Problem 134 - A Complete Resource Book in Chemistry for JEE Main

Question

The substances which contain atoms with unpaired electrons in their orbitals behave as paramagnetic substances. Such substances are weakly attracted by the magnetic field. The paramagnetism is expressed in terms of magnetic moment. The magnetic moment is related to the number of unpaired electrons according to the following relation. Magnetic moment, \(\mu=\sqrt{n(n+2)} B M\) Where, \(\mathrm{n}\) - number of unpaired electrons. BM stands for Bohr Magneton, a unit of magnetic moment. Which of the following ions has magnetic momen equal to that of \(\mathrm{Ti}^{3+}\) (a) \(\mathrm{Ni}^{2+}\) (b) \(\mathrm{Co}^{2+}\) (c) \(\mathrm{Fe}^{2+}\) (d) \(\mathrm{Cu}^{2+}\)

Step-by-Step Solution

Answer

The ion with a magnetic moment equal to Ti³⁺ is Cu²⁺.

Step 1: Determine unpaired electrons for Ti³⁺

Start with the electronic configuration of the neutral titanium atom, Ti, which is extit{3d² 4s²}. Removing three electrons (since it's Ti³⁺), we first remove from the 4s, then the 3d orbitals, resulting in: \[\mathrm{Ti}^{3+}:\; 3d^1\]Thus, Ti³⁺ has 1 unpaired electron.

Step 2: Calculate magnetic moment for Ti³⁺

Using the formula for magnetic moment, \(\mu = \sqrt{n(n+2)} BM\), calculate for \(n = 1\):\[\mu = \sqrt{1(1+2)} = \sqrt{3} \approx 1.73 \; BM\]This is the magnetic moment for Ti³⁺.

Step 3: Determine unpaired electrons for each ion

For each given ion, determine the electronic configuration and the number of unpaired electrons: - **Ni²⁺**: Starts with Ni: 3d⁸ 4s². Removing 2 electrons: 3d⁸. It has 2 unpaired electrons. - **Co²⁺**: Starts with Co: 3d⁷ 4s². Removing 2 electrons: 3d⁷. It has 3 unpaired electrons. - **Fe²⁺**: Starts with Fe: 3d⁶ 4s². Removing 2 electrons: 3d⁶. It has 4 unpaired electrons. - **Cu²⁺**: Starts with Cu: 3d¹⁰ 4s¹. Removing 2 electrons: 3d⁹. It has 1 unpaired electron.

Step 4: Calculate magnetic moment for each ion

Now calculate the magnetic moment for each ion using the same formula:- **Ni²⁺**: \[ \mu = \sqrt{2(2+2)} = \sqrt{8} = 2.83 \; BM \]- **Co²⁺**: \[ \mu = \sqrt{3(3+2)} = \sqrt{15} = 3.87 \; BM \]- **Fe²⁺**: \[ \mu = \sqrt{4(4+2)} = \sqrt{24} = 4.9 \; BM \]- **Cu²⁺**: \[ \mu = \sqrt{1(1+2)} = \sqrt{3} = 1.73 \; BM \]Cu²⁺ has a magnetic moment of 1.73 BM, which matches Ti³⁺.

Step 5: Conclusion

Among the given ions, Cu²⁺ matches the magnetic moment of Ti³⁺ at approximately 1.73 BM.

Key Concepts

Magnetic Moment

Understanding the concept of magnetic moment is crucial in grasping paramagnetism. The magnetic moment is a measure of the strength and direction of a magnetic source. It indicates how much an object will align with a magnetic field and how energetically it will behave in response to it. In the realm of chemistry, particularly for ions and molecules containing unpaired electrons, the magnetic moment plays a vital role in determining paramagnetic behavior.

For ions, the magnetic moment is calculated using the formula: \[ \mu = \sqrt{n(n+2)} \times BM \]where \(n\) represents the number of unpaired electrons, and \(BM\) stands for Bohr Magneton, the unit of magnetic moment. The square root indicates that the formula accounts for both the number of unpaired electrons and their interactions, producing a more accurate measure of an ion's magnetic character.

When examining ions like \( \text{Ti}^{3+} \) and \( \text{Cu}^{2+} \), they both exhibit a magnetic moment of approximately 1.73 BM due to their unpaired electrons. This means they will demonstrate similar paramagnetic behavior in a magnetic field.

Unpaired Electrons

An understanding of unpaired electrons is essential when exploring the magnetic properties of substances. Within an atom, electrons occupy orbitals, and according to the Pauli exclusion principle, each orbital can hold up to two electrons with opposite spins. An "unpaired electron" is one that does not have another electron of opposite spin in the same orbital.

Substances with unpaired electrons in their electronic structure usually display paramagnetism because the unpaired electrons have magnetic moments that can align with an external magnetic field. This alignment gives rise to a net magnetic moment.

For example, in our problem, we determined the number of unpaired electrons by evaluating the electronic configurations of some ions:

• \( \text{Ti}^{3+} \) leads to \(3d^1\), having 1 unpaired electron.
• \( \text{Co}^{2+} \) in \(3d^7\) has 3 unpaired electrons.
• \( \text{Fe}^{2+} \) with \(3d^6\) contains 4 unpaired electrons.

These unpaired electrons are central to calculating the magnetic moment and thus the magnitude of paramagnetic behavior.

Bohr Magneton

The Bohr Magneton, denoted by \( BM \), is a physical constant that describes the quantum of magnetic moment for an electron due to its orbital or intrinsic spin. It provides a standardized measure for calculating magnetic moments, particularly in atoms and subatomic particles. Scientifically, one Bohr Magneton is defined in terms of the electron's charge, mass, and spin, aligning with the equation:\[ \mu_B = \frac{e \hbar}{2m_e} \]where \( e \) is the elementary charge, \( \hbar \) is the reduced Planck's constant, and \( m_e \) is the electron mass.

In practice, the Bohr Magneton is often used to express the magnetic moment of atoms, specifically because it provides a scale that mirrors the energies involved in atomic magnetic interactions. This is why, in the formula for magnetic moment \( \mu = \sqrt{n(n+2)} \times BM \), \( BM \) serves as the unit. It standardizes how magnetic moments are compared across different ions and atoms, such as \( \text{Ti}^{3+} \) and \( \text{Cu}^{2+} \), which both have a magnetic moment of about 1.73 Bohr Magneton.