Question

Write a complete set of quantum numbers \(\left(n, \ell, \text { and } m_{\ell}\right)\) that quantum theory allows for each of the following orbitals: (a) \(2 p,\) (b) \(3 d,\) and \((c) 4 f\)

Short answer

(a) \(2p: n=2, \ell=1, m_\ell=-1,0,1\); (b) \(3d: n=3, \ell=2, m_\ell=-2,-1,0,1,2\); (c) \(4f: n=4, \ell=3, m_\ell=-3,-2,-1,0,1,2,3\).

Step-by-step solution

Understanding Quantum Numbers

Quantum numbers describe the properties of atomic orbitals and the electrons in those orbitals. They include the principal quantum number \(n\), the azimuthal (angular momentum) quantum number \(\ell\), and the magnetic quantum number \(m_\ell\).

Identify Quantum Numbers for 2p Orbital

For a \(2p\) orbital: - The principal quantum number \(n = 2\).- The azimuthal quantum number \(\ell = 1\) (since \(p\) orbitals correspond to \(\ell = 1\)).- The magnetic quantum numbers \(m_{\ell}\) range from \(-\ell\) to \(+\ell\), so \(m_{\ell} = -1, 0, 1\).

Identify Quantum Numbers for 3d Orbital

For a \(3d\) orbital: - The principal quantum number \(n = 3\).- The azimuthal quantum number \(\ell = 2\) (since \(d\) orbitals correspond to \(\ell = 2\)).- The magnetic quantum numbers \(m_{\ell}\) range from \(-\ell\) to \(+\ell\), so \(m_{\ell} = -2, -1, 0, 1, 2\).

Identify Quantum Numbers for 4f Orbital

For a \(4f\) orbital: - The principal quantum number \(n = 4\).- The azimuthal quantum number \(\ell = 3\) (since \(f\) orbitals correspond to \(\ell = 3\)).- The magnetic quantum numbers \(m_{\ell}\) range from \(-\ell\) to \(+\ell\), so \(m_{\ell} = -3, -2, -1, 0, 1, 2, 3\).

Key concepts

Principal Quantum Number

The principal quantum number, denoted as \( n \), is one of the key quantum numbers in quantum mechanics that helps describe atomic orbitals. It determines the overall size and energy level of an electron's orbit around a nucleus. The principal quantum number can be any positive integer (\( n = 1, 2, 3, \) etc.). As \( n \) increases, the electron is located further from the nucleus, and the energy of the electron also increases.
When we refer to different electron shells, they are actually defined by their principal quantum number:

• The first shell, closest to the nucleus, is \( n = 1 \).
• The second shell is \( n = 2 \).
• The third shell is \( n = 3 \), and so on.

For instance, in the case of the \(2p\) orbital mentioned in the original exercise, the principal quantum number is \( n = 2 \). This implies that the related orbital belongs to the second shell of the atom, at a moderate distance from the nucleus compared to higher shells.

Azimuthal Quantum Number

The azimuthal quantum number, also known as the angular momentum quantum number and represented by \( \ell \), further defines the specific shape of the electron's orbital within a given shell. The value of \( \ell \) can range from \( 0 \) to \( n-1 \), where \( n \) is the principal quantum number.
This quantum number is crucial for classifying the sublevel or subshell in which the electron resides:

• \( \ell = 0 \) is designated as an "s" orbital.
• \( \ell = 1 \) represents a "p" orbital.
• \( \ell = 2 \) indicates a "d" orbital.
• \( \ell = 3 \) corresponds to an "f" orbital, and so forth.

Each type of orbital (s, p, d, f) has a unique shape, affecting the distribution of the electron cloud. For example, a \(3d\) orbital has \( n = 3 \) and \( \ell = 2 \), showing that it is a "d" type orbital in the third shell, which has a characteristic double-lobed shape.

Magnetic Quantum Number

The magnetic quantum number, \( m_\ell \), offers even more specific information about an electron's position within an orbital. It determines the orientation of the orbital in space relative to an external magnetic field. The values it can take range from \( -\ell \) to \( +\ell \), including zero.
This means that for any given azimuthal quantum number (\( \ell \)), \( m_\ell \) can have \( 2\ell + 1 \) possible values. These values represent the different spatial orientations that an orbital type can take:

• For a "p" orbital (\( \ell = 1 \)), possible values for \( m_\ell \) are \( -1, 0, 1 \).
• For a "d" orbital (\( \ell = 2 \)), \( m_\ell \) can be \( -2, -1, 0, 1, 2 \).
• For an "f" orbital (\( \ell = 3 \)), \( m_\ell \) ranges from \( -3 \) to \( 3 \).

In the context of a \(4f\) orbital from the original exercise, \( m_\ell \) takes values \( -3, -2, -1, 0, 1, 2, 3 \), illustrating the seven possible orientations of an "f" orbital in spatial terms.