Question

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

Short answer

The correct set of quantum numbers for a 4f orbital electron is (c).

Step-by-step solution

Understanding Quantum Numbers

Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. The principal quantum number (\(n\)) describes the energy level/shell. The angular momentum quantum number (\(l\)) should be smaller than \(n\) and describes the shape of the orbital, where \(l = 0\) is s, \(l = 1\) is p, \(l = 2\) is d, and \(l = 3\) is f. The magnetic quantum number (\(m\)) ranges from \(-l\) to \(+l\), and the spin quantum number (\(s\)) can be \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

Examining 4f Orbital Quantum Numbers

For a 4f orbital, \(n = 4\) and \(l = 3\). The \(m\) values can range from \(-3\) to \(+3\), and \(s\) can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\). Given these rules, evaluate each set of quantum numbers provided in the options.

Evaluating Option (a)

Option (a) presents the quantum numbers: \(n = 4, l = 3, m = +4, s = +\frac{1}{2}\). Since \(m = +4\) is outside the allowable range of \(-3\) to \(+3\) for a 4f orbital, this set is incorrect.

Evaluating Option (b)

Option (b) presents the quantum numbers: \(n = 4, l = 4, m = -4, s = -\frac{1}{2}\). Here, \(l = 4\) is incorrect for a 4f orbital since \(l\) for f is 3. Thus, this set is incorrect.

Evaluating Option (c)

Option (c) presents the quantum numbers: \(n = 4, l = 3, m = +1, s = +\frac{1}{2}\). All these values are within the permissible ranges for a 4f orbital: \(n = 4\), \(l = 3\), \(m\) is within \(-3\) to \(+3\), and \(s\) is valid. Therefore, this set is correct.

Evaluating Option (d)

Option (d) presents the quantum numbers: \(n = 3, l = 2, m = -2, s = +\frac{1}{2}\). While \(l = 2\) is suitable for a d orbital, the principal quantum number is \(n = 4\) for a 4f orbital, rendering this option incorrect.

Key concepts

Principal Quantum Number

The principal quantum number, denoted by \( n \), serves as a key identifier of the electron's energy level or shell in an atom. It's a positive integer value that tells us the distance of the electron from the nucleus of the atom.

• Values of \( n \) can be 1, 2, 3, and so on.
• Higher \( n \) values correspond to higher energy levels, situated farther from the nucleus.

This number is pivotal because it defines the size and energy of the orbital where an electron resides. For a 4f orbital, the principal quantum number \( n = 4 \) indicates that the electron is in the fourth energy level. Understanding \( n \) helps us grasp how electrons are distributed across different shells and how these distributions influence atomic structure and behaviors.

Angular Momentum Quantum Number

The angular momentum quantum number, symbolized by \( l \), is essential in defining the shape of the electron orbital. It takes on integer values from 0 up to \( n-1 \).

• When \( l = 0 \), the orbital is spherically shaped (s-orbital).
• When \( l = 1 \), the orbital has a dumbbell shape (p-orbital).
• When \( l = 2 \), it represents more complex shapes (d-orbital).
• When \( l = 3 \), the orbital further increases in complexity, representing an f-orbital.

For the 4f orbital in our problem, \( l \) is 3. This number not only indicates the shape but also helps in understanding the regions where electrons are likely to be found within that orbital.

Magnetic Quantum Number

The magnetic quantum number, denoted by \( m \), further differentiates orbitals of the same shape and energy level by indicating their orientation in space. Its value ranges from \(-l\) to \(+l\), including zero.

• For an f-orbital where \( l = 3 \), \( m \) can be -3, -2, -1, 0, 1, 2, or 3.

This quantum number plays a crucial role when an atom is under the influence of an external magnetic field, affecting how orbitals are arranged. In the context of a 4f orbital, \( m = +1 \) from option (c) remains within the permissible range, showing a correct orientation possibility for the electron's position.

Spin Quantum Number

The spin quantum number, indicated by \( s \), reflects the electron's intrinsic angular momentum, or "spin." It assumes one of two possible values: \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

• These values represent the two possible spin states of an electron: "up" and "down."

Even though the electron spin is an abstract concept, it has measurable effects on magnetic properties and is essential for the Pauli exclusion principle, which states that no two electrons in an atom can have identical sets of quantum numbers. In our example, the spin quantum number values in both options (a) and (c) are \(+\frac{1}{2}\), correctly representing one of the valid orientations an electron can have in its orbital.