Page 21 GRAPHING OBJECTIVES: 1. To learn how to present data in graphical form manually (paper-and-pencil) and using computer software. 2. To learn how to interpret graphical data by, a. determining the slope and intercept of a straight-line graph b. interpolating and extrapolating from graphical data. INTRODUCTION: In many scientific experiments the value of one measurable property is deliberately varied and the resultant change in a second property is then measured. The general nature of the relationship between the two variables is often more apparent when it is presented pictorially as a graph, rather than as a table of numerical data. A properly drawn graph also allows predictions to be made about what would have been observed under conditions other than those used in the experiment, and it can be used to identify data points involving significant experimental error. There are certain general principles to remember when presenting data in graphical form. 1. Label the axes of the graph. Identify both the quantity being measured and its units. Generally, the independent variable, the quantity whose value you control, is plotted along the horizontal axis (also called the abscissa or x-axis). The dependent variable, whose value changes as a result of the change in the independent variable, is plotted along the vertical axis (also called the ordinate or y-axis). 2. Choose an appropriate and convenient scale. Generally, the graph should use as much of the available space, e.g., graph paper, as possible. Look at the range of values of each variable. The scale on each axis should start below the minimum data value for the corresponding variable and should extend beyond the maximum data value for that variable. It is generally NOT necessary, and may be undesirable, for the scale on either axis to start at a value of zero. The numerical scale should be indicated along each axis. Each grid space on an axis should represent the same value range. 3. Locate data points by a small dot with a circle drawn around it or at the center of a cross or X (the circle or cross is termed a point protector ). Use of a large dot makes it difficult to determine the actual coordinates of the data point and small dots without point protectors are difficult to see. 4. To emphasize the pattern in the data, draw a smooth line which best fits the data points. Do not simply draw straight lines from one data point to the next. In general chemistry, the relation between the variables will most often be linear and the graph will be a straight line, but other relationships are possible. When drawing the line that best fits the data, it should pass close to all of the data points, but may, in fact, not pass through any individual data point. 5. Choose a descriptive title for your graph. Generally, the title will indicate the use to which the graph will be put (e.g., the title of the graph which you will draw in the Density exercise might be Determination of the Density of a Solid ) or the variables being plotted (e.g., Mass vs. Volume for an Unknown Solid ). In the latter case, the dependent variable (y-axis) should be listed first.
Page 22 GRAPH The data below will be used to illustrate two different methods of creating a graph: plotting the graph on graph paper and using a computer with specialized software (we will use a program called Logger Pro). The following data were obtained when the volume of a gas was measured as the temperature was changed. Temperature ( C) Volume (mm 3 ) 97.0 301 21.0 246 12.0 238 0.0 231-5.0 221-78.0 165 Manual Graphing (using graph paper): Using the graph paper found on page 28, plot the data in the table above. First, choose an appropriate scale and label the axes. Temperature is the independent variable and volume the dependent variable. In order to plot all of the data points the temperature scale must run from below -78 C to above 97 C (a minimum range of 175 C) and the volume scale from below 165 mm 3 to above 301 mm 3 (a minimum range of 136 mm 3 ). To represent these data it is not necessary for either scale to begin at zero. In the example figure below the graph paper has 5 large units along the horizontal axis. Starting at -100 C and allowing each large unit to represent 50 C will spread our temperature scale over most of this axis. It will also make plotting temperatures relatively convenient, since each small unit will represent 10 C. Similar considerations suggest starting the volume scale at 125 mm 3 and allowing each large block along the vertical axis to represent 50 mm 3, thus making each small division equal to 10 mm 3. Volume of a Gas as a Function of Temperature: Determination of Slope 325 X Volume (mm³) 275 225 Y 2 175 1 125-100 -50 0 50 100 150 Temperature ( C)
GRAPH Page 23 After plotting the data points, notice that they lie, more or less in a straight line, so use a ruler to draw the straight line which best fits the data. Adjust the ruler so the edge passes near as many of the data points as possible with roughly as many points above the line as there are points below the line A straight line can be represented by an equation of the form Y = Ax + B, where A is the slope of the graph and B its intercept on the y-axis, i.e., the value of y when x = 0. The slope is defined as the change in the dependent variable (y) divided by the change in the independent variable (x). To find the slope, choose two points on the best fit straight line; for example, points 1 and 2 on the sample graph on page 22. The selected points should be as far apart as possible and should not be measured data points. Subtracting the y-value (in this case the volume) of point 1 from that of point 2 gives y = (280-187) mm 3 = 93 mm 3. Now subtracting the x-value (in this case temperature) of point 1 from that of point 2 gives x = [68 ( 53)] C = 121 C. Note that the subtraction must always be done in the order point 2 minus point 1. The slope is thus A = y x 3 93 mm 121 C mm 0.77 C The line crosses the y-axis (The y-axis is the vertical line corresponding to a value of zero for the independent variable, T = 0 C in our case) at 227 mm 3, which is B. The relation between our variables may now be written as V = 0.77 mm 3 C T + 227 mm 3 Following the instructions above, determine the slope and intercept of the straight line you have drawn on your graph. Show your calculations and record your results on the Homework sheet on page 27 (Item 1). The equation of the straight line that you draw will probably differ slightly from the one determined in the example. Computer (using Logger Pro) The use of a computer and graphing software can greatly facilitate the graphical presentation and analysis of experimental data. The computer can automate many of the mechanical steps of creating the graph, allowing you to focus on its interpretation. The following instructions will guide you in the use of the program Logger Pro to graph the same temperature-volume data used for the manual graph above. Start the Logger Pro program. On the left of the screen you should see a table labeled Data Set with two columns labeled X and Y and to the right you should see a blank graph grid. Manually enter the Temperature data into the X-column and the Volume data into the Y-column (data collected with the LabQuest 2 data logger can be imported into Logger Pro, see page 31 of the LabQuest 2 Data Logger exercise for directions). As you enter the data, you will begin to see it plotted on the graph. To present the graph properly, you must follow the same basic steps used in manual graphing. First, double click on the column heading X. A dialog box labeled Manual Column Options will appear. Click the Column Definition tab. In the Name box, enter: Temperature, in the Short Name box, enter: T, and in the Units box, enter: C (to enter the degree symbol, click the pull-down box to the right of the Units box and click on the degree symbol in the list that appears). Click the Options tab. In the Displayed Precision box, make sure that Decimal Places is selected, then click the blue) box and select 1 (corresponding to the precision of the collected temperature data), and click Done. Note that the X-axis is now properly labeled. Next, double click on the column heading Y. Click the Column Definition tab and enter into the Name box: Volume, into the Short Name box: V, and into the Units box: mm 3 (to enter the superscript, click the pull-down box to the right of the Units box then click on Superscript, and finally select 3 in the list that appears). Click the Options tab. Set Displayed Precision this time to 0 decimal place (corresponding to the precision of the collected volume data), change Color to Black (this makes the printed graph more legible), and click Done or OK. Now both axes are properly labeled. Next, you will need to adjust the way the data are displayed on the graph and to set the appropriate ranges for the scales on the two axes. Click anywhere in the graph to highlight it. On the menu bar, click Options, then Graph Options. Click the Graph Options tab and in the Title box enter: Volume of a Gas as a Function of Temperature. On this same tab, in the box labeled Plot Appearance, confirm that Point Symbols is checked and that Connect Points is 3
Page 24 GRAPH not checked. Click the Axes Options tab. You will notice that Top and Bottom values have automatically been set for the Y (Volume) axis and Left and Right values for the X (Temperature) axis. Although these values could be left as they are, reset them to the same limits used for the paper-and-pencil graph above. In the Scaling box, choose Manual and set the limits Bottom : 125 and Top : 325 for Volume, then set Left : 100 and Right : 150 for Temperature and click Done or OK. To determine the equation for the best fit line and to display it on the graph, click on the Linear Fit button on the tool bar (this button has a diagonal red line superimposed on a blue curve). The line of best fit will appear on the graph along with a box containing the equation for this line (slope and intercept) as well as two additional parameters: the Correlation and RMSE. Both of these parameters are measures of the goodness of the fit. If all the data points lie close to the calculated line, the value of the Correlation (often designated by the letter R) will be close to 1.0. The RMSE is effectively the average distance of the points above or below the line. Position the box so that data points are not obscured (click anywhere inside the box and drag it to where you want it to appear). You are now ready to print your data table and graph. On the menu bar, click on File and then on Page Setup. Check to see that the correct printer has been selected (your instructor will tell you what printer to use) and select Landscape icon as the orientation, then click OK. On the menu bar click on File again and then on Print. In the Print Options dialog box, be sure that Print Footer is checked, in the Name box fill in your name, and click OK and then click OK again or click Print. Record the value of the slope, intercept, and correlation coefficient on page 27 (Item 2). The printed graph and data table should be attached to the Homework sheet and turned in to your instructor. INTERPOLATION AND EXTRAPOLATION Consider the following questions: 1. What will be the value of V when T is 41.0 C? 2. At what temperature will V = 259 mm 3? 3. Theoretically, at what temperature would the gas have zero volume? Each of these questions can be answered by substituting the given value of T or V into the equation of the straight line on your graph and solving for the requested variable. Before continuing, carry out the suggested calculations. Show your work and record your answers on your Homework sheet (Items 3, 4, and 5). These questions can also be answered by interpolating or extrapolating directly on a graph of the sample data provided. Interpolation refers to the process of relating values of the dependent and independent variables which lie between the data points. Extrapolation refers to the process of relating values of the dependent and independent variables which lie beyond any of the data points, i.e., before the first data point or after the last data point. Manual Graphing: To answer the first question, draw a vertical line upward at a temperature of 41 C until it intersects the line you have drawn to fit the data points. Then move horizontally to the volume axis, and read the corresponding value of the volume. Similarly, to answer the second question, draw a horizontal line across from the volume of 259 mm 3 until it intersects the line, and then draw a vertical line from this point down to the temperature axis where we find the desired temperature. Record your answers in the appropriate spaces in Items 3 and 4 on page 27. To the number of significant figures justified by the data, are these results the same as those you calculated from the equation of the best fit straight line? Because answering the third question directly from the graph would require replotting the data, instructions for answering it will be provided only for the computer graphing (one advantage of using the computer is that changes such as this can be made quickly). Computer (using Logger Pro )
GRAPH Page 25 With Logger Pro open and the graph of the sample data that you previously created displayed, click Analyze on the menu bar and click Interpolate on the drop-down menu. A vertical line will appear on your graph with a small circle where it crosses the line of best fit. The coordinates of the center of the circle will be displayed in a box on the graph. The position of this line can be moved by dragging the mouse. Drag the line until the Temperature in the box is 41 C. Read and record the value of Volume (Item 3, page 27). Now drag the line until the Volume is 259 mm 3. Read and record the value of Temperature (Item 4, page 27). Now you are ready to answer the third question posed above. To do so, you will need to adjust the scale on your graph. Click anywhere in the graph to make it active, then click Options on the menu bar; next click Graph Options; and finally click the Axis Options tab. In the section for the y-axis (Volume) choose Manual in the Scaling box and set the limits Bottom : 0 (you want to know the temperature when the volume is zero) and Top : 325 (i.e., leave this limit alone). In the section for the x-axis (Temperature) again choose Manual for Scaling and set Left : 300 and Right : 150, then click Done or OK. The line of best fit will automatically be extended and you can read from the scale where it intersects the x-axis or you can drag the interpolation line as described above until Volume in the box is zero. Read and record the corresponding value of Temperature (Item 5, page 27).
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GRAPH Page 27 Name Section HOMEWORK: 1. Follow the instructions on pages 22-23 to plot the temperature-volume data in the table at the top of page 22 (use the graph paper on the next page). Be sure to label your axes. Use a straight edge/ruler to draw the straight line that you think best fits the data. In the space below, calculate the slope of straight line which you have drawn (show on the graph the two points used for this calculation). Record the value that you calculate for the slope as well as the value of the y-intercept read from the graph: Slope: Y-Intercept: 2. Follow the instructions on pages 23-24 to enter the data from the table at the top of page 22 into Logger Pro and determine the equation of the line of best fit (linear regression). Record the slope, intercept, correlation Root Mean Square Error (RMSE) below (Attach the print out of your graph and data table): Slope (m): Correlation: Intercept (b): RMSE: 3. Determination of V when T is 41 : Calculated from equation of best fit line: From Graph: From Logger Pro : 4. Determination of T when V is 259 mm 3 : Calculated from equation of best fit line: From Graph: From Logger Pro : 5. Determination of T when V is zero: Calculated from equation of best fit line: From Logger Pro :
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