I have a kinetic theory/Brownian motion/difussion problem that I need
to solve, pls take a look at the following..
 
 
 
consider a 1 litre closed cube/box, with side 10 cm. In the box are
two gases - A and B.

A is Nitrogen and B is Benzene. We can assume the mixture to behave as
ideal gas mixture with no interactions between the two gas molecules.
also Assume STP conditions. (Lets say total number of molecules in the
box ~ 10^22)
 
The mole fraction of A to B = 10^9 (1 billion) ... that is, there is 1
molecule of benzene per billon molecules of nitrogen... Or Number of
benzene molecules in the cube = 10^13
 
The property of the box is that it captures Benzene molecules. That is
to say, when a benzene molecule collides against the wall of the box
it sticks to the wall, or reacts with it etc, but does not rebound the
benzene molecule. It consumes benzene molecules that collide on the
walls.
 
How much time does it take for the population of Benzene molecules to
reduce by an order of magnitude ... that is from 10^13 (initially) to
10^12 molecules ?
 
How long does it take (as a statistical function in time) for all the
benzene molecules to stick to the walls of the box ?  ie as a function
of time - the concentration of benzene molecules remaining after 10
milli seconds , 1 second,  1000 seconds so on...
 
 
How do I approach this problem ? I am unable to get a clear idea..
 
I am sure this is a standard problem solved as an example in the
fileds of brownian motion / kinetic theory / reaction kinetics  etc..
but cannot find it in any of these texts.. any one has any idea where
to find this problem discussed ?
 
many thanks in advance guys..

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Clarification of Question by b2000-ga on 16 Dec 2005 11:39 PST

Taishar  : )

thanks for your attempt at analysis.. the way you are considering, and
I too initially, in first principle is highly involving in math - pure
Brownian motion kind of stochastic analysis.. i asked my friends and
found that the diffusion approach to solving this problem, with apt
boundary conditions, is easier

the approach i am taking now is solving the ficks second law PDE by
separation of variables... i think i got a good start from their
imputs.. so withdrawing this question..

and no i did not intend any one to solve this problem and give me the
solution.. It is quite time consuming to solve it rigorously, which is
what i am interested in.. i was asking for a reference, in way of text
book or publication etc where a similar one is solved.. and this is a
standard problem in chem engg.. so for such a reference I placed 10
$... and no this is not a take home exam.. encountered it and need to
solve it to analyze an aspect as part of my research (phd)... thanks
again for your comments though..

kottekoe - mean free path under ideal gas approximation is infinite.. ?

Hrm.  That's a difficult one.

The pressure in the box is caused by the repeated collisions the
particles of gas are having with the walls of the container.

If you could determine the amount of collisions per second, you could
then use that to find the decrease in particles with respect to time
since each time there was a collision, the # of particles and
correspondingly the pressure would decrease.

I'm sure you know that, I'm just mentally working it out.

I'm not sure where to start exactly, but by finding the root mean
square velocity of the benzene particles, and the mean free path, you
can determine the magnitude of the collision frequency: mean free
path/root mean square velocity.

The frequency of these collisions with other molecules of gas should
be the same as the frequency of collisions with the walls of the
container.

This frequency will vary with time, since the # of particles will
decrease with each collision.  Using this you should be able to solve
for those things that you desire.

Hopefully that helps you out with where to start.  If I was getting
paid for this, I'd bust out pen and paper and figure this sucker out
:)

Is this a take home exam or something ?  I wouldn't pay $10 for a homework problem!

If you really assumed an ideal gas there would be no interactions
between the gas molecules, the mean free path would be infinite, and
the benzene molecules would move in a straight line at thermal
velocities (about 300 meters per second) and the whole thing would be
over in a small fraction of a second (roughly 0.1m/(300 m/s) ~ less
than a millisecond.

More realistically, we can estimate the mean free path to be of order
100 nm at STP and you now have a diffusion problem. Everything within
100 nm of the edge of the box is virtually instantaneously depleted of
benzene molecules. Benzene molecules then diffuse toward the edge of
the box, where they are immediately depleted. For $10.00, I leave it
to you to set up a diffusion problem in a box geometry with perfectly
absorbing walls. You can estimate the appropriate diffusion constant
from the mean free path and thermal velocity. Have fun!

One correction. I mentioned the thermal velocity and gave an estimate
of ~300 m/s, which is from memory for air. I'm not sure of the actual
value and too lazy to calculate it, but it is about the same as the
sound velocity. For an ideal gas, the thermal velocity is ~
sqrt(3kT/m), for Benzene it is a bit smaller than Nitrogen or air,
since the atomic mass is 78 vs. 28 for Nitrogen. The conclusion is the
same. You'll have to plug in the right constants to get the diffusion
coefficient to use in the standard diffusion equation. It is the same
problem as taking a cube at a uniform high temperature and clamping
metal plates at zero degrees to the faces of the cube. The heat
diffuses out so the temperature of the center goes down,
asymptotically approaching zero degrees. It's not a hard problem to
do, but I think it will involve the error function. (integral of a
gaussian).