Which of the following are the transition metal ions responsible for colour in ruby and emerald respectively?

We begin by recalling a basic fact from crystal field theory: the brilliant colours seen in many gemstones originate from the presence of trace amounts of certain transition-metal ions. These ions replace some of the normal lattice cations, and their partly filled $$d$$-orbitals split into energy sub-levels inside the crystal field created by the surrounding anions. When visible light falls on the crystal, specific wavelengths are absorbed to promote electrons from lower to higher $$d$$-levels; the remaining transmitted or reflected light then appears to us as a characteristic colour.

First, let us examine ruby. Ruby is chemically a variety of corundum, whose ideal formula is $$Al_2O_3$$. Inside the corundum lattice we normally have $$Al^{3+}$$ ions. A very small fraction of these $$Al^{3+}$$ ions are replaced (substituted) by another trivalent transition-metal ion. Experimental spectroscopy and crystal chemistry confirm that the ion doing this job is $$Cr^{3+}$$. The presence of $$Cr^{3+}$$ introduces new energy levels inside the forbidden gap; electronic transitions between split $$d$$-levels of $$Cr^{3+}$$ absorb green and violet light, so the transmitted light appears deep red. Hence, the red colour of ruby is due to $$Cr^{3+}$$ ions.

Now we turn to emerald. Emerald is a green variety of the mineral beryl, whose ideal formula is $$Be_3Al_2Si_6O_{18}$$. Inside the beryl lattice we again have octahedrally coordinated $$Al^{3+}$$ sites. Just as in corundum, some of these $$Al^{3+}$$ sites are occupied by traces of $$Cr^{3+}$$ (sometimes accompanied by $$V^{3+}$$). The $$Cr^{3+}$$ ion has a $$3d^3$$ electronic configuration; under the octahedral field of surrounding oxygen ligands its $$d$$-orbitals split into the usual $$t_{2g}$$ and $$e_g$$ sets. The energy gap between these sets happens to correspond to the absorption of red and blue light, so the complementary colour, rich green, is observed. Thus, the green colour of emerald is also produced by $$Cr^{3+}$$ ions.

Therefore, both in ruby and in emerald the transition-metal ion responsible for colour is the same, namely $$Cr^{3+}$$. We match this inference with the alternatives given:

Option A: $$Co^{3+}$$ and $$Cr^{3+}$$ - incorrect, ruby is not coloured by $$Co^{3+}$$.
Option B: $$Co^{3+}$$ and $$Co^{3+}$$ - incorrect for both gemstones.
Option C: $$Cr^{3+}$$ and $$Cr^{3+}$$ - correct for both ruby and emerald.
Option D: $$Cr^{3+}$$ and $$Ag^+$$ - incorrect, emerald is not coloured by $$Ag^+$$.

Hence, the correct answer is Option C.