Problem 3 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The correct electronic configuration and spin-only magnetic moment (BM) of \(\mathrm{Gd}^{3+}(\mathrm{Z}=64)\), respectively, are: (a) \([\mathrm{Xe}] 4 f^{7}\) and \(8.9\) (b) \([\mathrm{Xe}] 4 f^{7}\) and \(7.9\) (c) \([\mathrm{Xe}] 5 f^{7}\) and \(8.9\) (d) \([\mathrm{Xe}] 5 f^{7}\) and \(7.9\)

Step-by-Step Solution

Answer

The correct answer is (b) \([ ext{Xe}] 4f^7\) and \(7.9\) BM.

Step 1: Determine Electron Configuration for Gd

Gadolinium (Gd) has a ground state electron configuration of \([\text{Xe}] \ 4f^7 \ 5d^1 \ 6s^2\). When it forms \(\text{Gd}^{3+}\), it loses three electrons. These electrons are removed from the outermost orbitals, so the electron configuration of \(\text{Gd}^{3+}\) is \[\text{Gd}^{3+}: [\text{Xe}] \ 4f^7\]. This means option (a) or (b) is correct for the electron configuration part.

Step 2: Calculate Spin-Only Magnetic Moment

The spin-only magnetic moment \(\mu_s\) can be calculated using the formula: \[ \mu_s = \sqrt{n(n + 2)} \text{ BM} \]where \(n\) is the number of unpaired electrons. For \(\text{Gd}^{3+}\), there are 7 unpaired electrons in the \(4f\) subshell.

Step 3: Apply Magnetic Moment Formula

Substitute \(n = 7\) into the formula:\[ \mu_s = \sqrt{7(7 + 2)} \text{ BM} = \sqrt{63} \text{ BM} \approx 7.94 \text{ BM} \]Thus, the spin-only magnetic moment is approximately \(7.94\) BM.

Step 4: Match Available Options

Comparing our calculations with the given options, we find that the configuration \([\text{Xe}] \ 4f^7\) and spin-only magnetic moment approximately \(7.9\) BM match option (b).

Key Concepts

Gadolinium

Gadolinium is an interesting rare-earth element with the atomic number 64. In the periodic table, it's positioned in the lanthanide series. One of its notable features is its electron configuration, which in its neutral state is \([\text{Xe}]\ 4f^7\ 5d^1\ 6s^2\). This configuration means Gadolinium has one electron in the 5d orbital and two in the 6s orbital, with seven in the 4f subshell.

When Gadolinium becomes a three-plus cation (\(\text{Gd}^{3+}\)), it loses three electrons. These electrons are typically removed from the outermost orbitals — initially, from the 6s then the 5d orbitals. As a result, the electron configuration for \(\text{Gd}^{3+}\) becomes \([\text{Xe}]\ 4f^7\). The interesting feature here is that all electrons are left in the 4f subshell. This makes Gadolinium's chemistry particularly intriguing, as these inner orbitals don't participate in chemical bonding but are crucial for its magnetic properties.

Spin-Only Magnetic Moment

The Spin-Only Magnetic Moment is an essential concept for understanding the magnetic properties of electron subshells. It is used to approximate the magnetic moment derived from the spin of unpaired electrons in a given atomic or molecular system.

The formula to calculate the spin-only magnetic moment \(\mu_s\) is:

• \(\mu_s = \sqrt{n(n + 2)}\) BM

Here, \(n\) represents the number of unpaired electrons. The unit BM stands for Bohr Magneton, which is the physical unit for expressing magnetic moments. The spin-only formula estimates the magnetic moment without considering orbital contributions, emphasizing electron spins' role.

In the case of \(\text{Gd}^{3+}\), since it has seven unpaired electrons in the 4f subshell, the magnetic moment is calculated as \(\mu_s = \sqrt{7(7 + 2)} \approx 7.94\) BM. This result clearly shows the direct contribution of the unpaired f-electrons in defining its magnetic behavior.

Unpaired Electrons

Unpaired electrons are the electrons that remain alone in an orbital of an atom or a molecule without a paired partner having the opposite spin. They are a fundamental aspect of understanding the magnetic properties of elements, as these electrons contribute significantly to the magnetic moment.

For Gadolinium in its \(\text{Gd}^{3+}\) state, the electron configuration is \([\text{Xe}]\ 4f^7\). This indicates there are seven unpaired electrons. Unpaired electrons create a magnetic dipole moment, essentially a measure of the magnetic strength and orientation of a magnet or other object that produces a magnetic field.

• Unpaired electrons contribute directly to the spin-only magnetic moment, which is significant in determining the paramagnetic properties of materials.
• Materials with unpaired electrons tend to show magnetic properties because of these electrons' magnetic moments.
• In contrast, elements or molecules with all paired electrons are usually diamagnetic, exhibiting no net magnetic moment.

Understanding the role of unpaired electrons is crucial in the study of magnetism, as they are key players in interactions with external magnetic fields.