kinmodel Introduction Lab

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course Phy 232

Experiments using the Kinmodel Simulation

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Introduction

Experiments and Activities

The default settings

Customized settings

'Research' questions

You may click here for the program

• kinmodel (DOS version) or

• billiard simulation (Windows version)

used in this simulation (read instructions first, though)

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Introduction

The program kinmodel_.EXE simulates in 2 dimensions the kinetic behavior of a user-specified number of spherical atoms with user-specified masses, colliding as hard elastic disks at a user-specified center-to-center distance. The initial positions and speeds of the particles are randomly generated by the computer and the simulation develops from the corresonding initial state. Information related to particle speeds, x- and y- kinetic energies, and energy distributions is provided in the form of unlabelled graphs on the screen.

The simulation can be stopped once it is running by striking the 's' key.

The default settings

The default settings are chosen to provide 30 atoms, one colored red, one blue and the others green. All atoms except the blue have equal masses, the blue atom having 10 times the mass of a green or red atom. The red and blue atoms each leave a marked trail, and the relative speed of the red atom is indicated on the screen. The simulated speed is appropriate for viewing on a typical 133-mhz Pentium with such goals as

estimating the distribution of atomic speeds and mean free path

equipartition of energy and the tendency for an ordered system to move toward disorder

the improbability of 30 particles being segregated on one side of the viewing area (unlikelihood of an ordered configuration)

the probability that a particle's speed will occur in a given range

the connection between relative particle mass and average speed

the development of empirical frequency vs. speed and frequency vs. energy histograms (order and disorder, this time with statistical order emerging from the disordered system)

images of 2-dimensional collisions

appreciation of time scale of kinetic interactions in a gas at typical pressures and temperatures (at medium default speed the simulation represents many of the features of a thin slice approximately 10 nanometers on a side and, say, a nanometer thick, of a monatomic gas at room temperature and several atmospheres pressure, with 1 second of real-world time corresponding to a few thousand years of simulation time).

and others.

Experiments and Activities

Preliminary Observation

Run the program kinmodel, accepting all defaults by using the 'Enter' key to answer the prompts on your computer (the computer will then automatically pick the selection with the asterisk) and observe the particles or 'billiard balls' bouncing around the screen and off one another.

• Watch the KEx and KEy values as they change with each collision, representing the total x and y kinetic energies of the particles.

• Watch the 'red' particle for a couple of minutes, estimating the average time between its collisions and its average speed (one of the speeds given near the top of the screen corresponds to that of the 'red' particle--which is it?).

The speed corresponding to the red particle is most likely the first number on the left.

• Watch the 'blue' particle, and speculate on what property of this particle is different from that of the other particles.

The blue particle turns the green particles yellow.

• Watch as the 'red' particle sometimes turns yellow. What causes this? What property does the particle have when it is yellow?

The red particle turns yellow as a green particle that has been turned yellow touches it. The yellow particle moves with a much greater velocity than before it was turned.

• What might the graphs represented at the right of the screen represent?

The graphs seem to be bar graphs that most likely represents speed.

• Strike the 'S' key to stop the simulation, and if you are done give the appropriate response to the prompt to quit the program. CTRL-ALT-DELETE will also stop the program, but if you're not careful it will reboot your computer so avoid that option if you can.

Before reading further email your instructor with your best answers to these questions. There are two good reasons for not reading ahead: If you get your answers by reading ahead your instructor will be able to tell, and if you read ahead you won't learn as much.

Experiment kinmodel_01: The Distribution of Atomic Speeds

When the speed of the simulation is moderate it is possible to watch a specific particle (the red particle or the blue particle in the default simulation) and obtain an intuitive feeling for the relative frequencies of various speeds.

Run the simulation kinmodel at the default settings--simply hit the 'Enter' key for each option presented.

• Observe the simulation long enough to get a feel for the maximum velocity you are likely to see. Then estimate how much time it spends at slow (less than 1/3 of max vel.), medium (between 1/3 and 2/3 of max. vel.) and fast (more than 2/3 of max. vel.) velocities.

• Express your estimates in percents of the total time spent in the three different velocity ranges.

The molecules seem to spend about 50% of the time in the medium range 12.5 % in the high range and 37.5% in the low range.

• Draw a histogram (a bar graph) of your estimates. Describe your histogram in your writeup.

The histogram looks like what is stated above. A section labeled mid is at a height of 50. A section labeled low is at a height of 37.5. A section labeled high is at a heigt of 12.5

• Now suppose you had estimated the percent of time spent in each of 10 velocity ranges (i.e., from 0 to .1 of max. vel., .1 to .2 of max. vel., etc, up to max. vel.). From your previous estimates, without further viewing the simulation, make a reasonably consistent estimate of the proportion of time spent in each of these ranges.

• Sketch a histogram of your estimates and describe the graph in your writeup.

I labeled my graphs based on the numbers for speed 0 to 1, 1 to 2, 3 to 4 etc.

My heights for each range were as follows

Range Height

1,2 2

2,3 17.5

3,4 20

4,5 25

5,6 25

6,7 6

7,8 2

8,9 2

9,10 1.5

• Watch the red particle for long enough to estimate the percent of time it spends colored yellow.

The red particle spends about 1 percent of the time colored yellow

• Watch the whole simulation to see what average percent of the particles are yellow at a given time (there are 30 particles, including the blue one).

About 10 percent of the particles are colored yellow on average.

• How would you expect the answers to these two questions to compare?

The red particle will spend less percent of time being yellow because it only turns yellow when it is hit by another yellow particle. This happens very rarely, however, the green particles turn yellow every time they touch the blue particle. Since there are a lot of green particles, the frequency of collision is pretty high.

• Watch the number corresponding to the speed of the 'red' particle.

• Close your eyes for a few seconds at a time and open them suddenly, and each time write down the velocity of the 'red' particle immediately when your eyes open. Record about 100 velocities in this manner.

5 5 7 5 2 9 4 2 2 2 8 4 4 5 5 5 5 5 5 5 5 2 2 2 3 6 2 1 1 1 4 2 8 1 1 3 3 3 3 4 3 2 2 2 2 2 2 3 8 9 10 8 2 5 5 5 5 5 6 6 7 5 3 2 7 2 2 1 1 3 7 7 2 3 6 6 7 7 7 3 3 3 3 2 7 7 7 3 4 6 8 2 2 2 2 3 5 5 5 6

• Tally your velocities to see how many of the 100 velocities were 0, how many were 1, how many were 2, etc.

Number How many were that number

0 0

1 8

2 25

3 16

4 6

5 19

6 7

7 11

8 5

9 2

• Construct a histogram of your results and compare to the histograms you predicted earlier.

My new histogram is a bit off from what I predicted earlier. I thought that the particle tended to stay in the mid-range, but it looks like it spends most of its time in the 2-3 range. My end estimates were pretty accurate, however.

• Observe the 'blue' particle for at least 5 minutes, monitoring its speed to see if it ever reaches 3 or more.

The blue particle reaches speeds of 3, but never more than three

• Based on your observations and your experience with other distributions, sketch a histogram of the speed distribution for the 'blue' particle, 'rounding' your results to the nearest .25 (i.e., use 0, .25, .5, .75, etc. for your velocities).

The histogram looks like 75% at 2 25% at 1 and 0% at 3 and 0% at 0

• Stop the simulation and quit it. Then run it again and watch the 'blue' graph at the right. Sketch this graph every minute or so, for about five minutes. Describe how the graph develops, and describe what you think it will look like after a long time.

The blue graph plots random points that are both high and low depending on the speeds of the blue particle. I believe that over time the graph will eventually stabilize fairly in the middle.

• What do you think this 'blue' graph represents?

I think the blue graph represent the speed of the blue particle over time.

• What do you think the 'yellow' rectangle is for, and what does the graph tell you in relation to this rectangle?

I believe that the rectangle is probably a reference point for max speed. The higher the points are on in relation to the rectangle, the greater the speed.

Experiment kinmodel_02: Mean free path; mean time between collisions

It is possible to observe a chosen particle (the red or the blue particle in the default simulation) for its mean free path between collisions. This observation becomes more involving if, for example, the observer is 'rooting' for the red particle (in the default simulation) to collide with the blue particle. The tracks left by these particles also provide a record of the path between collisions.

• First observe the 'red' particle for a few minutes and try to get a feel for how the distances traveled between collisions with other particles are distributed. Make your best estimate of what percent of the time the particle travels less than 1/2 inch between collisions, the percent of the time the distance rounds off to 1 inch, the percent of the time the distance rounds to 2 inches, etc.. When the particle collides with a 'wall', it doesn't count as a collision and distance keeps accumulating until it collides with another particle.

• Sketch a histogram of your estimates, and also document the distance on your monitor between the 'walls' that confine the particles.

The paricle spends approximately 50% of the time traveling .5 to 1 inch between collisions. 25% is spent between 0 and .5 inches. 12.5% is spent 2 inches, and 12.5 percent is spent 3 inches.

The length between walls is approximately 10 inches

Now take some data.

• Using the Pause/Break key on your computer, stop and start the particle motion as required in order to observe the distances traveled by the 'red' particle (the computer will stop when the key is depressed, and can be restarted using the 'Enter' key). Use a ruler to measure distances traveled. Don't leave any distances out, because this would bias the sample. Observe at least 100 distances.

• Describe how you obtained your data and report your data as a frequency distribution (i.e., the number of observations for which the distance rounded to 0, 1, 2, 3, ..., inches).

0 40

1 15

2 15

3 10

4 4

5 6

6 2

7 8

• Sketch a histogram of your results.

• Sketch the histogram you would expect from a large number of observations.

My data seems to show that the majority of the collision were closer to 0 inches. This was due to the fact that the red atom seemed to get caught up in a large portion of green balls and several collisions happened very quickly. I would expect that for a large number of trials, the numbers would average out better. I think that on average, however, the collisions favor a smaller distance simply due to the fact that there are so many particles to collide with. Because of this, I believe that the final histogram would be weighted towards the bottom end of the spectrum with very few collisions in the upper inches. This is somewhat consistent with my previous predictions except that I failed to take into account inches greater than 3.

• Describe your histograms, and how they compare with your previous predictions.

Experiment kinmodel_03: Equipartition of energy and the direction of disorder to (increasing or decreasing)

Observe the first several seconds of the simulation at the 'slow' default speed. You will see how the particles initially are all moving in or very close to the x direction, with little or no y component. Note the x and y kinetic energies, displayed near the top of the screen.

• Observe what happens to the directions of motion of the particles as they start colliding.

As collisions occur, the direction of the particles is apparently random with little or no common direction.

• Observe what happens to the x and y kinetic energies.

The x and y kinetic energies seem to balance out even though the y kinetic energy was initially lower than the x.

• Is the system more organized at the beginning of the simulation or after a couple of minutes?

The system is more organized after a couple of minutes because initially, the kinetic energy is skewed in the x direction.

• If the x and y kinetic energies were averaged for 100 years, starting a few minutes after the simulation began, which do you think would be greater?

I believe that the x and y would be the same approximately on average.

Run the simulation in this manner several times, and each time determine how long it takes before the total y kinetic energy is first greater than the total x kinetic energy. Report your results.

It took about 12 seconds 10 seconds 15 seconds and 16 seconds.

Now take some data:

• Running at the fastest default speed, stop the simulation with the pause/break key every few seconds, keeping your eyes closed for at least 2 seconds before stopping the motion.

• Write down the x and y kinetic energies each time.

X Y

594.8 651.7

588.3 656.2

543.6 697.8

733.9 551.9

533.2 721.4

543.5 686.0

642.4 572.1

785.9 465.8

527.3 719.7

654.5 573.3

623.5 636.2

758.5 504.7

597.6 654.5

546.6 587.5

533.2 724.5

433.5 665.4

568.8 674.7

465.5 762.2

737.9 533.4

607.8 613.7

333.3 906.0

693.3 584.4

579.8 666.2

767.2 478.0

632.0 607.8

555.1 703.1

584.3 662.1

728.5 485.2

641.2 636.4

759.9 492.8

• Do this at least 30 times.

• Find the average of all your x and all your y kinetic energies.

Avg of x: 611.1

Avg of y: 629.2

• Do you believe the difference in the averages is significant, in that the direction that has the higher average will always tend to have the higher average every time the simulation is run?

The values don’t vary significantly enough to believe that one value is always higher. I believe that the averages clearly

Averages clearly show that the trend is to meet at the same point for both x and y average kinetic energy.

Experiment kinmodel_04: The improbability of all particles being segregated on one side of the viewing area (order vs. disorder)

Any selected region of the screen can be selected for viewing by masking the rest of the screen. The viewer can estimate the probability of this region being vacated within an hour, within a day, within a year, ..., within the age of the universe. Results will differ with the size of the region, the number of particles and the speed of the simulation.

• Cut out a 1-inch square and watch the simulation for 2 minutes on the middle default speed. Observe how many times the square becomes 'empty' of particles. Estimate what percent of the time this square is empty.

The square was empty 23 times

• Enlarge the square to a 1-inch by 2-inch rectangle and repeat.

The rectangle was empty 14 times

• Enlarge to a 2-inch by 2-inch square and repeat.

The square was empty 6 times

• Enlarge this square to a 2-inch by 4-inch rectangle and repeat.

The rectangle was empty 2 times

• Enlarge to a 4-inch by 4-inch square and repeat.

The square was empty 1 time

• Mask all but 1/4 of the screen and repeat.

The area was empty 0 times

• How long do you think it would take, on the average, for 1/4 of the screen to become completely empty of particles?

A 4 by 4 square is about a fourth of the screen, and I observed that the square was empty in a 2 minute period, so I would say that on average, it would take about 2 minutes to be empty.

• How long do you think it would take, on the average, for 1/2 of the screen to become completely empty of particles?

To determine the time for ½ of the screen would be very hard to determine because I observed that .75 of the screen was never empty in 2 minutes, so I would venture to guess that It would have to be within 30 minutes just because it would be extremely difficult to fit all 30 atoms in just one half of the screen.

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If the 30 particles on the screen were randomly positioned, then any given particle would have a 50-50 chance of being on the right-hand side of the screen. So the probability that a random position would have all the particles on the right-hand side would be .5 ^ 30. That's roughly 1 in ten billion. The balls move far enough that we can consider them to have become repositioned in about a second. So every ten billion seconds we might expect to observe this. That's about once in 200 years.

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• A typical closet is about 100 million times as far across as the distance represented by the screen. Ignoring for the moment that the closet is three-dimensional and hence contains many more air molecules than would be represented by a 2-dimensional simulation, how long do you think you would have to wait for all the molecules to move to one side of the closet?

This would be almost impossible to determine. The likelihood of all particles being on one side would be even less than the screen because an increase in space would mean even more disorder and more undpredictablility. I would guess that it might occur in 24 hours, but I honestly have no idea.

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To get this probability you would have to raise .5 to the power of the number of particles present. That's a very, very large power. There is a very miniscule chance that this would every happen in the universe, even if the universe was filled with such closets.

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Experiment kinmodel_05: The probability that a particle's speed will occur in a given range

The default settings will not work with this experiment. This time when you run the program you need to choose to customize the settings. For everything but the numbers of different particles and their masses, and the 'yellow' marker, you may use the defaults.

For the number and masses of particles:

• When asked for the number of particles of type 1, enter 28. When asked for the mass of this type give 1.

• When asked for the number of particles of type 2, enter 1. When asked for the mass of this type give 10.

• When asked for the number of particles of type 3, enter 1. When asked for the mass of this type give 1.

Regarding the 'yellow' marker:

• You will choose the minimum and maximum speeds which will result in the particle being 'painted' yellow. This will allow you to observe the proportions of the particles in different velocity ranges.

If you wish you may also adjust the speed factor, which has default value 3. If you want the simulation to slow down to 1/3 the pace, you can enter 1 for the speed factor. If you want the simulation to go as fast as practical for the other default setting, you could use a speed factor up to 5. Only the pace of the simulation is affected by the speed factor; the speeds displayed on the screen are not affected.

Now try to observe the numbers of particles in various ranges:

• Run the simulation and use a 'yellow' range of 3 to 6 and attempt to observe the proportion of the particles falling within this range. You will be able to get a fairly good idea of the proportion, but it will be hard to get a really good estimate unless you repeatedly pause the program and count the 'yellow' particles.

• Run the simulation using a 'yellow' range of 4 to 4, which will give mark only particles whose velocity is 4. Determine to reasonable accuracy the average percent of particles with this velocity.

• Repeat for velocities 0, 1, 2, 3, 5, 6, 7, 8, 9 and 10.

What are the percentages corresponding to each of these velocities?

????? My program did not let me adjust the values for the atoms. It went through me being able to edit particle 3 and then it simply cut off. I was not able to actually do this experiment.

What therefore do you think is the average particle velocity?

Experiment kinmodel_06: The connection between relative particle mass and average speed; equality of average kinetic energies

Using default settings, answer the following:

• What percent of the time is the blue particle at each of the velocities 0, 1, 2, and 3?

The blue particle is at 2 for 80% of the time 2 for 19% and 0 and 3 both around .5%

• What therefore do you think is the average velocity of the blue particle?

The average velocity would have to be around 2

• The blue particle is 10 times more massive than the other particles. How do you think its average KE therefore compares with the average KE of the other particles?

The average KE of the large particle is on average less than the other particle because even though it has a greater mass, the KE equation is ½ mv^2, so the velocity has more of an effect than mass because it is squared. This means that the smaller particles might have a smaller mass, but they are moving with much greater velocity, so they have greater kinetic energy.

Experiment kinmodel_07: The development of empirical frequency vs. speed and frequency vs. energy histograms (more about order vs. disorder, with statistical order emerging from the disordered system)

The frequency vs. speed, frequency vs. squart root of energy, and frequency vs. energy histograms (it is left up to the student to determine which is which) are normalized to have a consistent total area. These distributions develop over time, eventually reaching a smooth distribution analogous to the Maxwell-Boltzmann distribution. This development occurs much more quickly if the settings are customized to encourage a maximal number of collisions.

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Very good. However if we measured everything that happens onscreen for, say, 20 minutes, we would find that the average kinetic energy of the blue particles is equal to that of the red. A particle with 10 times the mass will have 1 / sqrt(10), which is close to 1/3, of the average velocity.

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Experiment kinmodel_08: Images of 2-dimensional collisions

It is easy to customize the settings to obtain two large relatively slow particles. Any student who has watched air hockey pucks or billiard balls colliding will recognize the validity of the simulation.

If the particles leave 'tracks' then a 'snapshot' in which a single collision between the particles occurs will provide data sufficient to validate conservation of momentum.

Customized settings

The following parameters can be controlled by electing to customize settings:

The number of particles (default setting is 30, maximum is 1000, which shouldn't be much of a restriction in the near future of PC's).

The number of different particle types (more than 9 different particle types is not recommended because of restricted number of colors).

The speed factor that determines how fast the particles move across the screen. If the speed factor is too great, particles may occasionally (or frequently, depending on how great) miss collisions. This is not a big problem unless data is being taken that assumes no 'misses'.

The radius of a particle (default radius is 1% the width of the square viewing area).

The proximity of the centers of the particles within which collision will occur (default is 5 particle radii). A greater value here will result in more collisions, other parameters being equal.

The minimum and maximum speeds defining a speed 'window'. Any particle whose speed is in this 'window' will be colored bright yellow. This range of speeds will be indicated by a yellow rectangle on one of the graphs.

Whether all the particles leave 'tracks' or not. The last two particles usually leave 'tracks'.

The number of iterations before the screen is cleared and the various graphs are updated. An iteration consists of the calculation and display of the position of every particle. A fairly small number allows the viewer to observe the evolution of the graphs, while a somewhat greater number permits observation of a significant number and variety of particle 'tracks'. If the number is too great the particle 'tracks' will be obscured.

The last two particles specified will have velocities indicated onscreen; the last of these particles will be sampled to obtain the velocity distribution shown at the right of the screen.

'Research' questions

For which particle speeds is the time between collisions likely to be greatest, and for which will it be least?

Initial particle speeds are uniformly distributed. After a short time a specific nonuniform distribution of speeds takes over. How long does it take before the contribution of the initial uniform distribution to the graphs and histograms displayed on the screen become indistinguishable? How will the shape of the graph differ from the ideal distribution during the transition?

For the default settings, what is the 'peak' particle energy? What is the 'peak' velocity of the sampled particle?

A narrower speed range near the peak of the speed distribution can result in more instances of 'yellow' particles than a wider speed range away from the peak. At each possible integer speed v, it is possible to define a speed range (v0, vf) with v at the midpoint of that range, such that the average number of 'yellow' particles will be the same as for the 'unit' range around the peak of the distribution. The 'unit' range is a velocity range of width 1 unit centered at the 'peak' velocity.

What does it take to get a massive molecule surrounded by low-mass particles moving fast?

Does the presence of an even more massive particle give a medium-mass particle, surrounded by a greater number of low-mass particles, an advantage in achieving greater speeds? Does the presence of a more massive particle affect the energy distribution of the medium-mass particle?

At an advanced level: Derive Maxwell-Boltzmann distribution in 2 dimensions and compare the the empirical distribution.

More information on this model.

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Good work. Check out my notes.

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