Difference in Spin Only Magnetic Moments for Octahedral Iron Complexes

To find the absolute difference between the spin-only magnetic moments of the two octahedral complexes, we analyze the electronic configurations of $\text{Fe}^{3+}$ in both ligand environments.

1. Electronic Configuration of $\text{Fe}^{3+}$

• Atomic number of $\text{Fe} = 26$
• Ground-state electronic configuration of $\text{Fe}$: $[\text{Ar}] 3d^6 4s^2$
• Electronic configuration of $\text{Fe}^{3+}$: $[\text{Ar}] 3d^5$

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2. Complex with $\text{SCN}^-$ (Thiocyanato-S)

• $\text{SCN}^-$ is a weak field ligand according to the spectrochemical series.
• In an octahedral crystal field, the crystal field splitting energy ($\Delta_o$) is less than the electron pairing energy ($P$), i.e., $\Delta_o < P$.
• This results in a high-spin complex.
• Distribution of $d$-electrons: $t_{2g}^3 e_g^2$
• Number of unpaired electrons ($n_1$) = $5$

The spin-only magnetic moment ($\mu_1$) is given by: $\mu_1 = \sqrt{n_1(n_1+2)} = \sqrt{5(5+2)} = \sqrt{35} \approx 5.916\text{ BM}$

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3. Complex with $\text{CN}^-$

• $\text{CN}^-$ is a strong field ligand.
• In an octahedral crystal field, the crystal field splitting energy ($\Delta_o$) is greater than the electron pairing energy ($P$), i.e., $\Delta_o > P$.
• This results in a low-spin complex.
• Distribution of $d$-electrons: $t_{2g}^5 e_g^0$
• Number of unpaired electrons ($n_2$) = $1$

The spin-only magnetic moment ($\mu_2$) is given by: $\mu_2 = \sqrt{n_2(n_2+2)} = \sqrt{1(1+2)} = \sqrt{3} \approx 1.732\text{ BM}$

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4. Difference in Magnetic Moments

The absolute difference between their spin-only magnetic moments ($\Delta \mu$) is: $\Delta \mu = |\mu_1 - \mu_2| = |5.916 - 1.732| = 4.184\text{ BM}$

Rounding $4.184$ to the nearest integer value gives 4.