Page 87 - Modern Analytical Chemistry

1400-CH04 9/8/99 3:54 PM Page 70

70 Modern Analytical Chemistry

SOLUTION
Letting M a and M b represent the molarity of the final solutions from method
(a) and method (b), we can write the following equations

)
(
(. M 1000 mL)
10
.
.
M a = 0 0010 M =
1000 0 mL
.
(
( 1 0 M 20 00 mL 25 00 mL)
.
)
.
.
)(
.
M b = 0 0010 M =
.
(
.
( 1000 0 mL 500 0 mL)
)
Using the tolerance values for pipets and volumetric flasks given in Table 4.2,
the overall uncertainties in M a and M b are
2 2
æ ö æ . 0 006 ö æ . 03 ö
s R
.
ç ÷ = ç ÷ + ç ÷ = 0 006
è R ø è 1 000. ø è 1000 0. ø
M a
2 2 2 2
æ ö æ . 003 ö æ . 003 ö æ . 02 ö æ . 03 ö
s R
.
ç ÷ = ç ÷ + ç ÷ + ç ÷ + ç ÷ = 0 002
è R ø è20 00. ø è25 00. ø è 500 0. ø è 1000 0. ø
M b
Since the relative uncertainty for M b is less than that for M a , we find that the
two-step dilution provides the smaller overall uncertainty.
4 D The Distribution of Measurements and Results
An analysis, particularly a quantitative analysis, is usually performed on several
replicate samples. How do we report the result for such an experiment when results
for the replicates are scattered around a central value? To complicate matters fur-
ther, the analysis of each replicate usually requires multiple measurements that,
themselves, are scattered around a central value.
Consider, for example, the data in Table 4.1 for the mass of a penny. Reporting
only the mean is insufficient because it fails to indicate the uncertainty in measuring
a penny’s mass. Including the standard deviation, or other measure of spread, pro-
vides the necessary information about the uncertainty in measuring mass. Never-
theless, the central tendency and spread together do not provide a definitive state-
ment about a penny’s true mass. If you are not convinced that this is true, ask
yourself how obtaining the mass of an additional penny will change the mean and
standard deviation.
How we report the result of an experiment is further complicated by the need
to compare the results of different experiments. For example, Table 4.10 shows re-
sults for a second, independent experiment to determine the mass of a U.S. penny
in circulation. Although the results shown in Tables 4.1 and 4.10 are similar, they
are not identical; thus, we are justified in asking whether the results are in agree-
ment. Unfortunately, a definitive comparison between these two sets of data is not
possible based solely on their respective means and standard deviations.
Developing a meaningful method for reporting an experiment’s result requires
the ability to predict the true central value and true spread of the population under
investigation from a limited sampling of that population. In this section we will take
a quantitative look at how individual measurements and results are distributed
around a central value.