68
Graphical Representation
Frequently, in order to use a convenient scale and to construct a graph of a
reasonable size, one must place the origin of thex-axis away from the lower left
corner of the graph paper When you have such a graph, you cannot obtain the
value ofb, they intercept, directly from the graph This situation need cause no
difficulty, however, because m (the slope) still can be obtained from the graph
Once obtained, it can be used, along with any arbitrarily chosen point on the
line, to solve for the value of b The following problem illustrates this
PROBLEM:
The following data show how the density D of mercury vanes with the Kelvin
temperature T Plot these data From the resulting graph, determine the mathematical equation that shows how D vanes with T
T (°K)
263
273
283
293
303
313
D (g/ml)
13 6201
13 5955
13 5708
13 5462
13 5217
13 4971
SOLUTION.
The graph of these values of D and T is shown in Figure 6-3 Note that a greatly
expanded scale is needed along the> axis to show in a reasonable way the small
changes in D with temperature also note how far from absolute zero the scale
along the \ axis starts The slope m is calculated by taking two convenient points
on the line as shown, then taking the ratio of the two sides of the triangle formed
from these points in the figure it is the ratio 0 0550/-22 5 = -0 00244 - m Note
also that the sign of the slope is negatn e as it must always be when the line slopes
downward from left to nght—that is, when v decreases with increasing values of v
Because the mathematical equation for this line must be of the form
D - mT + b
we can take any point on the line (300 and 13 530 for convenience) and use it with
the value of m = -0 00244 to solve for the value of b
b - D ~ mT
= 13 530
(-0 00244X300) - 13 530 + 0 732
- 14 262
The equation that relates D to T for mercury over this range of temperatures
thus is
D - -0 00244T + 14 262
It often is desirable to recast data into a different form before making a graph,
in order to obtain a straight-line graph The data in Table 11-1 can be used to
illustrate this procedure With the data plottedjust as they are given in the table
(vapor pressure of water expressed m torr, and the corresponding temperature
in °C), we obtain the graph shown in Figure 6-4
Graphical Representation
Frequently, in order to use a convenient scale and to construct a graph of a
reasonable size, one must place the origin of thex-axis away from the lower left
corner of the graph paper When you have such a graph, you cannot obtain the
value ofb, they intercept, directly from the graph This situation need cause no
difficulty, however, because m (the slope) still can be obtained from the graph
Once obtained, it can be used, along with any arbitrarily chosen point on the
line, to solve for the value of b The following problem illustrates this
PROBLEM:
The following data show how the density D of mercury vanes with the Kelvin
temperature T Plot these data From the resulting graph, determine the mathematical equation that shows how D vanes with T
T (°K)
263
273
283
293
303
313
D (g/ml)
13 6201
13 5955
13 5708
13 5462
13 5217
13 4971
SOLUTION.
The graph of these values of D and T is shown in Figure 6-3 Note that a greatly
expanded scale is needed along the> axis to show in a reasonable way the small
changes in D with temperature also note how far from absolute zero the scale
along the \ axis starts The slope m is calculated by taking two convenient points
on the line as shown, then taking the ratio of the two sides of the triangle formed
from these points in the figure it is the ratio 0 0550/-22 5 = -0 00244 - m Note
also that the sign of the slope is negatn e as it must always be when the line slopes
downward from left to nght—that is, when v decreases with increasing values of v
Because the mathematical equation for this line must be of the form
D - mT + b
we can take any point on the line (300 and 13 530 for convenience) and use it with
the value of m = -0 00244 to solve for the value of b
b - D ~ mT
= 13 530
(-0 00244X300) - 13 530 + 0 732
- 14 262
The equation that relates D to T for mercury over this range of temperatures
thus is
D - -0 00244T + 14 262
It often is desirable to recast data into a different form before making a graph,
in order to obtain a straight-line graph The data in Table 11-1 can be used to
illustrate this procedure With the data plottedjust as they are given in the table
(vapor pressure of water expressed m torr, and the corresponding temperature
in °C), we obtain the graph shown in Figure 6-4