The Ideal Gas Law

Many of you have taken a chemistry or physics class that has talked about the relationship between volume and pressure. How was this relationship discovered? By taking lots of data, and fitting a mathematical model to it. Essentially, by regression.

Animation of Gases

Let's look at the following StarLogo simulations of an ideal gas.

gas-law2.slogo gas-law3.slogo

Each molecule of gas is represented as a turtle. The red walls define the volume of the container. Molecules bounce off the sides of the container, and we can count the number of impacts per unit time. "Pressure" we will define as an emergent phenomenon (remember that topic?) from the average number of impacts per unit time.

• You can adjust the volume (as a percentage of the whole field) using the slider and the "setup" button.
• The "single-step" button runs the simulation for just one time step, deltat, so each turtle takes one step forward in its current direction. Note that deltat is approximately 1/300, since the average velocity of molecules at room temperature is 300 meters per second.
• The "go for 5" button runs the simulation for that many time units. This takes a while!
• The "report-results" button prints the

Let's collect some data by running this system in the following conditions. All of these are with about 500 particles.

Everyone take a different value and we'll collect data in parallel. Then, we'll plot the data and analyze it.

Reviewing the Code

The underlying StarLogo model is that all the turtles are updated "in parallel" (but probably sequentially) at each timestep. You program a simulation by saying what the turtles do at each timestep. You do that by writing turtle procedures. You can also program the interface by writing observer procedures.

Question: Is StarLogo a Continuous Process simulation or a Discrete Event Simulation?

You can look at both sets of procedures by choosing Windows / Control Center. Let's look at the Observer procedures first, and then the Turtle procedures.

• First, we declare some "global" variables. These are variables that all of the procedures can use and modify.
• Then we declare some more global variables.
• StarLogo defines a procedure by the keyword to, so most of the procedures are verbs. The first procedure is draw-chamber. The keyword end marks the end of the procedure. This one is just two lines long. The first asks all the patches (positions on the screen) to set their color to green. The second asks one of the turtles to execute the draw-lines procedure.

  This illustrates a technique called abstraction that is very important in computer science as well as other fields.

• The next procedure sets the values of the relevant global variables based on the slider that indicates the chamber's size. The "volume" slider affects the "volume-pct" variable, which affects the calculations by this procedure. Notice that this procedure may call draw-chamber at the end.
• The setup procedure is the main procedure that does all the work of getting ready for a simulation. It sets some global variables, creates the turtles, and sets up the chamber.
• The go-a-step procedure is our fundamental state-change procedure: all the turtles move and a few globals are updated.
• The go-for-5 procedure is a straightforward use of the go-a-step procedure so that we can go for 5 seconds.
• The report-results procedure writes the min, mean, and max impact rate to the output log.
• The first part of the turtles procedures uses turtles-own to specify some personal variables that each turtle possesses. Each turtle gets its own copy of these, so each turtle can have a different heading (this is built-in), velocity, vx, vy and so forth.
• The first turtle procedure is print-wall-info which is a simple debugging procedure to print out the locations of the walls to the output stream. It isn't used, but easily could be.
• The set-velocity-components procedure determines the velocity components from the decomposes the velocity and heading information into velocity vector components: vx is the velocity of the turtle in the x direction, and similarly for vy.
• The init-turtle procedure does all the setup of a turtle when it is "born," including setting its color, heading, velocity, location, and velocity components.
• The draw-lines procedure is the special-purpose function for drawing the walls of the chamber. (The simulation doesn't actually need them in order to work.) This procedure has four loops in it, as the turtle draws each wall.
• The move-a-step procedure is crucial: this is how turtles update themselves each time step. It bears close scrutiny.

You can look up the commands, syntax and so forth of the StarLogo language by clicking on the button at that web site.

Extend Model (lab)

The StarLogo simulation crashes sometimes. It's also slow sometimes (almost entirely due to the graphics, which are nice but not essential). The following Extend model is incomplete, but can handle a lot more data.

gas.mox

Spend some time studying this model and understanding what it's doing.

• This model makes lots of use of attributes (much like the turtle variables). The "info" block records certain attributes of the items that pass through it. Run the simulation once and take a look. This model uses the following attributes
  • id: this numbers each item. It's not necessary, but it's convenient for certain observations
  • xcoor: this is the x coordinate of the particle
  • ycoor: this is the y coordinate of the particle
  • heading: the direction (in radians) where the particle is going
• The "value" attribute of an item is a special attribute: you can think of it as the "multiplicity" of the item: how many separate items this one stands for. Certain blocks, such as queues, turn a single item with a value of N into N items with a value of 1. Run the simulation with animation to see.
• What is the purpose of the first queue? Try removing it to see.
• What is the purpose of the "delay" block? Try removing it to see.
• Does the histogram of x coordinates seem right? What do you expect? Uniform? Non-uniform?
• Does the histogram of x,y coordinates seem right?
• Increase the number of items to 100. What other things need to change?
• Look at the changed histograms. Do they seem better or worse?
• Look at the DE Eqn blocks and understand what they're doing. It can help to have a known constant heading (instead of a random value) to check the computations.

When you feel you understand the model:

• Think about how to have the items bounce off the walls.
• Think about how to count the number of impacts
• Increase the number of items to give ourselves some good data.

Here are two solutions:

gas2.mox
gas3.mox

There's a lot to observe here.

• Additional attributes:
  • velocity
  • vx
  • vy
• The weird "data storage" setting in the Plotter window for the "x hits"
• Blocks to count impacts (x and y are done separately)
• Blocks to reflect particles that hit the walls

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