View Question
Q: Ideal gas kinetics / Brownian Motion ( No Answer,   3 Comments )
 Question
 Subject: Ideal gas kinetics / Brownian Motion Category: Science > Chemistry Asked by: b2000-ga List Price: \$10.00 Posted: 15 Dec 2005 11:52 PST Expires: 16 Dec 2005 11:40 PST Question ID: 606250
 ```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..``` 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.. ?```
 There is no answer at this time.

 ```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).```