Page 83 - Modern Analytical Chemistry

1400-CH04 9/8/99 3:54 PM Page 66

66 Modern Analytical Chemistry

Thus, we report the volume and its absolute uncertainty as 19.984 ± 0.008 mL.
The relative uncertainty in the total delivered volume is

0 0085
.
.
´ 100 = 0 043%
.
19 984

4 3 Uncertainty When Multiplying or Dividing
C.
When measurements are multiplied or divided, the relative uncertainty in the result
is the square root of the sum of the squares of the relative uncertainties for the indi-
vidual measurements. Thus, for the equations R = A ´B ´C or R = A ´B/C, or any
other combination of multiplying and dividing A, B, and C, the relative uncertainty
in R is
2 2 2
æ ö æ ö æ ö
s R s A s B s C
= ç ÷ + ç ÷ + ç ÷ 4.7
R è A ø è B ø è C ø

4
EXAMPLE .6
The quantity of charge, Q, in coulombs passing through an electrical circuit is
Q = I ´t
where I is the current in amperes and t is the time in seconds. When a current
of 0.15 ± 0.01 A passes through the circuit for 120 ± 1 s, the total charge is
Q = (0.15 A) ´(120 s) = 18 C
Calculate the absolute and relative uncertainties for the total charge.
SOLUTION

Since charge is the product of current and time, its relative uncertainty is
2 2
æ ö æ 1 ö
s R . 001
.
= ç ÷ + ç ÷ =±0 0672
R è . 015 ø è 120 ø
or ±6.7%. The absolute uncertainty in the charge is
s R = R ´0.0672 = (18) ´(±0.0672) = ±1.2
Thus, we report the total charge as 18 C ± 1 C.

4 4 Uncertainty for Mixed Operations
C.
Many chemical calculations involve a combination of adding and subtracting, and
multiply and dividing. As shown in the following example, the propagation of un-
certainty is easily calculated by treating each operation separately using equations
4.6 and 4.7 as needed.