A frictionless, perfectly insulating cylinder is fitted with a movable, massless piston. The cylinder contains an ideal monatomic gas at 298 K. The external pressure is suddenly increased so that the piston is pushed inward until the volume of the gas is exactly one-half of its original value. The temperature of the gas is maintained at 298 K by placing the cylinder in a large isothermal bath.
Answer the following.
(a) Qualitatively describe how the frequency of molecular collisions with the cylinder walls changes as the volume is halved at constant temperature. Support your answer with particle-level reasoning.
(b) Discuss how the average force per collision changes, if at all, under the conditions described. Justify your response in terms of molecular speed and kinetic energy.
(c) Combine your reasoning from parts (a) and (b) to predict the overall change in macroscopic pressure. State the numerical factor by which the pressure changes and explain each logical step.
(d) Suppose instead that the compression occurred so rapidly that no heat flowed to or from the bath and the temperature rose to a value higher than 298 K. Predict how this temperature rise would further affect the frequency and force of collisions, and hence the final pressure, compared with the isothermal compression discussed above.