Problem 161 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following sets of quantum numbers is correct for an electron in \(4 \mathrm{f}\) orbital? (a) \(\mathrm{n}=4, l=3, \mathrm{~m}=+4, \mathrm{~s}=+1 / 2\) (b) \(\mathrm{n}=4, l=4, \mathrm{~m}=-4, \mathrm{~s}=-1 / 2\) (c) \(\mathrm{n}=4, l=3, \mathrm{~m}=+1, \mathrm{~s}=+1 / 2\) (d) \(\mathrm{n}=3, l=2, \mathrm{~m}=-2, \mathrm{~s}=+1 / 2\)

Step-by-Step Solution

Answer

Option (c) is the correct set of quantum numbers for a 4f orbital.

Step 1: Understanding Quantum Numbers

Quantum numbers describe the properties of electrons in an atom. They include:1. Principal quantum number (\(n\)), which specifies the electron's energy level and distance from the nucleus.2. Azimuthal quantum number (\(l\)), which indicates the shape of the orbital. For \(f\) orbitals, \(l = 3\).3. Magnetic quantum number (\(m\)), which describes the orientation of the orbital. Its values range from \(-l\) to \(+l\).4. Spin quantum number (\(s\)), which tells the intrinsic spin of the electron. Its values are \(+1/2\) or \(-1/2\).

Step 2: Identifying Valid Quantum Numbers for 4f Orbital

For an electron in a \(4f\) orbital, the quantum numbers must follow:- \(n = 4\)- \(l = 3\) (since \(f\) corresponds to \(l = 3\))- \(m\) can be \(-3, -2, -1, 0, +1, +2, +3\)- \(s\) can be \(+1/2\) or \(-1/2\).

Step 3: Evaluate Option (a)

Option (a) provides: \(n=4\), \(l=3\), \(m=+4\), \(s=+1/2\).- \(m = +4\) is invalid because \(m\) should be within \(-3\) to \(+3\). Hence, option (a) is incorrect.

Step 4: Evaluate Option (b)

Option (b) provides: \(n=4\), \(l=4\), \(m=-4\), \(s=-1/2\).- \(l=4\) is invalid for a \(4f\) orbital because \(f\) corresponds to \(l=3\). Hence, option (b) is incorrect.

Step 5: Evaluate Option (c)

Option (c) provides: \(n=4\), \(l=3\), \(m=+1\), \(s=+1/2\).- All quantum numbers fit within the allowable range for a \(4f\) orbital. Therefore, option (c) is correct.

Step 6: Evaluate Option (d)

Option (d) provides: \(n=3\), \(l=2\), \(m=-2\), \(s=+1/2\).- \(n=3\) and \(l=2\) correspond to a \(3d\) orbital, not \(4f\). Hence, option (d) is incorrect.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by \( n \), is a pivotal part of understanding quantum mechanics and the behaviors of electrons within atoms. It represents the shell or energy level in which an electron resides.

• It has positive integer values, such as 1, 2, 3, and so forth. The higher the \( n \) value, the greater the energy level and the farther the electron is from the nucleus.
• This number helps define the electron cloud's overall size and energy magnitude. For example, a shell with \( n = 4 \) has more energy and is larger compared to one with \( n = 1 \).

Understanding \( n \) is crucial for predicting electron behavior and their energy state within an atom. A larger value of \( n \) implies a greater electron cloud, more nodes, and higher potential energy.

Azimuthal Quantum Number

The azimuthal quantum number, represented by \( l \), dives deeper into understanding an electron's environment by specifying the subshell and shape of the orbital.

• The values of \( l \) depend on the principal quantum number \( n \) and range from 0 to \( n-1 \).
• Each value of \( l \) corresponds to a particular type of orbital, such as \( s \) with \( l = 0 \), \( p \) with \( l = 1 \), \( d \) with \( l = 2 \), and \( f \) with \( l = 3 \).

With \( f \) orbitals specifically, this number is always 3, indicating a complex, flower-like shape that can host multiple electron pairs. This intricacy allows for more nuanced bonding and interactions with other atoms.

Magnetic Quantum Number

The magnetic quantum number, \( m \), provides insight into the orientation of the electron's orbital within a magnetic field.

• The values of \( m \) are determined by the azimuthal quantum number \( l \), and they range from \(-l\) to \(+l\).
• For example, if \( l = 3 \), such as in an \( f \) orbital, \( m \) can take on values like -3, -2, -1, 0, 1, 2, or 3.

This range allows each orbital type to have multiple orientations, giving rise to diverse electron configurations. This diversity in orientation aids in the prediction of electron occupancy across different magnetic fields, helping chemists and physicists understand and predict atomic behaviors more precisely.

Spin Quantum Number

The spin quantum number, \( s \), is essential for describing the electron's intrinsic angular momentum, often referred to simply as "spin."

• This quantum number can only hold values of \(+1/2\) or \(-1/2\), representing the two possible spin orientations of an electron.
• Spin quantum numbers are crucial for the Pauli Exclusion Principle, which states that no two electrons in an atom can have the same set of quantum numbers, differentiating pairs of electrons within the same orbital.

Understanding electron spin is critical in chemistry and physics as it ultimately affects magnetic properties and the formation of covalent bonds between atoms. Spin adds another layer to the already complex nature of electron behavior.

f Orbital

The \( f \) orbital is one of the more complex electron orbitals, associated with the azimuthal quantum number \( l = 3 \). This shape of the \( f \) orbital is intricate, allowing it to host up to 14 electrons across its multiple orientations.

• These orbitals are significant for elements with larger atomic numbers, particularly those in the lanthanide and actinide series.
• The shape and complex nodal structures make \( f \) orbitals integral in deeper chemical interactions, especially among transition metal and heavy element chemistry.

Understanding \( f \) orbitals aids in explaining phenomena such as specific light absorption properties and a material's magnetic attributes. Recognizing these can offer insights into advanced material design and atomic interactions.