# Connected Chemistry 2 Changing Pressure

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## WHAT IS IT?

This model explores the relationship between the number of gas particles in a fixed-volume container -- a bike tire -- and the pressure of the gas in that container. This model is part of the "Connected Chemistry" curriculum http://ccl.northwestern.edu/curriculum/ConnectedChemistry/ which explore the behavior of gases.

Most of the models in the Connected Chemistry curriculum use the same basic rules for simulating the behavior of gases. Each model highlights different features of how gas behavior is related to gas particle behavior.

In all of the models, gas particles are assumed to move and to collide, both with each other and with objects such as walls.

In this model, the gas container (a bike tire represented by a yellow box) has a fixed volume. The user can set the initial number of gas particles, and can also "pump" additional particles into the box through a valve on the left wall.

This model helps students study the representations of gas pressure in the model and the dynamics of the gas particles that lead to increases and decreases in pressure. In this model, students can also look at the relationship between number of particles and pressure. When the particles hit the walls they change their color temporarily. These models have been adapted from the model GasLab Pressure Box.

## HOW IT WORKS

Particles are modeled as perfectly elastic with no energy except their kinetic energy, due to their motion. Collisions between particles are elastic.

The exact way two particles collide is as follows:

- Two turtles "collide" when they find themselves on the same patch.
- A random axis is chosen, as if they are two balls that hit each other and this axis is the line connecting their centers.
- They exchange momentum and energy along that axis, according to the conservation of momentum and energy. This calculation is done in the center of mass system.
- Each turtle is assigned its new velocity, energy, and heading.
- If a turtle finds itself on or very close to a wall of the container, it "bounces," reflecting its direction but keeping its speed.

## HOW TO USE IT

Buttons: SETUP - sets up the initial conditions set on the sliders. GO/STOP - runs and stops the model. ADD-PARTICLES - "pumps" additional particles into the tire while the simulation is running.

Sliders: INITIAL-NUMBER - sets the number of gas particles in the box when the simulation starts. NUMBER-TO-ADD - the number of gas particles released into the box when the ADD-PARTICLES button is pressed.

Monitors: CLOCK - number of clock cycles that GO has run. NUMBER - the number of particles in the box. PRESSURE - the total pressure in the box.

Plots: PRESSURE VS TIME - plots the pressure in the box over time. NUMBER VS TIME - plots the number of particles in the box over time.

Initially, the particles are not moving. Therefore the initial pressure is zero. When the particles start moving, they repeatedly collide, exchange energy and head off in new directions, and the speeds are dispersed -- some particles get faster, some get slower. When they hit the wall they change their heading, but not their speed.

- Adjust the INITIAL-NUMBER slider.
- Press the SETUP button
- Press GO/STOP and observe what happens.
- Adjust the NUMBER-TO-ADD slider.
- Press the ADD PARTICLES button.
- Observe the relationship between the Number vs. Time graph and Pressure vs. Time.

## THINGS TO NOTICE

Can you relate what you can see happening to the particles in the box with changes in pressure?

Why does the pressure change over time, even when the number of particles is the same? How long does it take for the pressure to stabilize?

What happens to the wall hits per particle when particles are added to the box?

In what ways is this model an incorrect idealization of the real world?

What is the relationship between particle number and pressure? Is it reciprocal, linear, quadratic, or exponential?

## THINGS TO TRY

Try different settings, especially the extremes. Are the particles behaving in a similar way? How does this affect the pressure?

You can pen-down a particle through the command center or by using the turtle menus. What do you notice about a particle's path when there more and fewer particles in the box?

How can you make the pressure monitor read 0?

Why is there a delay between when particles are added (using ADD PARTICLES) and when the pressure goes up?

Can you make the pressure graph smooth? Can you do it in more than one way?

Sometimes, when going up in an elevator, airplane or up a mountain we feel a 'popping' sensation in our ears. This is associated with changes in pressure. Can you relate between this model and these changes in pressure? Are the temperature and volume constant in this situation?

Pressure waves (expanding wavefronts of gas particles) are used to describe other phenomena in nature. What changes in the model could be made to represent pressure waves from explosions, sounds, breaking the sound barrier, etc...

## EXTENDING THE MODEL

Add a histogram showing all the particles speeds throughout the run of the model.

Add a switch that would continuously add particles to the box?

Make a hole in the box and observe what happens to the pressure.

## NETLOGO FEATURES

The Connected Chemistry models include invisible dark particles (the "dark-particles" breed), which only interact with each other and the walls of the yellow box. The inclusion of dark particles ensures that the speed of simulation remains constant, regardless of the number of particles visible in the simulation.

For example, if a model is limited to a maximum of 400 particles, then when there are 10 visible particles, there are 390 dark particles and when there are 400 visible particles, there are 0 dark particles. The total number of particles in both cases remains 400, and the computational load of calculating what each of these particles does (collides, bounces, etc...) is close to the same. Without dark particles, it would seem that small numbers of particles are faster than large numbers of particles -- when in reality, it is simply a reflection of the computational load. Such behavior would encourage student misconceptions related to particle behavior.

## RELATED MODELS

See GasLab Models See other Connected Chemistry models.

## CREDITS AND REFERENCES

This model is part of the Connected Chemistry curriculum. See http://ccl.northwestern.edu/curriculum/chemistry.

We would like to thank Sharona Levy and Michael Novak for their substantial contributions to this model.

## HOW TO CITE

If you mention this model in a publication, we ask that you include these citations for the model itself and for the NetLogo software:

- Wilensky, U. (2004). NetLogo Connected Chemistry 2 Changing Pressure model. http://ccl.northwestern.edu/netlogo/models/ConnectedChemistry2ChangingPressure. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL.
- Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL.

To cite the Connected Chemistry curriculum as a whole, please use: Wilensky, U., Levy, S. T., & Novak, M. (2004). Connected Chemistry curriculum. http://ccl.northwestern.edu/curriculum/chemistry. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL.

## COPYRIGHT AND LICENSE

Copyright 2004 Uri Wilensky.

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu.

This model was created as part of the projects: PARTICIPATORY SIMULATIONS: NETWORK-BASED DESIGN FOR SYSTEMS LEARNING IN CLASSROOMS and/or INTEGRATED SIMULATION AND MODELING ENVIRONMENT. The project gratefully acknowledges the support of the National Science Foundation (REPP & ROLE programs) -- grant numbers REC #9814682 and REC-0126227.

## Comments and Questions

globals [ tick-delta ;; how much we advance the tick counter this time through max-tick-delta ;; the largest tick-delta is allowed to be box-edge ;; distance of box edge from axes delta-horizontal-surface ;; the size of the wall surfaces that run horizontally - the top and bottom of the box delta-vertical-surface ;; the size of the wall surfaces that run vertically - the left and right of the box total-particle-number labels? ;; show ID of particles collisions? instant-pressure ;; the pressure at this tick or instant in time pressure-history ;; a history of the three instant-pressure values pressure ;; the pressure average of the pressure-history (for curve smoothing in the pressure plots) wall-hits-per-particle ;; average number of wall hits per particle maxparticles particles-to-add ;; keeps track of particles to add after pressing ADD PARTICLES new-particles ;; agentset of particles added via add-particles-middle show-dark? ;; hides or shows the dark particles in the simulation. ;; see NetLogo features Info tab for full explanation of ;; what dark particles are and why they are used. ] breed [ particles particle ] breed [ flashes flash ] ;; a breed which is used to mark the spot where a particle just hit the wall flashes-own [birthday] ;; flashes only last for a short period and then disappear. ;; their birthday helps us keep track of when they were created and ;; when we need to remove them. particles-own [ speed mass ;; particle info wall-hits ;; # of wall hits during this clock cycle momentum-difference ;; used to calculate pressure from wall hits momentum-instant ;; used to calculate pressure last-collision ;; keeps track of last particle this particle collided with darkparticle? ;; is this particle visible and interacting with visible particles? ] to setup ca reset-ticks set show-dark? false set-default-shape particles "circle" set-default-shape flashes "square" set maxparticles 400 set tick-delta 0 ;; starting this at zero means that no particles will move until we've ;; calculated vsplit, which we won't even try to do until there are some ;; particles. set pressure-history [0 0 0] ;; plotted pressure will be averaged over the past 3 entries set pressure 0 set particles-to-add 0 set labels? false ;; box has constant size... set box-edge (max-pxcor - 1) ;;; the delta of the horizontal or vertical surface of ;;; the inside of the box must exclude the two patches ;; that are the where the perpendicular walls join it, ;;; but must also add in the axes as an additional patch ;;; example: a box with an box-edge of 10, is drawn with ;;; 19 patches of wall space on the inside of the box set delta-horizontal-surface ( 2 * (box-edge - 1) + 1) set delta-vertical-surface ( 2 * (box-edge - 1) + 1) draw-box set collisions? true make-particles maxparticles if labels? [turn-labels-on] reset-ticks end to go let old-clock 0 ask particles [ bounce ] ask particles [ move ] if collisions? [ ask particles with [darkparticle? = false] [ check-for-collision-regular ] ask particles with [darkparticle? = true] [ check-for-collision-dark ] ] set old-clock ticks tick-advance tick-delta calculate-instant-pressure if floor ticks > floor (ticks - tick-delta) [ ifelse any? particles with [darkparticle? = false] [ set wall-hits-per-particle mean [wall-hits] of particles with [darkparticle? = false] ] [ set wall-hits-per-particle 0 ] ask particles [ set wall-hits 0 ] calculate-pressure update-plots ] calculate-tick-delta ask flashes with [ticks - birthday > 0.4] [ die ] ifelse labels? [turn-labels-on] [ask turtles [ set label ""]] if (show-wall-hits? = false) [ ask flashes [ die ]] display end to calculate-tick-delta ifelse any? particles with [speed > 0] [ set tick-delta 1 / (ceiling max [speed] of particles) ] [ set tick-delta 1 ] end ;;; Pressure is defined as the force per unit area. In this context, ;;; that means the total momentum per unit time transferred to the walls ;;; by particle hits, divided by the surface area of the walls. (Here ;;; we're in a two dimensional world, so the "surface area" of the walls ;;; is just their delta.) Each wall contributes a different amount ;;; to the total pressure in the box, based on the number of collisions, the ;;; direction of each collision, and the delta of the wall. Conservation of momentum ;;; in hits ensures that the difference in momentum for the particles is equal to and ;;; opposite to that for the wall. The force on each wall is the rate of change in ;;; momentum imparted to the wall, or the sum of change in momentum for each particle: ;;; F = SUM [d(mv)/dt] = SUM [m(dv/dt)] = SUM [ ma ], in a direction perpendicular to ;;; the wall surface. The pressure (P) on a given wall is the force (F) applied to that ;;; wall over its surface area. The total pressure in the box is sum of each wall's ;;; pressure contribution. to calculate-instant-pressure ;; by summing the momentum change for each particle, ;; the wall's total momentum change is calculated set instant-pressure 15 * sum [momentum-instant] of particles with [darkparticle? = false] output-print precision instant-pressure 1 ask particles [ set momentum-instant 0 ] ;; once the contribution to momentum has been calculated ;; this value is reset to zero till the next wall hit end to calculate-pressure ;; by summing the momentum change for each particle, ;; the wall's total momentum change is calculated set pressure 15 * sum [momentum-difference] of particles with [darkparticle? = false] set pressure-history lput pressure but-first pressure-history ask particles [ set momentum-difference 0 ] ;; once the contribution to momentum has been calculated ;; this value is reset to zero till the next wall hit end to bounce ;; particle procedure let new-px 0 let new-py 0 ;; if we're not about to hit a wall (yellow patch), or if we're already on a ;; wall, we don't need to do any further checks if shade-of? yellow pcolor or not shade-of? yellow [pcolor] of patch-at dx dy [ stop ] ;; get the coordinates of the patch we'll be on if we go forward 1 set new-px round (xcor + dx) set new-py round (ycor + dy) ;; if hitting left or right wall, reflect heading around x axis if (abs new-px = box-edge) [ set heading (- heading) set wall-hits wall-hits + 1 ;; if the particle is hitting a vertical wall, only the horizontal component of the speed ;; vector can change. The change in velocity for this component is 2 * the speed of the particle, ;; due to the reversing of direction of travel from the collision with the wall set momentum-instant (abs (dx * 2 * mass * speed) / delta-vertical-surface) set momentum-difference momentum-difference + momentum-instant ] ;; if hitting top or bottom wall, reflect heading around y axis if (abs new-py = box-edge) [ set heading (180 - heading) set wall-hits wall-hits + 1 ;; if the particle is hitting a horizontal wall, only the vertical component of the speed ;; vector can change. The change in velocity for this component is 2 * the speed of the particle, ;; due to the reversing of direction of travel from the collision with the wall set momentum-instant (abs (dy * 2 * mass * speed) / delta-horizontal-surface) set momentum-difference momentum-difference + momentum-instant ] if (darkparticle? = false) [ ask patch new-px new-py [ sprout 1 [ set breed flashes set birthday ticks set color yellow - 3 ] ] ] end to move ;; particle procedure if patch-ahead (speed * tick-delta) != patch-here [ set last-collision nobody ] jump (speed * tick-delta) end to check-for-collision-regular ;; particle procedure let candidate 0 ;; Here we impose a rule that collisions only take place when there ;; are exactly two particles per patch. We do this because when the ;; student introduces new particles from the side, we want them to ;; form a uniform wavefront. ;; ;; Why do we want a uniform wavefront? Because it is actually more ;; realistic. (And also because the curriculum uses the uniform ;; wavefront to help teach the relationship between particle collisions, ;; wall hits, and pressure.) ;; ;; Why is it realistic to assume a uniform wavefront? Because in reality, ;; whether a collision takes place would depend on the actual headings ;; of the particles, not merely on their proximity. Since the particles ;; in the wavefront have identical speeds and near-identical headings, ;; in reality they would not collide. So even though the two-particles ;; rule is not itself realistic, it produces a realistic result. Also, ;; unless the number of particles is extremely large, it is very rare ;; for three or more particles to land on the same patch (for example, ;; with 400 particles it happens less than 1% of the time). So imposing ;; this additional rule should have only a negligible effect on the ;; aggregate behavior of the system. ;; ;; Why does this rule produce a uniform wavefront? The particles all ;; start out on the same patch, which means that without the only-two ;; rule, they would all start colliding with each other immediately, ;; resulting in much random variation of speeds and headings. With ;; the only-two rule, they are prevented from colliding with each other ;; until they have spread out a lot. (And in fact, if you observe ;; the wavefront closely, you will see that it is not completely smooth, ;; because some collisions eventually do start occurring when it thins out while fanning.) if count other particles-here with [darkparticle? = false] = 1 [ ;; the following conditions are imposed on collision candidates: ;; 1. they must have a lower who number than my own, because collision ;; code is asymmetrical: it must always happen from the point of view ;; of just one particle. ;; 2. they must not be the same particle that we last collided with on ;; this patch, so that we have a chance to leave the patch after we've ;; collided with someone. set candidate one-of other particles-here with [who < [who] of myself and myself != last-collision and darkparticle? = false] ;; we also only collide if one of us has non-zero speed. It's useless ;; (and incorrect, actually) for two particles with zero speed to collide. if (candidate != nobody) and (speed > 0 or [speed] of candidate > 0) [ collide-with candidate set last-collision candidate ask candidate [ set last-collision myself ] ] ] end to check-for-collision-dark ;; particle procedure let candidate 0 ;; Here we impose a rule that collisions only take place when there ;; are exactly two particles per patch. We do this because when the ;; student introduces new particles from the side, we want them to ;; form a uniform wavefront. ;; ;; Why do we want a uniform wavefront? Because it is actually more ;; realistic. (And also because the curriculum uses the uniform ;; wavefront to help teach the relationship between particle collisions, ;; wall hits, and pressure.) ;; ;; Why is it realistic to assume a uniform wavefront? Because in reality, ;; whether a collision takes place would depend on the actual headings ;; of the particles, not merely on their proximity. Since the particles ;; in the wavefront have identical speeds and near-identical headings, ;; in reality they would not collide. So even though the two-particles ;; rule is not itself realistic, it produces a realistic result. Also, ;; unless the number of particles is extremely large, it is very rare ;; for three or more particles to land on the same patch (for example, ;; with 400 particles it happens less than 1% of the time). So imposing ;; this additional rule should have only a negligible effect on the ;; aggregate behavior of the system. ;; ;; Why does this rule produce a uniform wavefront? The particles all ;; start out on the same patch, which means that without the only-two ;; rule, they would all start colliding with each other immediately, ;; resulting in much random variation of speeds and headings. With ;; the only-two rule, they are prevented from colliding with each other ;; until they have spread out a lot. (And in fact, if you observe ;; the wavefront closely, you will see that it is not completely smooth, ;; because some collisions eventually do start occurring when it thins out while fanning.) if count other particles-here with [darkparticle? = true] = 1 [ ;; the following conditions are imposed on collision candidates: ;; 1. they must have a lower who number than my own, because collision ;; code is asymmetrical: it must always happen from the point of view ;; of just one particle. ;; 2. they must not be the same particle that we last collided with on ;; this patch, so that we have a chance to leave the patch after we've ;; collided with someone. set candidate one-of other particles-here with [who < [who] of myself and myself != last-collision and darkparticle? = true] ;; we also only collide if one of us has non-zero speed. It's useless ;; (and incorrect, actually) for two particles with zero speed to collide. if (candidate != nobody) and (speed > 0 or [speed] of candidate > 0) [ collide-with candidate set last-collision candidate ask candidate [ set last-collision myself ] ] ] end ;; implements a collision with another particle. ;; ;; THIS IS THE HEART OF THE PARTICLE SIMULATION, AND YOU ARE STRONGLY ADVISED ;; NOT TO CHANGE IT UNLESS YOU REALLY UNDERSTAND WHAT YOU'RE DOING! ;; ;; The two particles colliding are self and other-particle, and while the ;; collision is performed from the point of view of self, both particles are ;; modified to reflect its effects. This is somewhat complicated, so I'll ;; give a general outline here: ;; 1. Do initial setup, and determine the heading between particle centers ;; (call it theta). ;; 2. Convert the representation of the velocity of each particle from ;; speed/heading to a theta-based vector whose first component is the ;; particle's speed along theta, and whose second component is the speed ;; perpendicular to theta. ;; 3. Modify the velocity vectors to reflect the effects of the collision. ;; This involves: ;; a. computing the velocity of the center of mass of the whole system ;; along direction theta ;; b. updating the along-theta components of the two velocity vectors. ;; 4. Convert from the theta-based vector representation of velocity back to ;; the usual speed/heading representation for each particle. ;; 5. Perform final cleanup and update derived quantities. to collide-with [ other-particle ] ;; particle procedure let mass2 0 let speed2 0 let heading2 0 let theta 0 let v1t 0 let v1l 0 let v2t 0 let v2l 0 let vcm 0 ;;; PHASE 1: initial setup ;; for convenience, grab some quantities from other-particle set mass2 [mass] of other-particle set speed2 [speed] of other-particle set heading2 [heading] of other-particle ;; since particles are modeled as zero-size points, theta isn't meaningfully ;; defined. we can assign it randomly without affecting the model's outcome. set theta (random-float 360) ;;; PHASE 2: convert velocities to theta-based vector representation ;; now convert my velocity from speed/heading representation to components ;; along theta and perpendicular to theta set v1t (speed * cos (theta - heading)) set v1l (speed * sin (theta - heading)) ;; do the same for other-particle set v2t (speed2 * cos (theta - heading2)) set v2l (speed2 * sin (theta - heading2)) ;;; PHASE 3: manipulate vectors to implement collision ;; compute the velocity of the system's center of mass along theta set vcm (((mass * v1t) + (mass2 * v2t)) / (mass + mass2) ) ;; now compute the new velocity for each particle along direction theta. ;; velocity perpendicular to theta is unaffected by a collision along theta, ;; so the next two lines actually implement the collision itself, in the ;; sense that the effects of the collision are exactly the following changes ;; in particle velocity. set v1t (2 * vcm - v1t) set v2t (2 * vcm - v2t) ;;; PHASE 4: convert back to normal speed/heading ;; now convert my velocity vector into my new speed and heading set speed sqrt ((v1t * v1t) + (v1l * v1l)) ;; if the magnitude of the velocity vector is 0, atan is undefined. but ;; speed will be 0, so heading is irrelevant anyway. therefore, in that ;; case we'll just leave it unmodified. if v1l != 0 or v1t != 0 [ set heading (theta - (atan v1l v1t)) ] ;; and do the same for other-particle ask other-particle [ set speed sqrt ((v2t ^ 2) + (v2l ^ 2)) if v2l != 0 or v2t != 0 [ set heading (theta - (atan v2l v2t)) ] ] ;; PHASE 5: final updates ;; now recolor, since color is based on quantities that may have changed end ;;; ;;; drawing procedures ;;; ;; draws the box to draw-box ask patches with [ ((abs pxcor = box-edge) and (abs pycor <= box-edge)) or ((abs pycor = box-edge) and (abs pxcor <= box-edge)) ] [ set pcolor yellow ] ask patches with [pycor = 0 and pxcor < (1 - box-edge)] [ set pcolor yellow - 5 ;; trick the bounce code so particles don't go into the inlet ask patch-at 0 1 [ set pcolor yellow ] ask patch-at 0 -1 [ set pcolor yellow ] ] end ;;; ;;; particle setup and addition procedures ;;; ;; creates initial particles to make-particles [number] create-particles number [ setup-particle set speed random-float 20 random-position recolornone ] set total-particle-number initial-number ask particles with [who < initial-number] [ set shape "circle" set darkparticle? false ] calculate-tick-delta end ;; adds particles from the left (the valve from a pump) to add-particles-side set particles-to-add number-to-add ifelse ((particles-to-add + total-particle-number ) > maxparticles) [user-message (word "The maximum number of particles allowed in this model is " maxparticles ". You can not add " number-to-add " more particles to the " count particles with [darkparticle? = false] " you already have in the model")] [ if particles-to-add > 0 [ ask particles with [who < (total-particle-number + particles-to-add) and who >= total-particle-number] [ set darkparticle? false set shape "circle" setxy (- box-edge) 0 set heading 90 ;; east rt 45 - random-float 90 set speed 10 recolornone ] set total-particle-number (total-particle-number + particles-to-add) set particles-to-add 0 calculate-tick-delta ] ] end to setup-particle ;; particle procedure set speed 10 set mass 1.0 set last-collision nobody set wall-hits 0 set momentum-difference 0 set darkparticle? true ifelse show-dark? [set shape "default"] [set shape "nothing" set color green] end ;; place particle at random location inside the box. to random-position ;; particle procedure setxy ((1 - box-edge) + random-float ((2 * box-edge) - 2)) ((1 - box-edge) + random-float ((2 * box-edge) - 2)) end ;;; ;;; visualization procedures ;;; to recolornone set color green - 1 end to turn-labels-on ;; [ask turtles [ set label who set label-color orange + 3 ]] end ; Copyright 2004 Uri Wilensky. ; See Info tab for full copyright and license.

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## Attached files

File | Type | Description | Last updated | |
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Connected Chemistry 2 Changing Pressure.png | preview | Preview for 'Connected Chemistry 2 Changing Pressure' | almost 7 years ago, by Uri Wilensky | Download |

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