A sealed syringe contains a gas at a pressure of 4.5 atm and a volume of 200 mL. The plunger is pushed, reducing the volume of the gas in three steps. The volume decreases in the following sequence: From 200 mL to 150 mL, with no temperature change. From 150 mL to 100 mL, with no temperature change. From 100 mL to 50 mL, with no temperature change. Questions: After each compression step, calculate the pressure in the syringe. If the syringe is then immersed in a cooler with the gas remaining at the same volume of 50 mL, how will the pressure change? Use Boyle’s Law to explain.
Answer (1)
Step 1: Calculating Pressure After the First Compression Boyle's Law states that for a fixed amount of gas at a constant temperature, the pressure and volume are inversely proportional. Mathematically, this is represented as: P_1V_1 = P_2V_2 Where: - P_1 is the initial pressure - V_1 is the initial volume - P_2 is the final pressure - V_2 is the final volume In the first step: - P_1 = 4.5 \, atm - V_1 = 200 \, mL - V_2 = 150 \, mL We solve for P_2: P_2 = \frac{P_1V_1}{V_2} = \frac{(4.5 \, atm)(200 \, mL)}{150 \, mL} = 6 \, atm Therefore, after the first compression, the pressure is 6 atm. Step 2: Calculating Pressure After the Second Compression For the second step: - P_1 = 6 \, atm - V_1 = 150 \, mL - V_2 = 100 \, mL P_2 = \frac{P_1V_1}{V_2} = \frac{(6 \, atm)(150 \, mL)}{100 \, mL} = 9 \, atm After the second compression, the pressure is 9 atm. Step 3: Calculating Pressure After the Third Compression For the third step: - P_1 = 9 \, atm - V_1 = 100 \, mL - V_2 = 50 \, mL P_2 = \frac{P_1V_1}{V_2} = \frac{(9 \, atm)(100 \, mL)}{50 \, mL} = 18 \, atm After the third compression, the pressure is 18 atm. Step 4: Effect of Cooling at Constant Volume When the syringe is immersed in a cooler, the temperature of the gas decreases while the volume remains constant at 50 mL. Boyle's Law doesn't directly address temperature changes. Instead, we need to use the combined gas law, which incorporates temperature: \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} Since the volume is constant (V_1 = V_2 = 50 \, mL), the equation simplifies to: \frac{P_1}{T_1} = \frac{P_2}{T_2} If the temperature (T_2) decreases, then the pressure (P_2) will also decrease proportionally. Therefore, cooling the gas at a constant volume will reduce the pressure in the syringe. The exact pressure change depends on the initial and final temperatures, which are not provided. Answer After each compression step: - Step 1: 6 atm - Step 2: 9 atm - Step 3: 18 atm Cooling the gas at constant volume will decrease the pressure.
