The correct option for the question is **D. 5P0**. ### Step-by-Step Explanation 1. **Understanding the Problem**: - We have a gas in a container with an initial volume \( V_0 \) and an initial pressure \( P_0 \). - The gas is compressed to one-fifth of its original volume, which means the new volume \( V \) is \( \frac{V_0}{5} \). - The temperature \( T \) of the gas remains constant throughout this process. 2. **Applying Boyle's Law**: - Since the temperature is constant, 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. Mathematically, this is expressed as: \[ P_1 V_1 = P_2 V_2 \] - Here, \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the final pressure and volume. 3. **Substituting Known Values**: - We know: - \( P_1 = P_0 \) - \( V_1 = V_0 \) - \( V_2 = \frac{V_0}{5} \) - Plugging these values into Boyle's Law gives: \[ P_0 \cdot V_0 = P_2 \cdot \frac{V_0}{5} \] 4. **Solving for Final Pressure \( P_2 \)**: - Rearranging the equation to solve for \( P_2 \): \[ P_2 = \frac{P_0 \cdot V_0}{\frac{V_0}{5}} = P_0 \cdot 5 \] - Thus, we find: \[ P_2 = 5P_0 \] 5. **Conclusion**: - The final pressure of the gas after it has been compressed to one-fifth of its original volume, while keeping the temperature constant, is \( 5P_0 \). ### Why Other Options Are Incorrect - **Option A: \( \frac{P_0}{5} \)**: - This option suggests that the pressure decreases significantly. However, since the volume is reduced, the pressure must increase, not decrease. - **Option B: \( \frac{4P_0}{5} \)**: - This option also implies a decrease in pressure. The relationship defined by Boyle's Law indicates that pressure increases as volume decreases when temperature is constant. - **Option C: \( P_0 \)**: - This option suggests that the pressure remains the same. However, since the volume is reduced to one-fifth, the pressure must increase significantly, as shown in our calculations. ### Common Pitfalls - **Ignoring Temperature**: It's crucial to remember that Boyle's Law applies only when the temperature is constant. If the temperature were to change, we would need to consider the ideal gas law instead. - **Misunderstanding Volume Change**: Some students may confuse the relationship between volume and pressure, thinking that a decrease in volume would lead to a decrease in pressure, which is incorrect. ### Revision Summary - Boyle's Law states that \( P_1 V_1 = P_2 V_2 \) for a gas at constant temperature. - When a gas is compressed to one-fifth of its volume, its pressure increases by a factor of five. - The final pressure is calculated as \( P_2 = 5P_0 \). - Always ensure to keep track of temperature conditions when applying gas laws.