Mathematics revision - Chemical Fundamentals of Geology and Environmental Geoscience - page 238

(a) It is easy to see how straight a straight-line graph
is. It is much harder to judge the curvature that a
curved relationship should have.
(b) It is a simple calculation to determine the values of
the constants in the equation (e.g. A , E a ) from a
straight-line graph. (A linear relation between y
and c may be written y = mx + c, where m is the
gradient and c is the y -axis intercept.) It is harder to
extract the values of these constants from a curved
graph.

X X

X

X

X

T /K

Figure A3 Sketch of variation of rate constant with
temperature.

Significant figures

Consider an alternative plot. The Arrhenius equa-
tion may be written in a logarithmic form by taking
natural logs of both sides:

How many decimal places should be quoted, for exam-
ple when tabulating a chemical analysis? The answer
depends on the precision (reproducibility) of the anal-
ysis: a precise analysis will justify quoting more digits
(e.g. 8.465%) than a less precise analysis (e.g. 8.5%).
The number of digits that a measurement's precision
will support is called the number of significant figures .
The value 1.57623 has 6 significant figures, whereas
1.58 has only 3 significant figures.
The accepted practice in tabulating numerical data
(e.g. Table 8.4) is to give the number of significant fig-
ures that most closely represents the precision of the
data. For example, a mass spectrometer may print out
a Sr isotope ratio as 0.704249 but, if the measured preci-
sion is ±0.0005, the ratio is more truthfully reported as
0.7042.
There is no point in writing down more significant
figures than the precision of a measurement will sup-
port (although a poorly managed spreadsheet may
often do this); thus 1.569821 ± 0.0187 is more objec-
tively written as 1.57 ± 0.02.
When two numbers are combined in arithmetical
operations, the answer should be given the same sig-
nificant figures as the less precise component:

E
RT

ln k A =+ − ln ln exp

a

E
RT

(A11)

a

=−
( =+ )

ln A

ycmx

It is clear that when ln k is plotted against 1/ T , the form
of this equation predicts a straight line, with slope
=−

E
R

a

and intercept = ln A (see Figure A4).

Thus there are two reasons for manipulating a
theoretical equation into linear form before attempting
to verify it against experimental results:

X

X

X

X

X

• 3.98595 + 3.2 should be written as 7.2, not 7.18595.
• 4.5 × 6.9877 = 31, not 31.4447.
• 0.79877 ÷ 2.9 = 0.28, not 0.27544.

1
T /K

Further reading

Figure A4 Sketch showing the linear variation of the
logarithm of rate constant versus inverse temperature
(in kelvins).

Waltham, D. (2000) Mathematics: A Simple Tool for Geologists ,
2nd edition. Oxford: Blackwell Science.