Problem 136 - A Complete Resource Book in Chemistry for JEE Main

Question

Match List I with List II and select the correct answer using the codes given below the lists: List-I (Metal ions) ]. \(\mathrm{Cr}^{3+}\) 2\. \(\mathrm{Fe}^{2+}\) 3\. \(\mathrm{Ni}^{2+}\) 4\. \(\mathrm{Mn}^{2+}\) List-II (Magnetic moment) (i) \(\sqrt{35}\) (ii) \(\sqrt{30}\) (iii) \(\sqrt{24}\) (iv) \(\sqrt{15}\) (v) \(\sqrt{8}\)

Step-by-Step Solution

Answer

1-iv, 2-iii, 3-v, 4-i

Step 1: Understand the Magnetic Moment Formula

The magnetic moment (abla) can be calculated using the formula: \[ \mu = \sqrt{n(n+2)} \ \text{B.M.} \] where \(n\) is the number of unpaired electrons in the metal ion. B.M. stands for Bohr Magneton.

Step 2: Determine Unpaired Electrons for Cr3+

For \(\text{Cr}^{3+}\), the electron configuration is \(\text{[Ar]}\,3d^3\). Therefore, \(\text{Cr}^{3+}\) has 3 unpaired electrons. Plug into the formula: \[ \mu = \sqrt{3(3+2)} = \sqrt{15} \]

Step 3: Determine Unpaired Electrons for Fe2+

For \(\text{Fe}^{2+}\), the electron configuration is \(\text{[Ar]}\,3d^6\). Therefore, \(\text{Fe}^{2+}\) has 4 unpaired electrons. Plug into the formula: \[ \mu = \sqrt{4(4+2)} = \sqrt{24} \]

Step 4: Determine Unpaired Electrons for Ni2+

For \(\text{Ni}^{2+}\), the electron configuration is \(\text{[Ar]}\,3d^8\). Therefore, \(\text{Ni}^{2+}\) has 2 unpaired electrons. Plug into the formula: \[ \mu = \sqrt{2(2+2)} = \sqrt{8} \]

Step 5: Determine Unpaired Electrons for Mn2+

For \(\text{Mn}^{2+}\), the electron configuration is \(\text{[Ar]}\,3d^5\). Therefore, \(\text{Mn}^{2+}\) has 5 unpaired electrons. Plug into the formula: \[ \mu = \sqrt{5(5+2)} = \sqrt{35} \]

Step 6: Match the Magnetic Moments

Using the solutions from previous steps, match the magnetic moments calculated:- \(\text{Cr}^{3+}\) matches \(\sqrt{15}\) (iv)- \(\text{Fe}^{2+}\) matches \(\sqrt{24}\) (iii)- \(\text{Ni}^{2+}\) matches \(\sqrt{8}\) (v)- \(\text{Mn}^{2+}\) matches \(\sqrt{35}\) (i)

Key Concepts

Bohr Magneton

Bohr Magneton is a fundamental concept in the study of magnetism at the atomic scale. A Bohr Magneton (B.M.) is the unit used to express the magnetic moment of an electron due to its orbital or spin angular momentum. The formula for the Bohr Magneton is \[ \mu_B = \frac{e\hbar}{2m_e} \]where \(e\) represents the electron charge, \(\hbar\) is the reduced Planck's constant, and \(m_e\) is the electron mass.
In practice, when calculating the magnetic moment of metal ions, the Bohr Magneton provides a standardized way to convey the influence of unpaired electrons in generating a magnetic field. The presence of unpaired electrons is crucial for determining the magnitude of the magnetic moment. Understanding how these basic physical constants affect magnetism allows students to grasp why some materials are magnetic and others are not.

Unpaired Electrons

Unpaired electrons play a vital role in determining the magnetic properties of an element. Simply put, they are electrons that do not have a partner with an opposite spin in the same orbital. These unpaired electrons are responsible for generating a magnetic moment within an atom or ion.
When you encounter an element with unpaired electrons, it can be diamagnetic, paramagnetic, or even ferromagnetic, depending on the arrangement of these electrons:

• Diamagnetic: All electrons are paired, resulting in no net magnetic moment.
• Paramagnetic: Has unpaired electrons, which create a magnetic moment that aligns with external magnetic fields.
• Ferromagnetic: Contains unpaired electrons that can align spontaneously, producing a strong permanent magnetization.

In the context of transition metals, unpaired electrons arise from partially filled d-orbitals, significantly affecting their magnetic characteristics. Calculating the number of unpaired electrons helps in determining the magnetic moment using the formula \( \mu = \sqrt{n(n+2)} \), where \( n \) refers to the number of unpaired electrons.

Electron Configuration

Electron configuration describes the distribution of electrons in an atom or ion in various orbitals. It is critical for predicting magnetic properties, chemical reactivity, and bonding behavior. In transition metal ions, knowing the electron configuration helps determine the number of unpaired electrons, which in turn affects their magnetic moment.
The general notation for electron configuration uses a combination of numbers and letters, such as \[ \text{[Ar]}\,3d^3 \] as in the case of \( \text{Cr}^{3+} \). The notation indicates the orbitals' filling order following the Aufbau principle, where electrons fill lower energy levels first.
When ions are formed, electrons are typically removed from the outermost orbitals, which alters the electron configuration and can affect the magnetic moment. For transition metals, this usually means electrons are removed from the 4s and 3d orbitals, highlighting the importance of electron configuration in understanding and predicting the behavior of metal ions.

Transition Metal Ions

Transition metal ions are elements that belong to the d-block of the periodic table and can form ions with partially filled d-orbitals. These ions often exhibit unique magnetic and chemical properties due to their electron configurations. Because the d-electrons in these ions can be unpaired, determining their configuration is essential for calculating the magnetic moment.
For example:

• \( \text{Cr}^{3+} \): electron configuration \( \text{[Ar]}\,3d^3 \)
• \( \text{Fe}^{2+} \): electron configuration \( \text{[Ar]}\,3d^6 \)
• \( \text{Ni}^{2+} \): electron configuration \( \text{[Ar]}\,3d^8 \)
• \( \text{Mn}^{2+} \): electron configuration \( \text{[Ar]}\,3d^5 \)

The partially filled d-orbitals result in magnetic properties that vary significantly across different ions. Exploring the electron configurations of transition metal ions gives insights into the material properties that are essential in fields like electronics, magnetics, and even biological systems.