Ideal Gas Laws

Today, we began with yet another neat presentation. We had a small beaker of water on the ground, and suspended in the water was long tube filled with water and stoppered at the other end. The question was once the stopper was removed, what would the water do and why?

We did lots of stuff today, so I'll break it into the experiemnts we did and what we learned (or should've.)

Presentation/Opening Question

We had a small beaker of water on the ground, and suspended in the water was long tube filled with water and stoppered at the other end. The tube was suspended in the water using a scale, which in addition to holding it up, measured the force (weight) the tube and water exerted on it. The question was once the stopper was removed, what would the scale read: heavier, lighter, the same, and why? After a suspensful 5 seconds while Prof. Mason pulled out the stopper, the water came gushing out and the scale.....showed it was less. Because the tube was filled with water and then suspended into the beaker of water, the water inside the stoppered tube was actually at a lower pressure than the atmosphere around it. When the stopper was removed, the pressure inside the tube equalized and the water came out. Initially, the tube and water were supported by the water in the beaker because of the pressure differential.

First project: Straw tube thing

In this experiment, we attached a straw to each end of a rubber tube and then filled it with a little bit of water. We then blew over one end of a straw to see what happened. The level of the water inside changed by some height y, which led us to believe that there was a change in pressure on our end, causing the water to move. The question in part b asks us to show that the addition pressure, ΔP, was independent of the cross sectional area.

First, we rationalize that as we blow into the straw, there are two different pressures. The side that we blow on (P1) and the other end (P2) and their different is ΔP. Now, a hint given is we need to start with the weight of the water, which is the mass of the water * gravity; so W = mg which equals force.
Next, lets take into account that Pressure = Force/A or P=F/A. Thus, force is PA.
We then take into account the mass of the liquid (water in this instance) Density = Mass/Volume or p=m/V => pV=m. Now, we know that volume is Area(A)*Height(h) so m=pAh.
Substituting the mass into the weight equation we get F=pAhg=PA and the area cancels, leaving us with F=P=phg.

Second Project - Syringe & Pressure Sensor

In this part, we hooked up a syringe to a pressure sensor which was then connected to the Macs.Using Logger Pro, we had the program take data entries whenever we entered them in manually. We adjusted the data graph to read Pressure as a function of the volume inside the syringe in cubic centimeters (cc). As we decreased the volume inside the syringe, we found that the pressure inside did not rise in a linear fashion; instead, it rose in an inverse function (1/x). What did this mean? As the volume decreased, the pressure inside increased!

Third Project (not really a project, but learning cool stuff)

From the previous project, we figured out the P=1/V. But what about other relations? What about Pressure vs Temperature? For this, Prof Mason took a well sealed syringe and put a piece of cotton inside. He then compressed the syringe very quickly, which caused the cotton to ignite! The temperature inside the syringe increased very rapidy as the pressure inside increased, to the point that it lit up. He repeated the experiment slower, and nothing happened. Because the syringe was not a perfect system, heat transfered from the inside to the glass to the air. Finally, Volume was inversely related to temperature. Using a program a friend of his designed, we had x atoms moving around in an enclosed box. By adding more atoms to the box, the energy levels fell which caused the temperature inside to fall.

Fourth Project - KE and Pressure

First we were split into non believers and believers. In what? Atoms. Non believers had to list down reasons atoms didn't exist, while believers wrote down reasons they do exist. While non believers used the lines "Becuase we can't see them or touch them, they can't be real" the believers said something more like "We have seen them, use and electron microsope or see the IBM picture."

We then visualized this using the Activ physics program. Here, we had a Java program of an atoms of gas inside an enclosed box. The program tracked the path, speed, collisions, and temperature of an atom of gas as it zoomed around. What did we find out.
1. Gas does not maintain a constant velocity.
2. Changes in velocity were caused by collisions with other atoms.
3. When the gas atom hit the wall, it bounced off in a nearly elastic collision. Becuase the gas atom is so small compared to the wall of the containers, and the atom is moving at such high speeds, we basically treat it as a perfectly elastic collision (or the closest we can ever get to one)
4. At any given instant, the atoms are all moving with different speeds. However, over a time invterval, we can find that they share an average kinetic energy.
5. Setting the temperature of the program at 100k, we tried to eyeball the average velocity. Then, we set the program to 400k (4x hotter) and observed. What did we find? The increase in avg speed was not 4x bigger but 16x bigger (my numbers were 250 m/s @ 100k to 4000 m/s @ 400K) so the relation between speed and temperature was and exponential number x2.

Proof:

Given any instant, the momentum of a particle is

Pm=mv (mass*velocity).

Divide both sides by time (t) will give us the force at that moment in time

F=mv/t

Now, if this particle was enclosed in a cube, how would we relate time to how quickly the particle travels thru the cube?

x=vt or t=x/v

Subsitute for t gives

F=(mv)/(x/v) = mv2/x

Now, Pressure = Force/Area

P=(mv2/x)/A

Area of the cube is x2

P=(mv2)/x3

Now, lets simplify this a little bit. If a particle is moving in some direction, it will have an x, y, and z component. If, in an idea direction, the particle moved in direction i+j+k, the x component would be 1/3 of the total velocity. Also, x3 = Volume

P=(mvx2)/3V

This then gives us

3PV=mvx2

If we multiply each side by 1/2, we get

(3/2)(PV)=(1/2)mvx2

Which now gives us

(3/2)(PV)=KE

Finally, if we have N molecules of stuff, we multiply KE by the number of molecules. Solving for P gives

P=(2NKE)/3V