I only need help for part c.. Kindly help pls. If you poke a hole in a container

Question

I only need help for part c.. Kindly help pls.

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 of the wall of a container full of gas. Show that the number of molecules colliding with this surface in a time interval is where P is the Pressure, m is the average molecule mass, and is the average x velocity of those molecules that collides with the wall. (b) It is not easy to calculate , but a good enough approximation is , where the bar now represents an average over all the molecules in the gas. Show that . (c) If we take away this small part 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: Solve this equation to obtain a formula of the form N(t) = No exp(t / ) where is the characteristic time, and No is the number of particles when t = 0.

Explanation / Answer

dN/dt =-A/2V sqrt (KT/m)

dN/N =-A/2V sqrt (KT/m) dt

ln(N) = [-A/2V sqrt (KT/m)] t +C1

ln(N) =-C *t + C1 (where C=[A/2V sqrt (KT/m)] )

so

Now at t=0 N=No

C1 =ln (No)

putting this

ln(N/No) =-C*t

N =No e ^(-Ct ) (where  C=[A/2V sqrt (KT/m)]

N= No e ^(-t/T ) [ where T =1/ C=1/[A/2V sqrt (KT/m)] (ans)

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