Question

Some species are given: \[ \mathrm{Ni^{2+},\ Fe^{2+},\ Co^{2+},\ V^{3+}\ \text{and}\ Ti^{2+}} \] How many species have magnetic moment (spin-only) less than \(3\) BM?

Hint:

For quick checks:

\(n = 1,2 \Rightarrow \mu<3\) BM
\(n \ge 3 \Rightarrow \mu>3\) BM
First find unpaired electrons, then apply the formula

Answer 1

Correct Answer: 3

Given:

• The following species are given:
  • Ni\(^{2+}\)
  • Fe\(^{2+}\)
  • Co\(^{2+}\)
  • V\(^{3+}\)
  • Ti\(^{2+}\)
• We are asked to determine how many of these species have a magnetic moment (spin-only) less than 3 BM (Bohr Magnetons).

Step 1: Magnetic Moment Formula

The magnetic moment (\( \mu \)) for a transition metal ion is given by the spin-only formula: \[ \mu = \sqrt{n(n+2)} \, \text{BM} \] where \( n \) is the number of unpaired electrons.

Step 2: Determine the number of unpaired electrons for each species

We need to determine the electronic configuration for each ion and find the number of unpaired electrons.

1. Ni\(^{2+}\) (Nickel ion)

Nickel (Ni) has an atomic number of 28, and its electronic configuration is: \[ \text{Ni:} \, [Ar] \, 3d^8 4s^2 \] When Ni forms a \( 2+ \) ion, it loses two electrons, typically from the 4s orbital. So, the electronic configuration for Ni\(^{2+}\) is: \[ \text{Ni}^{2+}: \, [Ar] \, 3d^8 \] This leaves 2 unpaired electrons in the \( 3d \) orbitals. \[ n = 2 \] Therefore, the magnetic moment is: \[ \mu = \sqrt{2(2+2)} = \sqrt{8} = 2.83 \, \text{BM} \] Since this is less than 3 BM, Ni\(^{2+}\) satisfies the condition.

2. Fe\(^{2+}\) (Iron ion)

Iron (Fe) has an atomic number of 26, and its electronic configuration is: \[ \text{Fe:} \, [Ar] \, 3d^6 4s^2 \] When Fe forms a \( 2+ \) ion, it loses two electrons, typically from the 4s orbital. So, the electronic configuration for Fe\(^{2+}\) is: \[ \text{Fe}^{2+}: \, [Ar] \, 3d^6 \] This leaves 4 unpaired electrons in the \( 3d \) orbitals. \[ n = 4 \] Therefore, the magnetic moment is: \[ \mu = \sqrt{4(4+2)} = \sqrt{24} = 4.90 \, \text{BM} \] This is greater than 3 BM, so Fe\(^{2+}\) does not satisfy the condition.

3. Co\(^{2+}\) (Cobalt ion)

Cobalt (Co) has an atomic number of 27, and its electronic configuration is: \[ \text{Co:} \, [Ar] \, 3d^7 4s^2 \] When Co forms a \( 2+ \) ion, it loses two electrons, typically from the 4s orbital. So, the electronic configuration for Co\(^{2+}\) is: \[ \text{Co}^{2+}: \, [Ar] \, 3d^7 \] This leaves 3 unpaired electrons in the \( 3d \) orbitals. \[ n = 3 \] Therefore, the magnetic moment is: \[ \mu = \sqrt{3(3+2)} = \sqrt{15} = 3.87 \, \text{BM} \] This is greater than 3 BM, so Co\(^{2+}\) does not satisfy the condition.

4. V\(^{3+}\) (Vanadium ion)

Vanadium (V) has an atomic number of 23, and its electronic configuration is: \[ \text{V:} \, [Ar] \, 3d^3 4s^2 \] When V forms a \( 3+ \) ion, it loses three electrons, typically from the 4s and \( 3d \) orbitals. So, the electronic configuration for V\(^{3+}\) is: \[ \text{V}^{3+}: \, [Ar] \, 3d^2 \] This leaves 2 unpaired electrons in the \( 3d \) orbitals. \[ n = 2 \] Therefore, the magnetic moment is: \[ \mu = \sqrt{2(2+2)} = \sqrt{8} = 2.83 \, \text{BM} \] Since this is less than 3 BM, V\(^{3+}\) satisfies the condition.

5. Ti\(^{2+}\) (Titanium ion)

Titanium (Ti) has an atomic number of 22, and its electronic configuration is: \[ \text{Ti:} \, [Ar] \, 3d^2 4s^2 \] When Ti forms a \( 2+ \) ion, it loses two electrons, typically from the 4s orbital. So, the electronic configuration for Ti\(^{2+}\) is: \[ \text{Ti}^{2+}: \, [Ar] \, 3d^2 \] This leaves 2 unpaired electrons in the \( 3d \) orbitals. \[ n = 2 \] Therefore, the magnetic moment is: \[ \mu = \sqrt{2(2+2)} = \sqrt{8} = 2.83 \, \text{BM} \] Since this is less than 3 BM, Ti\(^{2+}\) satisfies the condition.

Step 3: Count the species with magnetic moment less than 3 BM

From the calculations above, the species with a magnetic moment less than 3 BM are: \[ \text{Ni}^{2+}, \text{V}^{3+}, \text{Ti}^{2+} \] Thus, there are 3 species with magnetic moments less than 3 BM.

Final Answer:

The number of species with a magnetic moment (spin-only) less than 3 BM is \( \boxed{3} \).