For the situation described in Problem 46, show that the speed distribution of the escap...

For the situation described in Problem 46, show that the speed distribution of the escaping molecules is proportional to v³ exp(-βmv²) and that the mean energy of the escaping molecules is 2kT.

Problem 46

Consider a thin-walled, fi xed-volume container of volume V that holds an ideal gas at constant temperature T. It can be shown by dimensional analysis that the number of particles striking the walls of the container per unit area per unit time is given by , where as usual n is the particle number density. The container has a small hole of area A in its surface through which the gas can leak slowly. Assume that A is much less than the surface area of the container. (a) Assuming that the pressure inside the container is much greater than the outside pressure (so that no gas will leak from the outside back in), estimate the time it will take for the pressure inside to drop to half the initial value. Your answer should contain A, V, and the mean molecular speed . (b) Obtain a numerical result for a spherical container with a diameter of 40 cm containing air at 293 K, if there is a circular hole of diameter 1.0 mm in the surface.