To determine the number of complexes with an even number of electrons in $t_{2g}$ orbitals, we calculate the electronic configuration of the central metal ion in each complex:
$[\text{Fe}(\text{H}_2\text{O})_6]^{2+}$: Fe$^{2+}$ has $(3d^6)$ configuration. In an octahedral field:
\[t_{2g}^4e_g^2 \quad (4 \, \text{electrons in } t_{2g})\]
Even number of electrons in $t_{2g}$.
$[\text{Co}(\text{H}_2\text{O})_6]^{2+}$: Co$^{2+}$ has $(3d^7)$ configuration. In an octahedral field:
\[t_{2g}^5e_g^2 \quad (5 \, \text{electrons in } t_{2g})\]
Odd number of electrons in $t_{2g}$.
$[\text{Co}(\text{H}_2\text{O})_6]^{3+}$: Co$^{3+}$ has $(3d^6)$ configuration. In an octahedral field:
\[t_{2g}^6e_g^0 \quad (6 \, \text{electrons in } t_{2g})\]
Even number of electrons in $t_{2g}$.
$[\text{Cu}(\text{H}_2\text{O})_6]^{2+}$: Cu$^{2+}$ has $(3d^9)$ configuration. In an octahedral field:
\[t_{2g}^6e_g^3 \quad (6 \, \text{electrons in } t_{2g})\]
Even number of electrons in $t_{2g}$.
$[\text{Cr}(\text{H}_2\text{O})_6]^{2+}$: Cr$^{2+}$ has $(3d^4)$ configuration. In an octahedral field:
\[t_{2g}^3e_g^1 \quad (3 \, \text{electrons in } t_{2g})\]
Odd number of electrons in $t_{2g}$.
Complexes with even number of electrons in $t_{2g}$ orbitals are:
\[[\text{Fe}(\text{H}_2\text{O})_6]^{2+}, \, [\text{Co}(\text{H}_2\text{O})_6]^{3+}, \, [\text{Cu}(\text{H}_2\text{O})_6]^{2+}\]
Final Answer: 3 complexes.
Match the following List-I with List-II and choose the correct option: List-I (Compounds) | List-II (Shape and Hybridisation) (A) PF\(_{3}\) (I) Tetrahedral and sp\(^3\) (B) SF\(_{6}\) (III) Octahedral and sp\(^3\)d\(^2\) (C) Ni(CO)\(_{4}\) (I) Tetrahedral and sp\(^3\) (D) [PtCl\(_{4}\)]\(^{2-}\) (II) Square planar and dsp\(^2\)
Let \( y = f(x) \) be the solution of the differential equation
\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:
Find the IUPAC name of the compound.
If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32