Problem 1.22. lf you poke 11 hole in 11 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 suiciently small.) (a) Consider a small portion (area : 1'1) of the inside wall of a container full of gas. Show that the number of molecules colliding with this surface in a time interval A1! is 1' '11 At / (2111?) where 1\" is the pressure, 111. is the average molecular mass, and T11; is the average :1: velocity of those molecules that. collide with the wall. (b) lts not easvt o calculate 1.11;, but a good enough 111.'1pioximation is (-1.15 );)1f2. where the bar now represents an average ov'e1 all moletules 111 the gas. Show that ['13 _2}1;21 :ft':T/-11'1..1 (c) if we now take away this small part. of the wall of the container, the mole- cules that 11.11111td have collided with it will instead escape through the hole. Assuming that nothing cat-(31's through the hole, show that the number N of molecules inside the container as a function of time is governed by the differential equation d N :'l AT nit 2l1'\\f'tf N Solve this equation (assuming constant temperature) to obtain a formula of the form NU) : ATOMt\((1) Calculate the characteristic time for air at room temperature to escape from a l-liter container punctured by a 1-mm2 hole. (e) Your bicycle tire has a slow leak. so that it goes at within about an hour after being inated. 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 win- dow. Do you think they can do this quickly enough to prevent a signicant amount of air from escaping? Justify your answer with some rough esti- mates and calculations