Problem 30 - Chemistry and Chemical Reactivity

Question

A possible excited state for the 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

Possible sets: (5, 2, -2), (5, 2, -1), (5, 2, 0), (5, 2, 1), (5, 2, 2).

Step 1: Understanding Quantum Numbers

Quantum numbers are used to describe the position and energy of an electron in an atom. The four quantum numbers are: the principal quantum number (), the azimuthal quantum number (), the magnetic quantum number (}, and the spin quantum number (\(m_s\)). However, for this exercise, we are only concerned with the first three to describe the electron in the \(5d\) orbital.

Step 2: Determine the Principal Quantum Number (\(n\))

The principal quantum number \(n\) signifies the electron's energy level or shell. Given that the electron is in the \(5d\) orbital, the principal quantum number \(n\) is \(5\).

Step 3: Determine the Azimuthal Quantum Number (\(\ell\))

The azimuthal quantum number \(\ell\) corresponds to the subshell or orbital type (s, p, d, f) and ranges from 0 to \(n - 1\). For the \(d\) orbital, \(\ell\) equals 2. This is because the \(s\) orbital corresponds to \(\ell = 0\), \(p\) to \(\ell = 1\), \(d\) to \(\ell = 2\), and \(f\) to \(\ell = 3\).

Step 4: Determine the Possible Values for Magnetic Quantum Number (\(m_{\ell}\))

The magnetic quantum number \(m_{\ell}\) specifies the orientation of the orbital in space and can take integer values ranging from \(-\ell\) to \(\ell\). For \(\ell = 2\), \(m_{\ell}\) can be \(-2, -1, 0, 1, 2\).

Step 5: List All Possible Sets of Quantum Numbers

Combine the quantum numbers to list all possible sets: \(n = 5\), \(\ell = 2\), and \(m_{\ell} = -2, -1, 0, 1, 2\). Therefore, the possible sets of quantum numbers are: \((5, 2, -2)\), \((5, 2, -1)\), \((5, 2, 0)\), \((5, 2, 1)\), \((5, 2, 2)\).

Key Concepts

Principal Quantum Number

The principal quantum number, often represented as \( n \), is crucial for understanding the structure of an atom. It indicates the main energy level or shell of an electron within an atom. Think of \( n \) as a set of floors in a building, where each floor represents a higher energy level.
In any atom, \( n \) can take positive integer values: 1, 2, 3, and so on. The larger the \( n \), the higher the energy level and the farther the electron is from the nucleus.

• For example, in the exercise, the electron is in a \(5d\) orbital, where the principal quantum number \( n \) is 5. This tells us that the electron is in the fifth energy level of the hydrogen atom.
• This energy level dictates not only the distance from the nucleus but also influences the number of possible subshells \(( s, p, d, f )\) that can exist within that energy level.

Overall, \( n \) is vital for determining the potential energy of an electron and offers a broad view of where the electron may reside.

Azimuthal Quantum Number

The azimuthal quantum number, denoted by \( \ell \), adds an additional layer of detail by specifying the shape of the electron's subshell within a given energy level. It's like determining the shape of a room on a specific floor.
The values of \( \ell \) range from 0 to \( n-1 \) for each principal quantum number, signifying different subshells:

• When \( \ell = 0 \), it's called an \( s \)-orbital and spherical in shape.
• \( \ell = 1 \) signifies a \( p \)-orbital which is dumbbell-shaped.
• \( \ell = 2 \) corresponds to a \( d \)-orbital with more complex shapes.
• \( \ell = 3 \) represents an \( f \)-orbital.

In our exercise, for a \(5d\) orbital, \( \ell \) equals 2, indicating a \( d \)-subshell within the fifth energy level. Understanding \( \ell \) helps in predicting the electron's path and shape of the orbital it occupies which is essential for determining how these electrons might interact or bond with others.

Magnetic Quantum Number

The magnetic quantum number, represented as \( m_{\ell} \), describes the orientation of the orbital in space, and it helps differentiate between orbitals of the same type within a subshell. Imagine it as the specific direction a room is facing inside a building's floor plan.
For any given \( \ell \), \( m_{\ell} \) can take on integer values from \(-\ell\) to \(\ell\).

• For example, with \( \ell = 2 \) in the exercise, \( m_{\ell} \) can be \(-2, -1, 0, 1, 2\).
• This range signifies the five different orientations available for \( d \)-orbitals in three-dimensional space.

The magnetic quantum number is significant because it reflects the potential energy variations caused by a magnetic field and gives insights into the electron distribution in molecules. This concept is not only important for quantum chemistry but also plays a substantial role in spectroscopy and chemical bonding theory, offering insights into how electrons and orbitals interact.