2: LAB 2 - INTRODUCTION to GRAPHICAL ANALYSIS
- Page ID
- 505943
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The purpose of this experiment is to
- Gain experience in interpreting data from graphs.
- Given a data set, construct a graph and identify the type of relationship observed.
- Determine the relationship between pressure and volume, and temperature and volume, of a gas using graphical analysis.
- Experimentally determine the relationship between volume/boiling point and percent composition/density using graphical analysis.
Many of the experiments you will be conducting may require you to determine the mathematical relationship between two variables. One way to accomplish this is by constructing a graph. For example, consider the following pairs of data, which will be generically labeled as X and Y. A graph of this data is shown below.
Consider the following pairs of data, which will be generically labeled as X and Y. A graph of this data is shown below. Summarize the relationship.
X |
Y |
|---|---|
|
1 |
2 |
|
2 |
4 |
|
4 |
8 |
|
8 |
16 |
|
16 |
32 |
|
32 |
64 |
Figure 1: A Graph Indicating a Direct Relationship Between Two Variables. Y varies directly with X and can be summarized as: Y = 2x
Solution
As the graph below, plotted using the data from the above table of example 1, shows, Y varies directly with X and can be summarized as Y = 2X.
This is known as a direct relationship between the two variables.
As another example, consider the following generic data.
Consider the following generic data table and graph generated using the following data. Summarize the relationship indicated.
|
X |
Y |
|---|---|
|
32 |
20 |
|
2 |
10 |
|
4 |
5 |
|
8 |
2.5 |
|
16 |
1.25 |
|
32 |
0.625 |
Solution
The graph above, using data from Example 2, shows that as X increases, Y decreases, indicating an inverse relationship. The equation summarizing the data is Y = 20/x. Indicating an Inverse Relationship Between Two Variables. Y varies inversely with X and can be summarized as: Y = 20/x
In this experiment, you will construct graphs from data sets and be asked to identify the relationship between two variables. First, you will learn how to create graphs using certain programs.
Click on the link below (or copy and paste it into your browser) for information on using Excel to construct graphs:
https://www.ablebits.com/office-addins-blog/make-scatter-plot-excel/(opens in new window)
The link below walks you through creating graphs in Google Sheets:
https://www.jotform.com/blog/how-to-make-an-x-y-graph-in-google-sheets/(opens in new window)
If you are proficient with creating graphs using another program, feel free to use that to construct graphs in this experiment.
1) Always wear chemical splash goggles while working with chemicals in this experiment.
2) Be careful when handling hot plates, and use beaker tongs or hot mittens to remove beakers from hot plates.
3) Dispose of all waste in the appropriate waste containers as your instructor indicates.
4) When you finish the lab, clean your work area and return all chemicals and equipment to their appropriate places.
EQUIPMENT* AND CHEMICALS NEEDED
• 100 mL beaker
• 600 mL beaker
• Hot plate
• Celsius Thermometer
• 100 mL graduated cylinder
• 10 mL graduated cylinder
• Milligram balance
• Stirring rod
• Ethanol
• Tap water
• Graphing program of choice
• Non-halogenated waste container
* Images of equipment needed in this lab can be found in the appendix (the equipment may be subject to changes; follow your instructors' directions).
EXPERIMENTAL PROCEDURE
Part I: Exploring Volume and Pressure Relationships in Gases
An experiment was run to determine what happens to the pressure of a gas in a syringe when the volume is increased. Experimental data were given as follows:
Volume (X-axis) |
Pressure (Y-axis) |
|---|---|
|
2 L |
25.0 atm |
|
4 L |
12.5 atm |
|
6 L |
8.3 atm |
|
8 L |
6.3 atm |
|
10 L |
5.0 atm |
|
12 L |
4.2 atm |
The volume of the gas was measured in liters (L), and the pressure was measured in atmospheres (atm). Using your graphing program of choice, construct a graph of volume vs. pressure, include a line to connect the points, and generate an equation to summarize the data.
Part II: Exploring Temperature and Volume Relationships in Gases
An experiment was run to determine what happens to the volume of a gas when the temperature is increased. Data was given as follows:
| Temperature (X-axis) | Volume (Y-axis) |
|---|---|
| 298 K | 25.0L |
| 350 K | 29.4 L |
| 375 K | 31.5 L |
| 425 K | 35.7 L |
| 500 K | 42.0 L |
| 650 K | 54.5 L |
The gas temperature was expressed in kelvins (K), and the volume was measured in liters (L). Using your graphing program of choice, construct a graph of temperature vs. volume, include a line to connect the points, and generate an equation to summarize the data.
Part III. Exploring the Volume and Boiling Point of Water
1) The instructor will assign each group a certain volume of water to use. The volumes of water used will be as follows: 50.0 mL, 100.0 mL, 200.0 mL, 300.0 mL, 400.0 mL, and 500.0 mL. If your group is assigned 50.0 mL of water, use a 100 mL beaker. If your group is assigned one of the other volumes, use a 600 mL beaker.
2) Using a 100 mL graduated cylinder, add the assigned amount of water to the appropriate beaker. Heat the water on a hot plate and set it on high. Once the water boils, record its temperature. Remember to round temperature readings to one digit past the decimal point.
3) Turn off the hotplate and remove the beaker from it using beaker tongs or hot mittens. Once the hotplate is cool, please return it to the appropriate lab drawer.
4) Gather data from the other lab groups and construct a graph of volume (mL) on the x-axis and temperature (oC) on the y-axis.
Part IV. Exploring the Percent Composition and Density of an Ethanol Solution
1) Using a milligram balance, determine the mass of an empty 10 mL graduated cylinder.
2) Add 10.0 mL of water to the graduated cylinder and determine the total mass of the graduated cylinder and the water. (Note: This will be the 0% ethanol reading.)
3) Subtract the mass of the empty graduated cylinder from the total mass to obtain the mass of water.
4) Using the density equation learned in the previous lab, calculate the density of water.
5) Pour the water and add 2.0 mL of ethanol and 8.0 mL of water to the graduated cylinder. Using a glass stirring rod, gently stir the mixture in the graduated cylinder for 1 minute. Then, determine the mass of the mixture and the graduated cylinder. (Note: This mixture consists of 20% ethanol by volume. We determine this by dividing the volume of ethanol, which is 2.0 mL, by the total volume of the mixture, which is 10.0 mL, and multiplying the quotient by 100 to obtain the percent.
6) Subtract the mass of the empty graduated cylinder from the total mass to obtain the mass of the 20% mixture.
7) Calculate the density of the 20% ethanol mixture using the density equation. (We divide the mass of the mixture by 10.0 mL, the total volume.)
8) Pour the contents of the mixture into the non-halogenated waste container as directed by your instructor.
9) Repeat steps 5-8 using 4.0 mL of ethanol and 6.0 mL of water (40% ethanol mixture).
10) Repeat steps 5-8 using 6.0 mL of ethanol and 4.0 mL of water (60% ethanol mixture).
11) Repeat steps 5-8 using 8.0 mL of ethanol and 2.0 mL of water (80% ethanol mixture).
12) Repeat steps 5-8 using 10.0 mL of ethanol (100% ethanol mixture).
13) Construct a graph of percent ethanol (x-axis) vs. density (y-axis). Include a line connecting the points and an equation summarizing the data.
Include all the graphs generated as attachments
PRE-LAB QUESTIONS
Name: ____________________________________
1) Which program will you use to create graphs in this experiment? Briefly explain your decision.
2) Briefly describe the difference between a direct and inverse relationship between data pairs.
3) Consider the following generic data:
| X | Y |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 4 | 16 |
| 8 | 64 |
| 16 | 256 |
| 32 | 1024 |
Using your program of choice, construct a graph of X vs. Y, include a line to connect the points, and generate an equation to summarize the data. (Include your graph as an attachment.)
For parts I and II, include your graphs as attachments.
Part III: Exploring the Volume and Boiling Point of Water
Volume of Water (mL) |
Boiling Point (oC) |
|---|---|
|
50.0 |
|
|
100.0 |
|
|
200.0 |
|
|
300.0 |
|
|
400.0 |
|
|
500.0 |
Include your graph of volume vs. boiling point as an attachment.
Part IV: Exploring Percent Composition and Density of an Ethanol Solution
Mass of empty 10.0 mL graduated cylinder =
Percent Ethanol |
Mass of Graduated Cylinder and Solution |
Mass of Solution |
Volume of Solution |
Density of Solution (g/mL) |
|---|---|---|---|---|
|
0% |
10.0 mL |
|||
|
20% |
10.0 mL |
|||
|
40% |
10.0 mL |
|||
|
60% |
10.0 mL |
|||
|
80% |
10.0 mL |
|||
|
100% |
10.0 mL |
Include your graph of percent ethanol vs. density as an attachment.
1) What new skills did you learn from performing this experiment, and what relevance can this have in your everyday life?
2) Refer to the graph generated in part I. How would you describe the relationship between the volume and pressure of a gas?
3) Use the graph and equation generated in part I to predict the pressure that 3 L of the gas would exert.
4) Refer to the graph generated in part II. How would you describe the relationship between the temperature and volume of a gas?
5) Use the graph and equation generated in part II to predict the volume of the gas at a temperature of 100 K.
6) Refer to the graph generated in part III. How would you describe the relationship between water's volume and boiling point?
7) Refer to the graph generated in part IV. How would you describe the relationship between the percentage of ethanol in the mixture and the density of the mixture?
8) What is the difference between a linear and an exponential relationship? Out of the five graphs generated in this experiment (one from the pre-lab and four from the lab component), which of the graphs, if any, are linear? Which are exponential?