Page 31 - Modern Analytical Chemistry

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14 Modern Analytical Chemistry

significant figures are included. To avoid ambiguity we use scientific notation. Thus,
2
2
1 ´10 has one significant figure, whereas 1.0 ´10 has two significant figures.
For measurements using logarithms, such as pH, the number of significant
figures is equal to the number of digits to the right of the decimal, including all
zeros. Digits to the left of the decimal are not included as significant figures since
they only indicate the power of 10. A pH of 2.45, therefore, contains two signifi-
cant figures.
Exact numbers, such as the stoichiometric coefficients in a chemical formula or
reaction, and unit conversion factors, have an infinite number of significant figures.
A mole of CaCl 2 , for example, contains exactly two moles of chloride and one mole
of calcium. In the equality
1000 mL =1 L
both numbers have an infinite number of significant figures.
Recording a measurement to the correct number of significant figures is im-
portant because it tells others about how precisely you made your measurement.
For example, suppose you weigh an object on a balance capable of measuring
mass to the nearest ±0.1 mg, but record its mass as 1.762 g instead of 1.7620 g.
By failing to record the trailing zero, which is a significant figure, you suggest to
others that the mass was determined using a balance capable of weighing to only
the nearest ±1 mg. Similarly, a buret with scale markings every 0.1 mL can be
read to the nearest ±0.01 mL. The digit in the hundredth’s place is the least sig-
nificant figure since we must estimate its value. Reporting a volume of 12.241
mL implies that your buret’s scale is more precise than it actually is, with divi-
sions every 0.01 mL.
Significant figures are also important because they guide us in reporting the re-
sult of an analysis. When using a measurement in a calculation, the result of that
calculation can never be more certain than that measurement’s uncertainty. Simply
put, the result of an analysis can never be more certain than the least certain mea-
surement included in the analysis.
As a general rule, mathematical operations involving addition and subtraction
are carried out to the last digit that is significant for all numbers included in the cal-
culation. Thus, the sum of 135.621, 0.33, and 21.2163 is 157.17 since the last digit
that is significant for all three numbers is in the hundredth’s place.
135.621 +0.33 +21.2163 =157.1673 =157.17

When multiplying and dividing, the general rule is that the answer contains the
same number of significant figures as that number in the calculation having the
fewest significant figures. Thus,
22 91 ´ 0152
.
.
.
= 0 21361 = 0 214
.
.
16 302
It is important to remember, however, that these rules are generalizations.
What is conserved is not the number of significant figures, but absolute uncertainty
when adding or subtracting, and relative uncertainty when multiplying or dividing.
For example, the following calculation reports the answer to the correct number of
significant figures, even though it violates the general rules outlined earlier.
101
.
= 102
99