principal quantum number, represented as \( n \), is a fundamental component in understanding an electron's behavior within an atom. It indicates the electron's energy level or shell. When you think about \( n \), picture it as a staircase — each step is a different energy level, and electrons can "climb" up or down these steps.

The value of \( n \) can be any positive integer (1, 2, 3, etc.), and it has no upper bound. This endless range of values reflects the limitless number of possible energy levels for electrons in larger, more complex atoms:

• \( n = 1 \) represents the first energy level, closest to the atomic nucleus.
• \( n = 2 \) is the second level, slightly further out, and so on.
• The larger the \( n \), the higher the energy and the larger the atomic size of the electron's orbit.

Understanding \( n \) helps in grasping why some elements require more energy to remove an electron compared to others. As \( n \) increases, electrons experience weaker attraction from the nucleus and are easier to remove.

Azimuthal Quantum Number

Diving into the azimuthal quantum number, denoted as \( \ell \), we explore another fascinating aspect of atomic structure. This quantum number describes the shape of an electron's orbital. Imagine electrons not just moving in circles, but through complex, three-dimensional paths.

The azimuthal quantum number \( \ell \) ranges from 0 to \( n - 1 \), where \( n \) is the principal quantum number:

• If \( n = 1 \), then \( \ell = 0 \). The electron's orbital shape is spherical, termed as an \( s \)-orbital.
• If \( n = 2 \), \( \ell \) can be 0 or 1. Now, in addition to an \( s \)-orbital, there are \( p \)-orbitals, which appear dumbbell-shaped.
• For \( n = 3 \), \( \ell \) could be 0, 1, or 2, introducing \( d \)-orbitals, recognized by their more intricate shapes.

Each type of orbital (\( s, p, d, f \), etc.) has a unique spatial distribution, influencing how electrons interact and bond with other atoms.

The concept of \( \ell \) helps explain why different elements exhibit diverse chemical behaviors and shapes in molecular structures.

Magnetic Quantum Number

The magnetic quantum number \( m_\ell \) might sound a bit intimidating, but it essentially pinpoints the exact orientation of an electron's orbital around the nucleus. Just like providing a precise address, \( m_\ell \) tells us exactly where an electron is likely found within its orbital.

The values for \( m_\ell \) depend on the azimuthal quantum number \( \ell \), ranging from \(-\ell\) to \(+\ell\):

• For \( \ell = 0 \), \( m_\ell \) is 0. Here, we are dealing with an \( s \)-orbital, which has only one possible orientation.
• If \( \ell = 1 \), \( m_\ell \) can be \(-1, 0,\) or \(+1\). This translates to three different ways a \( p \)-orbital can orient itself in space.
• With \( \ell = 2 \), \( m_\ell \) can range from \(-2\) to \(+2\), allowing five orientations, typical for \( d \)-orbitals.

The various orientations highlight how electron clouds position themselves relative to external fields, affecting magnetic and electronic properties of materials.

In essence, \( m_\ell \) is crucial for understanding the subtleties of chemical bonding and molecular geometry, providing detailed insight into electron placement.