Consider the following concentration cell`:`
`Zn(s)|Zn^(2+)(0.024M)||Zn^(2+)(0.480M)|Zn(s)`
which of the following statements is `//` are correct?

Options

A. The `EMF` of the cell at `25^(@)C` is nearly `0.038V.`
B. The `EMF` of the cell at `25^(@)C` is nearly -`0.038V`.
C. If water is added in `LHE`, so that the` [Zn^(2+)]` is reduced to `0.012M` , the cell voltage increases.
D. If water is added in `LHE`, so that the `[Zn^(2+)]` is reduced to `0.12M` , the cell voltage remains same.

Explanation

To solve the problem regarding the concentration cell \( \text{Zn(s)} | \text{Zn}^{2+}(0.024M) || \text{Zn}^{2+}(0.480M) | \text{Zn(s)} \), we will analyze the cell and evaluate the statements provided.

### Step-by-Step Solution:

1. **Identify the Cell Components**:
- The cell consists of two half-cells with the same metal (Zn) but different concentrations of \( \text{Zn}^{2+} \) ions.
- The left half-cell has a concentration of \( 0.024M \) and the right half-cell has a concentration of \( 0.480M \).

2. **Determine the Standard Cell Potential**:
- For a concentration cell, the standard cell potential \( E^\circ \) is zero because the same species is involved in both half-reactions.
- Therefore, \( E^\circ_{\text{cell}} = 0 \).

3. **Calculate the Cell Potential Using the Nernst Equation**:
- The Nernst equation for the cell is given by:
\[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{0.059}{n} \log \frac{[\text{Zn}^{2+}]_{\text{anode}}}{[\text{Zn}^{2+}]_{\text{cathode}}} \]
- Here, \( n = 2 \) (2 electrons are transferred in the reaction).
- The concentrations are:
- \( [\text{Zn}^{2+}]_{\text{anode}} = 0.024M \) (left side)
- \( [\text{Zn}^{2+}]_{\text{cathode}} = 0.480M \) (right side)

4. **Substituting Values into the Nernst Equation**:
- Plugging in the values:
\[ E_{\text{cell}} = 0 - \frac{0.059}{2} \log \frac{0.024}{0.480} \]
- Calculate the logarithm:
\[ \log \frac{0.024}{0.480} = \log 0.05 = -1.301 \]
- Now substituting this back:
\[ E_{\text{cell}} = -\frac{0.059}{2} \times (-1.301) \approx 0.0384 \text{ V} \]

5. **Evaluate the Statements**:
- **Statement 1**: The emf of the cell at 25°C is nearly 0.038 V. **(Correct)**
- **Statement 2**: The emf of the cell is -0.038 V. **(Incorrect)**
- **Statement 3**: If water is added on the left side, reducing \( \text{Zn}^{2+} \) concentration to \( 0.012M \), the cell voltage increases. **(Correct)**
- **Statement 4**: If water is added and \( \text{Zn}^{2+} \) concentration is reduced to \( 0.12M \), the cell voltage remains the same. **(Incorrect)**

### Final Conclusion:
The correct statements are:
- Statement 1
- Statement 3