Common Mathematical Operations In Chemistry Worksheets With Answers Page 4
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14A-4
appendix a
•
Common Mathematical Operations in Chemistry
graphIng Data In thE ForM
oF a straIght lInE
Visualizing changes in variables by means of a graph is used throughout science. In
many cases, it is most useful if the data can be graphed in the form of a straight line.
Any equation will appear as a straight line if it has, or can be rearranged to have, the
following general form:
y mx 1 b
where y is the dependent variable (typically plotted along the vertical axis), x is the
independent variable (typically plotted along the horizontal axis), m is the slope of
the line, and b is the intercept of the line on the y axis. The intercept is the value of
y when x 0:
y m(0) 1 b b
The slope of the line is the change in y for a given change in x:
y
2 y
Dy
Slope (m) 5
2
1
5
x
2 x
Dx
2
1
The sign of the slope tells the direction of the line. If y increases as x increases, m
is positive, and the line slopes upward with higher values of x; if y decreases as x
increases, m is negative, and the line slopes downward with higher values of x. The
magnitude of the slope indicates the steepness of the line. A line with m 3 is
three times as steep (y changes three times as much for a given change in x) as a line
with m 1.
Consider the linear equation y 2x 1 1. A graph of this equation is shown in
14
y 2x 1 1
Figure A.1. In practice, you can find the slope by drawing a right triangle to the line,
12
using the line as the hypotenuse. Then, one leg gives y, and the other gives x. In
3 A 4
the figure, y 8 and x 4.
10
Slope
3 A 4
8
At several places in the text, an equation is rearranged into the form of a straight
⁄
2
4
y
8
line in order to determine information from the slope and/or the intercept. For exam-
ple, in Chapter 16, we obtained the following expression:
6
0
x
4
ln
5 kt
t
2
Based on the properties of logarithms, we have
Intercept
ln [A]
2 ln [A]
kt
0
t
0
2
4
6
2
Rearranging into the form of an equation for a straight line gives
2
ln [A]
2kt 1 ln [A]
Figure A.1
t
0
y
mx 1
b
Thus, a plot of ln [
A
]
vs. t is a straight line, from which you can see that the slope
t
is 2k (the negative of the rate constant) and the intercept is ln [
]
(the natural
A
0
logarithm of the initial concentration of
).
A
At many other places in the text, linear relationships occur that were not shown
in graphical terms. For example, the conversion of temperature scales in Chapter 1
can also be expressed in the form of a straight line:
9
°F
°C 1 32
5
y mx 1 b
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