🧲AP Physics 2
4. A rigid, sealed cylindrical container holds 0.030 mol of an ideal monatomic gas. The gas is initially in thermal equilibrium at temperature Ti = 300 K and pressure Pi = 1.20×10^5 Pa. The container is surrounded by insulation except for a flat wall made of aluminum that has thickness L = 2.0×10^-3 m and area A = 1.5×10^-2 m^2. The thermal conductivity of the aluminum is k = 205 W/(m·K). The outer surface of the aluminum wall is suddenly brought into good thermal contact with a large thermal reservoir at temperature TR = 450 K, as shown in Figure 1. The container is rigid, so the gas volume remains constant throughout the process, and the reservoir temperature remains constant.
Figure 1. Sealed cylinder with heated monatomic gas
A student claims: “As energy is transferred from the reservoir to the gas, the gas pressure increases because the gas atoms move faster and collide with the container walls more often and with greater impulse.”
Indicate whether the student’s claim is correct or incorrect. Without manipulating equations, justify your answer by describing (i) how the temperature of the gas is related to the motion of its atoms and (ii) how the pressure exerted by the gas is related to collisions of atoms with the container walls. Refer to the rigid container shown in Figure 1 in your reasoning.
Derive an expression for the time t needed for the gas temperature to increase from Ti = 300 K to Tf = 360 K due to conduction through the aluminum wall in Figure 1. Assume the inner surface of the aluminum wall is always at the instantaneous gas temperature T (so the temperature difference across the wall is TR − T), and assume the gas is spatially uniform in temperature at all times. Express your answer in terms of n, k, A, L, TR, Ti, Tf, and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Indicate whether your expression from part B is or is not consistent with the following claim: “If the aluminum wall thickness L is doubled while all other given quantities remain the same, the required time t for the gas to warm from Ti to Tf doubles.” Briefly justify your answer by referencing the functional dependence of your expression on L.
