At a constant temperature what should be the percentage increase in pressure for a `10%` decrease in the volume of gas ?
(a)`11%`
(b)`20%`
(c)`5%`
(d)`50%`

Options

A. `11%`
B. `20%`
C. `5%`
D. `50%`

Answer

To solve the problem of determining the percentage increase in pressure for a 10% decrease in the volume of a gas at constant temperature, we can follow these steps:

### Step 1: Define Initial Conditions
Let the initial volume of the gas be \( V \) and the initial pressure be \( P_0 \).

### Step 2: Calculate the New Volume
A 10% decrease in volume means the new volume \( V' \) can be calculated as:
\[ V' = V - 0.1V = 0.9V \]

### Step 3: Apply Boyle's Law
According to Boyle's Law, at constant temperature, the product of pressure and volume is constant:
\[ P_0 V = P' V' \]
where \( P' \) is the final pressure after the volume change.

### Step 4: Substitute the New Volume
Substituting \( V' \) into the equation gives:
\[ P_0 V = P' (0.9V) \]

### Step 5: Solve for Final Pressure
Rearranging the equation to solve for \( P' \):
\[ P' = \frac{P_0 V}{0.9V} = \frac{P_0}{0.9} = \frac{10}{9} P_0 \]

### Step 6: Calculate the Increase in Pressure
The increase in pressure can be calculated as:
\[ \Delta P = P' - P_0 = \frac{10}{9} P_0 - P_0 = \left(\frac{10}{9} - 1\right) P_0 = \frac{1}{9} P_0 \]

### Step 7: Calculate the Percentage Increase
To find the percentage increase in pressure:
\[ \text{Percentage Increase} = \left(\frac{\Delta P}{P_0}\right) \times 100 = \left(\frac{\frac{1}{9} P_0}{P_0}\right) \times 100 = \frac{1}{9} \times 100 \approx 11.11\% \]

### Conclusion
Thus, the percentage increase in pressure for a 10% decrease in volume at constant temperature is approximately \( 11\% \). Therefore, the correct answer is:
**(a) 11%**