1 INTRODUCTION A common challenge faced by an analytical chemist - - PowerPoint PPT Presentation

ADVANCED ANALYTICAL LAB TECH (Lecture) CHM 4130-0001

Spring 2013 Professor Andres D. Campiglia Textbook: “Principles of Instrumental Analysis” Skoog, Holler and Crouch, 5th Edition, 6th Edition or newest Edition

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INTRODUCTION

• A common challenge faced by an analytical chemist is the determination of target species in

complex samples

• Complex sample: sample with numerous species. Example of complex samples: physiological

fluids (blood, urine, saliva), environmental samples (air, water, soil), etc.

• Target species is the species of interest. It is also called analyte. Example: benzo[a]pyrene in

soil sample, PSA (prostate specific antigen) in physiological fluid, etc.

• Some possibilities:

Analyte: Other species = concomitants: Analyte is the main component in the sample with

• nly

two types of species Analyte is not the main component in the sample but sample contains

• nly two types of species

Analyte is not the main component in the sample and sample contains several types of species

3 Sample Collection Sample = Matrix Sample Preparation: Clean-up and/or Pre-concentration Analytical Sample Qualitative and Quantitative Analysis Statistical Analysis of Data

General Scheme

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Quantitative and Qualitative Analysis

• Classical and instrumental methods
• Classical methods = wet-chemical methods

Analyte separation: precipitation, extraction or distillation Qualitative analysis: chemical reactions yielding products of characteristic colors boiling or melting points solubility in a series of solvents

• dors, optical activities or refractive indexes

Quantitative analysis: Gravimetric or volumetric analysis

• Main disadvantages of classical methods:

Time consuming Numerous manual steps, which make them prone to indeterminate (random) errors

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Instrumental Methods

• Most instrumental methods require a source of excitation to stimulate a measurable

response from the analyte. See Figure 1-1.

• The first six entries in Table 1-1 involve interactions of the analyte with

electromagnetic radiation.

• The first characteristic response involves radiant energy produced by the analyte.
• The next five properties involve changes in electromagnetic radiation brought about

by its interaction with the sample.

• Four electrical properties and miscellaneous properties follow.
• The name of the corresponding instrumental method is given in the second column of

Table 1-1.

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Evaluation of Analytical Data (Appendix One)

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• Analytical chemists may be presented with two types of problems

1) Provide a qualitative answer Example: Does this distilled water contain any Boron? Is this soil sample contaminated with polycyclic aromatic hydrocarbons (PAH)? 2) Provide a quantitative answer Example: How much lead is in this water sample? This steel sample contains traces of chromium, tungsten and manganese; how much of each

• ne?
• Often, both types of questions are answered with quantitative methods

Example: B, Pb, Cr, W, Mn in H2O: AAS or AES PAH in H2O: HPLC

• In cases where a positive answer is obtained, the analyst will give the answer in terms of

analyte concentration Example: This water sample contain 1 mg/mL of B Most certainly, if the analyst repeats the experiment with the same sample using the same method he/she will find a different result Why? Because of inherent experimental errors

8 8 Random and systematic errors

• Example: Four students (A-D) each perform an analysis in which exactly 10.00mL of exactly

0.1M sodium hydroxide is titrated with exactly 10.00mL of exactly 0.1M hydrochloric acid. Each student performs five replicate titrations with the results shown in the following table Student Results (mL) A 10.08, 10.11, 10.09, 10.10, 10,12 B 9.88, 10,14, 10.02, 9.80, 10.21 C 10.19, 9.79, 9.69, 10.05, 9.78 D 10.04, 9.98, 10.02, 9.97, 10.04 A Results are all very close to each other (10.08-10.12) = highly reproducible All the results are too high (they are all higher than 10.00, which is the theoretical value) Two separate types of errors have occurred with this student: Random errors: these cause the individual results to fall on both sides of the average value (10.10mL) Systematic errors: these cause all the results to be in error in the same sense (too high) Random errors affect the reproducibility of an experiment or precision Systematic errors affect the proximity of the experimental value to the theoretical value or accuracy

9 9 B The average of the five results (10.01mL) is very close to the theoretical value = data is accurate, without substantial systematic error The spread of the results is very large (9.80 – 10.21) = data is imprecise, with the presence of substantial random errors Comparing A and B A: precise and inaccurate B: poor precision and accurate Random and systematic errors can occur independently of one another C His work is neither precise (range 9.69 – 10.19mL) nor accurate (average = 9.90mL) D Precise results (9.97 – 10.04mL) and accurate (average = 10.01mL)

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Distinction between random and systematic errors, and precision and accuracy

Student Results A Precise but inaccurate B Accurate but imprecise C Inaccurate and imprecise D Accurate and precise

Terms used to describe accuracy and precision of a set of replicate data

• Accuracy (systematic errors):

absolute error or relative error #1

• Precision (random errors):

standard deviation, variance or coefficient of variation

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Random Errors

• Whenever

analytical measurements are repeated on the same sample, a distribution of data similar to that in Table a1-1 is obtained.

• The variations among the individual results are

due to the presence of random (indeterminate) errors.

• The data can be organized into equal-sized,

adjacent groups or cells, as shown in Table a1- 2.

• Figure a1-1A shows the histogram of the data,

i.e. the relative frequency of occurrence of results in each cell.

• As the number of measurements increases, the

histogram approaches the shape of the continuous curve shown as plot B in Figure a1- 1.

• Plot B shows a Gaussian curve, or normal

error curve, which applies to an infinitely large set of data.

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Table a1-1 Table a1-2 Figure a1-1

Systematic Errors and the Gaussian Curve

• Systematic errors have a definite value

and an assignable cause and are of the same magnitude for replicate measurements made in the same way.

• Systematic errors lead to bias in

measurement results.

• Figure a1-2 shows the frequency

distribution of replicate measurements in the analysis of identical samples by two methods that have random errors

• f identical size.
• Method A has no bias so that the

mean (mA) corresponds to the true

• value. Method B has a bias that is

given by: bias = mB – mA

• The analyst should be able to identify

systematic errors and remove them from the method of analysis.

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Figure a1-2

Statistical Treatment of Random Errors

• Random errors can not be completely eliminated from experiments.
• Statistical treatment of random errors provide the means to evaluate their

contribution to final results.

• Definition of some terms:

Population Mean (m) #2 Sample Mean #3 Population Standard Deviation (s) and Population Variance (s2) #4 Sample Standard Deviation (s) and Sample Variance (s2) #5 Relative Standard Deviation (RSD) and Coefficient of Variation (CV) #6

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The Normal Error Law

• In Gaussian statistics, the results of replicate

measurements arising from indeterminate (random) errors distribute according to the normal error law, which states that the fraction of a population of observations, dN/N, whose values lie in the region x to (x+dx) is given by: #7

• The two plots in Figure a1-3a are plots of the

equation above. The standard deviation for the data in curve B is twice that for the data in curve A.

• (x – m) is the absolute deviation of the individual

values of x from the mean.

• Figure a1-3b plots the deviations from the mean

in terms of the variable z: z = x – m / s when x – m = s  z = 1 x – m = 2s  z = 2

x – m = 3s  z = 3 and so forth.

• The distribution of dN/N in terms of the single variable z

is given by: #8 15

Figure a1-3a Figure a1-3b

Characteristic Properties of the Normal Error Curve

• Zero deviation from the mean occurring with

maximum frequency.

• Symmetrical

distribution

• f

positive and negative deviations about this maximum

• Exponential decrease in frequency as the

magnitude of the deviation increases. Thus, small random errors are much more common than large random errors.

• The area under the curve in figure a1-3b is the

integral of equation #8, which is given by: #9

• The fraction of the population between any

specified limits is given by the area under the curve between these limits. Examples:

• 1  z  1  DN/N = 0.683 = 68.3% of a

population of data lie within 1s.

• 2  z  2  DN/N = 0.954 = 95.4% of a

population of data lie within 2s.

• 3  z  3  DN/N = 0.997 = 99.7% of a

population of data lie within 3s.

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Figure a1-3b

Confidence Intervals

Keep in mind the following:

• The equation for the Normal or Gaussian distribution is derived for a set of infinite measurements

(N = ∞).

• Derivation assumes no systematic errors.
• The infinite number of measurements is called the population.
• The mean obtained with an infinite number of measurements is the true value (m).
• The true value has a standard deviation denoted by s.
• In practical situations the number of measurements is far from infinite. For a finite number of

measurements, the set of results is called the sample.

• The mean (x) obtained with a sample is an estimate of the true (m) value and its standard

deviation (s) is an estimate of s.

• In other words: when N → ∞ ; x → m and s → s.
• In most of the situations encountered in chemical analysis, the true value of the mean (m) can not

be determined because a huge number of measurements (N = infinite) would be required.

• The best we can do is to establish an interval surrounding an experimentally determined mean (x)

within which the population mean (m) is expected to lie with certain degree of probability. This interval is known as the confidence interval.

• Example: assume an analysis for potassium gave concentration with the following confidence

interval: 7.25  0.15% K. The significance of this result is the following: 17

7.25 7.25 + 0.15 = 7.40 7.25 - 0.15 = 7.10 99% probability that m is within this Interval of experimental results.

Calculation of Confidence Intervals

• When the value of s is known:
• The general expression of the confidence interval

(CI) of the true mean of a set of measurements is

• btained via the following equation:

#10

• Values of z at various confidence levels are given

in Table a1-3.

• Note the following:

a) For the same number of repetitions (N = constant) and as the probability increases, the size of the confidence interval increases with the value of z. b) For the same probability (z = constant), the size of the confidence interval decreases as the number of repetitions increases.

• When the value of s is unknown:
• When the number of repetitions is far from infinite

(N  30), the confidence interval is calculated via the following equation: #11

• Table a1-5 summaries t values for various levels
• f probability.
• With the t value the confidence interval follows

the same trend as the one observed with the z value. 18

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CALIBRATION OF INSTRUMENTAL METHODS (Chapter 1)

• Unless a correlation between the analyte response and the analyte concentration is somehow

established by the analyst, Instrumentation by itself does not provide concentrations.

• A very important part of all analytical procedures is the calibration and standardization process.
• Calibration determines the relationship between the analytical response and the analyte

concentration.

• Calibration is usually accomplished with the use of chemical standards.
• Two types of chemical standards:

External standard is prepared separately from the sample. Internal standard is added to the sample itself.

• External standards are used when there is no interference effects from matrix components

(concomitants)

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The Calibration Curve Method

• The calibration curve method used when there

is no interference effects from matrix components (concomitants).

• Experimental procedure for the calibration curve

method: a) Several standards of known concentrations of analyte are introduced into the instrument and the instrumental response is recorded. b) The instrumental response is obtained with a blank. Blank = all the components of the analytical sample - analyte Analytical sample = is the sample presented to the instrument. In many cases, the analytical sample is different than the original sample.

• The signal intensity is plotted as a function of

analyte concentration. If the relationship between signal and analyte concentration is linear, the perfect calibration curve should look as the one in the figure.

• The unknown concentration can then be
• btained by interpolation.

Analyte Concentration* Signal Signal Average Zero I0,1; I0,2; I0,3 I0 ± s0 C1 I1,1; I1,2; I1,3 I1 ± s1 C2 I2,1; I2,2; I2,3 I2 ± s2 C3 I3,1; I3,2; I3,3 I3 ± s3 C4 I4,1; I4,2; I4,3 I4 ± s4 O = data points from external standards

• = data from unknown

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• The general behavior of experimental data is “far from ideal”



• This type of plot generates several questions:

a) Is the calibration graph linear?  b) What is the best straight line through these points?  c) When the calibration plot is used for the analysis of a test sample, what are the errors and the confidence limits for the determined concentration? 

You need to consider the following:

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Is the calibration graph linear?

• There are two ways of checking for linearity which are complementary rather than exclusive:

Correlation coefficient Graphically

• Correlation coefficient
• The correlation coefficient (r) is given by the equation:

#12

• r can take values in the range –1  r  1

r = +1 describes perfect positive correlation, i.e. all the experimental points lie on a straight line of positive slope r = -1 describes perfect negative correlation, i.e. all the experimental points lie on a straight line of negative slope r = 0 describes no linear correlation between y and x y

x

23 Example of calculation of r: Standard aqueous solutions of fluorescein are examined in a fluorescence spectrometer, and yield the following fluorescence intensities (in arbitrary units): Fluorescence intensity: 2.1 5.0 9.0 12.6 17.3 21.0 24.7 Concentration, pg/mL: 0 2 4 6 8 10 12



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• Keep in mind that the correlation coefficient equation will always generate an r value even if the

data are patently non-linear in character, i. e. experience shows that even quite poor-looking calibration plots give very high r values Lesson of this example: Calibration curve must always be physically plotted On graph paper or computer monitor, otherwise a straight-line relationship might wrongly be deduced from the calculation of r This example is a remainder that r = 0 does not mean That y and x are entirely unrelated; it only means that they are not linearly related Correlation coefficients are simple to calculate but they can lead to serious misinterpretation

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• Graphically
• Plot the data in a “x” versus “y” plot keeping always in mind the following convention:

Analytical response = y Concentration of external standard = x

• Visually, determine the linear dynamic range (LDR):



• Calculate the correlation coefficient of the LDR excluding the data points that do not

belong to the LDR. Include only the data points that belong to the LDR.



• Although you already know the graph is linear, the correlation coefficient gives you a

quantitative measure on how well the data points fit a straight line. 

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What is the best straight line through these points?

• What points? The points that belong to the LDR. The points that you used to

calculate the correlation coefficient.

• The mathematical expression that describes a straight line can be represented as:

y = mx + b Where m is the slope and b is the intercept.

y x

b

Blank signal Experimental data points

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Random Errors in Concentrations Obtained with the Calibration Curve Method

• The least squares method assumes that any deviation of

the individual points from the straight line arises from error in the measurement, i.e. only from the variation in the instrumental signal. In other words, the error in concentration is considered negligible in comparison to the instrumental signal.

• The difference between any given experimental value of

the signal and the corresponding signal fitted in the best straight line by the least-squares method is called the

• residual. The concept is shown in figure a1-6.
• The least-squares method minimizes the residuals for all

the experimental points to provide the best straight line within a given set of experimental data.

• The slope (m) of the best straight line is given by:

#13

• The intercept of the best straight line is given by:

#14

• The standard deviation of the slope (sm ) is given by:

#15

• The standard deviation of the intercept (sb) is given by:

#16

• The standard deviation of a concentration (sc) is given by:

#17

• Confidence interval is given by:

#18