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 4f 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) \(\mathrm{n}=4, l=3, \mathrm{~m}=+\mathrm{l}, \mathrm{s}=+1 / 2\) (d) \(\mathrm{n}=3, l=2, \mathrm{~m}=-2, \mathrm{~s}=+1 / 2\)

Step-by-Step Solution

Answer

Option (c) is correct: \(n=4, l=3, m=+3, s=+1/2\).

Step 1: Identify Quantum Number Rules

For an electron in any orbital, the quantum numbers must meet specific rules:- The principal quantum number (\(n\)) is a positive integer (\(n = 1, 2, 3, \ldots\)). For a 4f orbital, \(n = 4\).- The azimuthal quantum number (\(l\)) relates to the shape of the orbital and is an integer from 0 to \(n-1\). For an f orbital, \(l = 3\).- The magnetic quantum number (\(m\)) ranges from \(-l\) to \(+l\). Thus, for \(l = 3\), \(m\) can be any integer from \(-3\) to \(+3\).- The spin quantum number (\(s\)) can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

Step 2: Examine Option (a)

Option (a) provides \(n=4, l=3, m=+4, s=+1/2\). Here, \(m = +4\) is not valid because for \(l=3\), \(m\) must be between \(-3\) and \(+3\). Thus, option (a) has an incorrect \(m\) value.

Step 3: Examine Option (b)

Option (b) provides \(n=4, l=4, m=-4, s=-1/2\). For a 4f orbital, \(l\) must be 3, not 4. Therefore, option (b) has the incorrect \(l\) value.

Step 4: Examine Option (c)

Option (c) provides \(n=4, l=3, m=+3, s=+1/2\). All quantum numbers meet the rules for a 4f orbital: \(n=4\) is correct, \(l=3\) is correct for f orbitals, \(m=+3\) is valid as it ranges from \(-3\) to \(+3\), and \(s=+1/2\) is a valid spin value. Thus, option (c) is correct.

Step 5: Examine Option (d)

Option (d) provides \(n=3, l=2, m=-2, s=+1/2\). These quantum numbers describe a 3d orbital, not a 4f orbital. Therefore, option (d) is incorrect for specifying a 4f orbital.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by \( n \), is crucial in determining the energy level and size of an electron's orbital. It represents the shell or energy level in which the electron resides. The values of \( n \) are positive integers starting from 1: \( n = 1, 2, 3, \ldots \).
Each increase in \( n \) signifies a higher energy level and larger orbital size. It means that electrons in higher principal quantum numbers are farther away from the nucleus. For instance, an electron in a 4f orbital has \( n=4 \), indicating it is in the 4th energy level.
The principal quantum number is a two-fold indicator: it tells us both about the energy an electron possesses and its general locality within the atom.

Azimuthal Quantum Number

The azimuthal quantum number, represented as \( l \), defines the shape of the electron's orbital. It is dependent on the principal quantum number \( n \), and can take values from 0 up to \( n-1 \).
Each value of \( l \) corresponds to a different subshell within a principal energy level. For \( l = 0 \), the subshell is 's'; for \( l = 1 \), it's 'p'; for \( l = 2 \), it's 'd'; and for \( l = 3 \), it is 'f'. So, for a 4f orbital, \( n = 4 \) and \( l = 3 \).
Understanding \( l \) helps us predict the shape of orbitals and, consequently, the type of bonding an atom might undergo. Given that electrons follow certain shapes and orientations in space, the value of \( l \) is key in chemistry and atomic physics.

Magnetic Quantum Number

The magnetic quantum number, \( m \), provides insight into the orientation of an electron's orbital in space. It can take on integer values ranging from \(-l\) to \(+l\), inclusive.
For a given azimuthal quantum number \( l \), \( m \) determines the exact orbital within the subshell that the electron occupies. For instance, if \( l = 3 \), then \( m \) can be \(-3, -2, -1, 0, +1, +2, +3 \). Thus, in a 4f orbital, where \( l = 3 \), the valid values for \( m \) range from \(-3\) to \(+3\).
The magnetic quantum number reveals how orbitals are oriented concerning external magnetic fields, which is a fundamental consideration when examining atomic interactions or conducting spectroscopic studies.

Spin Quantum Number

The spin quantum number, symbolized as \( s \), explains an electron's intrinsic angular momentum. Every electron has a spin, which can be thought of as rotating on its axis, and it can take on two possible values: \(+\frac{1}{2}\) or \(-\frac{1}{2}\).
This property introduces the idea of magnetism into quantum mechanics. In essence, an electron behaves like a tiny magnet due to its spin. Electrons in the same orbital must have opposite spins, a requirement explained by the Pauli exclusion principle stating that no two electrons in an atom can have the same set of four quantum numbers.
The spin quantum number is crucial for understanding magnetic properties of materials and how electrons fill various orbitals, influencing atomic stability and chemistry.