Question

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

Short answer

(a) \(n=2, \ell=1, m_{\ell}=-1,0,+1\); (b) \(n=3, \ell=2, m_{\ell}=-2,-1,0,+1,+2\); (c) \(n=4, \ell=3, m_{\ell}=-3,-2,-1,0,+1,+2,+3\).

Step-by-step solution

Understanding Quantum Numbers

Quantum numbers are used to describe the properties of atomic orbitals and the electrons in those orbitals. There are four primary quantum numbers: the principal quantum number \(n\), the azimuthal (angular momentum) quantum number \(\ell\), the magnetic quantum number \(m_{\ell}\), and the spin quantum number (\(m_s\)), although the problem doesn't ask for \(m_s\).

Determine the Principal Quantum Number (n)

The principal quantum number \(n\) indicates the energy level or shell of the electron within the atom. It is derived directly from the orbital notation. In the given question, (a) for the orbital \(2p\), \(n = 2\); (b) for \(3d\), \(n = 3\); and (c) for \(4f\), \(n = 4\).

Determine the Azimuthal Quantum Number (ℓ)

The azimuthal quantum number \(\ell\) determines the shape of the orbital and corresponds to the subshell type. It ranges from 0 to \(n-1\). The values for different orbitals are: for "s", \(\ell = 0\); for "p", \(\ell = 1\); for "d", \(\ell = 2\); and for "f", \(\ell = 3\). Therefore, (a) for \(2p\), \(\ell = 1\); (b) for \(3d\), \(\ell = 2\); and (c) for \(4f\), \(\ell = 3\).

Determine the Magnetic Quantum Number (m_ℓ)

The magnetic quantum number \(m_{\ell}\) indicates the orientation of the orbital in space. It ranges from \(-\ell\) to \(+\ell\), including zero. Therefore: (a) for \(2p\), \(m_{\ell} = -1, 0, +1\); (b) for \(3d\), \(m_{\ell} = -2, -1, 0, +1, +2\); and (c) for \(4f\), \(m_{\ell} = -3, -2, -1, 0, +1, +2, +3\).

Key concepts

Principal Quantum Number

The principal quantum number, denoted by \( n \), is one of the key quantum numbers in quantum mechanics. It describes the primary energy level of an electron within an atom. This number has a direct impact on the energy and size of the electron orbitals. To put it simply, the higher the value of \( n \), the larger and more energetic the orbital is.
For any given element, you can determine \( n \) simply by looking at the period (or row) in which the element resides in the periodic table. It is a positive integer that starts from 1 and moves upward \( n = 1, 2, 3, \) and so on.

• An orbital like \( 2p \) has a principal quantum number of \( n = 2 \), indicating that it is on the second energy level or shell.
• For \( 3d \), \( n \) equals 3, placing the electron on the third shell.
• The \( 4f \) orbital has \( n = 4 \), therefore occupying the fourth energy level.

Understanding the principal quantum number helps in visualizing the general anatomy of an atom, which in turn enhances the comprehension of atomic behavior in various chemical reactions.

Azimuthal Quantum Number

The azimuthal quantum number, represented by \( \ell \), defines the shape of an electron's orbital. Additionally, \( \ell \) helps categorize the subshells found within an energy level. The azimuthal quantum number varies between 0 and \( n-1 \), where \( n \) is the principal quantum number. Each value of \( \ell \) corresponds to a different type of subshell:

• \( \ell = 0 \) for "s" orbitals,
• \( \ell = 1 \) for "p" orbitals,
• \( \ell = 2 \) for "d" orbitals, and
• \( \ell = 3 \) for "f" orbitals.

This hierarchical numbering effectively describes how electrons fill the orbitals:

• For an orbital such as \( 2p \), \( \ell = 1 \) indicates a p-shaped orbital.
• In a \( 3d \) orbital, \( \ell = 2 \) reflects a more complex d-shaped orbital.
• For \( 4f \), \( \ell = 3 \) designates an f-shaped orbital.

These subshells not only vary in shape but also possess unique energies, therefore influencing electron configuration and chemical bonding.

Magnetic Quantum Number

The magnetic quantum number, noted as \( m_\ell \), describes the orientation of an orbital within a subshell. The values of \( m_\ell \) range from \(-\ell\) to \(+\ell\), inclusive of zero. Essentially, \( m_\ell \) provides insight into how the individual orbitals within a subshell are further split in terms of energy in a magnetic field.

• For the \( 2p \) orbital, \( \ell = 1 \), which results in possible \( m_\ell \) values of -1, 0, and +1. Hence, there are three possible orientations.
• In the case of the \( 3d \) orbital, \( \ell = 2 \), and \( m_\ell \) can be -2, -1, 0, +1, and +2, yielding five potential orientations.
• For \( 4f \), with \( \ell = 3 \), \( m_\ell \) includes -3, -2, -1, 0, +1, +2, and +3, allowing for seven orientations.

Understanding \( m_\ell \) provides clarity on how electrons can fill the atomic orbitals, facilitating predictions about the atom's magnetic properties. Whether these orbitals will be aligned or misaligned depends largely on their interactions with magnetic fields, impacting how the electrons are distributed among them.