Problem 27 - Chemistry and Chemical Reactivity

Question

(a) When \(n=4,\) what are the possible values of \(\ell ?\) (b) When \(\ell\) is \(2,\) what are the possible values of \(m_{\ell} ?\) (c) For a \(4 s\) orbital, what are the possible values of \(n, \ell,\) and \(m_{\ell} ?\) (d) For a \(4 f\) orbital, what are the possible values of \(n, \ell,\) and \(m_{\ell} ?\)

Step-by-Step Solution

Answer

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

Step 1: Determine possible values of \( \ell \) for \( n=4 \)

The azimuthal quantum number \( \ell \) can take values from \( 0 \) to \( n-1 \). Therefore, for \( n=4 \), \( \ell \) can be \( 0, 1, 2, \) or \( 3 \).

Step 2: Determine possible values of \( m_{\ell} \) for \( \ell=2 \)

The magnetic quantum number \( m_{\ell} \) can take integer values ranging from \( -\ell \) to \( +\ell \). For \( \ell = 2 \), \( m_{\ell} \) can be \( -2, -1, 0, 1, \) or \( 2 \).

Step 3: Identify possible quantum numbers for a \( 4s \) orbital

For the \( 4s \) orbital, the principal quantum number \( n \) is \( 4 \), and the azimuthal quantum number \( \ell \) for an \( s \) orbital is \( 0 \). The only value for \( m_{\ell} \) is \( 0 \) since \( m_{\ell} \) ranges from \( -\ell \) to \( +\ell \).

Step 4: Identify possible quantum numbers for a \( 4f \) orbital

For the \( 4f \) orbital, the principal quantum number \( n \) is \( 4 \), and the azimuthal quantum number \( \ell \) for an \( f \) orbital is \( 3 \). Therefore, \( m_{\ell} \) can be \( -3, -2, -1, 0, 1, 2, \) or \( 3 \).

Key Concepts

Principal quantum number

The principal quantum number, represented by the symbol \( n \), is a fundamental concept in quantum mechanics. It plays a crucial role in determining the energy level of an electron in an atom. This quantum number indicates the main energy shell in which the electron resides.

• Higher values of \( n \) correspond to higher energy levels and are situated further from the nucleus.
• The principal quantum number can take on any positive integer value (1, 2, 3...).
• For each value of \( n \), there are \( n \) allowed values of the azimuthal quantum number \( \ell \).

An example from the exercise: When \( n = 4 \), the electron is situated in the fourth energy level or shell. This also determines the possible values for the azimuthal quantum number, which shows the shape of the electron’s orbital.

Azimuthal quantum number

The azimuthal quantum number, denoted by \( \ell \), describes the subshell or orbital shape of an electron within a given energy level \( n \). It can take integer values from 0 to \( n-1 \). Each value of \( \ell \) corresponds to a particular type of orbital shape, making it essential in understanding electron distribution.

• \( \ell = 0 \) indicates an \( s \)-orbital, spherical in shape.
• \( \ell = 1 \) represents a \( p \)-orbital, which is dumbbell-shaped.
• \( \ell = 2 \) corresponds to a \( d \)-orbital, which is typically cloverleaf-shaped.
• \( \ell = 3 \) denotes an \( f \)-orbital, which has an even more complex shape.

In the provided exercise, for \( n = 4 \), the possible values of \( \ell \) are 0, 1, 2, and 3, correlating to the allowed orbital types for that energy level.

Magnetic quantum number

The magnetic quantum number, symbolized by \( m_{\ell} \), defines the specific orientation of an orbital within a subshell. Each value of \( \ell \) can hold several orientations, depending on the magnetic quantum number. The values of \( m_{\ell} \) range from \( -\ell \) to \( +\ell \), including zero.

• Each subshell can contain multiple orientations, described by different \( m_{\ell} \) values.
• This quantum number is pivotal in explaining how electrons occupy specific regions around a nucleus.

In the original exercise, for \( \ell = 2 \), the possible values of \( m_{\ell} \) range from -2 to +2. This means the electron can occupy any of these five orientations within the \( d \)-orbital. Similarly, for the \( 4f \) orbital where \( \ell = 3 \), \( m_{\ell} \) can vary from -3 to +3, resulting in a total of seven possible orientations.