Let's take what we know from last time and expand on it. First, let's put our molecules in a cubic container. If you can, imagine one molecule boucing elastically against a wall like a very happy mental patient. Of course we know F = ma, but where can we go from that? Well, by following the link below this, very far. We can eventually determine that force also equals the change in momentum divided by the change in time. Why is this important? Well, we'll get there. But first, picture also that this is a nice molecule and it's only moving along the X-axis. And now I hope you realize that momentum equals the molecules mass times its volume (or, in the X direction only, Px = mv(x)). So what then is the change in momentum? Well, the final momentum, coming out of the wall, was mv(x). And the momentum going into the wall was -mv(x). Therefore,
total momentum = 2mv(x) remember this!
Next assume that this is a very fast moving molecule (as many are in a gas). Imagine that it has so much force that it bounces off the first wall elastically, goes over to the other side, bounces off of that one elastically, and comes all the way back to the first wall. Now in its journeys, our molecule has covered a distance of 2 times the length of the box, or 2L. and time, as we know, is simply distance over speed, so:
change in time = 2L / Vx
By rearranging numbers and combining equations (as seen through the link below!), we an arrive at the following equation:
PV = mv(x)^2 This equation works for one molecule in one direction.
Now, the odds that a molecule moves in the X direction are exactly the same as the odds that it moves in the Y or Z directions. That's what completely random motion is all about. Therefore, the velocity of all of the molecules in the box in the X direction equals the velocity of all of the molecules in the box in the Y direction equals the velocity of all of the molecules in the box in the Z direction. And thus the total velocity of all of the molecules is the square root of 3 times v(x)^2 (as seen in the link below, yadda yadda). So what can we learn? Well, we can get a nice equation:
P = 1/3 * the number of molecules * mv^2 * 1/volume
So why is this important? Well, look what we have and what we started with. We started with properties of individual molecules of gases. We were able to take these microscopic properties and successfully correlate them with macroscopic characteristics of gases! That's pretty impressive. It really is!
Well, that does it for this time around buckaroos. Enjoy the link below (when it's up), and stay tuned for further notes, ones which are going to begin to include information from the book. Enjoy!