Boyle's Law

Boyle's Law involves a relationship between three properties of a gas in a container.

The volume of the container -- denoted V. Volume is measured in units of length^3 -- for example, cubic centimeters or cubic inches.
The pressure of the gas -- denoted P. Pressure is measured in units of force per area -- for example, pounds per square inch.
The temperature of the gas -- denoted T. Temperature is in many ways more complicated than pressure or volume. It is usually measured using one of three units -- degrees Fahrenheit, degrees Celsius, or degrees Kelvin.
In the United States the Fahrenheit scale is most common. Using this scale water boils at 212 degrees and freezes at 32 degrees. The Celsius scale is more common in the rest of the world. Using this scale water boils at 100 degrees and freezes at zero degrees. Either of these scales is comfortable for everyday life but for scientific purposes the Kelvin scale is better.
The following formulas can be used to convert from one scale to another.
```
Degrees Kelvin = Degrees Celsius + 273

Degrees Celsius = Degrees Kelvin - 273

                      9 
Degrees Fahrenheit = --- * Degrees Celsius + 32
                      5
                   5
Degrees Celsius = --- * (Degrees Fahrenheit - 32)
                   9
```
Whichever scale you use, temperature reflects the random motion of molecules. We feel fast moving molecules as hot and slower moving molecules as cold. But temperature is a bit more complex than just the speed of random motion. You may know that it takes more energy to accelerate a car from 50 to 60 miles per hour than from 10 to 20 miles per hour. The same thing is true for the random motion of molecules. In general the energy required to accelerate an object from rest to the speed v is
All three temperature scales measure energy rather than raw speed. It takes the same energy to raise an object one degree regardless of its initial temperature. The difference between the Celsius scale and the Kelvin scale is that in the Celsius scale zero represents the freezing point of water but in the Kelvin scale zero represents zero energy -- the temperature of an object whose molecules have no random motion.

Boyle's Law states

PV = kT

where k is a constant, or

Pressure * Volume = Constant * Temperature

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.

If the temperature of a gas in a container increases but the volume stays fixed what happens to the pressure? answer
If the volume of a gas-filled container increases but the temperature remains fixed what happens to the pressure? answer
If the air pressure in a room drops but the temperature remains constant what happens to a filled balloon? answer
Does Boyle's Law work when temperature is measured in any one of the three scales -- Fahrenheit, Celsius, or Kelvin. Explain your answer.

We begin our exploration of Boyle's Law 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.

Missing picture

Your CAS window contains data that we collected keeping the temperature constant and measuring the pressure while we varied the volume from 5 to 20 cubic centimeters. The units in which the pressure are measured are unimportant. You can use the data in your CAS window or, better yet, you can collect your own data using the TI-CBL, pressure sensor, and a TI graphing calculator as described below.

The links below lead to programs and instructions for using four different TI graphing calculators with the CBL to collect data. Whichever calculator you use the basic idea is simple.

Begin by attaching the plunger-and-cylinder to the pressure sensor as shown in the picture above. Then connect the pressure sensor to channel 1 of the CBL and the CBL to your graphing calculator using the linking cable. Turn the CBL on.
Set the plunger so that the volume in the cylinder is 20 cubic centimeters. Then open and close the pressure release valve on top of the pressure sensor. The air inside is now at the same pressure as the air in the atmosphere. With the pressure release valve closed air should not leak in or out of the cylinder.
Following the instructions for your calculator, take and record a series of pressure readings with the plunger set so that the volume in the cylinder is 20, 19, 18, ... 5 cubic centimeters.

Click on the icon below for your graphing calculator.

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 Law.

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 Law 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 Law -- 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.

What do you think a gas "looks like."
One model of a gas in a container pictures a gas as made up of many very small molecules moving around inside the container. The molecules are so small that most of the container is "empty." Do you think this model is "right?" Why? Why not?
Pursuing the model described above, what feature of the model is measured by temperature?
Pursuing the model described above, what feature of the model is measured by pressure?

We will investigate Boyle's Law and the model described above together by seeing if Boyle's Law can be predicted from the model. Since we have experimental evidence supporting Boyle's Law, if the model predicts Boyle's Law then this evidence also is evidence in favor of the model.

Click here to open a new window with a Java applet. Arrange these two windows so they aoverlap and you can go back-and-forth between them by clicking on the inactive window to make it active. This JAVA applet shows the model in action. It is not a perfect rendition of the model because it is two-dimensional rather than three-dimensional.

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 molecules have 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.

What effect does the radius have on the number of times molecules hit the walls of the container per unit of time?
What effect does the radius have on the area of the container?
What effect does the radius have on the length of the wall of the container?
What do you think the pressure measures?
Based on your answers above, what relationship do you expect between the radius of the container and the pressure of the gas?
Based on your answers above, what relationship do you expect between the area of the container and the pressure of the gas?

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.

What effect does the radius have on the number of times molecules hit the walls of the container per unit of time?
What effect does the radius have on the volume of the container?
What effect does the radius have on the area of the wall of the container?
What do you think the pressure measures?
Based on your answers above, what relationship do you expect between the radius of the container and the pressure of the gas?
Based on your answers above, what relationship do you expect between the volume of the container and the pressure of the gas?

Do your answers above agree wth Boyle's Law?