In adiabatic expansion of monoatomic ideal gas, if volume increases by `24%` then pressure decreases by `40%`.
Statement-`2` : For adiabatic process `PV^(gamma)` = constant

Options

A. Statement -`1` is True , Statement -`2` is True , Statement -`2` is correct explanation for Statement - `1`
B. Statement -`1` is True , Statement -`2` is True , Statement -`2` is not correct explanation for Statement - `1`
C. Statement -`1` is True , Statement -`2` is False.
D. Statement -`1` is False , Statement -`2` is True.

Answer

To solve the problem, we need to analyze the adiabatic expansion of a monoatomic ideal gas and how changes in volume and pressure relate to each other.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We are given that the volume of a monoatomic ideal gas increases by 24%. This means if the initial volume \( V \) is considered, the final volume \( V' \) can be expressed as:
\[ V' = V + 0.24V = 1.24V \]

2. **Using the Adiabatic Condition**:
- For an adiabatic process, the relationship between pressure \( P \) and volume \( V \) is given by:
\[ PV^\gamma = \text{constant} \]
- For a monoatomic ideal gas, the value of \( \gamma \) (gamma) is \( \frac{5}{3} \).

3. **Expressing the Final Pressure**:
- According to the adiabatic condition, we have:
\[ P' (V')^\gamma = PV^\gamma \]
- Substituting \( V' = 1.24V \):
\[ P' (1.24V)^\gamma = PV^\gamma \]
- Rearranging gives:
\[ P' = P \left(\frac{V}{1.24V}\right)^\gamma = \frac{P}{(1.24)^\gamma} \]

4. **Calculating the Change in Pressure**:
- Now, we can express the change in pressure \( \Delta P \):
\[ \Delta P = P' - P = \frac{P}{(1.24)^\gamma} - P \]
- Factoring out \( P \):
\[ \Delta P = P \left(\frac{1}{(1.24)^\gamma} - 1\right) \]

5. **Substituting the Value of Gamma**:
- Substituting \( \gamma = \frac{5}{3} \):
\[ \Delta P = P \left(\frac{1}{(1.24)^{5/3}} - 1\right) \]

6. **Calculating \( (1.24)^{5/3} \)**:
- First, calculate \( (1.24)^{5/3} \):
\[ (1.24)^{5/3} \approx 1.302 \]
- Thus:
\[ \Delta P \approx P \left(\frac{1}{1.302} - 1\right) \approx P \left(0.769 - 1\right) \approx -0.231P \]

7. **Finding the Percentage Decrease**:
- The percentage decrease in pressure is given by:
\[ \frac{\Delta P}{P} \times 100 \approx -0.231 \times 100 \approx -23.1\% \]
- Therefore, the pressure decreases by approximately 23.1%, which is less than the stated 40%.

8. **Conclusion**:
- Since the calculated decrease in pressure (approximately 23.1%) does not match the given decrease of 40%, we conclude that Statement 1 is false.
- Statement 2, which states that \( PV^\gamma \) is constant for an adiabatic process, is true.

### Final Answer:
- Statement 1 is false, and Statement 2 is true.