This question tests your understanding of the electrochemical series and the concept of selective discharge of ions during electrolysis. The key idea is that during the electrolysis of an aqueous salt solution, you are not just dealing with the metal ions from the salt, but also with H⁺ and OH^– ions from water. The ion that gets reduced (gains electrons) at the cathode depends on its standard reduction potential (E°).

Ions with a higher (more positive) reduction potential are reduced more easily. Therefore, if a metal ion has a reduction potential that is lower (more negative) than that of the H⁺ ion, the H⁺ ions from water will be preferentially reduced instead of the metal ions. This means the metal will not be deposited at the cathode.

Let's analyze the standard reduction potentials (E°) for the given metals:

• Cu²⁺/Cu: $E^{°}=+0.34V$
• Cr³⁺/Cr: $E^{°}=-0.74V$
• Ag⁺/Ag: $E^{°}=+0.80V$
• Ca²⁺/Ca: $E^{°}=-2.87V$

The standard reduction potential for water is: 2H⁺/H₂: $E^{°}=0.0V$ (by definition).

Step-by-step reasoning:

1. Copper (Cu): E° = +0.34 V. This is higher than 0.0 V, so Cu²⁺ ions are more easily reduced than H⁺ ions. Copper metal is obtained.
2. Silver (Ag): E° = +0.80 V. This is much higher than 0.0 V, so Ag⁺ ions are more easily reduced than H⁺ ions. Silver metal is obtained.
3. Chromium (Cr): E° = -0.74 V. This is lower than 0.0 V. In theory, H⁺ should be reduced. However, in practice, chromium can be electroplated from aqueous solutions due to a high overpotential for hydrogen evolution on chromium surfaces, which alters the effective voltage required. This makes it a tricky case, but it is often considered obtainable.
4. Calcium (Ca): E° = -2.87 V. This is much lower than 0.0 V. H⁺ ions have a far greater tendency to be reduced. Therefore, at the cathode, hydrogen gas ($H_2$) is produced instead of calcium metal. Calcium metal cannot be obtained by electrolysis of its aqueous salts.

Final Answer: The metal that cannot be obtained by electrolysis of an aqueous solution of its salts is Ca (Calcium).

Related Topics & Formulae

Electrochemical Series: A list of elements arranged in order of their standard electrode potentials. It predicts the feasibility of redox reactions. Cations lower in the series (with more negative E°) are stronger reducing agents and harder to reduce.

Nernst Equation: Used to calculate the electrode potential of a half-cell under non-standard conditions. $E=E^{°}-\frac{RT}{nF}\ln Q$

Overpotential: The difference between the thermodynamically determined electrode potential and the actual potential at which a redox reaction begins. It is crucial for reactions like hydrogen evolution and is the reason some metals like Cr can be deposited from aqueous solutions.