Problem 29 - Chemistry and Chemical Reactivity

Question

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

Step-by-Step Solution

Answer

The sets are \((4, 1, -1)\), \((4, 1, 0)\), and \((4, 1, +1)\).

Step 1: Identify the principal quantum number (n)

The principal quantum number \( n \) signifies the main energy level of an electron. For an electron in a \(4p\) orbital, \( n = 4 \).

Step 2: Determine the azimuthal quantum number (ℓ)

The azimuthal quantum number \( \ell \) defines the shape of the orbital and for \(p\) orbitals, \( \ell = 1 \). Since the electron is in a \(4p\) orbital, \( \ell = 1 \) is confirmed.

Step 3: Identify the magnetic quantum number (m_ℓ)

The magnetic quantum number \( m_\ell \) relates to the orientation of the orbital in space. It can take values from \(-\ell\) to \(\ell\), providing integer possibilities. Thus, for \( \ell = 1 \), \( m_\ell = -1, 0, \) or \( +1 \). This means there are three possible values for \( m_\ell \).

Step 4: List the possible sets of quantum numbers

The possible sets of quantum numbers for an electron in the \(4p\) orbital are as follows: \((n, \ell, m_\ell) = (4, 1, -1)\), \((4, 1, 0)\), and \((4, 1, +1)\). Each set corresponds to a different spatial orientation of the \(4p\) electron.

Key Concepts

Principal Quantum Number

In quantum mechanics, the principal quantum number, denoted by \( n \), is a crucial quantum number that determines the primary energy level an electron occupies in an atom. It plays a fundamental role in defining the size and energy of an electron's orbit around the nucleus.
The principal quantum number is a positive integer value \((n = 1, 2, 3, \ldots)\) and directly associates with the electron's energy and distance from the nucleus. Higher \( n \) values mean the electron is further from the nucleus and possesses more energy.
For instance, if an electron is in a \(4p\) orbital, the principal quantum number is \(4\). This indicates that the electron resides in the fourth energy level, which is relatively high, suggesting the electron is more energetic and further away from the nucleus than if it were on a lower level, such as the first or second.

Azimuthal Quantum Number

The azimuthal quantum number, also known as the angular momentum quantum number and represented by \( \ell \), specifies the shape of an electron's orbital within a given energy level. This quantum number provides insights into the orbital's angular shape and its subshell classification.
Values of \( \ell \) range from \(0\) to \(n-1\), where \(n\) is the principal quantum number. Each value of \( \ell \) corresponds to a particular letter that describes the shape:

• \( \ell = 0 \) is an \(s\)-orbital (sphere shape)
• \( \ell = 1 \) is a \(p\)-orbital (dumbbell shape)
• \( \ell = 2 \) is a \(d\)-orbital (clover shape)
• \( \ell = 3 \) is an \(f\)-orbital (complex shapes)

For an electron in a \(4p\) orbital, \( \ell = 1 \), designating a \(p\)-orbital. This tells us that the electron's orbital is shaped like a dumbbell. It also indicates that within the fourth energy level, this electron resides in the \(p\) subshell, which holds up to six electrons across three orientations.

Magnetic Quantum Number

The magnetic quantum number, symbolized by \( m_\ell \), pertains to the orientation of an electron's orbital in three-dimensional space. It provides information on the spatial direction in which an orbital aligns within a magnetic field or around the nucleus.
The magnetic quantum number ranges from \(-\ell\) to \(+\ell\), including zero, offering a total of \(2\ell + 1\) possible values. These values represent the different orientations an orbital with a given \( \ell \) can have.
As an example, in a \(4p\) orbital where \( \ell = 1 \), \( m_\ell \) can take on values of \(-1\), \(0\), or \(+1\). This signifies three possible orientations for the electron's orbital, each contributing to the electron's behavior concerning spatial interactions and external fields. Each set of \((4, 1, m_\ell)\) quantum numbers presents a unique spatial orientation for the electron in the \(4p\) subshell.