In this module we discuss Boyle's model. This is usually known as Boyle's Law but that is a misnomer -- a common misnomer that we will encounter and correct throughout this course. We are working with models that are attempts by people to describe, understand, and predict real-world phenomena. We are not working with laws enacted by governments or handed down on stone tablets.
Boyle's Law appears in an appendix to the 1661 work -- New Experiments Physio-Mechanicall, Touching the Spring of Air and its Effects -- written by Robert Boyle. A brief biographical sketch of Boyle is available at the MacTutor History of Mathematics Archive.
The purpose of this module is to examine the way in which we build a "mechanical" model based on everyday experience to explain phenomena that are only indirectly visible and tangible. This mechanical model relies heavily on mathematics. Besides being important in chemistry and physics, Boyle's model is an excellent vehicle for discussing the meaning of the ingredients of mathematical formulas.
Boyle's model involves a relationship among three properties of a gas in a container.
Boyle's model states
where k is a constant, or
The value of the constant k depends on the units used for the other quantities but once the units are fixed, k is also fixed.
When any one of the quantities V, P, or T is changed, one or two of the others must change so that the equation above still holds.
We begin our exploration of Boyle's model with some experimentation. The picture below shows a pressure sensor attached to something that looks very much like a large hypodermic needle. It consists of a cylinder with a plunger. By pushing the plunger in or pulling it out you can vary the volume of the air in the cylinder. The side of the cylinder is marked with volume in cubic centimeters. The pressure sensor measures the air pressure inside the cylinder.
The links below lead to programs and instructions for using five different TI graphing calculators with the CBL to collect data. Click on the icon for your graphing calculator. The instructions for your calculator will appear in a new window. Arrange these two windows so that they overlap and you can move easily back-and-forth between them by clicking on the inactive window to make it active. Close the new window when you are done with it.
Whichever calculator you use the basic idea is simple.
Use your CAS window to examine either your own data or data supplied in the CAS window. Compare the data with the predictions made by Boyle's model.
where the constant C can be determined using any one of the data points. Because of the way we collected the data, the best data point to use to determine the value of the constant C is the first one, at normal atmospheric pressure. The other data points might be affected by any leaks in the apparatus.
You may want to use the ideas and tools developed so far to examine Boyle's model further -- for example, to look at the relationship between temperature and pressure with volume constant.
Next we want to look more closely at Boyle's model -- to understand what it is saying and to see if it can help us understand what is going on in a container filled with a gas. Before going on, think about the following questions.
We will investigate Boyle's model and the model described above together by seeing if Boyle's model can be predicted from the model. Since we have experimental evidence supporting Boyle's model, if the model predicts Boyle's model then this evidence also is evidence in favor of the model.
Click here to open a new window with a Java applet that shows the model in action. When the window is open arrange the two windows so that they overlap and it is easy to move back-and-forth between the windows by clicking on the inactive window to make it active. This applet is not a perfect rendition of the model because it is two-dimensional rather than three-dimensional. When you have completed this module close the window with the Java applet.
This JAVA applet performs a computer simulation of the possible model of a gas-filled container discussed above. As the applet runs, molecules inside the container move around. Each time a molecule hits the wall of the container and bounces off, the length of purple bar at the right of the display indicates how hard it hit the wall. The strength with which the molecule hits the wall depends on its speed and the angle at which it hits the wall -- molecules hitting the wall head on at a right angle hit harder than those hitting the wall with a "glancing blow." At the same time that the purple bar indicates the strength of the hit, the count is incremented by one. The count display indicates how many times a molecule has hit the wall. You can run this simulation with your choice of three possible radii for the container -- 1 unit, 2 units, and 4 units. You can also choose from two speeds for the molecules -- 1 unit and 2 units. Click in one of the six labeled boxes to run an experiment with the indicated radius and speed. Run several experiments to see how the size of the container and the speed of the molecules affects the number of hits and their strength. All the experiments run for the same simulated length of time. Because this is a random simulation, the results will vary somewhat if you duplicate the same experiment.
Answer the following questions based in part on your experimentation varying the radius and speed in the JAVA applet. Because the Java applet is a random simulation, the results of the applet will not be exact.
The questions and your answers above were based on the JAVA applet which showed a two-dimensional model. Think about a three dimensional model with a spherical container and answer the questions below based on your intuition and experience with the two dimensional model above.
Do your answers above agree wth Boyle's model?
The purpose of the next problem is to build and test a model of the relationship between the "intensity" of light and distance from the light source. We are interested in a bare bulb that radiates light in all directions. You should use an incandescent bulb for this problem because fluorescent bulbs flicker and this makes it more difficult to measure their light. As you do this problem, the units used to measure the quantities involved will be very important.
Most bulbs waste a lot of the power they consume, turning it into heat rather than light.
As light emanates from a light source it spreads out as shown in the movie below.
Think about this movie and build a model expressing the relationship between the light received by a measuring device at a distance R from a bare bulb and the distance. Make sure that the units in your formula work out correctly. Write a short paragraph explaining your model. Finally test your model using your CBL and TI graphing calculator. Click on the icon below for instructions for your graphing calculator. For this experiment you can use the CBL as a meter, making but not recording light intensity readings. Be careful about what your readings are meauring. What adjustments do you need to make if you are not working in a dark room?
