Question

(a) When \(n=4, \ell=2,\) and \(m_{\ell}=-1,\) to what orbital type does this refer? (Give the orbital label, such as 1s.) (b) How many orbitals occur in the \(n=5\) electron shell? How many subshells? What are the letter labels of the subshells? (c) How many orbitals occur in an \(f\) subshell? What are the values of \(m_{\ell} ?\)

Short answer

(a) 4d orbital; (b) 25 orbitals, 5 subshells: 5s, 5p, 5d, 5f, 5g; (c) 7 orbitals, \(m_\ell = -3 \) to \(3\).

Step-by-step solution

Understanding Quantum Numbers

Quantum numbers describe the properties of atomic orbitals. The principal quantum number \( n \) defines the energy level; \( \ell \) (azimuthal quantum number) determines the orbital shape; and \( m_\ell \) (magnetic quantum number) describes the orientation of the orbital. The possible value of \( \ell \) ranges from 0 to \( n-1 \) and \( m_\ell \) from \(-\ell\) to \(\ell\).

Determine Orbital Type for Part (a)

Given \( n=4 \), \( \ell=2 \), and \( m_\ell=-1 \), first identify the shape based on \( \ell \). The values \( (\ell=0, 1, 2, 3) \) correspond to \( (s, p, d, f) \) respectively. Thus, \( \ell=2 \) indicates a \( d \) orbital. Since \( n=4 \), the orbital type is labeled as \( 4d \).

Determine Orbitals and Subshells for Part (b)

For \( n=5 \), the value of \( \ell \) can range from 0 to 4. This results in subshells labeled as \( 5s \), \( 5p \), \( 5d \), \( 5f \), and \( 5g \). The total number of orbitals is given by \( n^2 = 5^2 = 25 \). The subshells are defined by the different values of \( \ell \).

Identify Orbitals in an 'f' Subshell for Part (c)

An \( f \) subshell corresponds to \( \ell=3 \). The number of orbitals in a subshell is given by \( 2\ell + 1 \). Thus, for \( \ell=3 \), the \( f \) subshell has \( 2 \times 3 + 1 = 7 \) orbitals. The possible values of \( m_\ell \) range from \(-3\) to \(3\): \(-3, -2, -1, 0, 1, 2, 3\).

Key concepts

Quantum Numbers

Quantum numbers are essential for understanding the structure of atomic orbitals. They act like an address, helping us locate and describe electrons in an atom.
Think of them as a set of coordinates that define distinctive properties:

• Principal Quantum Number ( ): Determines the electron shell or energy level.
• Azimuthal Quantum Number ( ): Describes the shape of the orbital.
• Magnetic Quantum Number ( ): Indicates the orientation of the orbital in space.

The combination of these numbers allows us to identify and describe any electron's position and behavior in an atom. This is crucial in fields like chemistry and physics, where electron configurations help predict an atom’s chemical properties.

Principal Quantum Number

The principal quantum number is symbolized by \(n\). It signifies the electron's energy level or shell, which can be any positive integer starting from 1: \(n = 1, 2, 3,\) and so on.
Each increment in \(n\) means a higher energy level and a larger orbital.

• Higher \(n\) values imply electrons that are further away from the nucleus.
• With increasing \(n\), the size of the orbital increases, impacting the atom's reactivity.

This is vital in determining the electron configuration and understanding how atoms bond with one another.

Azimuthal Quantum Number

The azimuthal quantum number, represented by \(\ell\), defines the shape of an electron's orbital.
The possible values of \(\ell\) range from 0 to \(n-1\) for each principal quantum number \(n\).

• \(\ell = 0\) represents an \'s\' orbital (spherical).
• \(\ell = 1\) represents a \'p\' orbital (dumbbell-shaped).
• \(\ell = 2\) represents a \'d\' orbital (cloverleaf-shaped).
• \(\ell = 3\) represents an \'f\' orbital (complex shapes).

This number directly influences an electron's spatial distribution and is used alongside other quantum numbers to classify subshells.

Magnetic Quantum Number

The magnetic quantum number, \(m_\ell\), describes the orientation of an orbital within a given subshell.
It can take any integer value between \(-\ell\) and \(\ell\), including zero.

• For example, if \(\ell = 1\), \(m_\ell\) can be -1, 0, or 1, accounting for three different orientations in space.
• This diversity in orientation allows for a rich variety of chemical bonding and interactions.
• \(m_\ell\) values help define the spatial distribution of electrons within a subshell, enabling precise identification of electron arrangements.

Understanding \(m_\ell\) is essential in fields that require precise electron placement, like spectroscopy and quantum mechanics.