A long, horizontal, rigid tube of length is completely filled with an ideal monatomic…
PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics
Problem
A long, horizontal, rigid tube of length L is completely filled with an ideal monatomic gas. External heaters and coolers keep the left half of the tube at a fixed temperature T1 and the right half at a lower temperature T2 (T1>T2). Because the tube walls are well insulated from each other, the temperature profile inside the gas can be approximated as a sharp step at the midpoint. Both ends of the tube are sealed, and the system is allowed to reach a steady state in which there is no bulk motion of the gas.
(a) In this steady state, what relationship, if any, must exist between the pressures in the two halves of the tube? State your expectation clearly.
(b) Suppose instead that the pressure in the hot half were slightly higher than in the cold half while the temperature difference remained. Describe qualitatively what molecular motions would occur at the interface and how they would act to change the hypothetical pressure difference.
(c) Using the ideal gas law, derive an expression for the ratio of the number of molecules in the hot half to that in the cold half in the actual steady state, assuming the volumes of the two halves are equal and the pressure is uniform.
(d) Explain how your results support the idea that pressure becomes spatially uniform even when a temperature gradient persists within an isolated gas.