The process of obtaining metals by electrolysis of their aqueous salts depends on the reduction potentials of the metal ions compared to the reduction of water. If the reduction potential of the metal ion is more negative than that of water, then water will be reduced instead of the metal, and the metal cannot be obtained from the aqueous solution.
Step 1: Compare Reduction Potentials.
Let's look at the reduction potentials of calcium (\( \text{Ca} \)) and water:
- The reduction potential for calcium:
\[
\text{Ca}^{2+} + 2e^- \rightarrow \text{Ca}, \quad E^\circ = -2.87 \, \text{V}.
\]
- The reduction potential for water:
\[
2\text{H}_2\text{O} + 2e^- \rightarrow \text{H}_2 + 2\text{OH}^-, \quad E^\circ = -0.83 \, \text{V}.
\]
Since the reduction potential for calcium (\( -2.87 \, \text{V} \)) is much more negative than that for water (\( -0.83 \, \text{V} \)), it is not favorable for calcium to be reduced from its aqueous solution. Instead, hydrogen gas will be preferentially produced at the cathode during electrolysis, and no calcium metal will be deposited.
Step 2: Other Metals in the Options.
Silver (Ag): The reduction potential for silver is positive, so it can easily be reduced from an aqueous solution.
Copper (Cu): Copper also has a positive reduction potential, making its reduction favorable in aqueous solutions.
Chromium (Cr): Chromium can be reduced under specific conditions, though it might require different electrolytic conditions.
Conclusion:
Since calcium has a highly negative reduction potential compared to water, it cannot be obtained by electrolysis of its aqueous salt solution. Therefore, the correct answer is \( \mathbf{(B)} \).
Thus, the metal that cannot be obtained by electrolysis of its aqueous salt solution is \( \mathbf{Ca} \).
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