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Chemistry Boyle’s Law Pressure-Volume Relationship in Gases

MATERIALS AND RESOURCES ABOUT THIS LESSON n this lesson, students will use a gas pressure data EACH GROUP collection system to collect pressure and volume data collection device I data pairs that can then be analyzed for their gas syringe, 20 mL mathematical relationship. The data collection is quick sensor, gas pressure and easy, and students typically get excellent results.

OBJECTIVES Students will: • Use a gas pressure sensor and a gas syringe to measure the pressure of an air sample at several different volumes. • Determine the relationship between pressure and volume of the gas. • Describe the relationship between gas pressure and volume in a mathematical equation. • Use the results to predict the pressure at other volumes.

LEVEL Chemistry

NEXT GENERATION ASSESSMENTS SCIENCE STANDARDS • AP Style Question: Gas Laws • Topic Assessment: States of Matter

REFERENCES

ANALYZING AND USING MATHEMATICS DEVELOPING AND INTERPRETING DATA USING MODELS Adapted from “Experiment 6: Boyles Law, Pressure- Volume Relationship in Gases.” Chemistry with Vernier, Vernier Software & Technology, 2008. Used with permission.

ENERGY AND MATTER SCALE, PROPORTION, CAUSE AND EFFECT AND QUANTITY

PS2: FORCES AND INTERACTION

CONNECTIONS TO AP

APAP CHEMISTRYCHEMISTRY 2

2.A.2 The gaseous state can be effectively modeled with a mathematical equation relating various macroscopic properties. A gas has neither a definite volume nor a definite shape; because the effects of attractive forces are minimal, we usually assume that the particles move independently.

*Advanced Placement® and AP® are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product.

COMMON CORE STATE STANDARDS

(LITERACY) RST.9-10.3 (MATH) HSF-LE.A.2 Follow precisely a multistep procedure when Construct linear and exponential functions, including carrying out experiments, taking measurements, or arithmetic and geometric sequences, given a graph, a performing technical tasks, attending to special cases description of a relationship, or two input-output pairs or exceptions defined in the text. (include reading these from a table).

(LITERACY) RST.9-10.7 (MATH) HSF-LE.B.5 Translate quantitative or technical information Interpret the parameters in a linear or exponential expressed in words in a text into visual form (e.g., a function in terms of a context. table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into (MATH) HSS-ID.B.6A words. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. (LITERACY) WHST.9-10.1 Fit a function to the data; use functions fitted to data Write arguments focused on discipline specific to solve problems in the context of the data. Use content. given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential (MATH) HSN-Q.A.2 models. Define appropriate quantities for the purpose of descriptive modeling. (MATH) HSS-ID.B.6C Represent data on two quantitative variables on a (MATH) HSA-CED.A.2 scatter plot, and describe how the variables are related. Create equations in two or more variables to represent Fit a linear function for a scatter plot that suggests a relationships between quantities; graph equations on linear association. coordinate axes with labels and scales. (MATH) HSS-ID.C.8 (MATH) HSA-CED.A.4 Compute (using technology) and interpret the Rearrange formulas to highlight a quantity of interest, correlation coefficient of a linear fit. using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

TEACHING SUGGESTIONS his activity provides a quick and simple LABQUEST KEYSTROKES way for students to develop and explore the 1. To start the experimental setup: relationship between gas pressure and volume, T a. Connect the gas pressure sensor to your data otherwise known as Boyle’s law. The use of a data collection device. collection device simplifies the data gathering process and allows students to focus on the development b. Adjust the volume of the syringe to 20 mL and understanding of the mathematical model that before connecting to the sensor. underlies this relationship. c. Choose “New” from the “File” menu. Boyle’s law describes that for a fixed quantity of gas 2. To set up the data-collection mode: the pressure and volume are inversely proportional. a. On the “Meter” screen, select “Mode.” While students may be familiar with linear graphs that Change the mode to “Events with Entry.” show a positive sloping, direct linear relationship, the negatively sloping curve generated by this inverse b. Enter the Name (“Volume”) and Units function and the subsequent manipulation required to (“mL”). Select “OK.” linearilize it may be new. Because the data collection 3. To collect data pairs. is so quick, you should spend time focusing on the development and utilization of the mathematical a. Move the piston so the front edge of the model so eloquently established by this real-wold data. inside black ring (Figure 2) is positioned at the specified line on the syringe. Hold the If your students need a refresher on the types of piston firmly in this position until the pressure relationships and their corresponding graphs, see the value displayed on the screen stabilizes. Middle Grades lesson, “Happiness is a Straight Line,” for discussion points and examples. b. Tap “Keep” and enter the gas volume (in mL) on the screen. Remember, to add 0.8 mL to the volume of the syringe for the total volume to account for the small volume of trapped air in the connection port. Select “OK” to store this pressure-volume data pair. c. Continue this procedure using each syringe volumes remembering to add the 0.8 mL to each volume before entering. d. Stop data collection by tapping the red square.

TEACHING SUGGESTIONS (CONTINUED) 4. To test different curve fits to the data: 6. To calculate regression statistics and to plot a best fit regression line on the graph: a. Choose “Curve Fit” from the “Analyze” menu. a. Choose “Graph Options” from the “Graph” menu. b. Test the direct hypothesis with “Linear Regression” where x is the volume and y is b. Select “Autoscale from 0”, and select “OK.” the pressure. c. Choose “Curve Fit” from the “Analyze” c. Test the inverse hypothesis with “Power as menu. the Fit Equation.” [this should provide the d. Select “Linear” as the Fit Equation. The better fit]. linear-regression statistics for these two data 5. To linearize the data by graphing pressure vs. columns are displayed in the form reciprocal volume: y = mx + b a. Tap the “Table” tab to display the data table. where b. Choose “New Calculated Column” from the x = 1/volume “Table” menu. y = pressure c. Enter the Name (“1/Volume”) and Units m = a proportionality constant (“1/mL”). Select the equation, “A/X”. Use b = y-intercept “Volume” as the column for X, and “1” as the value for A. e. Select “OK.” d. Select “OK.” Because the relationship between pressure and volume is an inverse relationship, the graph of pressure vs. 1/ volume should be direct; that is, the curve should be linear and pass through (or near) the origin. The slope of the line is equal to the proportionality constant k as established by Boyle’s law, PV = k.

DATA AND CALCULATIONS

Table 1. Volume vs. Pressure

Constant, k Volume (mL) Pressure (kPa) (P/V)(P × V)

5.8 175.9 30.3 1020

7.8 131.4 16.8 1025

9.8 105.1 10.7 1030

11.8 87.0 7.37 1027

13.8 74.4 5.39 1027

15.8 65.1 4.12 1029

17.8 57.6 3.24 1025

19.8 52.0 2.63 1030

PROCESSING THE DATA 1. If the volume is doubled from 5.0 mL to 10.0 mL, 7. What experimental factors are assumed to be what does your data show happens to the constant in this experiment? pressure? Show the pressure values in your The number of gas particles (n) and the answer. temperature (T) remain constant in this If the volume is doubled, the pressure decreases experiment. by approximately half. 8. One way to determine if a relationship is inverse 2. If the volume is halved from 20.0 mL to 10.0 or direct is to find a proportionality constant, k, mL, what does your data show happens to the from the data. If this relationship is direct, pressure? Show the pressure values in your k = P/V; if it is inverse, k = P × V. answer. Based on your answer to Question 4, calculate If the volume is halved, the pressure doubles. k for the seven ordered pairs in your data table (divide and multiply the P and V values). Show 3. If the volume is tripled from 5.0 mL to 15.0 mL, the answers in the third column of the Data and what does your data show happens to the Calculations table. pressure? Show the pressure values in your answer. Student answers should show that P × V yields an almost constant value. See Table 1. If the volume is tripled, the pressure decreases by one third. 9. How constant were the values for k that you obtained in Question 8? Good data may show 4. From your answers to the first three questions some minor variation but the values for k should and the shape of the curve in the plot of pressure be relatively constant. vs. volume, do you think the relationship between the pressure and volume of a confined gas is Students should calculate the range of values they direct or inverse? Explain your answer. obtained for k. It should be small. The relationship between pressure and volume is When the temperature and number of particles inverse. As one value increases, the other value are held constant, the volume of a gas varies decreases. inversely with the pressure (P × V = k). 5. Based on your data, what would you expect the 10. Using P, V, and k, write an equation representing pressure to be if the volume of the syringe was Boyle’s law. Write a verbal statement that increased to 40.0 mL? Explain or show work to correctly expresses Boyle’s law. support your answer. The product of pressure and volume for a gas at Answers will vary but should be half of the constant temperature is always the same value pressure obtained at the 20.0-mL reading. (P × V = k). 6. Based on your data, what would you expect the pressure to be if the volume of the syringe was decreased to 2.5 mL? Answers will vary but should be twice the pressure obtained at the 5.0-mL reading.

he primary objective of this experiment is to determine the relationship between the pressure and volume of a confined gas. The gas we use will T be air, and it will be confined in a syringe connected to a gas pressure MATERIALS sensor (Figure 1). When the volume of the syringe is changed by moving the piston, a change occurs in the pressure exerted by the confined gas. data collection device gas syringe, 20 mL This pressure change will be monitored using a gas pressure sensor. It is assumed sensor, gas pressure that temperature will be constant throughout the experiment. Pressure and volume data pairs will be collected during this experiment and then analyzed. From the data and graph, you should be able to determine what kind of mathematical relationship exists between the pressure and volume of the confined gas. Historically, this relationship was first established by Robert Boyle in 1662 and has since been known as Boyle’s law.

Figure 1. Gas pressure sensor

PROCEDURE 1. Prepare the gas pressure sensor and an air sample for data collection. a. Connect the gas pressure sensor to your data collection device. b. With the 20-mL syringe disconnected from the gas pressure sensor, move the piston of the syringe until the front edge of the inside black ring (indicated by the arrow in Figure 1) is positioned at the 10.0-mL mark. c. Attach the 20-mL syringe to the gas pressure sensor. 2. Set up the data-collection mode. a. Collect data as Events with Entry where pressure will be collected from the sensor and volume in mL will be manually entered.

Look at the syringe; 3. To obtain the best data possible, you will need to correct the volume readings its scale reports its from the syringe. To account for the extra volume in the system, you will own internal volume. However, that volume need to add 0.8 mL to your syringe readings. is not the total volume For example, with a 5.0 mL syringe volume, the total volume would be of trapped air in your system because 5.8 mL. It is this total volume that you will need for the analysis. there is a little bit of space inside the 4. You are now ready to collect pressure and volume data. It is easiest if one pressure sensor. person takes care of the gas syringe and another enters volumes. a. Start data collection. b. Move the piston so the front edge of the inside black ring (Figure 2) is positioned at the 5.0 mL line on the syringe. Hold the piston firmly in this position until the pressure value displayed on the screen stabilizes. c. Capture the pressure data point and enter “5.8”, the gas volume (in mL) as the corresponding value. Remember, you are adding 0.8 mL to the volume of the syringe for the total volume. Record this pressure-volume data pair. d. Continue this procedure using syringe volumes of 7.0 mL, 9.0 mL, 10.0 mL, 11.0 mL, 11.0 mL, 15.0 mL, 17.0 mL, and 19.0 mL. Remember to add the 0.8 mL to each volume before entering. e. Stop data collection. 5. When data collection is complete, a graph of pressure vs. volume will be displayed. To examine the data pairs on the displayed graph, tap any data point. As you tap each data point, the pressure and volume values are displayed to the right of the graph. Record these pressure and volume data values in your data table.

PROCEDURE (CONTINUED)

Figure 2. Inside black ring positioned at the 5.0 mL line

6. Based on the graph of pressure vs. volume, decide what kind of mathematical relationship exists between these two variables, direct or inverse. To see if you made the right choice: a. Analyze the data by trying different curve fits. b. If you decide that a direct relationship exists, select “Linear Regression” where x is the volume and y is the pressure. c. If you decide that an indirect relationship exists, select “Power” as the fit equation. The curve fit statistics for these two data columns are displayed for the equation in the form

The relationship y = AxB between pressure and volume can be where determined from the value and sign of x = volume the exponent, B. y = pressure A = a proportionality constant B = the exponent of x (volume) in this equation d. If you have correctly determined the mathematical relationship, the regression line should very nearly fit the points on the graph (that is, pass through or near the plotted points). e. Record the equation for the line or curve of best fit.

Optional 7. If directed by your instructor, proceed directly to the Extension that follows Processing the Data.

DATA AND CALCULATIONS

Table 1. Volume vs. Pressure

Constant, k Volume (mL) Pressure (kPa) (P/V)(P × V)

PROCESSING THE DATA 1. If the volume is doubled from 5.0 mL to 10.0 mL, what does your data show happens to the pressure? Show the pressure values in your answer.

2. If the volume is halved from 20.0 mL to 10.0 mL, what does your data show happens to the pressure? Show the pressure values in your answer.

3. If the volume is tripled from 5.0 mL to 15.0 mL, what does your data show happens to the pressure? Show the pressure values in your answer.

4. From your answers to the first three questions and the shape of the curve in the plot of pressure vs. volume, do you think the relationship between the pressure and volume of a confined gas is direct or inverse? Explain your answer.

PROCESSING THE DATA (CONTINUED) 5. Based on your data, what would you expect the pressure to be if the volume of the syringe was increased to 40.0 mL? Explain or show work to support your answer.

6. Based on your data, what would you expect the pressure to be if the volume of the syringe was decreased to 2.5 mL?

7. What experimental factors are assumed to be constant in this experiment?

8. One way to determine if a relationship is inverse or direct is to find a proportionality constant, k, from the data. If this relationship is direct, k = P/V; if it is inverse, k = P × V. Based on your answer to Question 4, calculate k for the seven ordered pairs in your data table (divide and multiply the P and V values). Show the answers in the third column of the Data and Calculations table.

PROCESSING THE DATA (CONTINUED) 9. How constant were the values for k that you obtained in Question 8? Good data may show some minor variation but the values for k should be relatively constant.

10. Using P, V, and k, write an equation representing Boyle’s law. Write a verbal statement that correctly expresses Boyle’s law.

EXTENSION To confirm that an inverse relationship exists between pressure and volume, a graph of pressure vs. reciprocal of volume (1/volume) may also be plotted. To do this you will need to create a third column in your data table that contains the values equal to 1/volume. Calculate regression statistics and plot a best fit regression line on your graph of pressure vs. 1/volume. The linear-regression statistics for these two data columns are displayed in the form

y = mx + b

where x = 1/volume y = pressure m = a proportionality constant b = y-intercept If the relationship between pressure and volume is an inverse relationship, the graph of pressure vs. 1/volume should be direct; that is, the curve should be linear and pass through (or near) the origin. Examine your graph to see if this is true for your data.