How much should the pressure be increased in order to decrease the volume of a gas 5% at a constant temprature ?

Options

A. 0.05
B. 0.0526
C. 0.1
D. 0.0426

Explanation

To solve the problem of how much pressure should be increased to decrease the volume of a gas by 5% 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 (P) and volume (V) is constant (PV = constant).

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- Let the initial pressure be \( P_1 \).
- Let the initial volume be \( V_1 \).

2. **Determine the Final Volume**:
- We need to decrease the volume by 5%.
- The final volume \( V_2 \) can be calculated as:
\[ V_2 = V_1 - 0.05V_1 = 0.95V_1 \]

3. **Apply Boyle’s Law**:
- According to Boyle's Law:
\[ P_1 V_1 = P_2 V_2 \]
- Substituting \( V_2 \) into the equation:
\[ P_1 V_1 = P_2 (0.95 V_1) \]

4. **Cancel Out \( V_1 \)**:
- Since \( V_1 \) is common on both sides, we can cancel it out:
\[ P_1 = P_2 (0.95) \]

5. **Solve for \( P_2 \)**:
- Rearranging gives:
\[ P_2 = \frac{P_1}{0.95} \]

6. **Calculate the Increase in Pressure**:
- The increase in pressure \( \Delta P \) is given by:
\[ \Delta P = P_2 - P_1 \]
- Substituting for \( P_2 \):
\[ \Delta P = \frac{P_1}{0.95} - P_1 \]
- Simplifying this:
\[ \Delta P = P_1 \left( \frac{1}{0.95} - 1 \right) = P_1 \left( \frac{1 - 0.95}{0.95} \right) = P_1 \left( \frac{0.05}{0.95} \right) \]

7. **Calculate the Fractional Increase**:
- Thus, the increase in pressure in terms of \( P_1 \) is:
\[ \Delta P = 0.05263 P_1 \]

8. **Convert to Percentage**:
- To express this as a percentage increase:
\[ \text{Percentage Increase} = \left( \frac{0.05263 P_1}{P_1} \right) \times 100 = 5.263\% \]

### Final Answer:
To decrease the volume of a gas by 5% at constant temperature, the pressure should be increased by approximately **5.26%**.