Question

An electron in an atom is in the \(n=3\) quantum level. List the possible values of \(\ell\) and \(m_{\ell}\) that it can have.

Short answer

Values of \( \ell \) are 0, 1, 2. Values of \( m_{\ell} \) are 0; -1, 0, 1; -2, -1, 0, 1, 2 for each \( \ell \).

Step-by-step solution

Identify Quantum Number n

The principal quantum number indicates the energy level of the electron. Here, we know that the electron is in the \( n=3 \) energy level.

Determine Possible Values of l

For a given principal quantum number \( n \), the azimuthal or angular momentum quantum number \( \ell \) can take integer values from 0 to \( n-1 \). Therefore, \( \ell \) can be 0, 1, or 2 for \( n = 3 \).

Determine Possible Values of ml for l = 0

The magnetic quantum number \( m_{\ell} \) can take integer values from \( -\ell \) to \( \ell \), including zero. When \( \ell = 0 \), then \( m_{\ell} = 0 \).

Determine Possible Values of ml for l = 1

When \( \ell = 1 \), \( m_{\ell} \) can be -1, 0, or 1 since \( m_{\ell} \) ranges from \( -1 \) to \( 1 \).

Determine Possible Values of ml for l = 2

When \( \ell = 2 \), \( m_{\ell} \) can be -2, -1, 0, 1, or 2, as \( m_{\ell} \) ranges from \( -2 \) to \( 2 \).

Key concepts

Principal Quantum Number

The principal quantum number, represented as \( n \), is a critical quantum number that depicts the energy level of an electron in an atom. In simple terms, it tells us about the electron's "distance" from the nucleus. The larger the value of \( n \), the further the electron is from the nucleus. For instance, in our exercise, the principal quantum number is \( n=3 \), which means the electron is in the third energy level. This energy level can accommodate more electrons because higher \( n \) values correspond to more extensive and higher energy electron shells.
Since \( n \) determines the overall size and energy of an electron's orbit, it directly influences the types of orbitals that are possible at that energy level. Higher energy levels allow for more complex orbital shapes and configurations, hence more possibilities for the angular momentum quantum number (\( \,\ell \, \)) and magnetic quantum number (\( \,m_{\ell} \, \)).
To sum up, \( n \) plays a pivotal role in determining the electron’s energy, its average distance from the nucleus, and the type of orbitals it can occupy, laying the foundation for the electron's other quantum numbers to define its specific position and orientation.

Angular Momentum Quantum Number

The angular momentum quantum number, symbolized as \( \,\ell \, \), enriches our understanding of an electron's behavior by describing the shape of the orbital it occupies. This number values range from 0 to \( n-1 \), where \( n \) is the principal quantum number. In the context of our exercise where \( n = 3 \), the possible values for \( \,\ell \, \) are 0, 1, or 2.
Here's what each value signifies:

• \( \ell = 0 \) represents an "s" orbital, which is spherical in shape.
• \( \ell = 1 \) corresponds to a "p" orbital, which has a dumbbell-like shape.
• \( \ell = 2 \) denotes a "d" orbital, with more complex four-lobed structures or similar forms.

The angular momentum quantum number essentially categorizes the electron cloud's shape around the nucleus, influencing how the electron interacts with both other electrons and external electric or magnetic fields.
Understanding \( \,\ell \, \) values is crucial because it steers us toward the possible spatial orientations and nodal patterns each electron can adopt, thereby impacting the chemical and physical properties of the atom.

Magnetic Quantum Number

The magnetic quantum number, denoted as \( \,m_{\ell} \, \), provides a further depth of insight by describing the orientation of an orbital in space. This number can be any integer between \( -\ell \) and \( \ell \), including zero. To put it simply, \( \,m_{\ell} \, \) specifies how many ways an orbital can be arranged around the nucleus.
For the different values of \( \,\ell \, \) in \( n = 3 \):

• When \( \ell = 0 \), \( \,m_{\ell} \, = 0 \) has only one orientation, aligning with the spherical nature of "s" orbitals.
• When \( \ell = 1 \), \( \,m_{\ell} \, \) can be -1, 0, or 1, corresponding to the three orientations possible for "p" orbitals.
• When \( \ell = 2 \), \( \,m_{\ell} \, \) can be -2, -1, 0, 1, or 2, reflecting the five possible orientations of "d" orbitals.

The magnetic quantum number becomes particularly significant in environments impacted by magnetic fields because it determines how orbitals from subshells split in energy levels when exposed. Therefore, \( \,m_{\ell} \, \) is crucial for understanding phenomena like the Zeeman effect, where spectral lines split or shift due to magnetic fields.