**Correct Option: A. The volume is halved** ### Detailed Explanation: **Understanding Boyle's Law:** Boyle's Law states that for a given mass of an ideal gas at constant temperature, the pressure (P) of the gas is inversely proportional to its volume (V). This relationship can be mathematically expressed as: \[ P \propto \frac{1}{V} \] or, more commonly, in the form of the equation: \[ P_1 V_1 = P_2 V_2 \] where: - \( P_1 \) and \( V_1 \) are the initial pressure and volume, - \( P_2 \) and \( V_2 \) are the final pressure and volume. **Scenario Analysis:** In this question, we are told that the pressure is doubled while the temperature remains constant. Let's denote the initial pressure as \( P_1 \) and the initial volume as \( V_1 \). If the pressure is doubled, we have: \[ P_2 = 2P_1 \] Now, we can use Boyle's Law to find the new volume \( V_2 \): 1. **Set up the equation using Boyle's Law:** \[ P_1 V_1 = P_2 V_2 \] 2. **Substituting \( P_2 \):** \[ P_1 V_1 = (2P_1) V_2 \] 3. **Dividing both sides by \( P_1 \) (assuming \( P_1 \neq 0 \)):** \[ V_1 = 2 V_2 \] 4. **Rearranging to find \( V_2 \):** \[ V_2 = \frac{V_1}{2} \] This shows that when the pressure is doubled, the volume is halved. ### Why Other Options Are Incorrect: - **Option B: The volume is doubled.** - This option suggests that increasing the pressure would lead to an increase in volume, which contradicts Boyle's Law. According to Boyle's Law, if pressure increases, volume must decrease, not increase. - **Option C: The volume remains the same.** - This option implies that volume is unaffected by changes in pressure, which is incorrect. Boyle's Law clearly indicates that volume and pressure are inversely related; thus, a change in pressure will result in a change in volume. - **Option D: The volume is increased by a factor of four.** - This option suggests a significant increase in volume with an increase in pressure, which is contrary to the principles of Boyle's Law. An increase in pressure leads to a decrease in volume, not an increase. ### Common Pitfalls: - **Misunderstanding Inverse Relationships:** Students often confuse direct and inverse relationships. Remember that in Boyle's Law, as one quantity increases, the other decreases. - **Ignoring Constant Temperature:** Boyle's Law only applies when the temperature is constant. If temperature changes, the relationship described by Boyle's Law does not hold. - **Forgetting to Rearrange Equations:** When manipulating equations, ensure that you correctly isolate the variable you are solving for. ### Revision Summary: - Boyle's Law states that pressure and volume of a gas are inversely related at constant temperature. - Doubling the pressure results in halving the volume. - The correct answer to the question is A: The volume is halved. - Always remember the relationship: \( P_1 V_1 = P_2 V_2 \) for calculations involving changes in pressure and volume.