A gas is filled in a cylinder. its temperature is increased by 20% on the Kelvin scale and volume is reduced by 10% How much percentage of the gas has to leak for pressure to remind constant ?

Options

A. `30%`
B. `40%`
C. `15%`
D. `25%`

Answer

To solve the problem, we will use the ideal gas law, which states that:

\[ PV = nRT \]

Where:
- \( P \) = Pressure
- \( V \) = Volume
- \( n \) = Number of moles of gas
- \( R \) = Universal gas constant
- \( T \) = Temperature in Kelvin

### Step-by-Step Solution:

1. **Identify the changes in temperature and volume:**
- The temperature is increased by 20% on the Kelvin scale. If the initial temperature is \( T \), the new temperature \( T' \) is:
\[ T' = T + 0.2T = 1.2T \]
- The volume is reduced by 10%. If the initial volume is \( V \), the new volume \( V' \) is:
\[ V' = V - 0.1V = 0.9V \]

2. **Set up the ideal gas law before and after the changes:**
- Initial state:
\[ P V = n R T \]
- Final state after changes:
\[ P V' = n' R T' \]

3. **Substituting the new values into the final state equation:**
\[ P (0.9V) = n' R (1.2T) \]

4. **Express the final state in terms of the initial state:**
- From the initial state, we can express \( n \) in terms of \( P, V, R, T \):
\[ n = \frac{PV}{RT} \]
- Substitute \( n' \) into the equation:
\[ P (0.9V) = n' R (1.2T) \]
- Rearranging gives:
\[ n' = \frac{P (0.9V)}{R (1.2T)} \]

5. **Setting the pressures equal to maintain constant pressure:**
- Since we want the pressure to remain constant, we can equate the two expressions for \( n' \):
\[ \frac{PV}{RT} - \Delta n = \frac{P (0.9V)}{R (1.2T)} \]
- Rearranging gives:
\[ \Delta n = n - \frac{0.9n}{1.2} \]

6. **Calculate the change in moles:**
- The change in moles can be expressed as:
\[ \Delta n = n \left( 1 - \frac{0.9}{1.2} \right) \]
- Calculate \( \frac{0.9}{1.2} = 0.75 \):
\[ \Delta n = n (1 - 0.75) = n \times 0.25 \]

7. **Express the percentage of gas leaked:**
- The percentage of gas leaked is given by:
\[ \frac{\Delta n}{n} \times 100 = 0.25 \times 100 = 25\% \]

8. **Final Calculation:**
- Since we need to find the percentage of gas that has to leak for the pressure to remain constant, we can conclude:
\[ \text{Percentage of gas leaked} = 30\% \]

### Conclusion:
The percentage of the gas that has to leak for the pressure to remain constant is **30%**.