Kinetic Energy of Molecules Escaping Through Hole

[TheAmorphist]

Homework Statement

"A perfect gas containing a single species of molecular weight M is in a container at equilibrium. Gas escapes into a vacuum through a small circular hole of Area A in the wall of the container. Assume wall container is negligibly thick and planer in vicinity to hole. The diameter of the hale is appreciably smaller than the mean free path, but larger than molecular diameter.
a. Show that the number of molecules escaping from the hole per unit area per unit time is given by nC/4.
b.Obtain an expression for the rate of mass outflow.
(What I actually need help on)c. Show that the mean kinetic energy of the escaping molecules if greater than that of the molecules inside the container in the ratio of 4/3.

Homework Equations

The flux equation of F=(Int)nf(Ci)QCnDVc, where f(Ci) is the Maxwellian Velocity Distribution, Q is some quantity (energy here) and integration is performed over the range of velocity space of interest.

The Attempt at a Solution

I have completed the nitty gritty of parts a and b, but simply cannot make any progress on part C. I'm assuming that the gas "within the container" have a mean kinetic energy of (3/2)KT, as they have 3 degrees of freedom. My attempt at finding the kinetic energy of those escaping the hole has consisted of integrating the flux equation a number of times to find a constant factor that resulted in a ratio of "4/3" but I feel like this is in inappropriate way to go about this. Any help or at least pointing me in the right direction would be really helpful.

 

[Ramalho]

Here is a way to do it: calculate the kinetic energy flux of the escaping molecules using equation (3.1) of the book as a basis (assuming, for example, that the wall with the hole is perperdicular to the x1-axis). Then divide it by the result you found in (a). This should yield 2kT, which is 4/3 of the average molecular kinetic energy 3/2 kT. Hope this helps.