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 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 correct: n=4, l=3, m=+1, s=+1/2.

Step 1: Understanding Quantum Numbers

Quantum numbers describe values of conserved quantities in the dynamics of a quantum system. For an electron, there are four types of quantum numbers: n (principal quantum number), l (azimuthal quantum number), m (magnetic quantum number), and s (spin quantum number). Each set of quantum numbers corresponds to an orbital in which an electron can be found.

Step 2: Identifying Orbital Type

The azimuthal quantum number, l, determines the shape of the orbital. For an f orbital, l is equal to 3. Therefore, for a 4f orbital, we must have n=4 and l=3.

Step 3: Evaluating Each Option

Review each option to ensure the quantum numbers conform to a 4f orbital and quantum number rules: (a) n=4, l=3 is possible, but m=+4 is not valid as m ranges from -3 to +3. (b) l=4 is invalid for any f orbital as l=3 is required. (c) n=4, l=3, m=+1, and s=+1/2 all conform to the 4f orbital requirements. (d) n=3 is not correct for a 4f orbital because it must be n=4.

Step 4: Selecting the Correct Option

Based on the evaluation, only option (c) provides the correct set of quantum numbers for an electron in the 4f orbital. The values are entirely within the allowed ranges for each quantum number with n=4, l=3, m=+1 being valid for a 4f orbital, and s=+1/2 is a standard electron spin.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by \( n \), is essential in understanding how electrons are arranged in an atom. This quantum number tells us about the energy level of the electron and its most probable distance from the nucleus. Essentially, \( n \) indicates the shell in which the electron resides.

• \( n \) is a positive integer and begins at 1, increasing indefinitely. As \( n \) increases, the electron is more distant from the nucleus and the energy level is higher.
• In any given atom, electrons with the same principal quantum number make up a shell.

For example, in a 4f orbital, the principal quantum number is 4, meaning the electron is in the fourth shell, with a higher energy than electrons in the third shell.

Azimuthal Quantum Number

The azimuthal quantum number, represented by \( l \), plays a crucial role in defining the shape of an electron's orbital. It is also referred to as the angular momentum quantum number.

• The range for \( l \) is from 0 to \( n-1 \) for each principal quantum number \( n \).
• The values of \( l \) correspond to different types of orbitals: 0 for s, 1 for p, 2 for d, and 3 for f orbitals.

In the context of our exercise, the azimuthal quantum number for an f orbital is 3, which fits the provided option (c) with \( l=3 \) for a 4f orbital.

Magnetic Quantum Number

The magnetic quantum number, designated as \( m \), provides information about the orientation of the orbital in space. It is within the range defined by the azimuthal quantum number.

• \( m \) can take on values from \(-l\) to \(+l\), including zero.
• This means if \( l = 3 \), as in a 4f orbital, \( m \) can be -3, -2, -1, 0, +1, +2, +3.

Therefore, option (c), with \( m = +1 \), is valid for an electron in a 4f orbital since +1 falls within the allowable range of -3 to +3.

Spin Quantum Number

The spin quantum number, represented by \( s \), accounts for the intrinsic spin of an electron. This aspect helps differentiate electrons within the same orbital.

• The possible values of \( s \) are +1/2 and -1/2, reflecting "spin-up" or "spin-down" orientations.
• Each orbital can hold two electrons with opposing spins, a principle that is fundamental to the Pauli Exclusion Principle.

In our exercise, option (c) uses \( s=+1/2 \), which is a valid representation for the spin orientation of one electron in an orbital.

Electron Orbitals

Electron orbitals are regions around the nucleus of an atom where electrons are most likely to be found. The shape and orientation of these orbitals are defined by the quantum numbers.

• s orbitals are spherical, p orbitals are dumbbell-shaped, d orbitals have more complex clover shapes, and f orbitals are even more complex.
• Each type of orbital corresponds to different azimuthal quantum numbers, influencing how electrons are distributed in each orbital type.

For a 4f orbital, the shape is complex, reflecting the high angular momentum associated with \( l=3 \), and typically, these orbitals start being filled in higher atomic number elements.