# GasLab Isothermal Piston

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## WHAT IS IT?

This model is one in a series of GasLab models. They use the same basic rules for simulating the behavior of gases. Each model integrates different features in order to highlight different aspects of gas behavior.

The basic principle of the models is that gas particles are assumed to have two elementary actions: they move and they collide --- either with other particles or with any other objects such as walls.

This model simulates the behavior of gas particles in a piston, or a container with a changing volume. The volume in which the gas is contained can be changed by moving the piston in and out. "Isothermal" means that the temperature of the gas is not changed by moving the piston.

This model is part of the Connected Mathematics "Making Sense of Complex Phenomena" Modeling Project.

## HOW IT WORKS

The particles are modeled as hard balls with no internal energy except that which is due to their motion. Collisions between particles are elastic. Particles are colored according to speed --- blue for slow, green for medium, and red for high speeds.

Coloring of the particles is with respect to one speed (10). Particles with a speed less than 5 are blue, ones that are more than 15 are red, while all in those in-between are green.

Particles behave according to the following rules:

- A particle moves in a straight line without changing its speed, unless it collides with another particle or bounces off the wall.
- Two particles "collide" if they find themselves on the same patch (NetLogo's View is composed of a grid of small squares called patches).
- 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" -- that is, reflects its direction and keeps its same speed.

Pressure is calculated as the force per unit area (or length in this two-dimensional model). Pressure is calculated by adding up the momentum transferred to the walls of the box by the particles when they bounce off and divided by the length of the wall, which they hit.

## HOW TO USE IT

Initial settings:

- NUMBER-OF-PARTICLES: number of particles
- INIT-PARTICLE-SPEED: initial speed of the particles
- PARTICLE-MASS: initial mass of the molecules
- BOX-WIDTH: width of the container
- BOX-HEIGHT: height of the container

Other settings:

- COLLIDE?: Turns collisions between particles on and off. It can be changed in the middle of the run.

The SETUP button will set the initial conditions.

The GO button will run the simulation.

Pushing the MOVE-PISTON button allows you to reposition the piston by clicking on the view with the mouse, hence changing the volume. When this button is pressed, the model stops. Once the reposition is done, push the GO button to continue.

The intention in this model is for the user to quickly pull the piston back thus simulating quickly removing a plate. This means no particles collide with the piston as it is removed. However, we have left in code that allows the user to push the piston in and compress the gas. In this model, though, the collisions of the piston with the particles are ignored. Note that there's a physical impossibility in the model here: in real life if you moved the piston in you would do work on the gas by compressing it, and its temperature would increase. In this model the energy and temperature are constant no matter how you manipulate the piston, hence the name "isothermal". Nonetheless, the basic relationship between volume and pressure is correctly demonstrated here.

The physically accurate version of piston compression is shown in the "Adiabatic Piston" model.

Monitors:

- PISTON POSITION: position of the piston with respect to the x-axis
- VOLUME: volume (or area) of the piston
- PRESSURE
- AVERAGE SPEED: average speed of the particles
- AVERAGE ENERGY: average energy of the particles, calculated as m*(v^2)/2.

Plots:

- PRESSURE: pressure in the piston over time.
- VOLUME: volume of the piston vs time.
- WALL HITS PER PARTICLE: the number of wall hits averaged for the particles during each time unit
- SPEED HISTOGRAM: particles' speed distribution
- ENERGY HISTOGRAM: distribution of energies of all the particles, calculated as m*(v^2)/2.

## THINGS TO NOTICE

How does the pressure change as you change the volume of the box by moving the piston? Compare the two plots of volume and pressure.

Measure changes in pressure and volume. Is there a clear quantitative relationship? Boyle's Law describes the relationship between pressure and volume, when all else is kept constant.

How can the relationship between volume and pressure be explained in terms of the wall hits? How does it relate to collisions among molecules?

What shapes do the energy and velocity histograms reach after a while? Why aren't they the same? Do the pressure and volume affect these shapes? How does changing the particles' mass or speed affect them?

Change different kinds of settings and observe the number of wall hits per particle. What causes this number to change? What changes do not affect this number? Can you connect these relationships with those between the number of particles and pressure? Volume and pressure?

## THINGS TO TRY

How would you calculate pressure? How does this code do it?

Change the number, mass, and initial velocity of the molecules. Does this affect the pressure? Why? Do the results make intuitive sense? Look at the extremes: very few or very many molecules, high or low volumes.

Figure out how many molecules there *really* are in a box this size --- say a 10-cm cube. Look up or calculate the *real* mass and speed of a typical molecule. When you compare those numbers to the ones in the model, are you surprised this model works as well as it does?

Observe the number of wall hits per particle with and without collisions. Does this number change? Why?

If you change the number of particles in the piston: will the pressure change? will the number of wall hits change? Why?

## EXTENDING THE MODEL

Are there other ways one might calculate pressure?

When the piston is moved out, the gas is not evenly distributed for a while. What's the pressure during this time? Does this ever happen in the real world? What does pressure mean when it's not the same throughout a gas?

## NETLOGO FEATURES

Notice how collisions are detected by the turtles and how the code guarantees that the same two particles do not collide twice. What happens if we let the patches detect them?

## CREDITS AND REFERENCES

This model was developed as part of the GasLab curriculum (http://ccl.northwestern.edu/curriculum/gaslab/) and has also been incorporated into the Connected Chemistry curriculum (http://ccl.northwestern.edu/curriculum/ConnectedChemistry/)

## 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. (1997). NetLogo GasLab Isothermal Piston model. http://ccl.northwestern.edu/netlogo/models/GasLabIsothermalPiston. 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.

## COPYRIGHT AND LICENSE

Copyright 1997 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 project: CONNECTED MATHEMATICS: MAKING SENSE OF COMPLEX PHENOMENA THROUGH BUILDING OBJECT-BASED PARALLEL MODELS (OBPML). The project gratefully acknowledges the support of the National Science Foundation (Applications of Advanced Technologies Program) -- grant numbers RED #9552950 and REC #9632612.

This model was developed at the MIT Media Lab using CM StarLogo. See Wilensky, U. (1993). Thesis - Connected Mathematics: Building Concrete Relationships with Mathematical Knowledge. Adapted to StarLogoT, 1997, as part of the Connected Mathematics Project. Adapted to NetLogo, 2002, as part of the Participatory Simulations Project.

This model was converted to NetLogo 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. Converted from StarLogoT to NetLogo, 2002.

## 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 raw-width raw-height ;; box size variables pressure pressure-history ;; lists previous pressure values, so that averaging can take place in plotting wall-hits-per-particle ;; average number of wall hits per particle length-horizontal-surface ;; the size of the wall surfaces that run horizontally - the top and bottom of the box length-vertical-surface ;; the size of the wall surfaces that run vertically - the left and right of the box init-avg-speed init-avg-energy ;; initial averages avg-speed avg-energy ;; current averages fast medium slow ;; current counts piston-position ;; xcor of piston wall run-go? ;; flag of whether or not its safe for go to run volume ] breed [ particles particle ] breed [ flashes flash ] flashes-own [birthday] particles-own [ speed mass energy ;; particle info wall-hits ;; # of wall hits during this cycle ("big tick") momentum-difference ;; used to calculate pressure from wall hits last-collision ;; sets identity of particle which is collided with, prevents colliding twice with the same particle if they remain on the same patch after moving away ] to setup clear-all set-default-shape particles "circle" set-default-shape flashes "plane" set run-go? true set max-tick-delta 0.1073 set raw-width round (0.01 * box-width * max-pxcor) set raw-height round (0.01 * box-height * max-pycor) set piston-position 0 ;;; the length 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 a edge of 10, is drawn with ;;; 19 patches of wall space on the inside of the box set length-horizontal-surface ( 2 * (raw-height - 1) + 1) set length-vertical-surface raw-width + piston-position set volume (length-horizontal-surface * length-vertical-surface * 1) ;;depth of 1 make-box draw-piston make-particles set pressure-history [0 0 0] ;; plotted pressure will be averaged over the past 3 entries update-variables set init-avg-speed avg-speed set init-avg-energy avg-energy reset-ticks end to update-variables set medium count particles with [color = green] set slow count particles with [color = blue] set fast count particles with [color = red] set avg-speed mean [speed] of particles set avg-energy mean [energy] of particles end to go if not run-go? [stop] ;; when the piston is moved, the model run is stopped ask particles [ bounce ] ask particles [ move ] ask particles [ if collide? [check-for-collision] ] tick-advance tick-delta if floor ticks > floor (ticks - tick-delta) [ ifelse any? particles [ set wall-hits-per-particle mean [wall-hits] of particles ] [ set wall-hits-per-particle 0 ] ask particles [ set wall-hits 0 ] calculate-pressure update-variables update-plots ] calculate-tick-delta ask flashes with [ticks - birthday > 0.4] [ die ] display end to calculate-tick-delta ;; tick-delta is calculated in such way that even the fastest ;; particle will jump at most 1 patch length in a tick. As ;; particles jump (speed * tick-delta) at every tick, making ;; tick length the inverse of the speed of the fastest particle ;; (1/max speed) assures that. Having each particle advance at most ;; one patch-length is necessary for it not to "jump over" a wall, ;; the piston or another particle. ifelse any? particles with [speed > 0] [ set tick-delta min list (1 / (ceiling max [speed] of particles)) max-tick-delta ] [ set tick-delta max-tick-delta ] 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 length.) 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 length 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-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 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 ;; get the coordinates of the patch we'll be on if we go forward 1 let new-patch patch-ahead 1 let new-px [pxcor] of new-patch let new-py [pycor] of new-patch ;; if we're not about to hit a wall, we don't need to do any further checks if ([pcolor] of new-patch != yellow and [pcolor] of new-patch != orange) [ stop ] ;; if hitting left or right wall, reflect heading around x axis if (abs new-px = raw-width or new-px = piston-position) [ 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-difference momentum-difference + (abs (dx * 2 * mass * speed) / length-vertical-surface) ] ;; if hitting top or bottom wall, reflect heading around y axis if (abs new-py = raw-height) [ 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-difference momentum-difference + (abs (dy * 2 * mass * speed) / length-horizontal-surface) ] ;; every time a particle hits the wall, it produces a short-living "flash" so assist in visualization ask patch new-px new-py [ sprout-flashes 1 [ set color pcolor - 2 set birthday ticks set heading 0 ] ] 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 ;; particle procedure ;; Here we impose a rule that collisions only take place when there ;; are exactly two particles per patch. if count other particles-here = 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. let candidate one-of other particles-here with [who < [who] of myself and myself != last-collision] ;; 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 ;;; PHASE 1: initial setup ;; for convenience, grab some quantities from other-particle let mass2 [mass] of other-particle let speed2 [speed] of other-particle let 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. let 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 let v1t (speed * cos (theta - heading)) let v1l (speed * sin (theta - heading)) ;; do the same for other-particle let v2t (speed2 * cos (theta - heading2)) let v2l (speed2 * sin (theta - heading2)) ;;; PHASE 3: manipulate vectors to implement collision ;; compute the velocity of the system's center of mass along theta let 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 ^ 2) + (v1l ^ 2)) set energy (0.5 * mass * speed ^ 2) ;; 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)) set energy (0.5 * mass * (speed ^ 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 recolor ask other-particle [ recolor ] end to recolor ;; particle procedure ifelse speed < (0.5 * 10) [ set color blue ] [ ifelse speed > (1.5 * 10) [ set color red ] [ set color green ] ] end ;;; ;;; drawing procedures ;;; ;; draws the box to make-box ask patches with [ ((abs pxcor = raw-width) and (abs pycor <= raw-height)) or ((abs pycor = raw-height) and (abs pxcor <= raw-width)) ] [ set pcolor yellow ] ;;color the left side of the wall gray: ask patches with [ ((pxcor = ( raw-width)) and (abs pycor <= raw-height))] [set pcolor gray] end ;; creates initial particles to make-particles create-particles number-of-particles [ setup-particle random-position recolor ] calculate-tick-delta end to setup-particle ;; particle procedure set speed init-particle-speed set mass particle-mass set energy (0.5 * mass * speed ^ 2) set last-collision nobody set wall-hits 0 set momentum-difference 0 end ;; place particle at random location inside the box ;; to the left of the piston to random-position ;; particle procedure setxy ((1 - raw-width) + random-float (raw-width + piston-position - 3)) ((1 - raw-height) + random-float (2 * raw-height - 2)) end ;; ------ Piston ---------- to move-piston set run-go? false ;;note: if user clicks too far to the left or right, nothing will happen if (mouse-down? and abs mouse-xcor < raw-width - 3 ) [ ifelse mouse-xcor >= piston-position [ piston-out ceiling (mouse-xcor - piston-position) ] [ piston-in (piston-position - mouse-xcor) ] set length-horizontal-surface ( 2 * (raw-width - 1) + 1) - (abs (piston-position - raw-width) - 1) set volume (length-horizontal-surface * length-vertical-surface * 1) ;;depth of 1 set run-go? true stop ] end ;; piston procedure to piston-out [dist] undraw-piston set piston-position round (piston-position + dist) draw-piston end ;; piston procedure to piston-in [dist] undraw-piston set piston-position round (piston-position - dist) ask particles with [xcor >= piston-position - 1] [ bounce-off-piston ] ask flashes with [xcor >= piston-position - 1] [ die ] draw-piston end ;; particle procedure to bounce-off-piston ifelse ((((2 * piston-position) - (xcor + 2)) < (1 - raw-width)) or (((2 * piston-position) - (xcor + 2)) > (piston-position - 2))) [ set xcor ((random (raw-width + piston-position - 2)) - (raw-width - 1)) ] [ set xcor ((2 * piston-position) - (xcor + 2)) ] end ;; piston procedure to draw-piston ask patches with [ ((pxcor = (round piston-position)) and ((abs pycor) < raw-height)) ] [ set pcolor orange ] ;; make sides of box that are to right right of the piston gray ask patches with [(pxcor > (round piston-position)) and (abs (pxcor) < raw-width) and ((abs pycor) = raw-height)] [set pcolor gray] end ;; piston procedure to undraw-piston ask patches with [ (pxcor = round piston-position) and ((abs pycor) < raw-height) ] [ set pcolor black ] ask patches with [(pxcor > (round piston-position)) and (abs (pxcor) < raw-width) and ((abs pycor) = raw-height)] [set pcolor yellow] ask flashes with [ (xcor = round piston-position) and ((abs ycor) < raw-height) ] [ die ] end ;; histogram procedure to draw-vert-line [ xval ] plotxy xval plot-y-min plot-pen-down plotxy xval plot-y-max plot-pen-up end ; Copyright 1997 Uri Wilensky. ; See Info tab for full copyright and license.

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## Attached files

File | Type | Description | Last updated | |
---|---|---|---|---|

GasLab Isothermal Piston.png | preview | Preview for 'GasLab Isothermal Piston' | almost 10 years ago, by Uri Wilensky | Download |

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