Transition metals and their ions often exhibit characteristic colors due to d-d transitions where electrons in the d-orbitals absorb specific wavelengths of light and move to higher energy levels. The color observed is complementary to the color of light absorbed. Here is the breakdown of each ion:
Set 1: \( V^{2+} \), \( Cr^{3+} \), \( Mn^{3+} \)
These ions are known to produce similar colors in solution. Specifically, they appear in shades of violet or blue-green. The colors arise from specific d-d transitions unique to each ion, but similar enough to result in perceived color similarity:
Set 2: \( Zn^{2+} \), \( V^{3+} \), \( Fe^{3+} \)
\( Zn^{2+} \) has a completely filled d-orbital, resulting in a colorless solution, thus not matching the others which may appear yellow to brown.
Set 3: \( Ti^{4+} \), \( V^{4+} \), \( Mn^{2+} \)
\( Ti^{4+} \) usually produces a colorless solution due to its electronic configuration, whereas the other ions have visible colors.
Set 4: \( Sc^{3+} \), \( Ti^{3+} \), \( Cr^{2+} \)
\( Sc^{3+} \) is colorless in solution due to its lack of d-electrons, and thus, this set does not have similar colors either.
Thus, the correct set of ions that produce similarly colored aqueous solutions is \( V^{2+} \), \( Cr^{3+} \), \( Mn^{3+} \), due to their ability to exhibit complementary colored transitions in solutions.
A transition metal (M) among Mn, Cr, Co and Fe has the highest standard electrode potential \( (M^{3+} / M^{2+}) \). It forms a metal complex of the type \( [M(CN)_6]^{4-} \). The number of electrons present in the \( e_g \) orbital of the complex is ________.
Give explanation for each of the following observations:
(a) With the same d-orbital configuration (d4), Mn3+ ion is an oxidizing agent whereas Cr2+ ion is a reducing agent.
(b) Actinoid contraction is greater from element to element than that among lanthanoids.
(c) Transition metals form a large number of interstitial compounds with H, B, C, and N.
If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then \(a + b + ab\) is equal to:
Given three identical bags each containing 10 balls, whose colours are as follows:
| Bag I | 3 Red | 2 Blue | 5 Green |
| Bag II | 4 Red | 3 Blue | 3 Green |
| Bag III | 5 Red | 1 Blue | 4 Green |
A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from Bag I is $ p $ and if the ball is Green, the probability that it is from Bag III is $ q $, then the value of $ \frac{1}{p} + \frac{1}{q} $ is:
If \( \theta \in \left[ -\frac{7\pi}{6}, \frac{4\pi}{3} \right] \), then the number of solutions of \[ \sqrt{3} \csc^2 \theta - 2(\sqrt{3} - 1)\csc \theta - 4 = 0 \] is equal to ______.