Problem 11 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following pairs of ions have same paramagnetic moment? (a) \(\mathrm{Cu}^{2+}, \mathrm{Ti}^{3+}\) (b) \(\mathrm{Ti}^{3+}, \mathrm{Ni}^{2+}\) (c) \(\mathrm{Ti}^{4+}, \mathrm{Cu}^{2+}\) (d) \(\mathrm{Mn}^{2+}, \mathrm{Cu}^{2+}\)

Step-by-Step Solution

Answer

(a) \( \mathrm{Cu}^{2+}, \mathrm{Ti}^{3+} \)

Step 1: Understand Paramagnetic Moment

The paramagnetic moment of an ion is determined by the number of unpaired electrons. The formula to calculate the magnetic moment (m) is given by \[ m = \sqrt{n(n+2)} \] where \( n \) is the number of unpaired electrons.

Step 2: Determine Electrons for Each Ion

Identify the electron configurations for each ion: - \( \mathrm{Cu}^{2+} \): For \( \mathrm{Cu} \), the configuration is \([\mathrm{Ar}] 3d^{10} 4s^{1}\). Losing two electrons results in \([\mathrm{Ar}] 3d^{9}\).- \( \mathrm{Ti}^{3+} \): For \( \mathrm{Ti} \), the configuration is \([\mathrm{Ar}] 3d^{2} 4s^{2}\). Losing three electrons gives \([\mathrm{Ar}] 3d^{1}\).- \( \mathrm{Ni}^{2+} \): For \( \mathrm{Ni} \), the configuration is \([\mathrm{Ar}] 3d^{8} 4s^{2}\). Losing two electrons gives \([\mathrm{Ar}] 3d^{8}\).- \( \mathrm{Mn}^{2+} \): For \( \mathrm{Mn} \), the configuration is \([\mathrm{Ar}] 3d^{5} 4s^{2}\). Losing two electrons gives \([\mathrm{Ar}] 3d^{5}\).

Step 3: Calculate Unpaired Electrons

Calculate the number of unpaired electrons:- \( \mathrm{Cu}^{2+} \): \( 3d^{9} \rightarrow 1 \) unpaired electron.- \( \mathrm{Ti}^{3+} \): \( 3d^{1} \rightarrow 1 \) unpaired electron.- \( \mathrm{Ni}^{2+} \): \( 3d^{8} \rightarrow 2 \) unpaired electrons.- \( \mathrm{Mn}^{2+} \): \( 3d^{5} \rightarrow 5 \) unpaired electrons.

Step 4: Compare Paramagnetic Moments

Use the magnetic moment formula for comparison:- \( \mathrm{Cu}^{2+} \) and \( \mathrm{Ti}^{3+} \) both have \( 1 \) unpaired electron: \ \( \mu = \sqrt{1(1+2)} = \sqrt{3} \).- \( \mathrm{Ni}^{2+} \) has \( 2 \) unpaired electrons: \ \( \mu = \sqrt{2(2+2)} = \sqrt{8} \).- \( \mathrm{Mn}^{2+} \) has \( 5 \) unpaired electrons: \ \( \mu = \sqrt{5(5+2)} = \sqrt{35} \).Pairs \( \mathrm{Cu}^{2+} \) and \( \mathrm{Ti}^{3+} \) have the same paramagnetic moment.

Key Concepts

Unpaired Electrons

In atoms, electrons are arranged in various energy levels or orbitals. Electrons have a property called "spin," which can be either up or down. When two electrons occupy the same orbital, their spins are opposite, effectively pairing each other. However, if an orbital contains only one electron, it is unpaired.
Unpaired electrons play a crucial role in determining the magnetic properties of an atom. This is because unpaired electrons create a net magnetic moment, as their magnetic fields do not cancel out.

• Electrons in a filled orbital pair up with opposite spins, resulting in no net magnetic moment.
• An unpaired electron contributes to the magnetism of the atom, resulting in what we call paramagnetism.

Thus, the more unpaired electrons an atom or ion has, the stronger its paramagnetic properties. This is why identifying the number of unpaired electrons in ions like \( \mathrm{Cu}^{2+} \) and \( \mathrm{Mn}^{2+} \) is essential for understanding their magnetic behavior.

Electron Configuration

Electron configuration describes how electrons are distributed in an atom's or ion’s orbitals. Understanding electron configurations is key to determining the atom's chemical properties, including magnetism.
The general rule for writing electron configurations is to fill orbitals from lower to higher energy levels. This is guided by the Aufbau principle, Hund's rule, and the Pauli Exclusion Principle.

• The Aufbau principle states that electrons occupy the lowest energy level first.
• Hund's rule suggests that electrons fill degenerate orbitals singly before pairing up.
• The Pauli Exclusion Principle explains that no two electrons can have identical quantum numbers within an atom, leading to opposing spins when in the same orbital.

For ions like \( \mathrm{Cu}^{2+} \) or \( \mathrm{Ti}^{3+} \), electrons are removed starting from the outermost shell. For instance, \( \mathrm{Cu}^{2+} \) originally has the configuration \([\mathrm{Ar}] 3d^{10}4s^{1}\). Losing two electrons results in \([\mathrm{Ar}] 3d^{9}\), and the presence of an unpaired electron is what gives it paramagnetic properties.

Magnetic Moment Formula

The magnetic moment is a measure of the strength of a magnetic source and, in the context of atoms or ions, is mainly due to unpaired electrons. To quantify this property, we use the formula: \[ \mu = \sqrt{n(n+2)} \] where \(n\) is the number of unpaired electrons.
This formula provides a way to calculate the magnetic moment in terms of the Bohr magneton, which is a physical constant. For example:

• For an ion with \(1\) unpaired electron, such as \( \mathrm{Cu}^{2+} \), the magnetic moment would be \( \sqrt{3} \).
• For \( \mathrm{Mn}^{2+} \), with \(5\) unpaired electrons, the moment significantly increases to \( \sqrt{35} \).

This relationship highlights that the more unpaired electrons present, the larger the magnetic moment and thus the stronger the paramagnetic property of the substance. By calculating these values, students can better understand which substances demonstrate similar magnetic behaviors.