Problem 30 - Chemistry and Chemical Reactivity

Question

A possible excited state for the \(\mathrm{H}\) atom has an electron in a \(5 d\) orbital. List all possible sets of quantum numbers \(n\) \(\ell,\) and \(m_{\ell}\) for this electron.

Step-by-Step Solution

Answer

Quantum number sets: \((5, 2, -2), (5, 2, -1), (5, 2, 0), (5, 2, 1), (5, 2, 2)\).

Step 1: Identify Principal Quantum Number (n)

The principal quantum number \( n \) corresponds to the electron's energy level or shell. Since the electron is in a \(5d\) orbital, \( n = 5 \).

Step 2: Identify Azimuthal Quantum Number (l)

The azimuthal quantum number \( \ell \) defines the shape of the orbital. For \(d\)-orbitals, \( \ell = 2 \).

Step 3: Determine Magnetic Quantum Number (m_l)

The magnetic quantum number \( m_\ell \) defines the orientation of the orbital in space. It can take on integer values from \(-\ell \) to \( \ell \). Hence, for \( \ell = 2 \), \( m_\ell \) can be \(-2, -1, 0, 1, 2 \).

Step 4: Combine Quantum Numbers

Each combination of \( n, \ell, \) and \( m_\ell \) that fits the criteria makes up a set of quantum numbers for the electron: \((5, 2, -2), (5, 2, -1), (5, 2, 0), (5, 2, 1), (5, 2, 2)\).

Key Concepts

Principal Quantum Number (n)

The principal quantum number, denoted as \( n \), is the foundation of understanding the electronic structure of an atom. It tells us about the main energy level of the electron, which can be imagined as the 'orbit' or 'shell' in which an electron resides.
To put it simply, \( n \) is a positive integer (1, 2, 3, ...), and it indicates the distance of the electron from the nucleus. The higher the value of \( n \), the higher the energy level and the larger the orbital. Thus, the electron in a \(5d\) orbital has a principal quantum number \( n = 5 \).
This means it resides in the fifth energy level, giving it certain characteristics and energy associated with this level. As \( n \) increases, electrons have more potential space to occupy, demonstrating why \( n \) is essential in defining the overall energy and size of the atom.

Azimuthal Quantum Number (ℓ)

The azimuthal quantum number, \( \ell \), is a key player in defining the shape and type of an electron's orbital. It is sometimes referred to as the angular momentum quantum number, as it also relates to the angular momentum carried by the electron within an atom.
Specifically, \( \ell \) can take values from 0 up to \( n - 1 \). For example, if \( n = 5 \), then \( \ell \) can be 0, 1, 2, 3, or 4. Each value of \( \ell \) corresponds to a specific type of orbital:

• \( \ell = 0 \) for s-orbitals
• \( \ell = 1 \) for p-orbitals
• \( \ell = 2 \) for d-orbitals
• \( \ell = 3 \) for f-orbitals

In a \(5d\) orbital, \( \ell = 2 \), indicating it's a d-orbital shape that's part of the fifth energy level. This quantum number is fundamental in determining not just the orbital's shape but also how it participates in bonding and chemical reactions.

Magnetic Quantum Number (mₗ)

The magnetic quantum number, \( m_\ell \), provides insight into the orientation of an electron's orbital, a concept which is pivotal when discussing the spatial arrangement of orbitals in an atom.
The possible values of \( m_\ell \) depend directly on the azimuthal quantum number, \( \ell \). For any given \( \ell \), \( m_\ell \) can take on integer values ranging from \( -\ell \) to \( +\ell \). This means there are \( 2\ell + 1 \) possible values for \( m_\ell \).
In the context of a \(5d\) orbital where \( \ell = 2 \):

• \( m_\ell \) can be \(-2, -1, 0, 1, 2 \)

These values signify that there are five different orientations possible for the d-orbitals. Thus, the magnetic quantum number is crucial in defining how these orbitals are positioned in three-dimensional space, affecting how they overlap with orbitals from other atoms to form chemical bonds.