Page 85 - Modern Analytical Chemistry

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68 Modern Analytical Chemistry

SOLUTION
+
The molar concentration of H for this pH is
–4
+
[H ]=10 –pH =10 –3.72 = 1.91 ´10 M
or 1.9 ´ 10 –4 M to two significant figures. From Table 4.9 the relative
+
uncertainty in [H ] is
s R
0
.
.
.
.
= 2 303 s ´ A =2 303 ´ 03 =069
0
R
and the absolute uncertainty is
–4
–5
(1.91 ´10 M) ´(0.069) = 1.3 ´10 M
+
–4
We report the [H ] and its absolute uncertainty as 1.9 (±0.1) ´10 M.
4 9
Table . Propagation of Uncertainty
for Selected Functions a
Function s R
R = kA s R = ks A
R = A + B s R = s 2 A s + 2 B
R = A – B s R = s 2 A s + 2 B
2 2
æ ö s æ ö
s R s A B
R = A ´ B = ç ÷ + ç ÷
R è A ø è B ø
2 2
A s R æ ö s æ ö
B
s A
R = = ç ÷ + ç ÷
B R è A ø è B ø
s A
R = ln( A) s R =
A
s A
R = log( A) s R = 0.4343 ´
0.4343
A
R = e A s R =s A
R
R =10 A s R =2 303 s A
.
R
ù
é s A
s R
k
R = A = k ú
ê
R ë A û
a These equations assume that the measurements A and B are
uncorrelated; that is, s A is independent of s B.
4 C.6 Is Calculating Uncertainty Actually Useful?
Given the complexity of determining a result’s uncertainty when several mea-
surements are involved, it is worth examining some of the reasons why such cal-
culations are useful. A propagation of uncertainty allows us to estimate an ex-