For the cell (at 298K)
`Ag(s) | AgCl(s) | Cl^(-)(aq)|| AgNO_(3)(aq) | Ag(s)`
Which of the following is correct?

Options

A. The cell emf will be zero when `[(Ag^(+))]` in anodic compartment =`[Ag^(+)]` in cathodic compartment
B. The amount of AgCl(s) precipitate in anodic compartment will decrease with the working of the cell
C. The concentration of `[Ag^(+)]`=constant, in anodic compartment during working of cell
D. `E_(cell)=E_(Ag^(+)|Ag)^(0)-E_(Cl^(-)|AgCl|Ag)^(0)-0.059/1"log"1/[Cl^(-)]_(a)`

Explanation

To solve the question regarding the electrochemical cell given, we will analyze the cell notation and apply the Nernst equation to determine the correctness of the statements provided.

### Step-by-Step Solution:

1. **Identify the Components of the Cell:**
The cell notation is given as:
\[ \text{Ag(s)} | \text{AgCl(s)} | \text{Cl}^-(\text{aq}) || \text{AgNO}_3(\text{aq}) | \text{Ag(s)} \]
This indicates that we have a silver/silver chloride electrode on the left and a silver electrode in a silver nitrate solution on the right.

2. **Determine the Half-Reactions:**
The half-reactions for the cell can be written as:
- Anode (oxidation):
\[ \text{AgCl(s)} + \text{e}^- \rightarrow \text{Ag(s)} + \text{Cl}^-(\text{aq}) \]
- Cathode (reduction):
\[ \text{Ag}^+(\text{aq}) + \text{e}^- \rightarrow \text{Ag(s)} \]

3. **Write the Nernst Equation:**
The Nernst equation for the cell can be expressed as:
\[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{0.0591}{n} \log \left( \frac{[\text{Ag}^+]_{\text{anode}}}{[\text{Ag}^+]_{\text{cathode}}} \right) \]
Here, \( n = 1 \) (one electron is transferred).

4. **Evaluate the Condition for EMF = 0:**
The EMF of the cell will be zero when the concentrations of silver ions in both compartments are equal:
\[ [\text{Ag}^+]_{\text{anode}} = [\text{Ag}^+]_{\text{cathode}} \]
In this case, the log term becomes log(1) = 0, leading to:
\[ E_{\text{cell}} = E^\circ_{\text{cell}} - 0 = E^\circ_{\text{cell}} \]
Therefore, the first statement is correct.

5. **Analyze the Other Statements:**
- **Statement 2:** "The amount of AgCl precipitate in the anodic compartment will decrease with the working of the cell."
- This statement is incorrect because as the cell operates, AgCl will dissolve, leading to a decrease in AgCl precipitate.
- **Statement 3:** "Concentration of Ag^+ is constant in the anodic compartment during working of the cell."
- This statement is incorrect because the concentration of Ag^+ will change as the cell operates.
- **Statement 4:** The last statement is also incorrect based on the analysis of the Nernst equation.

6. **Conclusion:**
The only correct statement is the first one. Therefore, the answer is:
\[ \text{Correct Answer: Option 1} \]