Part I: Equation of State

3 Motorized Molecules:  

    For this part, we observed the properties of a “gas” of motorized molecules in an enclosed area.  We used the  contained area described in the setup with two Force Sensors to record the movement.  For each system, we observed the motion 10 times, for 30 seconds each.  The equation of state was then calculated for this system.  This experiment was modeled after a previous study done by Jeffrey Prentis in 2000.

  The following graphs represent the force on the piston at a specific time.

Fig. 6. Graph of force vs. time for 3 balls in an enclosure of length 20 cm.

Fig. 7. Graph of force vs. time for 3 balls in an enclosure of length 50 cm.

Qualitatively, one can see that there is a dependence on the area of the enclosure and the force on the piston.  Molecules enclosed in a smaller area will exert force more frequently, but of lesser strength.  Molecules enclosed in a larger area exert force less frequently, but with greater strength.  This is due to the greater amount of momentum a molecule can achieve in a system of larger area.  If one looks at the amount and height of peaks in these two graphs, one clearly sees this relationship.  The graph below is a quantitative analysis of this dependence.

Fig. 8. Graph of time-average force versus position of wall for 3 balls

By collecting many sets of data relating force to position at different areas, we obtained the graph of time-average versus position for 3 motorized molecules in an enclosure.  Each error bar on the graph represents the standard deviation after 10 trials.  The trend line is our equation of state.  

The  equation of state for the same system found by a previous lab is in pink (Prentis, 246).  A power-law function was chosen to fit this data to make comparison between experimental and theoretical values easier.  The percent error for this part then is 2.5% for the coefficient and 6.4% for the power.  Because both of these errors are below 10% we can assume our data is relatively accurate.        

                                                                                                                                                                                                                        4 Motorized Molecules:                                                           

The data taken for 4 balls is shown below:

Fig. 9. Graph of force vs. time for 4 balls in a small enclosure

Fig. 10. Graph of force vs. time for 4 balls in a large enclosure

  Again, in these two graphs, one can see that there is a dependence on the area of the enclosure and the force on the piston.  If one compares Fig. 9 with Fig. 6 and Fig. 10. with Fig. 7, one can also see that there is a dependence on the amount of balls in an enclosure on the force on the piston.  The more balls there are, the more force there is. At the end of the section, this will be analyzed in greater depth.

Fig. 11. Graph of time-average force versus position of wall for 4 balls

  The equations of state for the above graph are the following:  

(Prentis, 246).  The percent error for this part then is 48.2% for the coefficient and 21.0% for the power.  This may seem odd because it appears that the two graphs look similar.  Perhaps our equations are off by so much due to the two somewhat extraneous data points at 30cm and 45cm. 

  5 Motorized Molecules:

              The data for 5 Molecules is shown below:

Fig. 12. Graph of force vs. time for 5 balls in a small enclosure

Fig. 13. Graph of force vs. time for 5 balls in a large enclosure

            Again, in the two graphs above, one sees a relationship between the area of enclosure and the number of balls enclosed (if compared to earlier graphs).

Fig. 14. Graph of time-average force versus position of wall for 5 balls

    The equations of state for the above graph are the following:  

(Prentis, 246).  The percent error for this part then is 55.5% for the coefficient and 8.08% for the power.  One can clearly see why the percent error for the coefficient is bigger, it appears that the first 4 data points are raising the plot slightly.  This is probably due to our setup.  In our experiment, rods were placed under the PVC pipes so that the balls would not get “stuck” on the walls.  These bars in effect served as gradually sloping potential walls.  The Squiggle Balls^TM were affected by these walls more at smaller areas.  At these areas, the base of the potential well was on the piston.  So, the time average force at smaller areas was bigger.  This could be one reason why our results differed so greatly from the results from the previous experiment.  However, we did not run into these problems with 3 or 4 balls.  So, this may not have been the problem. 

6 Motorized Molecules:

     The data taken for 6 balls is shown below: 

Fig. 15. Graph of force vs. time for 6 balls in a small enclosure

Fig. 16. Graph of force vs. time for 6 balls in a large enclosure

  Again, in the two graphs above, one sees a relationship between the area of enclosure and the number of balls enclosed (if compared to earlier graphs).

Fig. 17. Graph of time-average force versus position of wall for 6 balls

    The equations of state for the above graph are the following:

(Prentis, 246).  The percent error for this part then is 140.2% for the coefficient and 21.5% for the power.  Again, the percent error for the coefficient could have been due to our setup.  For smaller areas, the bars under the PVC pipes caused the balls to hit the piston more often.  This was probably one problem.  However, the error is so big that there must be another.   It was noticed earlier in the experiment that all of the Squiggle Balls^TM did not function well.  The motor of some of the balls would stop rotating when they hit a wall. With 3 and 4 ball systems, this problem was easy to eliminate.  One merely did not use the balls that were not good.  However, with 6 balls, all of the balls were used, even those that did not function well.  So, when a ball would stop functioning, one merely tapped it to keep it going.  This slight tapping could have caused the ball to have a higher than average momentum.  Because change in momentum is related to force, these balls would subsequently have a greater force against the wall.  This may account for much of our percent error.  Perhaps the previous experiment did not have problems with their Squiggle Balls^TM

Comparison of the 4 Systems:

     Below a graph and a table are shown to aid in comparing the several systems.

Fig. 18 Graph of time-average force vs. length for 3 and 6 balls

      One can clearly see from the above graph that there is a distinct difference between the time average force of 3 balls compared with the 6 balls.  For 6 balls, the force is much larger than it is for 3.  However, it is also clear that both depend on the area of the enclosure.  At larger areas, the average force is smaller whereas for smaller areas, the force is bigger.    

  Table 1. Time-Average force on the piston due to a gas of N molecules in an enclosure of length x.

      The above table shows that again, the time-averaged force depends on the area of the enclosure and the number of molecules.  The force (pressure * area) increases as the number of molecules increases (moving left to right across the row) and decreases as the size of the container increases (moving down a column).  These results are in full agreement with the equation of state for a gas in a three dimensional container.

According to this equation, the pressure (or force in our experiment) is inversely dependent on the volume of the container and proportional to the number of molecules.  Again, the pressure increases as the number of molecules increase and decreases as the volume increases.  Below is a table of our equations of state versus their equation of states.

Table 2. Equations of State for our experiment and for a previous experiment (Prentis 245).

    The percent error between our experimental average and their experimental average is for the coefficient a, 41.9% and for the power b, 12.5%.  There are many explanations for this error.  As explained above, the rods underneath the mat and the defective balls could have caused the force on the piston to be greater.  Also, the position of our Force Sensor was different from theirs in that we had two while the Prentis experiment placed one Force Sensor in the center of the piston.  Perhaps the Force Sensor measures a greater force when tilted at an angle as in our experiment rather than when held parallel to the piston as in their experiment.  For N=4, 5, and 6, we taped the Force Sensor down to prevent it moving when hit by a ball.  This also may have caused it to read a higher force than the previous experiment because it felt the force from the collisions with the other three walls.

            Because of the many differences between our experiments, it might be unrealistic to compare them.  Comparing our average equation with each individual equation might be the best way to analyze our data.  For N=3 our percent error is 74.3% for N=4 balls 10.3%, N=5 balls 14.6%, and N=6 balls 42.9%.  Again, these errors could be due to inconsistencies throughout our experiment.  At N=3 balls, for example, the force sensor was held parallel to the piston and it was not taped down.  For N=4, 5, and 6 balls this was not the case.  Also, the amount of “defective” balls in N=6 was much greater than N=3 and N=4.  This could be one cause for the amount of error seen in this experiment.  However, no matter how much error between our equations of state, qualitatively we saw the results expected.