## Lab: Boyle’s Law

### Overview

This lab is designed to demonstrate one of the fundamental laws of gas behavior, Boyle’s Law. This law is about the relationship between the pressure and volume of an ideal gas when the number of moles and the temperature are held constant. Furthermore, the law holds true regardless of the identity of the gas. Credit for the first publication about this law regarding gas behavior belongs to Robert Boyle who wrote up his results in 1662. You will use the Vernier Gas Pressure Sensor and a gas syringe to vary volume and measure pressure to generate data to graph in order to determine the nature of the proportion between the pressure and volume of a gas. By collecting data and graphing it you will determine whether Boyle’s Law is a direct proportion or an inverse proportion. A direct proportion is one in which two variables are related in such a way that the value of the dependent variable (usually plotted on the y-axis) equals the independent variable (on the x-axis) multiplied by a constant. The graph of such a proportion has the form of a simple straight line and the equation for the line (in y = mx + b form) shows how one variable is multiplied by a constant (the slope of the line) to calculate the other. A sample graph of a direct proportion is shown at right. An inverse proportion is one in which two variables are related in such a way that the value of the dependent variable (y) equals a constant times the inverse of the independent variable (1/x). The graph of an inverse proportion has the form of a simple inverse curve and the equation of the curved line has the form y = m(1/x). The problem with this is that several functions have graphs that look a lot alike. When using real data from a lab it can be hard to be sure that the data are following the equation y = m(1/x) or y = m(1/x2). These two functions are hard to tell apart—try graphing them using a graphing calculator. In order to confirm that your data do follow a simple inverse proportion, just plot the inverse of your independent variable values versus the dependent variable. In other words, plug in the value of 1/x instead of x and graph it again. If it turns out to be a simple straight line, then you can confirm a simple inverse proportion.

### Objective

This lab is designed to allow students to discover the nature of the proportion between the pressure and volume of a gas by collecting data and analyzing it using computer software. Students will determine whether Boyle’s Law is a direct proportion or an inverse proportion. They will also learn to do simple proportional calculations.

### Materials

• 20 mL plastic syringe
• Vernier Gas Pressure Sensor and Interface
• air
• Computer with MS Excel

### Safety

• This lab is really very nearly risk-free

### Pre-lab Exercises

Boyle’s Law is a useful proportion that can be put to work to answer questions about changes in volume and pressure.

Some Common Units and Conversions
Pressure
1 atmosphere (1 atm) is the average air pressure at sea level on Earth.
1 atm = 101.3 kPa = 1.013 × 105 Pa = 14.7 lb/in2
A typical car tire will have a pressure of about 35 psi (lb/in2)
Volume
1 cubic meter (m3) is the SI unit of volume
1 m3 = 1,000 LĀĀĀĀĀĀĀĀĀĀĀĀ1 L = 1,000 mL

Example

If a gas’s pressure is reduced by half at constant temperature then its volume doubles. Here is how to use the Boyle’s Law proportion to calculate such changes.

First, let’s define the initial pressure and volume as P1 and V1 and the pressure and volume after a change as P2 and V2. According to Boyle’s Law:

```P1V1 = k1     and     P2V2 = k2
```

Let’s assume those constants are the same (and they will be as long as we do not add or remove any gas or change the temperature). In that case:

```P1V1 = P2V2
```

page break

Now, here is a question we can answer using this proportional equation: What is the final volume of a gas when its pressure is reduced by half and its initial pressure is 1.0 atm and its volume is 5.0 L?

```P1 = 1.0 atm          P2 = 0.50 atm                P1V1 = P2V2
V1 = 5.0 L            V2 = ?          (1.0 atm)(5.0 L) = (0.50 atm)(V2)
```

Solving for V2 gives the answer 10 L. This is exactly what we expected based on the idea that this is an inverse proportion: when one variable is cut in half, the other doubles. Use this example to help you to answer the questions in the exercises below.

#### Exercises

Do the following exercises neatly on a separate piece of paper. Assume that temperature is constant for all of the changes described in these problems. Also, assume that pressure at the Earth’s surface is equal to 1 atm or 101.3 kPa.

All problems can be solved using the correct form of Boyle’s Law. This can be written as either PV = k or, more usefully, as P1V1 = P2V2.

1. If a gas in a volume of 25 mL with a pressure of 1 atmosphere (atm) is compressed to 5 mL what is its pressure?
2. If a gas with a pressure of 2 atm is confined in a volume of 10 L what will its pressure be if the volume is made to be 20 L?
3. A balloon with an internal pressure of 1.3 atm and a volume of 2.5 L is placed into a vacuum chamber. What is the balloon’s volume if the internal pressure is reduced to 0.17 atm?
4. What is the new volume of a gas if the pressure of the gas is reduced from 220 kPa to 100 kPa and the initial volume was 1.5 m3?
1. At the bottom of the Challenger Deep in the Pacific Ocean the pressure due to all that water overhead is 1091 atm. A bubble of gas with a volume of 1 mL is released by an advanced submarine research vessel. What is the volume of the gas bubble when it pops at the surface of the ocean where the pressure is 1 atm?
2. A gas is confined in a bottle with a pressure of 5.2 atm. The volume of the bottle is 40 L. What volume would the gas have if it were stored at a pressure of 1 atm instead?
3. What happens to the pressure of a gas when its volume is changed from 14 mL to 27 mL?
4. Why does the volume of helium in a weather balloon increase as it rises from the ground to the upper atmosphere?

### Procedure

Remember to record your observations in your lab notebook before you leave class.

1. Connect the LabQuest Mini to the computer and connect the Gas Pressure Sensor to it. Start the Logger software.
2. Pull out the plunger of the syringe until the volume reads 5 mL. Read the volume at the bottom-most black ring on the rubber plunger where it touches the inside of the barrel.
3. Attach the syringe to the sensor by gently screwing it into place.
4. Set up data collection as follows:
1. On the “Experiment” menu, select “Data Collection…”
2. Change Mode from “Time Based” to “Events with Entry”.
3. Make the “Column Name” Volume and the units mL. Then click “Done”. (Pressure will be recorded using the SI unit of pressure, the kilopascal (kPa): 1 atm = 101.3 kPa.
5. Right-click on the graph and select “Graph Options…” at the top of the menu that pops up.
6. On the “Axes Options” tab change the box marked “Right” to 20. This sets the maximum value for the volume measurements which will not exceed 20 mL.
7. If you have a lab partner then have one partner manage the syringe and have the other enter the data. Click the big green “Collect” button to begin.
1. Move the piston of the syringe so that the volume is exactly 2.5 mL and hold it in place.
2. Wait for the pressure reading to stabilize then click the “Keep” button.
3. Enter the volume and click OK.
4. Repeat these steps at volumes of 5.0 mL, 10.0 mL, 12.5 mL, 15.0 mL, 17.5 mL, and 20.0 mL.
5. Click the big red “Stop” button.
8. Click the big “Save” button and write a descriptive name for the data file. Store it on your personal network drive. Alternatively, save it to the desktop and then email it to yourself or upload it to your personal document service.

### Calculations and Graphing

Now that you have generated some data and taken a look at a graph of it on your screen you will put that data in Excel to produce a well-formatted graph and perform an analysis to determine the exact nature of the proportion between the pressure and volume of a gas.

1. Highlight the data table on the LoggerLite or LoggerPro software screen. Copy by using Ctrl-C.
2. Paste the data into a spreadsheet program, leaving an empty row at the top to add labels. Label the column on the left Volume (mL) and label the column on the right Pressure (kPa).
3. Insert a Scatter-plot graph of the data and format it according to your teacher’s directions. The online version of this lab has a link to a sample set of graphs.
4. Right-click the column heading for the column containing your Pressure data (probably where it says ‘B’) and select “Insert” from the menu that pops up. Label this column “Inverse Volume (1/mL)”. In the cells of this column use a formula to calculate the inverse of the Volume data. One possible formula is “=A2^-1”. Copy the formula to all the cells in the column next to Volume data.
5. You are about to create a graph to determine the exact nature of the proportion. The first graph you made should give you an idea of whether the data are likely to be directly or inversely proportional. By graphing P vs. 1/V you will see whether a straight line results. If it does then it will confirm the proportion as a simple inverse proportion that has the form P = k/V or PV = k. Create a scatter-plot of the 1/V and P data and format it according to your teacher’s instructions.
6. If the data do not turn out to be a nice straight line on your graph then consult with your teacher immediately to get help trouble-shooting.
7. Use the spreadsheet software to generate a line of best fit or trendline. Be sure to show the equation and the R-squared value on the graph. The slope of this line is the constant of the proportion (k).
8. To the right of the Pressure column of data add another column heading: “Pressure times Volume (kPa × mL)”.
9. Set up a calculation of pressure times volume. A likely formula is “=A2*C2”. Copy the formula to all the cells in the column. The result of this calculation would be a constant for an inverse proportion.
10. At the bottom of your column of P × V values calculate the average. Type: “=AVERAGE(D2:D7)”.
11. Add yet another column to the right: “Pressure divided by Volume (kPa/mL)”.
12. Set up a calculation of pressure divided by volume. A likely formula is “=C2/A2”. Copy the formula to all the cells in the column. The result of this calculation would be a constant for an direct proportion.
13. At the bottom of your column of P ÷ V values calculate the average. A formula you might use: “=AVERAGE(E2:E7)”.
14. Format your data table for printing by making the headings different from the cell contents and adding borders.
15. Arrange to print each graph and the data table. Make sure each lab group member will get a copy to hand in with their answers to the post-lab questions. Students may hand in the same graphs and table but must do their own work in answering the questions.

### Post-lab Questions

Answer the following questions using complete sentences in a professional-quality typed document. Submit them along with your spreadsheet with neatly labeled data and graphs.

1. What is the equation of the line for your second graph in the form y = mx + b? Identify each of the letters in [y = mx + b] with a quantity from the lab and clearly identify the units of each quantity. Give decimal values for the slope and y-intercept.
2. What is the average value of P × V? Is this product a constant or nearly constant? Justify your answer.
3. What is the average value of P ÷ V? Is this quotient a constant or nearly constant? Justify your answer.