Problem 119 - A Complete Resource Book in Chemistry for JEE Main

Question

Some of the following sets of quantum numbers are correct for a \(4 \mathrm{~d}\) electron. Which are correct sets? (a) \(4,3,2,+\frac{1}{2}\) (b) \(4,2,1,0\) (c) \(4,2,-2,+\frac{1}{2}\) (d) \(4,2,1,-\frac{1}{2}\)

Step-by-Step Solution

Answer

Correct sets: (c) and (d).

Step 1: Understand Quantum Numbers

There are four quantum numbers: principal quantum number \(n\), azimuthal quantum number \(l\), magnetic quantum number \(m_l\), and spin quantum number \(m_s\). For a \(4d\) electron: \(n = 4\), \(l = 2\) (since \(l\) for \(d\) orbital equals 2), allowed values of \(m_l\) are -2, -1, 0, 1, 2, and \(m_s\) is \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

Step 2: Assess Option (a)

The given quantum numbers are \(4,3,2,+\frac{1}{2}\). Here, \(n = 4\) is correct, but \(l = 3\) corresponds to an 'f' orbital, not a 'd' orbital. Therefore, this set is incorrect for a \(4d\) electron.

Step 3: Assess Option (b)

The quantum numbers are \(4,2,1,0\). \(n = 4\), \(l = 2\) is correct for a \(d\) subshell, \(m_l = 1\) is valid as \(m_l\) ranges from -2 to 2, and \(m_s = 0\) is invalid as it must be \(+\frac{1}{2}\) or \(-\frac{1}{2}\). Therefore, this set is incorrect.

Step 4: Assess Option (c)

The quantum numbers are \(4,2,-2,+\frac{1}{2}\). Here, \(n = 4\), \(l = 2\) is correct for a \(d\) subshell, \(m_l = -2\) is within the valid range, and \(m_s = +\frac{1}{2}\) is valid. Therefore, this set is correct.

Step 5: Assess Option (d)

The quantum numbers are \(4,2,1,-\frac{1}{2}\). \(n = 4\), \(l = 2\) is correct for a \(d\) orbital, \(m_l = 1\) is valid, and \(m_s = -\frac{1}{2}\) is acceptable. Therefore, this set is also correct.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted as \(n\), is one of the four quantum numbers used to describe the unique quantum state of electrons in an atom. It primarily determines the energy level and size of an electron's orbit. The value of \(n\) is always a positive integer (1, 2, 3, ...), and it also indicates the number of shells in which the electron can exist. For instance, if \(n = 4\), the electron is part of the fourth energy level or shell.

• Energy Level: Higher \(n\) means a higher energy level.
• Orbital Radius: A larger \(n\) value implies a larger orbital radius.

When analyzing a particular electron, like those in the options provided, \(n\) gives us a starting point to understand where the electron is primarily located in terms of energy and size.

Azimuthal Quantum Number

The azimuthal quantum number, or angular momentum quantum number, is represented by \(l\). This quantum number defines the shape of the electron's orbital and is related to the subshells within a given principal quantum level. It takes integer values from 0 to \(n-1\) for each principal quantum number \(n\).

• s subshell: \(l = 0\)
• p subshell: \(l = 1\)
• d subshell: \(l = 2\)
• f subshell: \(l = 3\)

For a \(4d\) electron, where \(n = 4\), \(l = 2\) is correct as it correlates with the 'd' orbital. Understanding \(l\) allows you to know the orbital type or shape the electron occupies.

Magnetic Quantum Number

The magnetic quantum number, symbolized as \(m_l\), signifies the orientation of the orbital in space relative to the other orbitals. It can be any integer between \(-l\) and \(+l\), including zero. This number provides details on the specific orbital the electron is in within the subshell.

• For \(l = 2\), values can be \(-2, -1, 0, 1, 2\)

Choosing the correct \(m_l\) is crucial when determining the correct set of quantum numbers. For example, a \(4d\) subshell allows \(m_l\) values ranging from \(-2\) to \(+2\). Options like \(m_l = 1\) and \(m_l = -2\) would be considered valid within this subshell.

Spin Quantum Number

The spin quantum number, or \(m_s\), represents the spin of an electron, which is an intrinsic form of angular momentum. Electrons can spin in two possible directions, so the spin quantum number can either be \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

• Spin Configuration: Determines the magnetic behavior of electrons in a magnetic field.
• Pauli Exclusion Principle: No two electrons can have the same set of all four quantum numbers, which includes the spin quantum number.

Recognizing valid \(m_s\) values is crucial for explaining electron configuration. Both \(+\frac{1}{2}\) and \(-\frac{1}{2}\) are valid options for any electron, making certain sets of quantum numbers correct or incorrect during analysis, as seen in the provided options.