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Unformatted text preview: PHYS 333 Winter 2009 Assignment 2 All of these problems are from Schroeder’s book. 1. Schroeder, 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 sufficiently small.) (a) Consider a small portion (area = A) of the inside wall of a container full of gas. Show that the number of molecules colliding with this surface in a time interval Δ t is P A Δ t/ (2 m ¯ v x ), where P is the pressure, m is the average molecular mass, and ¯ v x is the average x-velocity of those molecules that collide with the wall. This expression is really the same as the relation between (average) pressure and rate- of-change of momentum that we saw in class. But, now, we assume that it is true for even th esmall area A . Thus: P = N A ¯ P = N A | Δ p | A Δ t = N A 2 m ¯ v x A Δ t and therefore N A = P A Δ t/ 2 m ¯ v x (b) 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 N of molecules inside the container as a function of time is governed by the differential equation dN dt =- A 2 V s k B T m N Use the approximation that ¯ v x ' ( ¯ v 2 x ) 1 / 2 = p k B T/m . Solve this equation (assuming constant temperature) to obtain a formula of the form N = N (0) e- t/τ , where τ is a “characteristic time” for N (and P ) to drop by a factor e . From part (a), the number Δ N of molecules escaping in time Δ t is Δ N ≡ N A = P A Δ t/ 2 m ¯ v x . Thus, Δ N = A 2 V s k B T m N Δ t 1 where we have used the result from part (b) and also the ideal gas law. (The pressure P depends on N , the number of molecules in the container, and so to find a differential equation for N we must make this dependence explicit.) To turn the expression for Δ N into a differential relationship, we must recognise that Δ N is a decrease in N , so that Δ N → - dN and Δ t → dt and, finally dN =- A 2 V s k B T m Ndt ≡ - N τ dt which can be integrated to give N = N (0) e- t/τ , where τ = 2 V A q m k B T . (c) Calculate the characteristic time for air at room temperature to escape from a 1-litre container punctured by a 1-mm 2 hole. About 7 seconds....
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This note was uploaded on 04/11/2010 for the course PHYS 333 taught by Professor Harris during the Winter '09 term at McGill.
- Winter '09