A gas is enclosed in a vessel at a constant temperature and at a pressure of 2 atm and volume 4 liters. Due to a leak in the vessel after some time the pressure is reduced to 1.5 atm as a result if `5P%` of the gas escaped out then find the value of P.

Explanation

To solve the problem step by step, we can use the ideal gas law and the relationship between pressure and the number of moles of gas.

### Step-by-Step Solution:

1. **Understand the Initial Conditions:**
- Initial pressure \( P_1 = 2 \, \text{atm} \)
- Initial volume \( V = 4 \, \text{liters} \) (constant)
- Initial temperature \( T \) is constant (not given, but we know it's constant).

2. **Determine the Final Conditions:**
- Final pressure \( P_2 = 1.5 \, \text{atm} \)

3. **Use the Ideal Gas Law:**
- According to the ideal gas law, at constant temperature and volume, the pressure is directly proportional to the number of moles of gas:
\[ \frac{P_2}{P_1} = \frac{n_2}{n_1} \]
- Where \( n_1 \) is the initial number of moles and \( n_2 \) is the final number of moles.

4. **Calculate the Ratio of Pressures:**
- Substitute the known values into the equation:
\[ \frac{1.5 \, \text{atm}}{2 \, \text{atm}} = \frac{n_2}{n_1} \]
- Simplifying gives:
\[ \frac{3}{4} = \frac{n_2}{n_1} \]

5. **Relate the Change in Moles to the Escape of Gas:**
- The fraction of gas that remains is \( \frac{n_2}{n_1} = \frac{3}{4} \), meaning that \( \frac{1}{4} \) of the gas has escaped.
- Therefore, the percentage of gas that escaped is:
\[ \text{Percentage escaped} = \left(1 - \frac{n_2}{n_1}\right) \times 100 = \left(1 - \frac{3}{4}\right) \times 100 = 25\% \]

6. **Relate the Escape Percentage to \( P \):**
- According to the problem, the percentage of gas that escaped is given as \( 5P\% \).
- Set up the equation:
\[ 5P = 25 \]

7. **Solve for \( P \):**
- Divide both sides by 5:
\[ P = \frac{25}{5} = 5 \]

### Final Answer:
The value of \( P \) is \( 5 \).