ANALYTICAL CHEMISTRY FOR
ENGINEERS – CHE 257
1
Notes
1.
2.
3.
4.
Assignments are due one week after they are given.
Late turn-ins will not be accepted.
Cell phones must be turned off in class-no flashing,
texting, or any use of cell phone.
No form of intimidation in class
2
Course Outline
Contents
Remark
Part I
Ch.1
INTRODUCTION AND DATA HANDLING
Introduction to Analytical Chemistry
Lecture
Ch. 2
Data handling in Analytical Chemistry
Lecture
Part II
Ch. 1
TITRIMETRY
Concept of Stoichiometry
Lecture
Ch. 2
Acid-base Equilibria
Lecture
Ch. 3
Acid-base titration
Lecture
3
Course Outline cont’d
Contents
Remark
Ch. 4
Complexometric reactions and titrations
Lecture
Ch. 5
Precipitation reactions & Titrations
Lecture
Ch. 6
Redox Reactions & Potentials
Lecture
Ch. 7
Redox & Potentiometric titrations
Lecture
Part III
INTRODUCTION TO INSTRUMENTATION METHODS
Lecture
Ch. 1
Calibration and Standardization
Lecture
4
5
Introduction to Analytical Chemistry
Definition – Chemical characterization of matter.
ANALYZE
•
What substances are present? Qualitative
•
How much is present? Quantitative
DEVELOPMENT OF NEW PRODUCTS
• Possibility of impurities or contaminants.
• Composition of mixtures
6
Applications
 Agriculture
 Clinical
 Environmental
 Forensic
 Manufacturing
 Metallurgical
 Pharmaceutical Chemistry
7
Qualitative and Quantitative Analysis
• Analytical
chemistry involves the use laboratory
methods to determine the composition of chemical
samples. Results from analysis are usually presented
as reports.
• Types of Reports
Quantitative
Qualitative
8
The Analytical Process
•Defining the process (problem)
•The information needed
•The source of information
•The purpose
•Type of sample to be analyzed
9
The Analytical Process
•Obtaining a representative sample
•Chemical analysis is performed on small
portions of the material to be characterized
•Material to be sampled may be in a solid,
liquid or gaseous state
•Homogeneous or heterogeneous in
composition
•Homogenous : “Grab sample”
•Heterogeneous : Gross sample
10
The Analytical Process
• Preparing the sample for analysis
• Measure the amount being analyzed (e.g. volume or
weight of sample)
• Produce replicate samples for more reliable results
• Solid samples must usually be put into solution
• Analyses must be nondestructive in nature
• Solution conditions must be adjusted e.g. pH, state
etc.
• Run a blank.
11
The Analytical Process
•Performing chemical separations
•Eliminate interferences
•Provide suitable selectivity
•Minimize losses of the analyte by separating
it from the sample matrix
•Separation steps may include precipitation,
extraction into an immiscible solvent,
chromatography and distillation
12
The Analytical Process
• Performing the measurement
• Quantitative measurement of
analyte depends on
the amount of analyte present and the accuracy and
precision required
• Gravimetric analysis : selective separation of the
analyte by precipitation
• Volumetric or titrimetric analysis : Analyte reacts
with a measured volume of reagent of known
concentration.
• Instrumental analysis
13
The Analytical Process
• Performing the measurement
• Quantitative measurement of analyte depends on
the amount of analyte present and the accuracy and
precision required
• Gravimetric analysis : selective separation of the
analyte by precipitation
• Volumetric or titrimetric analysis : Analyte reacts
with a measured volume of reagent of known
concentration.
• Instrumental analysis
14
Cont’d
Table. 1.1 Comparison of different analytical Methods
Method
Approx.
range
Approx.
Precision (%)
Selectivity
Speed
Cost
Principal uses
Gravimetry
10-1 – 10-2
0.1
Poormoderate
Slow
Low
Inorg
Titrimetry
10-1 – 10-4
0.0-1
Poormoderate
Moderate
Low
Inorg, org
Potentiometry
10-1 – 10-6
2
Good
Fast
Low
Inorg
Electrogravimetry
10-1 – 10-4
0.01-2
Moderate
Slowmoderate
Moderate
Inorg
Chromatography
10-3 – 10-9
2-5
Good
Fastmoderate
ModerateHigh
Org,
multicomp.
Spectrophotometry
10-3 – 10-6
2
Goodmoderate
Fastmoderate
LowModerate
Inorg, org
Atomic spectroscopy
10-3 – 10-9
2-10
Good
Fast
ModerateHigh
Inorg,
multicomp.
15
Cont’d
•Instruments are more selective and sensitive
than volumetric and gravimetric methods.
•Examples are spectrophotometry, atomic
spectroscopy (AS), mass spectroscopy (MS)
•Various methods for determining an analyte
can be classified as absolute or relative.
•The instrumentation must be calibrated. A
calibration curve is the instrument response
as a function of concentration.
16
The Analytical Chemist’s Job
•Analyte : the species being measured in a
chemical analysis (chemical substance).
* Stages in Chemical Analysis
Decide first level of results needed
Level of Accuracy.
Economical aspects.
17
Range
• Analytical methods are often classified according to size of
sample.
• Analysis may be classified as meso, semimicro, micro, or
ultramicro
• Classification of Analytical Methods
Method
Sample weight (mg)
Sample Volume (µm)
Meso
>100
>100
Semimicro
10-100
50-100
Micro
1-10
<50
Ultramicro
<1
18
The Analytical Chemist’s Job
Figure 1.1: Steps in chemical analysis
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Case Study – How much caffeine in a chocolate bar?
Harris, Quantitative Chemical
Analysis, 8e © 2011 W. H. Freeman
Harris, Quantitative Chemical Analysis, 8e © 2011 W. H.
Freeman
1. Sampling
Harris, Quantitative Chemical Analysis, 8e © 2011 W. H.
Freeman
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2. Sample Preparation
Harris, Quantitative Chemical Analysis, 8e © 2011 W. H.
Freeman
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Harris, Quantitative Chemical Analysis, 8e © 2011 W. H.
Freeman
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3. Chemical Analysis
Harris, Quantitative Chemical Analysis, 8e © 2011 W. H.
Freeman
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Harris, Quantitative Chemical Analysis, 8e © 2011 W. H.
Freeman
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Harris, Quantitative Chemical Analysis, 8e © 2011 W. H.
Freeman
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Tools of Analytical Chemistry
•Chemicals, Apparatus, and Unit
Operations of Analytical Chemistry.
•Data Handling (Statistics) & Spreadsheets
in Analytical Chemistry.
•Calculations in Analytical Chemistry.
26
Spreadsheets in Analytical Chemistry
• Most popular spreadsheet programs include Microsoft Excel, Lotus 1-2-3
and Quattro Pro.
• Making Complex Calculations with Excel:
Equation for
computing the %
chloride in samples A,
B, and C.
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Spreadsheets in Analytical Chemistry
Fundamentals of Analytical Chemistry, 8e © 2011 Skoog
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Other Applications:
•Area under a Gaussian curve.
•Determination of confident intervals
•T-test
•Slope and Intercepts
•Least squares etc.
29
Data Handling in Analytical Chemistry
•Knowledge of statistical analysis is required for
data handling and processing as experiments
are performed in the laboratory.
•Statistics are necessary to understand the
significance of the data that are collected.
• The design of experiments is determined from
a proper understanding of what the data will
represent.
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Accuracy and Precision
Accuracy

Accuracy refers to the closeness of such measurements to the
“true” magnitude concerned.

Accuracy measures agreement between a result and the accepte
d true value.

The accuracy to which the value of the standard sample is
known is dependent on some measurement with a given limit of
certainty.

Accuracy is expressed in terms of either absolute or relative error.
31
Cont’d
Precision
Precision
(or reproducibility) refers to the agreement among repeated
measurements of a given sample.
Precision shows
only how closely many measurements agree.
Generally,
the precision of a measurement is readily determined by simply
repeating the measurement on replicate samples.
Precision may
be expressed as the
•
Standard deviation
•
Coefficient of variation
•
Range of the data
•
Confidence interval (e.g., 95%) about the mean value.
32
Illustration of Precision and Accuracy
imprecise
precise
33
Ways of Expressing Accuracy
Absolute error = the margin of uncertainty associated with a
measurement
-
Absolute error() = E = xm xt
where,
xm = Measured value
xt = True value
• Mean
error is where the measured value corresponds to the average
of several measurements.
34
Ways of Expressing Accuracy Cont’d
•Relative
error refers to the absolute or
mean error expressed as a percentage of
the true value.
•It
can be expressed in units other than
percentage.
- Relative
error(%) = Er = {(xi xt) / xt }  100
35
Types of Error
Systematic Error (Determinate)
 Random Error (Indeterminate)
 Gross error

36
Systematic error
They
are determinable and can be presumably
corrected or avoided
It
is possible to avoid or eliminate systematic errors if
their causes are known.
Determinate
or systematic errors can be assigned to
definite causes.
Such
errors are characterized often than not as being
unidirectional. Example is the solubility loss of a
precipitate.
37
Systematic error

Errors can also be random in sign
Their
existence and magnitude characterize the
accuracy of a result of measurement.
Systematic
errors decreases the accuracy of
results.
38
Sources of systematic errors
Instrumental errors
• Includes faulty equipment, uncalibrated weights e
tc.
• Can be corrected or minimized by proper calibration
Errors of the method
•Errors due to no ideal physical or chemical behavior
•Can be reduced by proper method development.
Personal Errors
− Occur where measurements require judgment
− Can be minimized by proper training and experience.
39
Effect of systematic errors
Constant systematic error:
The amount of a systematic error is independent of analyte, which
leads to a parallel displacement of the matrix calibration line. The cause
of this error may be the co-detection of a matrix component.
Proportional systematic errors : The amount of a systematic error
increases or decreases with the amount of analyte. Examples : method
bias, laboratory bias, instrumental bias.
2 (constant systematic error)
1 (ideal pure standard)
Signal
3 (proportional systematic error)
Analyte concentration
Representation of systematic errors. W. Funk, V. Dammann, G. Donnevert, Quality Assurance in Analytical Chemistry, VCH,
1995.
40
Detection of systematic instrument and
personal errors

Periodic calibration of equipment

Care and self discipline
Detection of systematic methods errors

Analysis of standard samples (standard reference materials:
SRM)

Independent analysis

Blank determinations

Variation in sample size
41
Random Error

The difference between the characteristic values obtained from the analysis
and the expected value (the mean result obtained by continuously repeated
experiments)

They represent the experimental uncertainty that occurs in any measurement.

This error is randomly distributed to higher and lower values. Thus it follows a
normal distribution, or Gaussian curve

It is brought about by the effects of uncontrolled variables. Random errors
cannot be eliminated by corrections.
42
Random Error

However, their influence on the result can be lessened by using a mean value
obtained from several independent determinations.

Random errors determine the reproducibility of measurements and therefore
their precision.

The precision of the results decreases, the scatter increases.
Examples : noise of radiation and voltage source, inhomogeneities of solids.
43
Random Error
• Indeterminate
errors should follow a normal
distribution, or Gaussian curve. Such a curve is
shown in figure 2.2.
• The symbol σ represents the standard deviation of an
infinite population of measurements
Figure 2.2: Normal error curve
44
Results from six replicate determinations of iron in aqueous samples of a
standard solution containing 20.0 ppm iron(III).
E
= 19.8  20.00 =  0.2
Er
= {(19.8  20.00)/20.0} ×100%
= 1%
45
Method 1
Method 2
Method 3
True value
Effect of systematic and random errors upon analytical results
46
Systematic error
Mean
True value
gross error
Outlier
Range of random errors
Schematic representation of systematic and random errors.
Helmut Gunzzler(Ed.) ; Accreditation and Quality Assurance in Analytical Chemistry, Springer, Berlin, 1994, p.106.
47
Statistical Treatment of Random Errors
Definition of Terms
The Population and the Sample:
• Population is an infinite number of observations
• The sample is a finite number of observations
representative of the population.
48
Statistical Treatment of Random Errors
Properties of a Gaussian Curve:
• Population mean, μ
• Population standard deviation, σ
a. Population mean. In the absence of systematic
error, μ, is the true value for the measurement.
The sample mean, 𝑥 , approaches μ and the
number of observations approach infinity. The
sample mean is defined as:
𝑥=
𝑛
𝑖 𝑥𝑖
𝑁
49
Statistical Treatment of Random Errors
b. Population standard deviation. One population standard deviation
contains ±34.15% of the most frequent values, while between 1 and 2
standard deviations contains ±13.6% of the next most frequent values.
Thus, 95.5% of all the values are found within ±2 standard deviations
in a Gaussian distribution.
The population standard deviation is defined as:
𝜎=
𝑥𝑖 −µ 2
𝑁
where xi represent the individual measurements
and 𝜇 the mean of the infinite number of
measurements (which should represent the
“true” value).
50
Statistical Treatment of Random Errors
EXAMPLE
Calculate the mean and standard deviation of the following set
of analytical results: 15.67, 15.69, and 16.03 g.
SOLUTION
xi
xi - 𝑥
(xi - 𝑥)2
15.67
-0.13
0.0169
15.69
-0.11
0.0121
16.03
47.39
0.23
0.0529
0.0819
The mean, 𝑥 is calculated
as 𝑥 = 15.80
𝑠=
0.0819
= 0.20 𝑔
3−1
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The central tendency of a set of results data
1) Mean
The mean value is the sum of the measured values divided by the total
number of values. For n sample determinations
x1, x2, x3, , xn.
The sample mean, can be calculated by :
This sample mean is an estimate of , the actual mean (true value) of
the population.
52
2) Median
The median(M) is defined as the middle value of data points arranged
in order of magnitude. Median is the value above and below which
there are an equal number of data points.
For an odd number of points, the median is the middle one.
For an even number of points, the median is halfway between the two
center values.
The advantage of M over the mean is that a gross error in one result
cause a large error in the mean, but not in M.
Other central tendencies
Geometric mean, Harmonic mean, Mode
53
Expressing precision
Deviation :
d = | xi  x |
Note that deviations from the mean are calculated without regard
to sign.
Standard deviation
The standard deviation measures how closely the data are
about the mean.
clustered
A small s is more reliable (precise) than large standard deviation.
54
Expressing precision
Standard deviation of the mean
• Sometimes referred to as the standard error
• It’s expressed as the relative standard deviation (rsd)
• Usually it is given as the percentage of the mean (% rsd)
𝑆𝑚𝑒𝑎𝑛 =
𝑆
𝑁
55
The Variance
• The
2
𝑆 =
variance
is the square of the standard deviation; V =
𝑁
2
𝑖
𝑥𝑖 −𝑥
𝑁−1
Advantage
Variances from independent sources can be
summed to obtain the total variance for a
measurement.
56
Gross error

Gross error usually occur only occasionally

They cause an experimental value to be discarded

Gross errors lead to outliers

These are results that appear to differ significantly
from all other data in a set of replicate measurements
57
Categories of Errors










sample errors
reagent errors
reference material error
method errors
calibration errors
equipment errors
signal registration and recording errors
calculation errors
transmission errors
errors in the reporting of result
58
Significant figures
The number of significant figures is the number of digits needed to write a giv
en value in
scientific notation without loss of accuracy.
8.25 × 104
3 significant figures
8.250 × 104
4
8.2500 × 104 5
0.801
3
0.0801
3
0.8010
4
59
Rules for determining the number of significant figures

Discard all initial zeros

Disregard all final zeros unless they follow a decimal point

All remaining digits, including zeros between nonzero digits, are
significant
60
Type of
Example
Standard deviation of y
y=a+b–c
𝑆𝑦 =
𝑆𝑎2 + 𝑆𝑏2 + 𝑆𝑐2
𝑆𝑦
=
( 𝑎𝑎 )2 +( 𝑏𝑏 )2 +( 𝑐𝑐 )2
= 𝑥.
𝑆𝑎
𝑎
Calculation
Addition or
(2.8)
Subtraction
Multiplication
y = a.b/c
𝑦
𝑆
𝑆
𝑆
(2.9)
Or division
Exponential
y = ax
Logarithm
𝑎
𝑦 = 𝑙𝑜𝑔10
Antilogarithm
𝑎
𝑦 = 𝑎𝑛𝑡𝑖𝑙𝑜𝑔10
𝑆𝑦
𝑦
𝑆𝑦 = 0.434
𝑆𝑦
𝑦
(2.10)
𝑆𝑎
𝑎
= 2.303. 𝑆𝑎
(2.11)
(2.12)
a, b and c are experimental variables with Sa, Sb and Sc as standard deviations respectively
61
Absolute and Relative uncertainty
Absolute uncertainty
• It
expresses the margin of uncertainty associated with a
measurement. For example ±0.02 mL
Relative uncertainty
• Compares the size of the absolute uncertainty with that
of its associated measurement. For example
12.35±0.02 mL
𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦
0.02 𝑚𝐿
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 =
=
= 0.002
𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 12.35 𝑚𝐿
62
SAMPLE QUESTION
Calculate the uncertainty in the number of
millimoles of chloride contained in 250.0
mL of a sample when three aliquots of
25.00 mL are titrated with silver nitrate
with the following results: 36.78, 36.82,
36.75 mL. The molarity of the AgNO3
solution is 0.1167±0.0002 M.
63
SOLUTION
• The absolute mean volume is
36.78 + 36.82 + 36.75
= 36.78 𝑚𝐿
3
• The standard deviation
Volume, xi
𝒙𝒊 − 𝒙
𝒙𝒊 − 𝒙
36.78
0.00
0.0000
36.82
0.04
0.0016
36.75
-0.03
0.0009
𝑥𝑖 − 𝑥
2
𝟐
𝑠=
0.0025
= ±0.035
3−1
Therefore, the mean volume =
36.78 ± 0.04 mL
= 0.0025
64
SOLUTION CONT’D
mmol of Cl- titrated = (0.1167 ± 0.0002 mmol/mL of Ag)( 36.78 ± 0.04 mL) = 4.29 ± ?
• Using table 2.1
y = 4.29
𝑆𝑦
=
𝑦
𝑆𝑦
=
4.29
𝑆𝑏
+
𝑏
2
0.1167
±0.0002
2
𝑆𝑎
𝑎
2
Therefore,
𝑆𝑐
+
𝑐
2
36.78
+
±0.04
𝑆𝑦 = 4.29 × ±0.0019 = ±0.0082
2
The absolute uncertainty in mmol of Cl- is 4.29
± 0.0082 mmol
Hence, mmol of Cl- in 250 mL = 10(4.292 ±
0.0082) = 42.90 ± 0.08 mmol
𝑆𝑦
=
4.29
±3.8 × 10−6 = ±1.9 × 10−3
65
Confidence Limit
• Calculation of the standard deviation for a set of data
provides an indication of the precision
• Statistical theory allows us to estimate the range
within which the true value might fall, within a given
probability
• This range is called confidence interval, and the
limits of this range are referred to as confidence limit
• The likelihood that the true value falls within the
range is termed confidence level
66
Confidence Limit
• The confidence limit is given by:
Confidence limit = 𝑥 ±
𝑡𝑠
𝑁
where t is the statistical factor which depends on the
number of degrees of freedom and the confidence
level desired.
• Values of statistical factor for 90, 95, 99 and 99.5%
are presented in appendix A.
67
SAMPLE QUESTION
A soda ash sample is analyzed in the analytical
chemistry laboratory by titration with standard
hydrochloric acid. The analysis is performed in
triplicate with the following results: 93.50, 93.58, and
93.43% Na2CO3. Within what range is the analyst 95%
confident that the true value lies?
68
SOLUTION
𝑥 = 93.50%
• Standard deviation, s = 0.075%
• At 95% confidence level, and two degrees of
freedom, t = 4.303
• Confidence limit
4.303 × 0.075
= 93.50 ±
= 93.50 ± 0.19%
3
• Mean,
69
Test of Significance
• In
developing a new analytical method, it is often
desirable to compare the results of that method with
those of accepted (perhaps standard) methods.
• Deciding whether one set of results is significantly
different from another depends not only on the
difference in means but also on the amount of data
available and the spread.
• The F test evaluates differences between the spread
of results, while the t test looks at differences
between means.
70
Test of Significance
Student T test
• It
is used to determine if two sets of measurements
are statistically different.
• The two cases of T-test are as follows.
(i) When the accepted value, μ, is known.
μ=x±
t. s
N
If 𝑡𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 > 𝑡𝑡𝑎𝑏𝑢𝑙𝑎𝑡𝑒𝑑 , then it is
statistically significant.
±𝑡 = 𝑥 − μ
𝑁
𝑠
71
Test of Significance
Student T test
(ii) When comparing the means of two samples.
±𝑡 =
𝑆𝑝 =
𝑁1
𝑖
𝑥𝑖 − 𝑥1
𝑥1 −𝑥2
𝑆𝑝
2
+
𝑁2
𝑗
𝑁1 𝑁2
𝑁1 +𝑁2
𝑥𝑗 − 𝑥2
2
+
𝑁3
𝑘
𝑥𝑘 − 𝑥3
2
+⋯
𝑁1 + 𝑁2 + 𝑁3 … − 𝑁𝑡
Degree of freedom = N1+N2-2
72
Test of Significance
Student T test (Example to be solved in class)
(iii) Paired t test.
• When the two methods are paired, the difference
between each of the paired measurements on each
sample is computed.
• The t value is calculated from
𝐷
𝑡=
𝑁
𝑠𝑑
𝑠𝑑 =
𝐷𝑖 − 𝐷
𝑁−1
2
where Di is the individual difference between the two methods for each sample
73
Test of Significance
Comparison of precision of measurement (F-test)
• Designed to indicate whether there is a significant
difference between two methods based on their
standard deviations.
• It is defined in terms of variances of the two methods.
𝐹=
𝑠12
,
2
𝑠2
where 𝑠12 > 𝑠22
74
Test of Significance
F-test (Example to be solved in class)
• If 𝐹𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 > 𝐹𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 , method 1 (S1) is significantly
different from method 2 (S2).
75
Test of Significance
Rejection of results: The Q-test
• The Q-test is used to determine if an “outlier” is due
to determinate error
• If not, then it falls within the expected random error
and should be retained.
• 𝑄𝑒𝑥𝑝
=
𝑥𝑞 −𝑥𝑛
𝑤
Xq is the questionable data or the data you want to get rid of, Xn is data in
value that is closest Xq and w is the range of the measurement.
76
Test of Significance
Rejection of results: The Q-test
• If 𝑄𝑒𝑥𝑝 > 𝑄𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 , then reject Qexp
• Qcritical values can be obtained from table 2.1 in
appendix C
77