To analyze the behavior of different ions in an acidic medium, consider the following: 1. Stability of \( \text{MnO}_4^- \): - \( \text{MnO}_4^- \) is stable in acidic conditions and retains its characteristic purple color because it does not undergo any further reactions under these conditions. 2. Behavior of \( \text{MnO}_2^{2-} \): - \( \text{MnO}_2^{2-} \) undergoes disproportionation in an acidic medium to form \( \text{MnO}_4^- \) (purple) and \( \text{Mn}^{2+} \) (colorless). 3. Conversion of \( \text{CrO}_4^{2-} \): - \( \text{CrO}_4^{2-} \) converts to \( \text{Cr}_2\text{O}_7^{2-} \), resulting in a color change from yellow to orange due to the equilibrium shift in acidic conditions. 4. Decomposition of \( \text{FeO}_4^{2-} \): - \( \text{FeO}_4^{2-} \) decomposes into \( \text{Fe}^{3+} \), which is colorless or pale yellow, depending on the concentration. Therefore, among the given species, \( \text{MnO}_4^- \) retains its purple color in an acidic medium.
Final Answer: \[ \boxed{\text{MnO}_4^-} \]
Given below are two statements:
Statement (I): The first ionization energy of Pb is greater than that of Sn.
Statement (II): The first ionization energy of Ge is greater than that of Si.
In light of the above statements, choose the correct answer from the options given below:
Given, the function \( f(x) = \frac{a^x + a^{-x}}{2} \) (\( a > 2 \)), then \( f(x+y) + f(x-y) \) is equal to