This document outlines the key topics in Analytical Chemistry I including significant figures, types of errors, propagation of uncertainty, and systematic vs random errors. It discusses how measurements have uncertainty and errors. There are two main types of errors - systematic errors which affect accuracy and can be discovered and corrected, and random errors which cannot be eliminated and have equal chances of being positive or negative. The document also describes how to calculate the propagation of uncertainty through calculations using addition, subtraction, multiplication, division and other operations. It emphasizes keeping extra digits in calculations to properly account for uncertainty.

1. Analytical Chemistry I
(CHEM 231)
Spring 2016
Dr. Marwa Elazazy

3. 3

4. 4
Some laboratory errors are more obvious than others.
There is error associated with EVERY measurement.
 There is no way to measure the “true” value of anything. The
best we can do in a chemical analysis is to carefully apply a
technique that experience tells us is reliable.
Repetition of one method of measurement several times tells
us the precision (reproducibility) of the measurement.
If the results of measuring the same quantity by different
methods agree with one another, then we become confident
that the results are accurate, which means they are near the
“true” value.

5. Chapter Outline
3-1 Significant Figures
3-2 Significant Figures in Arithmetic
3-3 Types of Error
3-4 Propagation of Uncertainty from Random
Error
3-5 Propagation of Uncertainty from
Systematic Error
5

6. Chapter’s Learning Objectives
Review the rules of significant figures and
emphasize of their importance in chemical
analysis.
Understand the types of error and how they are
propagated in calculating final results.
Know the importance of propagation of
uncertainty and discuss how it is commuted in
different chemical calculations.
6

7. 3-1 and 3-2 Significant Figures
Significant figures: minimum number of digits
required to express a value in scientific
notation without loss of precision.
Review the rules of significant figures and
rounding off numbers.
Remember that the last digit in any number is
uncertain. The minimum uncertainty is ± 1 in
the last digit.
7

8. Significant figures are important in scientific calculation and practice
because they show us the accuracy (and the uncertainty) of the number
we are calculating
How many significant figures are in each of the
following measurements?
24 mL 2 significant figures
3001 g 4 significant figures
0.0320 m3 3 significant figures
6.4 x 104 molecules 2 significant figures
560 kg 2 or 3 significant figures
Significant Figures

9. Significant Figures
Addition or Subtraction:
89.332
1.1
+
90.432 round off to 90.4
one significant figure after decimal point
3.70
-2.9133
0.7867
two significant figures after decimal point
round off to 0.79
In addition and subtraction, the last significant figure is determined by the
number with the fewest decimal places (when all exponents are equal).

10. Significant Figures
Multiplication or Division
The number of significant figures in the result is set by the original
number that has the smallest number of significant figures
4.51 x 3.6666 = 16.536366 = 16.5
3 sig figs round to
3 sig figs
6.8 ÷ 112.04 = 0.0606926
2 sig figs round to
2 sig figs
= 0.061

11. Logarithms and Antilogarithms:
 Remember that for n = 10a means that log n = a
n is said to be the antilogarithm of a.
 A logarithm is composed of a characteristic and a mantissa.
 Number of digits in mantissa of log x (ANSWER) = number of
significant figures in x
11
Significant Figures

12.  In the conversion of a logarithm into its antilogarithm, the
number of significant figures in the antilogarithm should equal
the number of digits in the mantissa.
Exercises: What is the pH of a solution that is 0.0255 M
in H+?
12
5
1.
pH
figures
t
signicican
3
]
M
0255
.
0
log[
pH
]
H
log[
pH
593




 

13. Write the answer with the correct number of
significant digits:
log(3.456 × 107)
a) 7.53 8
b) 7.54
c) 7.538 6
d) 0.538 6
e) 0.539

14. Precision and Accuracy
Precision: reproducibility
o Reproducing the same measurement over and over and over.
o Nothing to do with being right.
Accuracy: nearness to the “truth”
o Getting it Right.
 A measurement might be precise (reproducible), but wrong.
 Poorly reproducible measurements may produce a correct value
(accurate).
 Producing “true” values requires experience and a well-tested
procedure or procedures.
14

15. 15
Three
targets with
three
arrows each
to shoot.
Can you hit the bull's-eye?
Both
accurate
and
precise
Precise
but not
accurate
Neither
accurate
nor
precise
How do
they
compare?

16. 16

17. 17
Precision Accuracy

18. 3.3 Types of Error
Every measurement has some uncertainty
(experimental error).
Results can be expressed with a high or a low
degree of confidence, but never with
complete certainty.
Types of experimental error:
 Systematic errors
 Random errors
18

19. Systematic error (determinate error) arises from a
flaw in the SYSTEM (equipment or the design of an
experiment).
Affects the accuracy (nearness to the “true” value).
 In principle, systematic errors can be discovered and
corrected.
KEY FEATURE: Reproducible
• Systematic error may always be positive in some
regions and always negative in others (One-Sided).
• With care and cleverness, systematic errors can
detected and corrected.
19

20. Random Errors (indeterminate errors), caused by
uncontrolled (and maybe uncontrollable) variables
in the measurement.
have equal chances of being positive or negative
(TWO SIDED _ Fluctuating around the mean).
always present and cannot be corrected.
reading a scale or an instrument produces random
errors as people reading the same instrument
several times might report several different readings.
random errors also result from electrical noise in an
instrument.
20

21. # Mass (g)
1 2.84
2 2.85
3 2.86
4 2.87
5 2.88
21
Ex. Imagine that you have a piece of metal and you tried to take its mass 5
times using an electronic balance. The results were as follows:
Q1. Why, and though it is the same balance for
the same piece of metal, you are getting 5
different values?
2.84 2.85 2.86 2.87 2.88
-
-
+ +
Random Error _Precision

22. 22
2.84 2.85 2.86 2.87 2.88
-
-
+ +
2.92
Random Error _Precision
Systematic Error _Accuracy

23. Examples of systematic errors
Ex1: Experimental design:
a pH meter that has been standardized
incorrectly.
You think that the pH of the buffer used to standardize the
meter is 7.00, but it is really 7.08. Then all your pH readings will
be 0.08 pH unit too low. pH reading of 5.60 is actually 5.68.
Solution: (can be discovered by using a second buffer of known
pH to test the meter).
23

24. Ex2. Glassware: A 50 mL uncalibrated buret has
a manufacturerʼs tolerance of ±0.05 mL. Hence,
if you deliver 29.43 mL, the real volume could be
anywhere from 29.38 to 29.48 mL.
Solution: make a calibration curve of volume as
a function of mass to obtain a correction factor.
24
FIGURE 3-3 Calibration curve
for a 50-mL buret. The volume
delivered can be read to the
nearest 0.1 mL. If your buret
reading is 29.43 mL, you can find
the correction factor accurately
enough by locating 29.4 mL on
the graph. The correction factor
on the ordinate (y-axis) for 29.4
mL on the abscissa (x-axis) is
−0.03 mL (to the nearest 0.01
mL).

25. Ways to detect systematic error:
1. Analyze a known sample (e.g. certified reference
material). Your method should reproduce the known
answer.
2. Analyze blank samples. If you observe a nonzero
result, your method responds to more than you intend.
3. Use different analytical methods to measure the
same quantity. If results do not agree, there is error in
one (or more) of the methods.
4. Round robin experiment: Different people in several
laboratories analyze identical samples by the same or
different methods. Disagreement beyond the estimated
random error is systematic error.
25

26. 26
Examples of Random error

27. 28

28. 29
Which of the following is not a characteristic
of random (or indeterminate) error?
a) Arises from uncontrolled variables in the measurement
b) Cannot be eliminated completely
c) Arises from a flaw in equipment or the design of an
experiment
d) Might be reduced by a better experiment
e) Has an equal chance to be positive or negative

29. Random Error large
Systematic Error small

30. Absolute and Relative Uncertainty
Absolute uncertainty expresses the margin of
uncertainty associated with a measurement
(e.g. a calibrated buret may produce a
reading with ±0.02 absolute uncertainty).
Relative uncertainty compares the size of the
absolute uncertainty with the size of its
associated measurement.
31

31. 32

32. 33
Which measurement is more precise?
0.25 g ± 0.005 100.00 g ± 0.05
Absolute uncertainty =
% Relative uncertainty =
± 2 %
± 0.05
± 0.005
± 0.05 %

33. 34
Example: If the absolute uncertainty in
reading a buret is constant at ±0.02 mL, the
%relative uncertainty is
0.2% for a volume of 10 mL and
0.1% for a volume of 20 mL.

34. 35

35. 36

36. 37
Propagation of Uncertainty

37. 3.4 Propagation of Uncertainty from
Random Error
For an arithmetic operation on several
numbers (each of which has a random error),
the uncertainty in the result is not the sum of
individual errors (as some are positive and
others are negative. Here there may be some
cancellation of errors).
38
𝑨 = 𝑿 ∓ 𝒙 + 𝒀 ∓ 𝒚 × 𝒁 ∓ 𝒛 ⤇ error ≠ 𝒙 + 𝒚 + 𝒛

38. For Addition and Subtraction
40
3.06 ± 0.04 (3.06 ± 0.041)
Final result can be written as:
3.06 (±0.04) (absolute uncertainty)
3.06 (±1%) (relative uncertainty)

39. 41

40. For Multiplication and Division
first convert all uncertainties into percent relative
uncertainties
then calculate the error of the product or quotient as
follows:
Advice Retain one or more extra insignificant figures
until you have finished your entire calculation. Then
round to the correct number of digits. When storing
intermediate results in a calculator, keep all digits
without rounding.
42

41. 43

42. For Mixed Operations
44
[𝟏.𝟕𝟔 ∓𝟎.𝟎𝟑 − 𝟎.𝟓𝟗(∓𝟎.𝟎𝟐)]
𝟏.𝟖𝟗(∓𝟎.𝟎𝟐)
= 0.6190 ± ?
Online uncertainty calculator:
http://web.mst.edu/~gbert/JAVA/uncertainty.HTML

43. 45

44. The Real Rule for Significant Figures
The 1st digit of the absolute uncertainty is the
last significant digit in the answer.
uncertainty occurs in the 4th decimal place. The answer 0.094 6 is
properly expressed with 3 significant figures, even though the
original data have 4 figures.
expressed with four significant figures because the uncertainty
occurs in the fourth place
46

45. In multiplication and division, keep an extra digit
when the first digit of answer lies between 1 and
2.
Example: 82/80 is better written as 1.02 than 1.0.
* If the uncertainties in 82 and 80 are in the
ones place, the uncertainty is of the order of
1%, which is in the second decimal place of
1.02.
* If written 1.0, it can assumed that the
uncertainty is at least 1.0 ± 0.1 = ±10%, which is
much larger than the actual uncertainty.
47

46. Exponents and Logarithms
Example: if 𝒚 = 𝒙𝟏/𝟐
, a 2% uncertainty in x will
yield a 0.5x2% = 1% uncertainty in y. If y = x2, a 3%
uncertainty in x leads to a 2 x3% = 6% uncertainty
in y.
48
𝒚 = 𝒙𝒂
⤇ %𝒆𝒚 = 𝒂(%𝒆𝒙)
𝒖𝒏𝒄𝒆𝒓𝒕𝒂𝒊𝒏𝒕𝒚 𝒇𝒐𝒓
powers and roots

47. 49
𝒚 = 𝐥𝐨𝐠 𝒙 ⤇ 𝒆𝒚 =
𝟏
𝐥𝐧 𝟏𝟎
𝒆𝒙
𝒙
⋍ 𝟎. 𝟒𝟑𝟒 𝟐𝟗
𝒆𝒙
𝒙
𝒖𝒏𝒄𝒆𝒓𝒕𝒂𝒊𝒏𝒕𝒚 𝒇𝒐𝒓
logarithm:

48. Example: Uncertainty in H+ Concentration
Consider the function pH = −log[H+], where [H+]
is the molarity of H+. For pH = 5.21 ± 0.03, find
[H+] and its uncertainty.
Exercise: If uncertainty in pH is doubled to
±0.06, what is the relative uncertainty in [H+]?
50

49. [H+] = 10−pH = 10−(5.21±0.03)
In Table 3-1, the relevant function is y = 10x, in
which y = [H+] and x = −(5.21 ± 0.03). For y =
10x, the table tells us that ey/y = 2.302 6 ex.
51
Inserting the value y = 10−5.21 = 6.17 × 10−6 into Equation 3-12
gives the answer:

50. 52
The concentration of H+ is 6.17 (±0.426) × 10−6 = 6.2 (±0.4) ×
10−6 M. An uncertainty of 0.03 in pH gives an uncertainty of
7% in [H+].

51. 3.5 Propagation of Uncertainty from
Systematic Error
Systematic error occurs in some common
situations and is treated differently from
random error.
53

52. Uncertainty in Atomic Mass: The Rectangular
Distribution
O atomic mass = 15.999 4 ± 0.000 3 g/mol.
• The uncertainty is not mainly from random error, but
it is predominantly from isotopic variation in samples
of oxygen from different sources. Example:
• source 1. O = 15.999 1, source 2. O = 15.999 7,
• so O mass can be relatively constant at 15.999 1 or
15.999 7 or any thing in between depending on the
source.
54

53. 55
FIGURE 3-4 Rectangular distribution for atomic mass. The standard
uncertainty interval (standard deviation) shown in color is equal to the
uncertainty given in the periodic table divided by √𝟑 . The atomic mass of
oxygen in the periodic table is 15.999 4 ± 0.000 3. The standard uncertainty is
±0.000 3/ √3 = ±0.000 17.
* There is approximately equal probability of finding any atomic
mass between 15.999 1 and 15.999 7 and negligible probability of
finding an atomic mass outside of this range.

54. Uncertainty in Molecular Mass
What is the uncertainty in molecular mass of
O2?
The uncertainty of the mass of n atoms is n ×
(standard uncertainty of one atom) = 2 × (±0.000 17)
= ±0.000 34.
The uncertainty is not For systematic
uncertainty, we add the uncertainties of each term in
a sum or difference.
Calculate the standard uncertainty in molecular
mass of C2H4?
Note: Use the rule for propagation of random uncertainty
for the sum of atomic masses of different elements
because uncertainties for different elements are
independent.
56

55. 57

56. Multiple Deliveries from One Pipet: The
Triangular Distribution
Example: a 25-mL Class A volumetric pipet is
certified by the manufacturer to deliver 25.00
± 0.03 mL (i.e. 24.97 - 25.03 mL).
58

57. 59
FIGURE 3-5 Triangular
distribution for volumetric
glassware including
volumetric flasks and
transfer pipets. The
standard uncertainty
interval (standard
deviation shown in color is
a/√𝟔 .
* delivering 25.00 mL has the highest probability.
* the probability falls off approximately in a linear manner as the
volume deviates from 25.00 mL.
* there is negligible probability that a volume outside of 25.00 ± 0.03
mL will be delivered.
* The standard uncertainty (standard deviation) in the triangular
distribution is ∓
𝒂
𝟔
= ∓
𝟎.𝟎𝟑
𝟔
= ∓𝟎. 𝟎𝟏𝟐 𝒎𝑳 .

58. Example: If you use an uncalibrated 25-mL Class A
volumetric pipet 4 times to deliver a total of 100 mL,
what is the uncertainty in 100 mL?
Note: For calibrating volumetric glassware refer to
section 2-9, page 42.
Calibration improves certainty by removing
systematic error.
• If a calibrated pipet delivers a mean volume of 24.991
mL with a standard uncertainty of ±0.006 mL, and you
deliver 4 aliquots, the volume delivered is 99.964 ±
0.012 mL.
• Uncalibrated pipet volume = 100.00 ± 0.05 mL
60

59. If you use an uncalibrated 25-mL Class A
volumetric pipet four times to deliver a total of
100 mL, what is the uncertainty in 100 mL?
The uncertainty is a systematic error, so the
uncertainty in four pipet volumes is like the
uncertainty in the mass of 4 mol of oxygen: The
standard uncertainty is ±4 × 0.012 = ±0.048 mL,
not ??.
61
Uncalibrated pipet volume = 100.00 ± 0.05 mL

60. If a calibrated pipet delivers a mean volume of
24.991 mL with a standard uncertainty of
±0.006 mL, and you deliver four aliquots, the
volume delivered is 4 × 24.991 = 99.964 mL and
the uncertainty is :
62
Calibrated pipet volume = 99.964 ± 0.012 mL
Uncalibrated pipet volume = 100.00 ± 0.05 mL

61. END OF CHAPTER 3
Terms to Understand page 64
Summary page 64
Exercises page 65
Problems page 65
63