At constant temperature if the pressure of an ideal gas is increased by 10% then its volume must decrease by

Options

A. 0.1
B. 0.0901
C. 0.1525
D. 0.1101

Answer

To solve the problem of how much the volume of an ideal gas must decrease when its pressure is increased by 10% at constant temperature, we can use Boyle's Law, which states that for a given mass of gas at constant temperature, the product of pressure and volume is constant (P1V1 = P2V2).

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- Let the initial pressure be \( P_1 \).
- The initial volume is \( V_1 \).
- According to the problem, the pressure is increased by 10%, so the new pressure \( P_2 \) can be expressed as:
\[ P_2 = P_1 + 0.1P_1 = 1.1P_1 \]

2. **Apply Boyle's Law**:
- According to Boyle's Law:
\[ P_1V_1 = P_2V_2 \]
- Substituting \( P_2 \) into the equation gives:
\[ P_1V_1 = (1.1P_1)V_2 \]

3. **Rearranging the Equation**:
- We can cancel \( P_1 \) from both sides (assuming \( P_1 \neq 0 \)):
\[ V_1 = 1.1V_2 \]
- Rearranging this gives:
\[ V_2 = \frac{V_1}{1.1} \]

4. **Calculate the Change in Volume**:
- The change in volume \( \Delta V \) can be calculated as:
\[ \Delta V = V_1 - V_2 \]
- Substituting for \( V_2 \):
\[ \Delta V = V_1 - \frac{V_1}{1.1} = V_1 \left(1 - \frac{1}{1.1}\right) \]
- Simplifying this:
\[ \Delta V = V_1 \left(\frac{1.1 - 1}{1.1}\right) = V_1 \left(\frac{0.1}{1.1}\right) = \frac{0.1V_1}{1.1} \]

5. **Calculate the Percentage Decrease in Volume**:
- The percentage decrease in volume can be calculated as:
\[ \text{Percentage Decrease} = \left(\frac{\Delta V}{V_1}\right) \times 100 = \left(\frac{0.1V_1/1.1}{V_1}\right) \times 100 \]
- This simplifies to:
\[ \text{Percentage Decrease} = \frac{0.1}{1.1} \times 100 \approx 9.09\% \]

### Final Answer:
The volume must decrease by approximately \( 9.09\% \).