Problem 1.22 (Schroeder’s Intro to Thermal Physics)

Problem 1.22

If you poke a hole in a container full of gas, the gas will start leaking out. In this problem you will make a rough estimate of the rate at which gas escapes through a hole. (This process is called effusion, at least when the hole is suciently small.)

(a) Consider a small portion of the inside wall of a container full of gas. Show that the number of molecules colliding with this surface in a time interval is , where is the pressure, is the average molecular mass, and is the average velocity of those molecules that collide with the wall.

(b) It’s not easy to calculate , but a good enough approximation is , where the bar now represents an average over all molecules in the gas. Show that .

(c) If we now take away this small part of the wall of the container, the molecules that would have collided with it will instead escape through the hole. Assuming that nothing enters through the hole, show that the number of molecules inside the container as a function of time is governed by the differential equation . Solve this equation (assuming constant temperature) to obtain a formula of the form , where is the “characteristic time” for (and ) to drop by a factor of .

(d) Calculate the characteristic time for air at room temperature to escape from a 1-liter container punctured by a hole.

(e) Your bicycle tire has a slow leak, so that it goes flat within about an hour after being inflated. Roughly how big is the hole? (Use any reasonable estimate for the volume of the tire.)

(f) In Jules Verne’s Round the Moon, the space travelers dispose of a dog’s corpse by quickly opening a window, tossing it out, and closing the window. Do you think they can do this quickly enough to prevent a significant amount of air from escaping? Justify your answer with some rough estimates and calculations.

Solution:

Problem 1.22

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