The correct option is **D**: an increase in the pressure of a gas is proportional to the decrease in volume at constant temperature. ### Detailed Explanation: To understand why option D is correct, we need to refer to **Boyle's Law**, which states that for a given mass of gas at constant temperature, the pressure (P) of the gas is inversely proportional to its volume (V). This relationship can be expressed mathematically as: \[ P \propto \frac{1}{V} \] or \[ PV = k \] where \( k \) is a constant for a given amount of gas at a constant temperature. #### Step-by-Step Breakdown: 1. **Understanding Boyle's Law**: - Boyle's Law indicates that if the volume of a gas decreases (meaning the gas is compressed), the pressure increases, provided the temperature remains constant. Conversely, if the volume increases, the pressure decreases. - This relationship is due to the fact that gas molecules collide with the walls of their container. If the volume is smaller, the molecules have less space to move around, leading to more frequent collisions with the walls, which increases pressure. 2. **Example Calculation**: - Suppose we have a gas at a pressure of 2 atm and a volume of 4 L. If we decrease the volume to 2 L, we can find the new pressure using Boyle's Law: \[ P_1V_1 = P_2V_2 \] \[ (2 \, \text{atm})(4 \, \text{L}) = P_2(2 \, \text{L}) \] \[ 8 = P_2(2) \] \[ P_2 = \frac{8}{2} = 4 \, \text{atm} \] - This shows that as the volume decreased from 4 L to 2 L, the pressure increased from 2 atm to 4 atm, confirming the inverse relationship. 3. **Why Other Options Are Incorrect**: - **Option A**: "An increase in the temperature of a given mass of gas increases the number of gas molecules." - This statement is incorrect because the number of gas molecules in a closed system does not change with temperature. Temperature affects the kinetic energy of the molecules, not their quantity. - **Option B**: "An increase in the temperature of the gas does not affect the kinetic energy." - This statement is also incorrect. According to the kinetic molecular theory, an increase in temperature results in an increase in the average kinetic energy of the gas molecules. The kinetic energy is directly proportional to the temperature of the gas. - **Option C**: "An increase in the pressure of the gas is proportional to the increase in volume." - This statement is incorrect because it contradicts Boyle's Law. An increase in pressure is inversely related to volume, not directly proportional. If pressure increases, volume must decrease if temperature is constant. ### Revision Summary: - Boyle's Law states that pressure and volume of a gas are inversely related at constant temperature (P ∝ 1/V). - Decreasing the volume of a gas increases its pressure, and vice versa, as shown in calculations. - Temperature affects the kinetic energy of gas molecules, not their quantity. - Understanding the relationships between pressure, volume, and temperature is crucial for solving gas law problems.