Problem 29 - Chemistry and Chemical Reactivity

Question

A possible excited state of the \(\mathrm{H}\) atom has the electron in a 4porbital. List all possible sets of quantum numbers \(n, \ell\) and \(m_{\ell}\) for this electron.

Step-by-Step Solution

Answer

The sets of quantum numbers are (4, 1, -1), (4, 1, 0), and (4, 1, 1).

Step 1: Identify the principal quantum number (n)

The principal quantum number, denoted as \( n \), indicates the energy level of the electron. For a 4p orbital, \( n = 4 \).

Step 2: Determine the angular momentum quantum number (l)

The angular momentum quantum number, \( \ell \), describes the shape of the orbital. For p orbitals, \( \ell = 1 \).

Step 3: Identify the magnetic quantum number (m_l)

The magnetic quantum number, \( m_\ell \), describes the orientation of the orbital in space. It can have integer values between \(-\ell\) and \(+\ell\), inclusive. Therefore, for \( \ell = 1 \), the possible values of \( m_\ell \) are \(-1, 0, \text{ and } 1\).

Step 4: List all sets of quantum numbers

The possible sets of quantum numbers for the electron in the 4p orbital are: \( (n, \ell, m_\ell) = (4, 1, -1), (4, 1, 0), \text{ and } (4, 1, 1) \).

Key Concepts

Principal Quantum Number

The principal quantum number, represented by \( n \), is fundamentally important in quantum mechanics as it sets the stage for an electron's residence in an atom. It primarily determines the electron's energy level or shell, which can be thought of as the electron's orbital 'home'.
For example, in a hydrogen atom with a 4p orbital, the principal quantum number is \( n = 4 \). This reflects the electron's shell or energy level, and equates to the energy it possesses.
The principal quantum number must be a positive integer (1, 2, 3...), and as \( n \) increases, the electron's energy increases, thereby affecting its distance from the nucleus.
In essence, \( n \) helps define the size and energy of an orbital, with higher values indicating greater energy and distance from the nucleus.

Angular Momentum Quantum Number

The angular momentum quantum number, often denoted as \( \ell \), provides insight into the shape of an electron's orbital within a given shell or energy level. It is instrumental in classifying the orbital type.
For any principal quantum number \( n \), \( \ell \) can range from 0 up to \( n-1 \). Each integer value of \( \ell \) corresponds to a specific orbital type:

• \( \ell = 0 \): s-orbital
• \( \ell = 1 \): p-orbital
• \( \ell = 2 \): d-orbital
• \( \ell = 3 \): f-orbital

In the context of a 4p orbital, \( \ell = 1 \), which indicates a p-shaped orbital.
This number doesn't stand alone; it is always linked with the principal quantum number which determines the broader context or energy level.

Magnetic Quantum Number

The magnetic quantum number, represented by \( m_\ell \), reflects the orientation of the orbital within a magnetic field, or in simple terms, how the orbital is oriented in three-dimensional space.
The possible values of \( m_\ell \) range between \(-\ell\) and \(+\ell\) inclusive, encompassing all integers in this range. For instance, if \( \ell = 1 \) (typical for p-orbitals), \( m_\ell \) could be \(-1, 0, \) or \( 1 \).
These values specify how many orientations a specific type of orbital can have. Hence, a p-orbital can appear in three orientations.
This quantum number is key for understanding magnetic properties and electron distribution in an atom, particularly when considering how electrons are influenced by external magnetic fields.