Problem 44 - Chemistry and Chemical Reactivity

Question

A given orbital has a magnetic quantum number of \(m_{\ell}=-1 .\) This could not be a(n) (a) \(f\) orbital (c) \(p\) orbital (b) \(d\) orbital (d) \(s\) orbital

Step-by-Step Solution

Answer

(d) An \(s\) orbital cannot have \(m_{\ell} = -1\).

Step 1: Understand Magnetic Quantum Number

The magnetic quantum number, denoted as \(m_{\ell}\), can have integer values ranging from \(-\ell\) to \(+\ell\), where \(\ell\) is the azimuthal quantum number corresponding to the orbital type.

Step 2: Determine Quantum Number Ranges

For different types of orbitals: - \(s\) orbital (\(\ell = 0\)) implies \(m_{\ell} = 0\).- \(p\) orbital (\(\ell = 1\)) implies \(m_{\ell} = -1, 0, +1\).- \(d\) orbital (\(\ell = 2\)) implies \(m_{\ell} = -2, -1, 0, +1, +2\).- \(f\) orbital (\(\ell = 3\)) implies \(m_{\ell} = -3, -2, -1, 0, +1, +2, +3\).

Step 3: Identify Unanswerable Orbital Type

The only orbital type that cannot have a magnetic quantum number \(m_{\ell} = -1\) is the \(s\) orbital, as it only allows for \(m_{\ell} = 0\).

Key Concepts

Magnetic Quantum Number

The magnetic quantum number, represented as \( m_{\ell} \), is a fundamental component in the quantum mechanical description of an electron's state within an atom. It arises from the solution of the Schrödinger equation and helps define the orientation of an orbital in space.

In more straightforward terms, \( m_{\ell} \) lets us know how an electron's path, specifically its orbit, is tilted or oriented around the nucleus relative to an external magnetic field. The value of \( m_{\ell} \) is dependent on the azimuthal quantum number, \( \ell \), and can take any integer value from \(-\ell\) to \(+\ell\).

For example, if \( \ell \) equals 2 (like in a \( d \) orbital), the possible \( m_{\ell} \) values range from -2 through +2. This means the \( m_{\ell} \) can be -2, -1, 0, 1, or 2, allowing five different orientations for \( d \) orbitals.

Azimuthal Quantum Number

The azimuthal quantum number, symbolized as \( \ell \), determines the shape of an electron's orbital. It significantly influences not only the orbital's shape but also the energy level, when placed inside the same principal quantum level.

For every principal quantum number \( n \), \( \ell \) ranges from 0 to \( n-1 \). Each \( \ell \) value correlates with a specific type of orbital.

• \( \ell = 0 \): This corresponds to an \( s \) orbital, spherical in shape.
• \( \ell = 1 \): This matches a \( p \) orbital, which is dumbbell-shaped.
• \( \ell = 2 \): Represents a \( d \) orbital, having a more complex clover shape.
• \( \ell = 3 \): Identifies an \( f \) orbital, with even more intricate shapes.

This classification not only reflects the shape but also the possible values of the magnetic quantum number \( m_{\ell} \) that are contained within each orbit.

Orbital Types

Orbitals are regions around the nucleus where electrons are most likely to be found. Each type of orbital is defined by specific values of quantum numbers and thus represents particular energy levels and shapes.

• \( s \) Orbitals: Defined by \( \ell = 0 \), these have the simplest spherical shape and only one orientation (\( m_{\ell} = 0 \)).
• \( p \) Orbitals: With \( \ell = 1 \), \( p \) orbitals have a dumbbell shape and can understand three orientations (\( m_{\ell} = -1, 0, +1 \)).
• \( d \) Orbitals: These are more complex with \( \ell = 2 \), including five potential orientations (\( m_{\ell} = -2, -1, 0, +1, +2 \)), forming a clover pattern.
• \( f \) Orbitals: The most intricate, having \( \ell = 3 \), results in seven possible orientations (\( m_{\ell} = -3, -2, -1, 0, +1, +2, +3 \)).

Understanding these can provide insight into electron arrangements, chemical bonding, and the physical properties of elements.