GasLab Moving Piston

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Uri_dolphin3 Uri Wilensky (Author)

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WHAT IS IT?

This model simulates the behavior of gas particles as the volume changes. In this model, the volume is slowly changing over time by a piston that is rising and falling. As the piston lowers, the volume of the box decreases and as the piston rises, the volume of the box increases. This systematic motion of the piston does no work on the particles inside the box. The piston only serves a mechanism to change the volume of the box.

The particles start with the same mass and speed upon the start of the simulation. The mass of the particles stays constant throughout the simulation, whereas, the speeds will change once particles start to collide. Particles are in constant motion colliding with other particles and the walls. All collisions are modeled as elastic collisions, in that the total kinetic energy before and after the collision is conserved. For example, when a fast moving particle collides with a slow moving particle, the fast moving particle will give some of its speed to the slow moving particle. Therefore, the fast moving particle will leave the collision moving slower then when it entered the collision. And the slow moving particle will speed up a bit. The speed in a particle to particle collision is still conserved. The collisions between a particle and a wall is modeled the same way. When the particles hit the wall they transfer momentum to the wall. After this transfer occurs, the particles then bounce off the wall with a different direction and speed. The system's pressure is calculated by averaging the number of collisions the particles have with the walls at each time step.

The Moving Piston model is one of a collection of GasLab models that use the same basic rules for expressing what happens when gas particles collide. Each model in this collection has different features to show the different aspects of the Gas Laws.

Multiple adaptations of this model can be found in the Chemistry folder of the Curricular Models section under the names Chem Volume 1 and 2. It is part of a suite of models used to teach students about the chemistry of the Gas Laws.

HOW IT WORKS

The particles are modeled as single particles, all with the same mass and initial velocity. Molecules are modeled as perfectly elastic particles with no internal energy except that which is due to their motion. Collisions with the box and between molecules are elastic. Particles are colored according to speed -- blue for slow, green for medium, and red for high speeds.

The exact way two particles collide is as follows:

  1. Two turtles "collide" if they find themselves on the same patch.
  2. A random axis is chosen, as if they were two billiard balls that hit and this axis was the line connecting their centers.
  3. They exchange momentum and energy along that axis, according to the conservation of momentum and energy. This calculation is done in the center mass system.
  4. Each turtle is assigned its new speed, energy and heading.
  5. 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.

HOW TO USE IT

Buttons

SETUP - puts in the initial conditions you have set with the sliders. Be sure to wait till the SETUP button stops before pushing GO.
GO - runs the code again and again. This is a "forever" button.

Sliders

BOX-HEIGHT - height of the container
BOX-WIDTH - width of the container
NUMBER - number of particles
PISTON-SPEED - rate of the piston
SCALE - number of clock cycles over which to average the pressure

Switch

HISTOGRAM? - turns histograms on or off

Plots

VOLUME - plots the volume over time
PRESSURE - plots the pressure over time
PRESSURE VS. VOLUME - plots pressure over volume
PRESSURE * VOLUME - plots the value of pressure * volume over time
TEMPERATURE - plots the average temperature
SPEED HISTOGRAM - illustrates the number of particles at their various speeds
ENERGY HISTOGRAM - illustrates the number of particles at their various energy levels

How to use it

Adjust the BOX-HEIGHT, BOX-WIDTH, NUMBER, and PISTON-SPEED variable before pressing SETUP. The SETUP button will set the initial conditions. The GO button will run the simulation.

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 down 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. Nonetheless, the basic relationship between volume and pressure is correctly demonstrated here.

THINGS TO NOTICE

How does the pressure change as the volume of the box changes? Compare the two plots of volume and pressure.

How does the pressure change as the shape of the box changes?

Measure changes in pressure and volume. Is there a clear quantitative relationship?

How can the relationship between pressure and volume be explained in terms of the collisions of molecules?

How does more particles change the relationship between pressure and volume?

What shapes do the energy and speed histograms reach after a while? Why aren't they the same? Do the pressure and volume affect these shapes?

THINGS TO TRY

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

Change the number, mass, and initial velocity of the particles. 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?

EXTENDING THE MODEL

Are there other ways one might calculate pressure?

Create an isothermal piston example where the user can manually move the piston to any level in the box.

Add in a temperature variable that allows for the particles to move the piston to the appropriate volume.

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:

COPYRIGHT AND LICENSE

Copyright 2002 Uri Wilensky.

CC BY-NC-SA 3.0

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 and associated activities and materials were created as part of the project: MODELING ACROSS THE CURRICULUM. The project gratefully acknowledges the support of the National Science Foundation, the National Institute of Health, and the Department of Education (IERI program) -- grant number REC #0115699. Additional support was provided through the projects: PARTICIPATORY SIMULATIONS: NETWORK-BASED DESIGN FOR SYSTEMS LEARNING IN CLASSROOMS and/or INTEGRATED SIMULATION AND MODELING ENVIRONMENT -- NSF (REPP & ROLE programs) grant numbers REC #9814682 and REC-0126227.

Comments and Questions

Click to Run Model

globals [
  fast average slow     ;; current counts
  avg-speed avg-energy  ;; current averages
  vsplit vclock         ;; clock variables
  raw-width raw-height  ;; box size variables
  piston-position       ;; position of the piston at any given time
  volume area           ;; because this is 2D, area is the perimeter and volume is the area
  total-pressure        ;; pressure per unit area
  pressure-history      ;; list of 'scale' previous total-pressures
  avg-pressure          ;; mean of the pressure-history
  initspeed initmass    ;; initial speed and initial mass of the all the particles-particle mass always stays the same, but the speed changes.
  going-down?           ;; flag for whether or not the piston is moving up or down
]

turtles-own [
  speed mass energy new-speed            ;; Turtle Info
  v1t v1l tmp-turtle                     ;; Turtle 1 -- Collide
  heading2 mass2 speed2 v2t v2l turtle2  ;; Turtle 2 -- Collide
  theta                                  ;; Turtles 1 + 2 -- Collide
  pressure                               ;; pressure variable
]

;; procedure that setup up initial variables

to setup
  clear-all
  set going-down? true
  set pressure-history []
  set initspeed 10.0
  set initmass 1.0
  set raw-width  round (0.01 * box-width  * max-pxcor)
  set raw-height round (0.01 * box-height * max-pycor)
  set piston-position 0.75 * raw-height
  set area ((4 * raw-width) + (2 * (piston-position + raw-height)))
  set volume ((2 * raw-width) * (piston-position + raw-height))
  make-box
  draw-piston
  ;;set vclock 0
  ;; create the gas molecules
  crt number [
     set new-speed initspeed
     set mass initmass
     random-position
     set shape "circle"
     recolor
     set pressure 0
  ]
  update-variables
  reset-ticks
  setup-histograms
end 

;; updates variables after every time tick

to update-variables
  ;; Gas Variables
  ask turtles
    [ set speed new-speed
      set energy (0.5 * mass * speed * speed) ]
  set average count turtles with [ color = green ]
  set slow    count turtles with [ color = blue  ]
  set fast    count turtles with [ color = red   ]
  set avg-speed  mean [ speed  ] of turtles
  set avg-energy mean [ energy ] of turtles

  ;; System Variables
  calculate-pressure
  set vsplit (round (max [speed] of turtles * 1.2))
end 

;; procedure that runs the model

to go
  ask turtles [bounce]
  ask turtles [move]
  ask turtles [check-for-collision]
  ;; control the piston's motion
  if piston-position < (-0.75 * raw-height)
  [ set going-down? false ]
  if piston-position > (0.75 * raw-height)
  [ set going-down? true ]
  ifelse going-down?
  [ piston-down piston-speed / vsplit ]
  [ piston-up piston-speed / vsplit ]

  set vclock vclock + 1
  ifelse (vclock = vsplit)
  [
    tick
    set vclock 0
    update-variables
    do-plotting
    do-histograms
  ]
  [ display ]
end 


;; turtle procedure for bouncing off of the walls

to bounce
  ; if we're not about to hit a wall (yellow patch)
  ; or the piston (gray+2 patch),
  ; we don't need to do any further checks
  if ([pcolor] of patch-ahead 1 != yellow) and
     ([pcolor] of patch-ahead 1 != gray + 2) [ stop ]
  ; get the coordinates of the patch we'll be on if we go forward 1
  let new-px [pxcor] of patch-ahead 1
  let new-py [pycor] of patch-ahead 1
  ; check: hitting left or right wall?
  if (abs new-px = raw-width)
    ; if so, reflect heading around x axis
    [ set heading (- heading)
      set pressure pressure + abs (dx * mass * speed)
    ]
  ; check: hitting piston or bottom wall?
  if (abs new-py = raw-height) or (new-py = round piston-position)
    ; if so, reflect heading around y axis
    [ set heading (180 - heading)
      set pressure pressure + abs (dy * mass * speed)
    ]
end 

;; turtle procedure that moves all the particles

to move
  jump (speed / vsplit)
end 

;; turtle procedure to check to see if two particles collide

to check-for-collision
  if count other turtles-here = 1
    [ set tmp-turtle one-of other turtles-here
      if ((who > [who] of tmp-turtle) and (turtle2 != tmp-turtle))
        [ collide ]
    ]
end 

;; turtle procedure for when two particles collide

to collide
  get-turtle2-info
  calculate-velocity-components
  set-new-speed-and-headings
end 

;; turtle gets mass and speed info from turtle it is colliding with

to get-turtle2-info
  set turtle2 tmp-turtle
  set mass2 [mass] of turtle2
  set speed2 [new-speed] of turtle2
  set heading2 [heading] of turtle2
end 

;; calculates new turtle velocity after the collision

to calculate-velocity-components
  set theta (random-float 360)
  set v1l (new-speed * sin (theta - heading))
  set v1t (new-speed * cos (theta - heading))
  set v2l (speed2 * sin (theta - heading2))
  set v2t (speed2 * cos (theta - heading2))
  let vcm (((mass * v1t) + (mass2 * v2t)) / (mass + mass2))
  set v1t (vcm + vcm - v1t)
  set v2t (vcm + vcm - v2t)
end 

;; set new speed and headings of each turtles that has had a collision

to set-new-speed-and-headings
  set new-speed sqrt ((v1t * v1t) + (v1l * v1l))
  set heading (theta - (atan v1l v1t))

  let new-speed2 sqrt ((v2t * v2t) + (v2l * v2l))
  let new-heading (theta - (atan v2l v2t))
  ask turtle2 [
    set new-speed new-speed2
    set heading new-heading
  ]

  recolor
  ask turtle2 [ recolor ]
end 

to recolor  ;; turtle procedure
  ifelse new-speed < (0.5 * initspeed)
    [ set color blue ]
    [ ifelse new-speed > (1.5 * initspeed)
        [ set color red ]
        [ set color green ] ]
end 

;; patch procedure to make a 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 ]
end 

;; turtle procedure to give turtles a random position within the confined area

to random-position
  setxy ((1 - raw-width)  + random-float (2 * raw-width - 2))
        ((1 - raw-height) + random-float (raw-height + piston-position - 2))
end 


;; ------ Piston ----------

to piston-up [dist]
  if (dist > 0)
  [ ifelse ((piston-position + dist) < raw-height - 1)
    [ undraw-piston
      set piston-position (piston-position + dist)
      draw-piston ]
    [ undraw-piston
      set piston-position (raw-height - 1)
      draw-piston ]
    set volume ((2 * raw-width) * (piston-position + raw-height))
    set area ((4 * raw-width) + (2 * (piston-position + raw-height)))
  ]
end 

to piston-down [dist]
  if (dist > 0)
  [ ifelse (piston-position - dist) > (2 - raw-height)
    [ undraw-piston
      set piston-position (piston-position - dist)
      ask turtles
      [ if (ycor >= (piston-position - 1))
        [ bounce-off-piston ] ]
      draw-piston ]
    [ undraw-piston
      set piston-position (3 - raw-height)
      ask turtles
      [ if (pycor >= 3 - raw-height)
        [ bounce-off-piston ] ]
      draw-piston ]
    set area ((4 * raw-width) + (2 * (piston-position + raw-height)))
    set volume ((2 * raw-width) * (piston-position + raw-height))
  ]
end 

to draw-piston
  ask patches with [ ((pycor = (round piston-position)) and ((abs pxcor) < raw-width)) ]
    [ set pcolor gray + 2 ]
end 

to undraw-piston
  ask patches with [ (pycor = round piston-position) and ((abs pxcor) < raw-width) ]
    [ set pcolor black ]
end 

to bounce-off-piston  ;; Turtles procedure particle bounces off piston
  ifelse ((((2 * piston-position) - (ycor + 2)) < (1 - raw-height)) or
          (((2 * piston-position) - (ycor + 2)) > (piston-position - 2)))
   [ set ycor ((random (raw-height + piston-position - 2)) - (raw-height - 1)) ]
   [ set ycor ((2 * piston-position) - (ycor + 2)) ]
end 

to calculate-pressure  ;; Observer procedure
  set total-pressure 100 * (sum [pressure] of turtles) / area
  ifelse (length pressure-history < scale)
  [ set pressure-history fput total-pressure pressure-history ]
  [ set pressure-history fput total-pressure but-last pressure-history ]
  set avg-pressure mean pressure-history
  ;; rezero pressures in preparation for the next cycle
  ask turtles [ set pressure 0 ]
end 

;;; plotting procedures

to setup-histograms
  ;; Speed Histogram
  set-current-plot "Speed histogram"
  set-plot-x-range 0 (initspeed * 2)
  set-plot-y-range 0 ceiling (number / 6)
  set-current-plot-pen "average"
  set-histogram-num-bars 45
  set-current-plot-pen "fast"
  set-histogram-num-bars 45
  set-current-plot-pen "slow"
  set-histogram-num-bars 45

  ;; Energy histogram
  set-current-plot "Energy histogram"
  set-plot-x-range 0 (0.5 * (initspeed * 2) * (initspeed * 2) * initmass)
  set-plot-y-range 0 ceiling (number / 6)
  set-current-plot-pen "average"
  set-histogram-num-bars 45
  set-current-plot-pen "fast"
  set-histogram-num-bars 45
  set-current-plot-pen "slow"
  set-histogram-num-bars 45
end 

;; does actual plotting (called in Go)

to do-plotting
  set-current-plot "Volume"
  plot volume
  set-current-plot "Pressure"
  plot avg-pressure
  set-current-plot "Temperature"
  plot avg-energy
  set-current-plot "Pressure vs. Volume"
  plotxy volume avg-pressure
  set-current-plot "Pressure * Volume"
  plot avg-pressure * volume / 1000
end 

;; does actual histograms plotting (called in Go)

to do-histograms
  if (histogram?)
    [ histo-energy
      histo-speed ]
end 

;; draw energy histogram

to histo-energy
  set-current-plot "Energy histogram"
  set-current-plot-pen "average"
  histogram [ energy ] of turtles with [ color = green ]
  set-current-plot-pen "slow"
  histogram [ energy ] of turtles with [ color = blue ]
  set-current-plot-pen "fast"
  histogram [ energy ] of turtles with [ color = red ]
  set-current-plot-pen "avg-energy"
  plot-pen-reset
  draw-vert-line avg-energy
end 

;; draw speed histogram

to histo-speed
  set-current-plot "Speed histogram"
  set-current-plot-pen "average"
  histogram [ speed ] of turtles with [ color = green ]
  set-current-plot-pen "slow"
  histogram [ speed ] of turtles with [ color = blue ]
  set-current-plot-pen "fast"
  histogram [ speed ] of turtles with [ color = red ]
  set-current-plot-pen "avg-speed"
  plot-pen-reset
  draw-vert-line avg-speed
end 

; draws a vertical line at xval on the current-plot with the current plot-pen

to draw-vert-line [xval]
  plotxy xval plot-y-min
  plot-pen-down
  plotxy xval plot-y-max
  plot-pen-up
end 


; Copyright 2002 Uri Wilensky.
; See Info tab for full copyright and license.

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Uri Wilensky almost 10 years ago GasLab Moving Piston Download this version

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