Step-by-Step Solution:

1. Understanding the Borax Bead Test Chemistry: The borax bead test is a qualitative analysis method used to identify certain metal ions. The test starts with borax, which is sodium tetraborate decahydrate, $Na_2B_4O_7 \cdot 10H_2O$. When heated on a platinum loop, borax first loses its water of crystallization and swells up. Upon further heating, it melts into a clear, colorless, glassy bead. The chemical reactions are: $Na_2B_4O_7 \cdot 10H_2O \xrightarrow{\Delta} Na_2B_4O_7 + 10H_2O$ Upon strong heating, sodium tetraborate decomposes to form sodium metaborate ($NaBO_2$) and boric anhydride ($B_2O_3$). $Na_2B_4O_7 \xrightarrow{\Delta} 2NaBO_2 + B_2O_3$ The resulting glassy bead is a mixture of sodium metaborate and boric anhydride.

2. Reaction with Chromium (III) Salt: When a small amount of a chromium (III) salt is introduced to the hot bead and heated again, the salt decomposes to form chromium (III) oxide, $Cr_2O_3$. For example, if the salt is chromium (III) sulfate, the decomposition is: $Cr_2(SO_4)_3 \xrightarrow{\Delta} Cr_2O_3 + 3SO_3$ Chromium (III) oxide itself is a green solid.

3. Formation of the Colored Bead: The chromium (III) oxide ($Cr_2O_3$) is a basic oxide. It reacts with the acidic boric anhydride ($B_2O_3$) present in the glassy bead to form a colored compound. The reaction is: $Cr_2O_3 + 3B_2O_3 \rightarrow 2Cr(BO_2)_3$ The product formed is chromium (III) metaborate, $Cr(BO_2)_3$. This compound imparts a characteristic green color to the borax bead. This color is stable in both oxidizing and reducing flames.

4. Evaluating the Options:

   • A: $Cr_2O_3$: While chromium (III) oxide is green and is formed as an intermediate, the final colored species in the borax bead is the product of its reaction with boric anhydride. So, this is not the final cause of the color in the bead.
   • B: CrB: This represents chromium boride. Borides are not formed under the conditions of the borax bead test.
   • C: $Cr(BO_2)_3$: This is chromium (III) metaborate, which is the green-colored compound formed by the reaction of $Cr_2O_3$ with $B_2O_3$ in the bead. This is the correct answer.
   • D: $Cr_2(B_4O_7)_3$: This represents chromium (III) tetraborate. The reaction chemistry of the borax bead test is best described by the formation of metaborates from the reaction of the metal oxide with the acidic component, $B_2O_3$. Therefore, the formation of metaborate is the accepted explanation.

Conclusion:

The green color observed in the borax bead test for a chromium (III) salt is due to the formation of chromium (III) metaborate, $Cr(BO_2)_3$.