Which of the following equations represents the reaction that takes place at the cathode during the electrolysis of aqueous silver nitrate with carbon electrodes ?

Options

A. `Ag^(+)("aq")+e^(-) to Ag(s)`
B. `2H^(-)("aq")+2e^(-) to H_(2)(l)`
C. `2N^(-)("aq") to N_(2)("aq") + 6e^(-)`
D. `4OH^(-)("aq") to O_(2)("g") + 2H_(2)O("l") +4e^(-)`

Explanation

To determine the reaction that takes place at the cathode during the electrolysis of aqueous silver nitrate with carbon electrodes, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Electrolyte**: The electrolyte in this case is aqueous silver nitrate (AgNO₃). When dissolved in water, it dissociates into its ions:
\[ \text{AgNO}_3 \rightarrow \text{Ag}^+ + \text{NO}_3^- \]

2. **Identify the Electrodes**: In this setup, we have carbon electrodes. The positive terminal of the battery is connected to the anode, and the negative terminal is connected to the cathode.

3. **Determine the Ions Present**: In addition to the ions from silver nitrate, water also dissociates into:
\[ \text{H}_2\text{O} \rightarrow \text{H}^+ + \text{OH}^- \]
Therefore, the ions present in the solution are:
- \(\text{Ag}^+\) (from silver nitrate)
- \(\text{NO}_3^-\) (from silver nitrate)
- \(\text{H}^+\) (from water)
- \(\text{OH}^-\) (from water)

4. **Identify the Reaction at the Cathode**: At the cathode, reduction occurs. The cations (\(\text{Ag}^+\) and \(\text{H}^+\)) will migrate towards the cathode. We need to determine which cation will be reduced.

5. **Compare Reduction Potentials**: The standard reduction potentials indicate that \(\text{Ag}^+\) has a higher tendency to be reduced compared to \(\text{H}^+\). The reduction half-reaction for silver is:
\[ \text{Ag}^+ + e^- \rightarrow \text{Ag (s)} \]
This means that \(\text{Ag}^+\) will be reduced to solid silver at the cathode.

6. **Write the Final Equation**: The reaction that takes place at the cathode during the electrolysis of aqueous silver nitrate is:
\[ \text{Ag}^+ + e^- \rightarrow \text{Ag (s)} \]

### Conclusion:
The correct equation representing the reaction at the cathode during the electrolysis of aqueous silver nitrate with carbon electrodes is:
\[ \text{Ag}^+ + e^- \rightarrow \text{Ag (s)} \]