Problem 28 - Chemistry and Chemical Reactivity

Question

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

Step-by-Step Solution

Answer

(a) 4d; (b) 25 orbitals, 5 subshells: 5s, 5p, 5d, 5f, 5g; (c) 7 orbitals, \( m_\ell = -3, -2, -1, 0, 1, 2, 3 \).

Step 1: Determine the Orbital Type (a)

The quantum numbers provided are \( n=4 \), \( \ell=2 \), and \( m_\ell=-1 \). The principal quantum number \( n \) indicates the main energy level and is labeled numerically as 4. For \( \ell=2 \), this corresponds to a 'd' subshell. Thus, the orbital type is labeled as 4d.

Step 2: Count Total Orbitals in Shell (b)

For the shell associated with \( n=5 \), the formula to find the total number of orbitals in a shell is given by \( n^2 \). Therefore, \( 5^2 = 25 \) orbitals occur in the \( n=5 \) shell.

Step 3: Determine Number of Subshells in Shell (b)

The number of subshells in any shell is equal to \( n \). So, for \( n=5 \), there are 5 subshells: \( \ell = 0, 1, 2, 3, 4 \).

Step 4: Identify Subshell Labels (b)

The letter labels of subshells corresponding to \( \ell \) values are: \( \ell=0 \) (s), \( \ell=1 \) (p), \( \ell=2 \) (d), \( \ell=3 \) (f), and \( \ell=4 \) (g). So, the subshells in \( n=5 \) are labeled as 5s, 5p, 5d, 5f, and 5g.

Step 5: Determine Number of Orbitals in an f Subshell (c)

An \( f \) subshell has \( \ell=3 \). The number of orbitals in a subshell is given by \( 2\ell + 1 \). For \( \ell=3 \), the number of orbitals is \( 2(3) + 1 = 7 \) orbitals.

Step 6: Identify Possible \( m_\ell \) Values for f Subshell (c)

\( m_\ell \) values range from \( -\ell \) to \( +\ell \) in integer steps. Therefore, for \( \ell=3 \), the \( m_\ell \) values are: -3, -2, -1, 0, 1, 2, 3.

Key Concepts

Electron Orbitals

Electron orbitals are regions around the nucleus where electrons are most likely to be found. Each orbital can hold a maximum of two electrons, which have opposite spins. These orbitals are grouped based on their energy levels, denoted by the principal quantum number \( n \), and their shapes, determined by the azimuthal quantum number \( \ell \).
For example, when \( n=4 \) and \( \ell=2 \), we have a 4d orbital. Here, '4' is the principal quantum number indicating the fourth energy level, and 'd' represents the type of orbital shape.
Understanding these orbitals helps in predicting how atoms will interact and bond with each other, which is crucial in chemistry.

Subshells

Subshells are divisions within an electron shell, defined by the azimuthal quantum number \( \ell \). They indicate different shapes and energies of orbitals within the same principal energy level. The azimuthal quantum number \( \ell \) can take values from 0 to \( n-1 \), where each value corresponds to a specific subshell label:

• \( \ell=0 \): s subshell
• \( \ell=1 \): p subshell
• \( \ell=2 \): d subshell
• \( \ell=3 \): f subshell
• \( \ell=4 \): g subshell

For instance, the \( n=5 \) shell contains 5 subshells, labeled as 5s, 5p, 5d, 5f, and 5g. These labels help identify the specific sets of orbitals available at a given energy level.

Magnetic Quantum Number

The magnetic quantum number, \( m_\ell \), defines the orientation of an orbital in space relative to the other orbitals around it. The values of \( m_\ell \) range from \(-\ell\) to \(+\ell\), including zero. So for a given subshell labeled \( \ell \), there are \( 2\ell + 1 \) possible values for \( m_\ell \).
For example, in an \( f \) subshell where \( \ell = 3 \), the possible values for \( m_\ell \) are -3, -2, -1, 0, 1, 2, and 3. This results in 7 different orientations of the orbitals in the \( f \) subshell.
By understanding \( m_\ell \), you can predict how electrons occupy these orbitals and gain insight into the magnetic properties of atoms.