GATE 2025 CY – Question 40

Inorganic Chemistry · Transition Metals: Coordination chemistry – structure and isomerism, theories of bonding (VBT, CFT, and MOT). Metal-metal multiple bonds. Energy level diagrams in various crystal fields, CFSE, applications of CFT, Jahn-Teller distortion. Electronic spectra of transition metal complexes: spectroscopic term symbols, selection rules, Orgel and Tanabe-Sugano diagrams, nephelauxetic effect and Racah parameter, charge-transfer spectra. Magnetic properties of transition metal complexes. Ray-Dutt and Bailar twists, Reaction mechanisms: kinetic and thermodynamic stability, associative, dissociative substitution and redox reactions. · 2 marks · Multiple choice

The effective magnetic moment, $\mu_{eff}$ value for $[\mathrm{Cr(H_2O)_6}]^{3+}$ taking into account for spin-orbit coupling is closest to [Given: Atomic number of Cr = 24, spin-orbit coupling constant $\lambda=92\ \mathrm{cm^{-1}}$, and $\Delta_o=17400\ \mathrm{cm^{-1}}$]

$3.79\mu_B$
$3.87\mu_B$
$4.05\mu_B$
$3.60\mu_B$

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Correct answer: (A) $3.79\mu_B$

Explanation

For an octahedral $d^3$ ion such as $[\text{Cr(H}_2\text{O)}_6]^{3+}$ (ground term $^4A_{2g}$), the magnetic moment is slightly reduced below the spin-only value by spin-orbit coupling. The formula (for $A_{2g}$ ground terms) is
$$\mu_{eff}=\mu_{s.o.}\left(1-\frac{4\lambda}{\Delta_o}\right),\qquad\mu_{s.o.}=\sqrt{n(n+2)}=\sqrt{15}\ \mu_B\quad(n=3).$$

**Numbers.** $\lambda=92\ \text{cm}^{-1}$, $\Delta_o=17\,400\ \text{cm}^{-1}$:
$$\frac{4\lambda}{\Delta_o}=\frac{368}{17\,400}=0.02115$$
$$\mu_{eff}=3.873\times(1-0.02115)=3.873\times0.97885=\mathbf{3.79\ \mu_B}.$$