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Graphical Representation of Data Flipbook PDF
A complete graph consists of a series of such data points, as shown in Figure 3(a). When all the points have been plotte
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Graphical Representation of Data prepared by NORMAN E. GRISWOLD, Nebraska Wesleyan University edited by H. A. NEIDIG and J. N. SPENCER adapted by L. S. Quimby and M. Morgan The Nature of This Investigation: The basic principles of graphing and the accompanying terminology are described. Instructions are given for the preparation and interpretation of different types of plotted data.
INTRODUCTION Science is devoted to finding answers to questions by carrying out experiments which provide information about various phases of any given problem. After considerable information has been collected, attempts are made to correlate the data into simple generalizations. Very frequently, these attempts to correlate experimental data involve making a graph. If properly plotted, graphs are very useful for discovering fundamental relations in science. Graphs can also be used to predict values that are difficult to determine experimentally. This module describes the basic principles of graph-making, the terminology, and some of the uses of graphs. A graph is a diagram which shows the relative sizes of numerical quantities in picture form so that they can be more easily understood. There are several types of graphs (see Figure 1), but the simplest and the most widely used type is the line graph. This module is concerned only with 10.0
15 12 9 6 3 0 -3 -6 -9 -12 -15
12% 3%
7.5
13%
5.0
72%
2.5 0
2 4 6 8 (a) Line graph
10
0 (b) Bar graph or histogram
preparing and interpreting line graphs. Figure 1. Three types of graphs.
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(c) Circle graph
TERMINOLOGY Line graphs are started by drawing two lines at right angles to each other such that one line is vertical and the other is horizontal. Each of these lines is called an axis. By convention, the vertical axis is called the y-axis or ordinate and the horizontal axis is called the x-axis or abscissa, as shown in Figure 2. These conventions correspond to the Cartesian coordinates of mathematics. The length of an axis is divided into a number of scale unites, each of which is of identical size. However, the size of the scale units on the x-axis may be different from the size of the scale units on the y-axis, as will be shown in some later examples. The numbers for one kind of data are displayed on one axis and numbers for the other kind of data are displayed on the other axis. The point at which the zero scale unite of the y-axis intersects the zero scale unit of the x-axis is called the origin. Thus, at the origin, x = 0 and y = 0. Data points are plotted by placing a dot at a point directly across from the y-axis value and directly above (or below) the corresponding x-axis value. For example, in Figure 2, the data point P represents a plot of y = 20 and x = 25. Point P is located by beginning at the origin, counting horizontally along the x-axis 25 units, and then vertically 20 units. Negative numbers can also be plotted on line graphs. Point Q in Figure 2 represents a plot of y = -15 and x = -20.
y-axis or ordinate
} data point, P 20 origin
-20
-10
10
20
30
-10 data point, Q -20
Figure 2. Terminology for a line graph.
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}
-30
10 x-axis or abscissa
A complete graph consists of a series of such data points, as shown in Figure 3(a). When all the points have been plotted, either a smooth curve or, if possible, a straight line is drawn which best represents all the points. This is shown in Figure 3(b). It can been seen that the line for the graph in Figure 3(b) does not pass through the origin, but instead crosses the y-axis at the value y = 32 when x = 0. This point is known as the y-intercept. Temperature, °C
100 75 50 25 0
0
Temperature, °C
Data
75
150
225
100
°F
°C
212
100
176
80
140
60
104
40
68
20
32
0
300
75 50 25 0
0
75
150
225
300
Temperature, °F
Temperature, °F
(a) Data plotted
(b) Line drawn to represent data.
Figure 3. Obtaining a graph from plotted data. How does one decide what type of data to plot on each axis? Experiments are normally planned so that the experimenter varies one property of the system under study and the measures the corresponding effect on another property. For example, if a study is being made of the effect of temperature on the solubility of a salt, the experimenter would vary the temperature of the system and would determine the solubility of the salt at each new temperature. The quantity which is changed by the experimenter is referred to as the independent variable and the other variable, the measured one, is called the dependent variable. It is customary to use the y-axis of a line graph for the dependent variable and the x-axis for the independent variable. A line graph representing the measured solubility (dependent variable) of potassium nitrate at several temperatures (independent variable) is shown in Figure 4.
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100
Solubility, g per 100 g H2O
90 80
Data
70
Temp.
Sol’y
0
15
10
22
20
30
30
46
40
66
10
50
86
0
60 50 40 30 20
0
10
20
30
40
50
60
Temperature, °C
Figure 4. The change in solubility of potassium nitrate with temperature. PLOTTING LINE GRAPHS Below are some general rules and the steps to follow when plotting a line graph. 1. Allow plenty of space for the graph. The first rule for drawing graphs is to make them as large as possible. Large graphs are easier to prepare and also are easier to interpret after the graph is drawn. A whole sheet of graph paper should be used for each graph. 2. Draw the axes. Axes should be drawn using a ruler to obtain straight lines. Space should be allowed along the left-hand side and along the bottom for labeling the axes. 3. Determine which data to plot on each axis. Recall that the dependent variable is usually represented by the y-axis and the independent variable is represented by the x-axis. 4. Determine how to display numbers on the axes. The next step is to figure out how to display the numerical quantities on each axis. Should each axis contain values from zero to the highest value? Should one interval on both axes be used to represent the same size unit? Some sample data may be used to answer these questions. Suppose that the following data are to be plotted: x y
40 50
41 55
42 60
page 4
43 65
44 70
If each axis were started at zero and the intervals on both axes were the same, the graph in Figure 5(a) would be obtained. Clearly, in Figure 5(a) a lot of space is wasted by requiring that each axis be numbered starting from zero. Thus, it seems much more appropriate to begin numbering each axis near the lowest value to be plotted. If this is done for the sample data, the result shown in Figure 5(b) is obtained. The graph in Figure 5(b) represents a considerable improvement, but might be even easier to read if the units on the x-axis were spread out as shown in Figure 5(c). Thus, it is often unnecessary to have identical intervals on both axes, or to start numbering each axis at zero. These conclusions make use of the rule which urges that graphs me made as large as possible. It should be pointed out that the intervals on each axis should be chosen so as to make plotting as easy as possible. For example, if numerical values ranging from 200 to 375 units are to be plotted along an axis which contains 25 intervals, the first inclination might be to determine the size of each interval by
(375 - 200) units = 7 units/interval. 25 intervals Data: x
41
42
43
44
45
y
50
55
60
65
70
(b) Better
(a) Bad
(c) Best
70
70
70
56
65
65
60
60
55
55
42 28 14 0
0
10 20 30 40 50
50
40
45
50
50
40 41 42 43 44 45
Figure 5. Some good and bad ways to plot data. However, 7 is not a convenient unit to use as a base because it is difficult to divide an interval into seven equal parts for plotting purposes. It would be much better to number the axis with 10 units per interval, especially if the graph paper being used is ruled one centimeter per interval, as is often the case. A metric ruler can then be used to measure out fractional parts of a numbered interval, as shown in Figure 6. As show in Figures 2 through 6, it is better not to try to number each scale unit because of lack of space. Instead, some convenient multiple is usually selected and numbers are placed at these intervals only.
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20 24 units
10
1
2
3
10
20
30
0
40
50
Figure 6. Using a metric ruler to measure fractional parts of numbered intervals 5. Label axes. When the axes are numbered properly, they should be labeled clearly with the type of property being plotted and the units being used. An example of the use of proper labels is shown in Figure 4, where the y-axis is labeled as “Solubility, g per 100 g H2O” and the x-axis is labeled as “Temperature, °C.” 6. Plot the points and draw the line or curve. When the axes are properly numbered and labeled, the data can be plotted as described earlier. When all the points have been plotted, either a smooth curve or, if possible, a straight line is drawn which best represents all the points. It is seldom useful to draw a broken line from point to point on a line graph. If some points cannot be made to fall on a straight line or smooth curve, the line or curve is drawn in such a way that there are about as many stray points on one side of the line as on e the other. This is also illustrated in Figure 4. Thus, a line graph represents an "averaging" of data. After the graph has been draw, how can it be interpreted? The next section describes the interpretation of a straight-line graph and then considers some graphs which are not linear. INTERPRETATION OF GRAPHS Straight-Line Graphs It is easiest to work with a plot which is a straight line, not only because such a plot is easier to draw but also because a straight line can be easily extended or extrapolated to points not determined experimentally. An example of extrapolation is shown in Figure 7.
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25 20 Volume, Liters
Extrapolation 15
Data: T V 200 10 300 15 400 20 500 25
10 5
0
100
200
300
400
500
Temperature, K Figure 7. Extrapolation of a linear graph. All straight-line graphs can be described by a general equation of the form y = ax + b
(Eq. 1)
in which x and y are variables and a and b are constants. For example, if a = 4 and b = 3, the equation becomes y = 4x + 3
A graph for this equation can be plotted by assigning values such as x = 0, 1, 2, and 3 and then solving for y in each case, as shown in Table 1. A plot of this equation is shown in Figure 8. Table 1. Values for x and y in the equation y = 4x +3. if x =
the equation becomes
and y =
0
y = 4(0) + 3
3
1
y = 4(1) + 3
7
2
y = 4(2) + 3
11
3
y = 4(3) + 3
15
page 7
As can be seen in Figure 8 and in Table 1, y has a value of 3 when x = 0. It is also true that b = 3 in this equation. This relationship between the numerical value of b and the value of y at x = 0 turns to be true for all straight-line graphs. Thus, the constant bin Equation (1) is said to represent the intercept on the y-axis. The y-intercept is the value of y when x = 0 for a straightline graph. In Equation (1) the constant a is called the slope of the line and is found by dividing the difference between any two values of y read the graph by the difference between the corresponding values for x. Values that are on the line, rather than data points, should be used for determining the slope because the line is the best representation of all the data, while individual data points may not even be on the line. By using some values from the graph in Figure 8, !y
12.0 – 8.0
slope =
= !x
4.0 =
2.3 – 1.3
!y
1.0
14.0 – 4.0
slope =
= !x
= 4.0,
10.0 =
2.8 – 0.3
= 4.0. 2.5
The latter calculation is diagrammed in Figure 8. Notice that the slope of this line is the same for any set of points. Normally, however, the slope is determined by using widely separated points on the graph for greater accuracy.
y
14
slope =
10 =
!x
= 4 2.5
12 10 8 6 4 2
!x = 2.8 – 0.3 = 2.5 1
2
!y = 14 – 4 = 10
!y 16
3
Figure 8. Plot of the equation y = 4x + 3
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x
To summarize, any straight-line graph can be represented by an equation of the form y = ax + b where a represents the slope of the line and b represents the intercept on the y-axis. Now, what are some of the linear equations which are found in a study of chemistry? The gas laws are usually studied quite early in chemistry. One of thee is Charles’ Law. This law states that the volume of a gas is directly proportional to the absolute temperature of the gas. Mathematically, this relationship can be expressed as
V = kT
(Eq. 2)
This equation has the form y = ax + b where y corresponds to V, x = T, a = k and b = 0. Thus, a graph of Charles’ Law would involve a plot of volume (V) versus absolute temperature (T), and the line would have a slope k and would intersect at the origin where V = 0 and T = 0 (see Figure 7). Where linear relationships such as this exist, it is not necessary to carry out experiments under extreme conditions. Volumes at very high temperatures or very low temperatures can easily be determined from a graph of Equation (2) by extrapolating (extending) the line. This technique was illustrated in Figure 7. This equation which describes the density of a substance is expressed as
D = m/V
or
m = DV
(Eq. 3)
A graph of values of the mass (m) of a solid substance and the corresponding volumes (V) would provide a straight line, the slope of which is the density (D). Another equation of the same form as Equation (2) is the equation which describes the rates of certain simple reactions as a function of changes in concentration. For example, for simple onestep reactions such as PCl5(g) ---> PCl3(g) + Cl2(g), the rate (R) of the reaction is directly proportional to the molar concentration of PCl5 (written as [PCl5]), so the rate equation for this reaction is a linear equation which is expressed as R = k [PCl5] The use of graph to determine the rate constant (k) of this simple reaction and of other kinds of reactions is often studied under the topic reaction mechanisms or reaction kinetics. Not all graphs in chemistry have lines with a y-intercept at the origin. The conversion of temperatures on the Fahrenheit scale (F) to temperatures on the Celsius scale (C) can be accomplished using the equation
F = (9/5)C + 32. (Eq. 5)
page 9
Equation (5) has the same form as Equation (1) with F corresponding to y and C corresponding to x. Here the y-intercept is at 32 (see Figure 3). The relationship between an equilibrium constant (K) and the absolute temperature (T) is an equation that appears somewhat more complicated.
log K = −
ΔH ⎛ 1 ⎞ ΔS , ⎜⎝ ⎟⎠ + 2.3R T 2.3R
(Eq. 6)
but this equation also has the same form as Equation (1). A graph of log K versus 1/T might be expected to result in a straight line if delta H and delta S are constant. To summarize, Equations (2), (3), and (4) can be described by straight-line graph in which the line passes through the origin. Equations (5) and (6) can be described by straight-line graph in which the line does not pass through the origin. A determination of the slope and/or y-intercept of these graphs can lead to quantities which have definite physical meaning. Graph with Curved Lines Some of the important relationships in chemistry do not have the form of Equation (1) and therefore do not give straight-line graphs. An example is provided by the gas law known as Boyle’s Law, expressed mathematically as
PV = k
(Eq. 7)
A plot of this equation has a slope known as hyperbola, as shown in Figure 9 (a). A large number of data points is required to plot this graph accurately and such a graph is not very easily extrapolated to other values, so chemists usually try to convert nonlinear functions of this type into linear relationships if possible. For example, Boyle’s Law can be converted into a linear function by writing Equation (7) as
V = k (1/P).
(Eq. 8)
Equation (8) has the same form as Equation (2), (3), and (4), and the plot of V versus 1/P describes a straight-line graph in which the line passes through the origin and has a slope equal to k. This us shown in Figure 9.
page 10
40
30
30 Volume, liters
Volume, liters
40
20
20
10
10
0
0 0
2
4
6
0.0
8
0.2
0.4
Pressure, atm
0.6
0.8
1/Pressure, atm
(a) Graph of V vs P.
1.0
1.2
-1
(b) Graph of V vs 1/P
Figure 9. A plot of Boyle’s Law for 1 mole of hydrogen at 0°C. Another equation, which appears quite complex because it involves an exponential term, is
k = Ae− Ea
RT
(Eq. 9)
This relates the rate constant (k) of a reaction to the absolute temperature (T). Equation (9) can be changed into the form
log k = −
Ea ⎛ 1 ⎞ ⎜ ⎟ + B 2.303R ⎝ T ⎠
(Eq. 10)
Equation (10) is a linear relationship similar to equation (6) and provides a graphical method for determining the activated energy (Ea) of the reaction. Not all nonlinear graphs must be converted to linear forms, however. Some examples of this situation are shown in Figure 4 and Figure 10. These graphs represent chemical behavior with changes in temperature, time or acidity, and do not require extrapolation or the calculation of a slope in order to yield their most important information. Thus, there is no advantage to be gained by having them in linear form. Nonetheless, the same rules for plotting good graphs apply to these situations.
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120
14 12 10
80
8
pH
Temperature, °C
100
60
6 40
4
20
2 0
0 0
5
10
15
20
25
30
0
10
20
30
40
Time, sec
Volume titrant, mL
(a) Graph of V vs P.
(b) Graph of V vs 1/P
50
Figure 10. Two common graphs with curving lines. Summary of Procedure for Plotting Graphs 1. Allow plenty of space for the graph. 2. Draw the axes. The vertical or y-axis is called the ordinate; the horizontal or x-axis is called the abscissa. 3. Determine which type of data to plot on each axis. Normally the dependent variable is plotted on the ordinate and the independent variable is plotted on the abscissa. 4. Determine how to display numerical quantities along each axis. (a) It is not necessary to start at zero. (b) Numerical intervals on one axis need to be the same as on the axis. (c) Intervals should be selected to be of a convenient size for ease of plotting and graph interpretation. 5. Label each axis clearly with the type of data being plotted and the units being used. 6. Plot the points. 7. Draw the straight line or curve which best represents the plotted points. 8. Interpret the graph.
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60
EXERCISES A sample of gas was trapped in a cylinder by a piston. The volume of the gas was changed and the resulting pressure was measured. The results are shown in Table 1 below:
Table 1: Data for Problems 1 and 2 Volume (mL)
Pressure (Torr)
33.3
600
25.0
800
20.0
1000
16.7
1200
14.3
1400
12.5
1600
11.1
1800
10.0
2000
1. Prepare a graph of the data given in the table above. Include a best fit curve and determine the equation of that curve.
2. What could be done to represent these data in a straight line? Try it. Include a best fit line and determine the equation of that line. This sample of gas was then held at a constant volume while its temperature was increased. As the temperature rose, the pressure of the gas was measured. The results are shown in Table 2.
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Table 2: Data for Problems 3 and 4 Temperature (K)
Vapor Pressure (Torr)
273
4.6
293
17.5
313
55.0
333
149.2
353
355.5
373
760.0
3. Prepare a graph of the vapor pressure of water at the temperatures given in the Table 2.
4. Now prepare a graph of log P vs. 1/T, where P stands for Pressure and T stands for Temperature. Include a best fit line or curve and determine the equation of that line or curve.
Table 3: Data for Problem 5 Temperature
Solubility KClO3
(°C)
(g/100 g H2O)
0
3
20
8
40
17
60
27
80
39
100
57
5. Prepare a graph of the solubility KClO3 at the temperatures given in Table 3 above. Connect your data points with a smooth curve.
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Below are some data concerning the rate of a chemical reaction. The letter C represents the concentration of a reactant.
Table 4: Data for Problems 6 and 7 Time (min)
C (mol/L)
Log C
0
2.00
0.301
1
1.41
0.149
2
1.00
0.000
3
0.70
-0.155
4
0.49
-0.310
5
0.35
-0.456
6. Prepare a graph of C vs. time. Connect your data points with a smooth curve.
7. Prepare a graph of log C vs. time and include a best fit line or curve and determine the equation of that line or curve.
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