Analytical Chemistry
Analytical Chemistry
• Analytical Chemistry?
– “Science of Chemical Measurements”
WHAT IS ANALYTICAL CHEMISTRY
- The qualitative and quantitative characterization of matter
- The scope is very wide and it is critical to our understanding of
almost all scientific disciplines
Characterization
- The identification of chemical compounds or elements present
in a sample (qualitative)
- The determination of the amount of compound or element present
in a sample (quantitative)
Four primary Areas of Analytical
Chemistry
• Detection:
– Does the sample contain substance X?
• Identification:
– What is the identity of the substance X in the sample?
• Separation:
– How can the species of interest be separated from the
sample matrix for better quantitation and identification?
• Quantitation:
– How much of substance X is in the sample?
Four primary Areas of Analytical
Chemistry
• Detection:
– Does the sample contain substance X?
• Identification:
– What is the identity of the substance X in the sample?
• Separation:
– How can the species of interest be separated from the
sample matrix for better quantitation and identification?
• Quantitation:
– How much of substance X is in the sample?
What are the roles of Analytical
Chemists?
Analytical Chemist:
1. Applies known measurement techniques to well defined
compositional or characterization questions.
2. Develops new measurement methods on existing principles
to solve new analysis problems.
What is Analytical Science?
• Analytical Chemistry provides the methods and
tools needed for insight into our material
world…for answering four basic questions about a
material sample?
• What?
• Where?
• How much?
• What arrangement, structure or form?
Different methods provide a range of precision, sensitivity, selectivity,
and speed capabilities.
Sample size is very important when choosing a particular
analytical technique.
The Analytical Chemistry Language
Analyte
- A substance to be measured in a given sample
Matrix
- Everything else in the sample
Interferences
- Other compounds in the sample matrix that interfere
with the measurement of the analyte
The Analytical Chemistry Language
Homogeneous Sample
- Same chemical composition throughout
(steel, sugar water, juice with no pulp, alcoholic beverages)
Heterogeneous Sample
- Composition varies from region to region within the sample
(pudding with raisins, granola bars with peanuts)
- Differences in composition may be visible or invisible to
the human eye (most real samples are invisible)
- Variation of composition may be random or segregated
The Analytical Chemistry Language
Analyze/Analysis
- Experimental evaluation of the sample under study
Determine/Determination
- Measurement of the analyte in the sample
Multiple Samples
- Identical samples prepared from another source
Replicate Samples
- Splits of sample from the same source
The Analytical Chemistry Language
General Steps in Chemical Analysis
1. Formulating the question or defining the problem
- To be answered through chemical measurements
2. Designing the analytical method/ protocol (selecting techniques)
- Find appropriate analytical procedures
3. Sampling and sample storage
- Select representative material to be analyzed
4. Sample preparation
- Convert representative material into a suitable form for analysis
The analytical chemistry approach for
sample analysis
General Steps in Chemical Analysis
5. Analysis (performing the measurement)
- Measure the concentration of analyte in several
identical portions
6. Assessing the data
7. Method validation
8. Documentation
Basic steps to
develop an
analytical
procedure.
An analysis involves several
steps and operations which
depend on:
•the particular problem
• your expertise
• the apparatus or
equipment available.
The analyst should be
involved in every step.
Integrity of Analytical method
Once an analytical method if conducted Statistical
Operations are used to determine the integrity of
the test method and results.
Integrity of Analytical method
Once an analytical method if conducted Statistical
Operations are used to determine the integrity of
the test method and results.
All measurement provide information about its magnitude
and its uncertainty.
Statistical Operations
- Statistics are needed in designing the correct experiment
An Analyst must
- select the required size of sample
- select the number of samples
- select the number of replicates
- obtain the required accuracy and precision
Analyst must also express uncertainty in measured values to
- understand any associated limitations
- know significant figures
Statistical Operations
- Statistics are needed in designing the correct experiment
An Analyst must
- select the required size of sample
- select the number of samples
- select the number of replicates
- obtain the required accuracy and precision
Analyst must also express uncertainty in measured values to
- understand any associated limitations
- know significant figures
Statistical Operations
Rules For Reporting Results
Significant Figures =
digits known with certainty + first uncertain digit
- The last sig. fig. reflects the precision of the measurement
- Report all sig. figs such that only the last figure is uncertain
- Round off appropriately
(round down, round up, round even)
Statistical Operations
Rules For Reporting Results
- Report least sig. figs for multiplication and division
of measurements (greatest number of absolute uncertainty)
- Report least decimal places for addition and subtraction
of measurements (greatest number of absolute uncertainty)
- The characteristic of logarithm has no uncertainty
- Does not affect the number of sig. figs.
- Discrete objects (absolute numbers) have no uncertainty
- Considered to have infinite number of sig. figs.
Accuracy and Precision
- Accuracy is how close a measurement is to the true
(accepted) value
- True value is evaluated by analyzing known standard samples
- Precision is how close replicate measurements on the
same sample are to each other
- Precision is required for accuracy but does not
guarantee accuracy
- Results should be accurate and precise
(reproducible, reliable, truly representative of sample)
Errors
- Two principal types of errors
- Determinate (systematic) and indeterminate (random)
Determinate (Systematic) Errors
- Caused by faults in procedure or instrument
- Fault can be determined and corrected
- Results in good precision but poor accuracy
Examples;
- constant (incorrect calibration of pH meter or mass balance)
- variable (change in volume due to temperature changes)
- additive or multiplicative
Errors
Examples of Determinate (Systematic) Errors
- Improperly calibrated volumetric flasks and pipettes
- Analyst error (misreading or inexperience)
- Incorrect technique
- Malfunctioning instrument (voltage fluctuations, alignment, etc.)
- Contaminated or impure or decomposed reagents
- Interferences
Errors
To Identify Determinate (Systematic) Errors
- Use of standard methods with known accuracy and precision
to analyze samples
- Run several analysis of a reference analyte whose concentration
is known and accepted
- Run Standard Operating Procedures (SOPs)
Errors
Indeterminate (Random) Errors
- Sources cannot be identified, avoided, or corrected
- Not constant (biased)
Examples
- Limitations of reading mass balances
- Electrical noise in instruments
Errors
- Random errors are always associated with measurements
- No conclusion can be drawn with complete certainty
- Scientists use statistics to accept conclusions that have high
probability of being correct and to reject conclusions that have
low probability of being correct
Efforts to Eliminate Errors
- Random errors follow random distribution and analyzed
using laws of probability (and statistics)
- Statistics deals with only random errors
- Systematic errors should be detected and eliminated
Review
Please see calculation examples in text
Review
Please see calculation examples in text
Statistical Operations…
Sample MEAN
- Arithmetic mean of a finite number of observations
- Also known as the average
- Is the sum of the measured values divided by the number
of measurements
N
_
x
x
i 1
N
i

1
x1  x 2  x 3  .....  x N 
N
∑xi = sum of all individual measurements xi
xi = a measured value
N = number of observations
Population MEAN (µ)
- The limit as N approaches infinity of the sample mean
lim
μ 
N
N
xi

i 1 N
Population Mean versus Sample mean?
Sample Mean is the mean of sample values collected.
Population Mean is the mean of all the values in the
population.
If the sample is random and sample size is large then the
sample mean would be a good estimate of the population
mean.
Quantifying Random Error
Median
- The middle number in a series of measurements
arranged in increasing order
- The average of the two middle numbers if the
number of measurements is even
Mode
- The value that occurs the most frequently
Range
- The difference between the highest and the lowest values
Error
Error (E)  the difference between T and either x i or x
E  x i  T or E  x  T
Absolute error  Absolute value of E
E abs  x i  T or E  x  T
Total error = sum of all systematic and random errors
Relative error = absolute error divided by the true value
E rel
E abs

T
%E rel
E abs

x 100%
T
Standard Deviation
Absolute deviation (d i )  x i  x
Relative deviation (D) = absolute deviation divided by mean
D 
di
_
x
Percent Relative deviation [D(%)]
D(%) 
di
_
x
x 100%  D x 100%
Standard Deviation
Sample Standard Deviation (s)
- A measure of the width of the distribution
- Small standard deviation gives narrow distribution curve
For a finite number of observations, N
N
s
d
i 1
 x
2
i
N 1

i 1

2
N
i
x
N 1
xi = a measured value
N = number of observations
N-1 = degrees of freedom
Standard Deviation
Standard Deviation of the mean (sm)
- Standard deviation associated with the mean
consisting of N measurements
s
sm 
N
Population Standard Deviation (σ)
- For an infinite number of measurements
2
N
σ
lim
N
 x
i
 μ
i 1
N
Standard Deviation
Percent Relative Standard Deviation (%RSD)
%RSD 
s
_
x 100
x
Variance
- Is the square of the standard deviation
- Variance = σ2 or s2
- Is a measure of precision
- Variance is additive but standard deviation is not additive
- Total variance is the sum of independent variances
Quantifying Random Error (using Gaussian
or Bell curve distribution)
- The Gaussian distribution and statistics are used to determine how
close the average value of measurements is to the true value
- The Gaussian distribution assumes infinite number of measurements
As N increases x  μ approaches zero
x μ
for N > 20
Random error  x  μ
- The standard deviation coincides with the point of inflection
of the curve (2 inflection points since curve is symmetrical)
Quantifying Random Error (Standard
Bell Curve Method)
Population mean (µ) = true value (T or xt)
x=µ
f(x)
a
Points of inflection
-3σ
-2σ
-σ
μ
σ
2σ 3σ
x
Quantifying Random Error
Probability
- Range of measurements for ideal Gaussian distribution
- The percentage of measurements lying within the given range
(one, two, or three standard deviation on either side of the mean)
Range
Gaussian Distribution (%)
µ ± 1σ
µ ± 2σ
µ ± 3σ
68.3
95.5
99.7
Quantifying Random Error
- The average measurement is reported as: mean ± standard deviation
- Mean and standard deviation should have the same number
of decimal places
In the absence of determinate error and if N > 20
- 68.3% of measurements of xi will fall within x = µ ± σ
- (68.3% of the area under the curve lies in the range of x)
- 95.5% of measurements of xi will fall within x = µ ± 2σ
- 99.7% of measurements of xi will fall within x = µ ± 3σ
Quantifying Random Error
x=µ±σ
f(x)
a
68.3%
known as the confidence level
(CL)
-3σ
-2σ
-σ
μ
σ
2σ 3σ
x
QUANTIFYING RANDOM ERROR
x = µ ± 2σ
f(x)
a
95.5%
known as the confidence level
(CL)
-3σ
-2σ
-σ
μ
σ
2σ 3σ
x
QUANTIFYING RANDOM ERROR
x = µ ± 3σ
f(x)
a
99.7%
known as the confidence level
(CL)
-3σ
-2σ
-σ
μ
σ
2σ 3σ
x
CONFIDENCE LIMITS
- Refers to the extremes of the confidence interval (the range)
- Range of values within which there is a specified probability
of finding the true mean (µ) at a given CL
- CL is an indicator of how close the sample mean lies
to the population mean
µ = x ± zσ
CONFIDENCE LIMITS
µ = x ± zσ
If z = 1
we are 68.3% confident that x lies within ±σ of the true value
If z = 2
we are 95.5% confident that x lies within ±2σ of the true value
If z = 3
we are 99.7% confident that x lies within ±3σ of the true value
CONTROL CHARTS ACCESS DATA
INTEGRITY
https://www.qimacros.com/free-excel-tips/control-chart-limits/
CONTROL CHARTS ACCESS DATA
INTEGRITY
https://www.flickr.com/photos/93642218@N07/8637804092
Tests to further evaluate statistical data
1. F - test
2. T - test
3. Q - test
CONFIDENCE LIMITS
- For N measurements CL for µ is
μ  x  zs m
- s is not a good estimate of σ since insufficient replicates are made
- The student’s t-test is used to express CL
- The t-test is also used to compare results from
different experiments
t
x  μ 
s
CONFIDENCE LIMITS
_
ts
μ  x
N
- That is, the range of confidence interval is
– ts/√n below the mean and + ts/√n above the mean
- For better precision reduce confidence interval by increasing
number of measurements
Example:
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: 43.51, 43.58, and 43.43% Na2CO3. Within what range are you
95% confident that the true value lies?
Example:
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: 43.51, 43.58, and 43.43% Na2CO3. Within what range are you
95% confident that the true value lies?
_
ts
μ  x
N
T- table
Example:
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: 43.51, 43.58, and 43.43% Na2CO3. Within what range are you
95% confident that the true value lies?
_
μ  x
ts
N
S = 0.0618
DOF = 2, So using two tailed test t = 4.303 at 95% CL
Using the Null Hypothesis to evaluate
experimental results
A hypothesis is proposed for the statistical relationship between
the two data sets.
Random Error occurs when the null hypothesis is Accepted.
Systematic Error occurs when the null hypothesis is Rejected.
Null Hypothesis uses the three statistical tests;
1. F - test
2. T - test
3. Q - test
t – Test
To test for comparison of Means – Typically three different ways or
scenarios.
- Calculate the standard deviation or the pooled standard deviation
(spooled) depending on the tests
- Calculate t
- Compare the calculated t to the value of t from the table
- The two results are significantly different if the calculated t
is greater than the tabulated t at particular confidence level
(that is tcal > ttab at the CL chosen)
t – Test
Scenario 1. t test when an Accepted (True) value is known.
- Testing of your sample mean against the population or
accepted mean (True value).


N
 t  x μ
s
- A known valid method is used to determine µ for a known sample.
- The mean and standard deviation of your test is then determined.
- t value is calculated for a given CL
- Systematic error exists in the new method if
tcal > ttab for the given CL – Null Hypothesis REJECTED
t – Test
- Systematic error exists in your method if
tcal > ttab for the given CL – Null Hypothesis REJECTED
- Systematic Errors are involved in your results.
- GROSS ERROR EXIST IN DATA SET.
tcal < ttab for the given CL – Null Hypothesis ACCEPTED
- Difference in results between sample mean and population
mean in purely RANDOM and can be ignored.
- GROSS ERROR NOT MANIFESTED IN DATA SET
t – Test
Scenario 2. t test for two sets of data with
- N1 and N2 measurements
 averages of x1 and x 2
- standard deviations of s1 and s2
s pooled 
s12 N1  1  s22 N 2  1
N1  N 2  2
x1  x2
t 
spooled
N1N 2
N1  N 2
Degrees of freedom = N1 + N2 - 2
t – Test
Scenario 3. Paired t test
- Testing of your sample mean against an accepted mean (from
another method) by analyzing several different samples of slightly
varying composition (within physiological range).
- Where di is the individual difference between the two methods for each sample, with
regards to sign and
is the mean of all the individual differences.
F-TEST
- Used to compare two methods (method 1 and method 2)
- Determines if the two methods are statistically
different in terms of precision
- The two variances (σ12 and σ22) are compared
F-function = the ratio of the variances of the two sets of numbers
σ12
F 2
σ2
F-TEST
- Ratio should be greater than 1 (i. e. σ12 > σ22)
- F values are found in tables (make use of two degrees of freedom)
Fcal > Ftab implies there is a significant difference between
the two methods
Fcal = calculated F value
Ftab = tabulated F value
REJECTION OF RESULTS
Outlier
- A replicate result that is out of the line
- A result that is far from other results
- Is either the highest value or the lowest value in a set of data
- There should be a justification for discarding the outlier
- The outlier is rejected if it is > ±4σ from the mean
- The outlier is not included in calculating the mean and
standard deviation
- A new σ should be calculated that includes outlier if it is < ±4σ
REJECTION OF RESULTS
Q – Test
- Used for small data sets
- Arrange data in increasing order
- Calculate range = highest value – lowest value
- Calculate gap = |suspected value – nearest value|
- Calculate Q ratio = gap/range
- Reject outlier if Qcal > Qtab
- Q tables are available
Performing The Experiment
Two types of Analytical Methods
1. Classical
2. Non-classical or Instrumental
PERFORMING THE EXPERIMENT
Detector
- Records the signal (change in the system that is related to the
magnitude of the physical parameter being measured)
- Can measure physical, chemical or electrical changes
Transducer (Sensor)
- Detector that converts nonelectrical signals to electrical signals
and vice versa
PERFORMING THE EXPERIMENT
Signals and Noise
- A detector makes measurements and detector response
is converted to an electrical signal
- The electrical signal is related to the chemical or physical
property being measured, which is related to the amount of analyte
- There should be no signal when no analyte is present
- Signals should be smooth but are practically not smooth
due to noise
PERFORMING THE EXPERIMENT
Signals and Noise
Noise can originate from
- Power fluctuations
- Radio stations
- Electrical motors
- Building vibrations
- Other instruments nearby
PERFORMING THE EXPERIMENT
Signals and Noise
- Signal-to-noise ratio (S/N) is a useful tool for comparing
methods or instruments
- Noise is random and can be treated statistically
- Signal can be defined as the average value of measurements
- Noise can be defined as the standard deviation
S
x
mean
 
N
s standard deviation
PERFORMING THE EXPERIMENT
Types of Noise
1. White Noise
- Two types
Thermal Noise
- Due to random motions of charge carriers (electrons)
which result in voltage fluctuations
Shot Noise
- When charge carriers cross a junction in an
electrical circuit
PERFORMING THE EXPERIMENT
Types of Noise
2. Drift (Flicker) Noise (origin is not well understood)
3. Noise due to surroundings (vibrations)
Improving s/n
- Signal is enhanced or noise is reduced for better results
- Hardware and software approaches are available to improve s/n
- Another approach is the use of Fourier Transform (FT) or
Fast Fourier Transform (FFT) which discriminates
signals from noise (FT-IR, FT-NMR, FT-MS)
CALIBRATION CURVES
Calibration
- The process of establishing the relationship between the
measured signals and known concentrations of analyte
- Calibration standards: known concentrations of analyte
- Calibration standards at different concentrations are
prepared and measured
- Magnitude of signals are plotted against concentration
- Equation relating signal and concentration is obtained and
can be used to determine the concentration of unknown
analyte after measuring its signal
CALIBRATION CURVES
- Many calibration curves have a linear range with the
relation equation in the form y = mx + b
- The method of least squares or the spreadsheet may be used
- m is the slope and b is the vertical (signal) intercept
- The slope is usually the sensitivity of the analytical method
- R = correlation coefficient (R2 is between 0 and 1)
- Perfect fit of data (direct relation) if R2 is closer to 1
BEST STRAIGHT LINE
(METHOD OF LEAST SQUARES)
The equation of a straight line
y = mx + b
m is the slope (y/x)
b is the y-intercept (where the line crosses the y-axis)
BEST STRAIGHT LINE
(METHOD OF LEAST SQUARES)
The method of least squares
- finds the best straight line
- adjusts the line to minimize the vertical deviations
Only vertical deviations are adjusted because
- experimental uncertainties in y values > in x values
- calculations for minimizing vertical deviations are easier
BEST STRAIGHT LINE
(METHOD OF LEAST SQUARES)
xi
yi
xiyi
xi2
∑xi =
∑yi =
∑(xiyi) =
∑xi2 =
BEST STRAIGHT LINE
(METHOD OF LEAST SQUARES)
m 
b 
N  x i y i    x i  y i
D
 x  y
2
i
i
  x i y i  x i
D
D  N  x i2    x i 
2
- N is the number of data points
Knowing m and b, the equation of the best straight line can
be determined and the best straight line can be constructed
ASSESSING THE DATA
A good analytical method should be
- both accurate and precise
- reliable and robust
- It is not a good practice to extrapolate above the highest
standard or below the lowest standard
- These regions may not be in the linear range
- Dilute higher concentrations and concentrate lower
concentrations of analyte to bring them into the working range
ASSESSING THE DATA
Limit of Detection (LOD)
- The lowest concentration of an analyte that can be detected
- Increasing concentration of analyte decreases signal
due to noise
- Signal can no longer be distinguished from noise at a point
- LOD does not necessarily mean concentration can be
measured and quantified
ASSESSING THE DATA
Limit of Detection (LOD)
- Can be considered to be the concentration of analyte that gives
a signal that is equal to 2 or 3 times the standard
deviation of the blank
- Concentration at which S/N = 2 at 95% CL or S/N = 3 at 99% CL
- 3σ is more common and used by regulatory methods (e.g. EPA)
ASSESSING THE DATA
Limit of Quantification (LOQ)
- The lowest concentration of an analyte in a sample that can be
determined quantitatively with a given accuracy and precision
- Precision is poor at or near LOD
- LOQ is higher than LOD and has better precision
- LOQ is the concentration equivalent to S/N = 10/1
- LOQ is also defined as 10 x σblank