A constant temperature , the volume of a gas is to be decreased by 4 % The pressure must be increased by

Options

A. 0.04
B. 0.0416
C. 0.08
D. 0.0386

Explanation

To solve the problem of how much the pressure must be increased when the volume of a gas is decreased by 4% at constant temperature, we can use the ideal gas law, which states that \( PV = \text{constant} \).

### Step-by-Step Solution:

1. **Understand the relationship**: At constant temperature, the product of pressure (P) and volume (V) remains constant. This can be expressed as:
\[ P_1 V_1 = P_2 V_2 \]
where \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the final pressure and volume.

2. **Express the volume change**: The problem states that the volume is decreased by 4%. Therefore, the new volume \( V_2 \) can be expressed as:
\[ V_2 = V_1 - 0.04 V_1 = 0.96 V_1 \]

3. **Substitute into the equation**: Substitute \( V_2 \) into the equation \( P_1 V_1 = P_2 V_2 \):
\[ P_1 V_1 = P_2 (0.96 V_1) \]

4. **Cancel \( V_1 \)**: Since \( V_1 \) is common on both sides (assuming \( V_1 \neq 0 \)), we can cancel it:
\[ P_1 = 0.96 P_2 \]

5. **Rearranging the equation**: Rearranging gives us:
\[ P_2 = \frac{P_1}{0.96} \]

6. **Calculate the change in pressure**: The change in pressure can be found by calculating \( P_2 - P_1 \):
\[ P_2 - P_1 = \frac{P_1}{0.96} - P_1 \]
\[ = P_1 \left( \frac{1}{0.96} - 1 \right) \]
\[ = P_1 \left( \frac{1 - 0.96}{0.96} \right) = P_1 \left( \frac{0.04}{0.96} \right) \]

7. **Calculate the fractional change in pressure**: The fractional change in pressure is given by:
\[ \frac{P_2 - P_1}{P_1} = \frac{0.04}{0.96} = \frac{1}{24} \]

8. **Convert to percentage**: To find the percentage increase in pressure:
\[ \text{Percentage Increase} = \left( \frac{1}{24} \right) \times 100 \approx 4.16\% \]

### Conclusion:
Thus, the pressure must be increased by approximately **4.16%**.