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An ideal gas, initially at 300 K, is compressed to half its original volume - AQA - A-Level Physics - Question 7 - 2022 - Paper 2
Question 7
An ideal gas, initially at 300 K, is compressed to half its original volume. It is then cooled at constant volume until the pressure is restored to its initial value... show full transcript
Worked Solution & Example Answer:An ideal gas, initially at 300 K, is compressed to half its original volume - AQA - A-Level Physics - Question 7 - 2022 - Paper 2
Step 1
Calculate the final temperature after compression
Answer
For an ideal gas, we can use the ideal gas law, which states that ( PV = nRT ). When the gas is compressed to half its volume while keeping the amount of gas and the temperature constant, the pressure doubles. Initially, the gas is at a volume ( V ) and temperature ( T_1 = 300 \text{ K} ). After compression to volume ( \frac{V}{2} ), the pressure becomes ( P_2 = 2P_1 ). Thus, we can express the state after compression using:\n[ P_1V = nRT_1 ]\n[ 2P_1 \left( \frac{V}{2} \right) = nRT_2 ]\nFrom which we find that the final temperature after compression ( T_2 ) will equal the initial temperature ( T_1 = 300 \text{ K} ).
Step 2
Cooling at constant volume
Answer
After compression, the gas is cooled at constant volume until the pressure is restored. At constant volume, temperature and pressure have a direct relationship. Using the ideal gas law again, we can set up the relationship between the parameters before and after cooling: [ \frac{P_1}{T_1} = \frac{P_2}{T_2} ] Since we are restoring to the original pressure ( P_1 ) and we previously found ( P_2 = 2P_1 ), we rewrite as: [ \frac{P_1}{300} = \frac{2P_1}{T_2} ] From this, we can solve for ( T_2 ): [ T_2 = 2 \times 300 = 600 \text{ K} ] However, since we are seeking the temperature after cooling in relation to the restored pressure, we reconsider that the cooling would mean reducing the temperature by half: [ T_{final} = \frac{T_2}{2} = 150 \text{ K} ] Thus, the final temperature of the gas is 150 K.