Page 84 - Modern Analytical Chemistry

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Chapter 4 Evaluating Analytical Data 67

EXAMPLE .7
4
For a concentration technique the relationship between the measured signal
and an analyte’s concentration is given by equation 4.5

S meas = kC A + S reag
Calculate the absolute and relative uncertainties for the analyte’s concentration
–1
if S meas is 24.37 ± 0.02, S reag is 0.96 ± 0.02, and k is 0.186 ± 0.003 ppm .
SOLUTION

Rearranging equation 4.5 and solving for C A
S meas - S reag 24 37 -0 96
.
.
C A = = =125 9 . ppm
k 0 186 ppm -1
.
gives the analyte’s concentration as 126 ppm. To estimate the uncertainty in
C A , we first determine the uncertainty for the numerator, S meas – S reag , using
equation 4.6
2
0
0
s R = (0 . ) 2 +(0 . ) 2 = .028
02
The numerator, therefore, is 23.41 ± 0.028 (note that we retain an extra
significant figure since we will use this uncertainty in further calculations). To
complete the calculation, we estimate the relative uncertainty in C A using
equation 4.7, giving
2 2
æ . 0 028 ö æ . 0 003 ö
s R
.
= ç ÷ + ç ÷ = 0 0162
R è 23 .41 ø è 0 186. ø
or a percent relative uncertainty of 1.6%. The absolute uncertainty in the
analyte’s concentration is
s R = (125.9 ppm) ´(0.0162) = ±2.0 ppm
giving the analyte’s concentration as 126 ± 2 ppm.

4 5 Uncertainty for Other Mathematical Functions
C.
Many other mathematical operations are commonly used in analytical chemistry,
including powers, roots, and logarithms. Equations for the propagation of uncer-
tainty for some of these functions are shown in Table 4.9.

4
EXAMPLE .8
The pH of a solution is defined as
+
pH = –log[H ]
+
+
where [H ] is the molar concentration of H . If the pH of a solution is 3.72
+
with an absolute uncertainty of ±0.03, what is the [H ] and its absolute
uncertainty?