Question

Write a complete set of quantum numbers \((n, \ell,\) and \(\left.m_{\ell}\right)\) for each of the following orbitals: (a) \(5 f,\) (b) \(4 d,\) and (c) \(2 s\)

Short answer

(a) 5f: \((n,\ell,m_{\ell}) = (5,3,-3\ ext{to}\ +3)\), (b) 4d: \((n,\ell,m_{\ell}) = (4,2,-2\ ext{to}\ +2)\), (c) 2s: \((n,\ell,m_{\ell}) = (2,0,0)\)

Step-by-step solution

Understanding Quantum Numbers

Quantum numbers are used to describe the properties and locations of electrons in atoms. The principal quantum number \(n\) defines the energy level, the azimuthal quantum number \(\ell\) defines the shape of the orbital, and the magnetic quantum number \(m_{\ell}\) defines the orientation of the orbital in space.

Identifying Quantum Numbers for 5f

For the orbital \(5f\):- The principal quantum number \(n = 5\).- The azimuthal quantum number for an f orbital is \(\ell = 3\) (f orbitals correspond to \(\ell = 3\)).- The magnetic quantum number \(m_{\ell}\) can range from \(-\ell\) to \(+\ell\), so \(m_{\ell} = -3, -2, -1, 0, 1, 2, 3\).

Identifying Quantum Numbers for 4d

For the orbital \(4d\):- The principal quantum number \(n = 4\).- The azimuthal quantum number for a d orbital is \(\ell = 2\) (d orbitals correspond to \(\ell = 2\)).- The magnetic quantum number \(m_{\ell}\) can range from \(-\ell\) to \(+\ell\), so \(m_{\ell} = -2, -1, 0, 1, 2\).

Identifying Quantum Numbers for 2s

For the orbital \(2s\):- The principal quantum number \(n = 2\).- The azimuthal quantum number for an s orbital is \(\ell = 0\) (s orbitals correspond to \(\ell = 0\)).- The magnetic quantum number \(m_{\ell}\) is 0, since \(m_{\ell}\) can only be \(0\) when \(\ell = 0\).

Key concepts

Understanding the Principal Quantum Number

The principal quantum number, often denoted as \( n \), is a foundational piece of the puzzle when discussing atomic structure. It primarily describes the energy level of an electron in an atom. The larger the value of \( n \), the higher the energy level and the farther the electron is from the nucleus. This number can be any positive integer (1, 2, 3, ...), meaning there are no negative or fractional principal quantum numbers.

• Higher \( n \) values indicate an electron's position further away from the nucleus.
• The principal quantum number also defines the size of the orbital; larger \( n \) values correspond to bigger orbitals.

For example, in the 5f orbital of Uranium, the principal quantum number is \( n = 5 \). Thus, it indicates an electron in the fifth energy level, which is larger and higher in energy than the first four levels.

Exploring the Azimuthal Quantum Number

The azimuthal quantum number, represented by \( \ell \), helps in understanding the shape of an orbital. Sometimes it is referred to as the angular momentum quantum number. This number can take any integer value from 0 up to \( n-1 \), where \( n \) is the principal quantum number. The shape of an orbital is critical because it influences many properties of the electron.

• \( \ell = 0 \): Spherical s orbitals
• \( \ell = 1 \): Dumbbell-shaped p orbitals
• \( \ell = 2 \): Cloverleaf-shaped d orbitals
• \( \ell = 3 \): Complex-shaped f orbitals

For instance, in a 4d orbital, the azimuthal quantum number is \( \ell = 2 \). This indicates that the orbital has a cloverleaf shape, which is characteristic of d orbitals. This shape is crucial in determining how electrons interact in bonding scenarios.

Delving into the Magnetic Quantum Number

The magnetic quantum number, \( m_{\ell} \), primarily explains the orientation of an electron's orbital around the nucleus. This number can range from \(-\ell\) to \(+\ell\), including zero. It indicates the number of orbitals possible within a subshell and how they are aligned in space. Understanding this quantum number is essential for visualizing electron distribution in a magnetic field.

• For \( \ell = 0 \), \( m_{\ell} = 0 \), indicating a single orientation as with the s orbital.
• For \( \ell = 1 \), \( m_{\ell} \) can be \(-1, 0, \text{or} +1\), corresponding to three orientations (or orbitals) in the p subshell.
• For \( \ell = 2 \), \( m_{\ell} \) ranges from \(-2\) to \(+2\), encompassing five possible orientations for d orbitals.

Take the 2s orbital as an example; since it belongs to the s subshell, it has \( \ell = 0 \). Consequently, the magnetic quantum number \( m_{\ell} = 0 \), representing just one orbital orientation, aligning without any specific direction preference.