Wet Labs and Simulations and Simulations

We have a wide spectrum of "experiments" available including both "Wet-labs" and simulations. Wet labs can be immediate and direct or they can be mediated by instrumentation and data-processing, so that the actual experience is not much different from a simulation.

Simulations can be simple and crude or they can be more detailed and scientifically sound. We can expect many very powerful and compelling simulations based on available virtual reality technology like the Nintendo 64 and even more powerful technology in the near future. Because these simulations are so compelling -- so lifelike -- we must be very careful that they are scientifically sound. We also need to reflect about the educational advantages and disadvantages of simulation as and simulations.

We illustrate some of these ideas with an example -- Boyle's Model.

• From Project Links

• Boyle's Model -- From Mathematical Modeling in a Real and Complex World

What happens when the speed of the molecules increases by a factor of V?

• The number of "hits" per unit of time goes up by a factor of V.

• The "strength" of each hit goes up by a factor of V.

• The result is that the pressure (force per unit length of the perimeter) goes up by a factor of V².

• This is "why" pressure is proportional to temperature (kinetic energy).

• Notice this same analysis works in the real three dimensional world.

What happens when the radius of the circular vessel increases by a factor of R?

• The average length of the path between hits goes up by a factor of R. Thus, the number of "hits" per unit time changes by the factor 1/R.

• The perimeter of the circle goes up by a factor of R. Thus, the number of "hits" per unit length of the perimeter changes by the factor 1/R.

• The net result is that the number of hits per unit length of the perimeter per unit of time changes by the factor 1/R².

• This is "why" the pressure is inversely proportional to the area of the circular vessel in two-dimensions.

• A similar analysis shows that in the real three-dimensional world pressure is inversely proportional to the volume of a spherical vessel.

Is this a "good" simulation?

• Consider a rectangle whose area is one and whose sides are x and 1/x. The perimeter of this rectangle is
  2 x + 2 / x

  Can we argue convincingly that the average length of time between hits is inversely proportional to the perimeter, so that the pressure does not depend on x?

  This is the reason that a circular vessel was used in the applet.

• The simulation omits collisions between gas molecules. Is this a significant omission?

  It was easier to write the applet this way.

• Does this focus attention (in fact, does Boyle's Model) focus attention in the wrong direction?

Boyle's Model (Traditional)		Boyle's Model (Revised)

The picture above shows a small section of the wall of the vessel under very high maginfication. Notice that the wall looks flat. Notice also that the two important factors are the speed of the gas molecules and their density. The facts below argue that this picture is more relevant than the global picture in the Java applets (in the simulations).

For gases at ordinary temperature and pressure

• There are roughly 10¹⁹ molecules per cubic centimeter.

• Each modecule has roughly 10⁹ to 10¹⁰ collisions per second.

• Average molecular speed is roughly 10⁴ to 10⁵ centimeters per second.

• The mean free path is roughly 10^-5 to 10^-4 centimeters.