INTRODUCTION to

ANALYTICAL CHEMISTRY

Chapter 1Introduction

Analytical Chemistry deals with methods for determining the chemical composition of samples.• Qualitative Analysis (identification)

provides information about the identity of species or functional groups in the sample (an analyte can be identified).

• Quantitative Analysis provides numerical information of analyte (quantitate the exact amount or concentration).

Analytical Methods

•Classical Methods: Wet chemical methods such as precipitation, extraction, distillation, boiling or melting points, gravimetric and titrimetric measurements.

• Instrumental Methods: Analytical measurements (conductivity, electrode potential, light absorption or emission, mass-to-charge ratio, fluorescence etc.) are made using instrumentation.

Types of Instrumental Methods

1. Spectroscopic methods:

a. Atomic spectroscopy

b. Molecular spectroscopy

2. Chromatographic methods (separations):

3. Electrochemistry:

Block diagram of an instrumental measurement

Block diagram of a fluorometer

Applications of Instrumental Methods

1. Bioanalytical: biological molecules and/or biological matrices (e.g., proteins, amino acids, blood, urine)

2. Environmental: pesticides, pollution, air, water, soil

3. Material science: polymers, characterization of new materials

4. Forensic science (application of science to the law): body fluids, DNA, gun shot residue, hair, fibers, elemental analysis, drugs, alcohols, poisoning, fingerprints, etc.

Analytical Methodology1. Plan: Qualitative or quantitative or both; what

kind of information have; which technique is suitable etc.

2. Sampling: Accuracy depends on proper sampling, characteristic of sample is very important, required good representative sample (from top, middle and bottom and mix up and take average sample).

3. Sample preparation: depends on analytical techniques.

4. Analytical measurement:

5. Data Analysis: Whether the data make sense or not.

Selecting an Analytical Method In order to select an analytical method intelligently, it is

essential to define clearly the nature of the analytical problem. In general, the following points should be considered when choosing an instrument for any measurement.

1. Accuracy and precision required

2. Available sample amount

3. Concentration range of the analyte

4. Interference in sample

5. Physical and chemical properties of the sample matrix

6. Number of sample to be analyzed

7. Speed, ease, skill and cost of analysis

Figures of Merit

Precision Bias Sensitivity Detection limit Concentration range (Dynamic range) Selectivity

Precision: How close the same measurements are to one another. The degree of mutual agreement among data that have been obtained in the same way. Precision provides a measure of the random or indeterminate error of an analysis.

Accuracy: How close the measurement approaches the real value.

Bias: Bias provides a measure of the systematic, or determinate error of an analytical method.

bias = - xt, where, is the population mean and

xt is the true value

Sensitivity: Sensitivity of an instrument is a measure of its ability to discriminate between small differences in analyte concentration. The change in signal per unit change in analyte concentration. The slope of the calibration curve at the concentration of interest is known as calibration sensitivity.

S = mc + Sbl

S = measured signal; c= analyte concentration;

Sbl = blank signal; m = sensitivity (Slope of line)

Analytical sensitivity () = m/ss

m = slope of the calibration curve

ss = standard deviation of the measurement

Detection Limit (Limit of detection, LOD): The minimum concentration of analyte that can be detected with a specific method at a known confidence level.

LOD is determined by S/N, where, S/N = Signal-to-noise ratio = (magnitude of the signal)/(magnitude of the noise)

• Noise: Unwanted baseline fluctuations in the absence of analyte signal (standard deviation of the background)

• The detection limit is given by,

Cm = (Sm – Sbl)/m, where, Cm = minimum concentration i.e., LOD, Sm = minimum distinguishable analytical signal (i.e., S/N = 2 or S/N = 3), Sbl = mean blank signal

m = sensitivity (i.e., slope of calibration curve)• The amount of analyte necessary to yield a net signal equal

to 2 or 3x the standard deviation of the background.

Dynamic Range: The lowest concentration at which quantitative measurements can be made (limit of quantitation, or LOQ) to the concentration at which the calibration curve departs from linearity (limit of linearity, or LOL).

The lower limit of quantitative measurements is generally taken to be equal to ten times the standard deviation of repetitive measurements on a blank or 10 Sbl. Dynamic range is the range over which detector still responds to changing concentration (at high concentrations – usually saturates – quits responding)An analytical method should have a dynamic range of at least two orders of magnitude, usually 2-6 orders of magnitude.

Selectivity: Selectivity of an analytical method refers to the degree to which the method is free from interference by other species contained in the sample matrix. No analytical method is totally free from interference from other species, and steps need to be taken to minimize the effects of these interferences. Selectivity coefficient gives the relative response of the method to interfering species as compared with analyte. Selectivity coefficient can range from zero (no interference) to values greater than unity. A coefficient is negative when the interference caused a reduction in the intensity of the output signal of the analyte.

Calibration of Instrumental Methods

All types of analytical methods require calibration for quantitation. Calibration is a process that relates the measured analytical signal to the concentration of analyte. We can’t just run a sample and know the relationship between signal and concentration without calibrating the response

The three most common calibration methods are:• Calibration curve• Standard addition method• Internal standard method

Calibration Curves• Several standards (with different concentration) containing exactly

known concentrations of the analyte are measured and the responses recorded.

• A plot is constructed to give a graph of instrument signal versus analyte concentration.

• Sample (containing unknown analyte concentration) is run, if response is within the LDR of the calibration curve then concentration can be quantitated.

• Calibration curve relies on accuracy of standard concentrations.• It depends on how closely the matrix of the standards resemble that of

the sample to analyzed.• If matrix interferences are low, calibration curve methods are OK.• If matrices for sample and standards are not same calibration curve

methods are not good. • Need to consider the linear part of the curves.

Standard Addition Methods Better method to use when matrix effects can be substantial Standards are added directly to aliquots of the sample,

therefore matrix components are the same. Procedure:

• Obtain several aliquots of sample (all with the same volume).

• Spike the sample aliquots ==> add different volume of standards with the same concentration to the aliquots of sample

• Dilute each solution (sample + standard) to a fixed volume• Measure the analyte concentration

Standard Addition Methods Instrumental measurements are made on each solutions to

get instrument response (S). If the instrument response is proportional to concentration, we may write,

S = (kVsCs)/Vt + (kVxCx)/Vt

Where, Vx =Volume of sample = 25 mL (suppose)

Vs = Volume of standard = variable (5, 10, 15, 20 mL)

Vt = Total volume of the flask = 50 mL

Cs = Concentration of standard

Cx = concentration of analyte in aliquotk = proportionality constant

A plot of S as a function of Vs is a straight line of the form,

S = mVs+b

Where, slope, m = (kCs)/Vt and intercept, b = (kVxCx)/Vt

Now, b/m = (kVxCx)/Vt x Vt/(kCs)

Cx = bCs /mVx

Standard Addition Method

Another approach to determine Cx

• Extrapolate line on plot to x-intercept

• Recall: At Vs = 0 instrument response (relating to concentration of x in sample)

• At x-intercept, you know the volume of analyte added to (i.e., inherent in) the sample.

• Another way: This value S = 0 (no instrument response) no analyte present in sample

In any case, Since S = 0,

Therefore, S = (kVsCs)/Vt + (kVxCx)/Vt = 0

Solve for Cx,

Cx = - (Vs)oCs / Vx

Standard Addition Methods• In the interest of saving time or sample, it is possible to

perform standard addition analysis by using only two increments of sample. A single addition of Vs mL of standard would be added to one of the two samples and we can write, S1 = (kVxCx)/Vt and S2 = (kVxCx)/Vt + (kVsCs)/Vt S

S

k V C V C

VX

V

kV CV C

V CV C

V C

S S

S

CS V C

V S S

x x s s

t

t

x x

s s

x x

s s

x x

xs

x

2

1

2 1

1

1

2 1

1

( )

( )

Internal standard Method

An Internal Standard is a substance that is added in a constant amount to all samples, blanks and calibration standards in an analysis.

Calibration involves plotting the ratio of the analyte signal to the internal standard signal as a function of analyte concentration of the standards.

This ratio for the samples is then used to obtain their analyte concentrations from a calibration curve.

Internal standard can compensate for several types of both random and systematic errors.

Sampling and

Method of Least Squares

Sampling is one of the most important operations in a chemical analysis.

Chemical analyses use only a small fraction of the available sample. The fractions of the samples that collected for analyses must be representative of the bulk materials.

Knowing how much sample to collect and how to further subdivide the collected sample to obtain a laboratory sample is vital in the analytical process.

All three steps of sampling, standardization, and calibration require a knowledge of statistics.

Analytical samples and methods

Sample size Type of analysis

> 0.1g Macro

0.01~0.1g Semimicro

0.0001~0.01g Micro

< 10–4 g Ultramicro

Analytical level Type of constituent

1%~100% Major

0.01%(100ppm)~1% Minor

1ppb~100ppm Trace

<1 ppb Ultratrace

Interlaboratory error as a function of analyte concentration. Note that the relative standard deviation dramatically increase as the analyte concentration decreases.

W. Horowitz, Anal. Chem., 1982, 54, 67A-76A.

Real Samples

Matrix is the medium containing analyte.

A matrix effect is a change in the analytical signal caused by anything in the sample other than analyte.

Sample are analyzed, but constituents or concentrations are determined.

Sampling is the process by which a sample population is reduced in size to an amount of homogeneous material that can be conveniently handled in the lab and whose composition is representative of the population (unbiased estimate of population mean).

Ex.

Population : 100 coins

Each coin is a sampling unit or an increment

Gross sample: 5 coins

the collection of individual sampling units or increments

Lab sample : the gross sample is reduced in size and made homogeneous

Identify the population

Collect a gross sample

Reduce the gross sample to a lab sample

Steps in obtaining a lab sample ( a few grams ~ a few hundred grams).

Lab sample may constitute as little as 1 part in 107 or 108 of the bulk material.

QUARTERING SAMPLES

A method of obtaining a representative sample for analysis or test of an aggregate with occasional shovelsful, of which a heap or cone is formed, This is flattened out and two opposite quarter parts are rejected. Another cone is formed from the remainder which is again quartered, the process being repeated until a sample of the required size is left.

The procedures vary somewhat, depending upon the size of the sample.

Quartering Method

Mix samples thoroughly. Pour it onto a large flat surface.

Divide the sample into four equal parts.

Save the 2 opposite quarters. Be sure to save the fine material at the bottom of the saved quarter. If the sample is still to large, divide the sample again.

Save

Save

Discard

Discard

SamplingSampling is the process of extracting from a large quantity of material a small portion, which is truly representative of the composition of the whole material.

1) Three main group of sampling:

1>   Census : all the material is examined impracticable

2>   Casual sampling on an ad hoc basis unscientific

3>  Statistical sampling

2) Sampling procedure

Bulk sample homogeneous or heterogeneous

    Increment

Gross sample

Sub sample

Analysis sample

Census vs Random sampling

CensusA complete enumeration, usually of a population, but also businesses and commercial establishments, farms, governments, and so forth.

A complete study of the population as compared to a sample.

Random samplingA commonly used sampling technique in which sample units are selected so that all combinations of n units under consideration have an equal chance of being selected as the sample.

(Statistical sampling meaning) A sampling method in which every possible sample has the same chance of being selected.

3) Sampling statistics :

Total error = sampling error + analytical error

sT = ( sS2 + sA

2 )1/2

In designing a sampling plan the following points should be considered.

1>   the number of samples to be taken

2>   the size of the sample

3>   should individual samples be analysed

or should a sample composed of two or more increments (composite) be prepared.

How much should be analyzed ?

mR2 KS

where m = mass of each sample analyzed, R= desired relative SD.

How many portions should be analyzed ?

e = (tsS) / (n)1/2

n = t2sS2 / e2

where n = the number of samples needed

t = Student’s t for the 95% confidence level and n1

degree of freedom.

Statistics of sampling segregated materials

sS2 = [A / mn] + [ B / n]

where sS is the standard deviation of n samples, each of mass m.

The constant A and B are properties of the bulk material and must be measured in preliminary experiments.

Sampling uncertainties

Both systematic and random errors in analytical data can be traced to instrument, method, and personal causes. Most systematic errors can be eliminated by exercising care, by calibration, and by the proper use of standards, blanks, and reference materials.

For random and independent uncertainties, the overall standard deviation s0 for an analytical measurement is related to the standard deviation of the sampling process ss and the the standard deviation of the method sm by the relationship

s02 = ss

2 + sm2

When sm< ss/3, there is no point in trying to improve the measurement precision.

Size of the gross sample

Basically, gross sample weight is determined by

(1) The uncertainty that can be tolerated between the composition of the gross sample and that of the whole,

(2) The degree of heterogeneity of the whole,

(3) The level of particle size at which heterogeneity begins.

To obtain a truly representative gross sample, a certain number N of particles must be taken. The number of particles required in a gross sample ranges from a few particles to 1012 particles.

Sampling homogeneous solutions of liquids and gases

Well mixed solutions of liquids and gases require only a very small sample because they are homogeneous down to the molecular level.

Homogeneous ?

Which portion?

Flowing stream?

Gas sampling : Sampling bag (Tedlar® bag)

with a Teflon fitting

Trap in a liquid

Adsorbed onto the surface of a solid

SPME

Tedlar® baghttp://www.tedlarbag.com/tedlarbag.asp#tedlar

Depth profile of nitrate in sediment from freshwater Lake Sobygard in Denmark.

Lake stratification from spring to summer

15m

18m

Epilinion 17oC

Thermocline 5~17oC

Hypolimnion 5~6oC

SAMPLE PREPARATION

(1) Is the sample a Solid or a Liquid?Liquids

(2) Are you interested in all sample components or only one or a few?If only a few then separation is necessary by extraction or chromatography.

(3) Is the concentration of the analytes appropriate for the measurement technique?If not, dilute or concentrate with extraction, evaporation, lyophilization.

(4) Is sample unstable ?If yes, derivatize, cool, freeze, store in dark

(5) Is the liquid or solvent compatible with the analytical method?If not, do solvent exchange with extraction, distillation, lyophilization.

http://www.trincoll.edu/~dhenders/textfi~1/Chem%20208%20notes/sample_preparation.htm

Sampling particulate solids

Identify the population to be analyzed

Randomly collect N particles to give a gross sample

Reduce particle size of gross sample and homogenize

Randomly collect N particles

Is this sample of a suitable size for the lab

Store the lab sample

Remove portions of the lab sample for analysis

No

Steps in sampling and measurement of salt in a potato chip. Step 1 introduces the sampling variance. Steps 2 to 4 introduce the analytical variance.

Linear regression

Linear regression uses the method of least squares to determine the best linear equation to describe a set of x and y data points. The method of least squares minimizes the sum of the square of the residuals - the difference between a measured data point and the hypothetical point on a line. The residuals must be squared so that positive and negative values do not cancel. Spreadsheets will often have built-in regression functions to find the best line for a set of data. A common application of linear regression in analytical chemistry is to determine the best linear equation for calibration data to generate a calibration or working curve. The concentration of an analyte in a sample can then be determined by comparing a measurement of the unknown to the calibration curve.

Calibration curve for the determination of isooctane in a hydrocarbon mixture.

The slope-intercept form of a straight line.

Least-square curve fitting.

Linear least-squares analysis gives you the equation for the best straight line among a set of x, y data points when the x data contain negligible uncertainty.

Determining the best line for calibration data is done using linear regression.

Equation of calibration line:

y(sy) = [m(sm)]x + [b(sb)]

The Least squares method finds the sum of the squares of the residuals ssresid and minimizes these according to the minimization technique of calculus.

ssresid = [yi (b+mxi)]2

A total sum of the squares :

sstot = syy = (yi y)2 = yi 2 yi 2/N

The coefficient of determination (R2) is a measure of the fraction of the total variation in y that can be explained by the linear relationship between x and y.

R2 = 1 (ssresid / sstot)

The closer R2 is to unity, the better the linear model explains the y variations.

Correlation coefficient

Pearson correlation coefficient :

r = [(xi x)(yiy) / nsxsy ]

= [xiyi(nxy)] / [(xi2 nx2)(yi

2ny2)] 1/2

= (nxiyixiyi) / [{nxi2(xi)

2}{nyi2(yi)

2}]1/2

rmax = 1, rmin = 1, r = 0 (when xy=0),

0.90 < r < 0.95 linearity

0.95 < r < 0.99 good linearity

0.99 < r excellent linearity

For the linear equation: y = mx + b Useful quantities:

Where: N is the number of calibration data points.L is the number of replicate measurements of the unknown.and is the mean of the unknown measurements.

Standard deviation of the residuals:

Standard deviation of the intercept:

Standard deviation of the slope:

Standard deviation of a unknown read from a calibration curve:

Chapter 2Chemical Apparatus, and Unit Operation of

Analytical ChemistryClassifying Chemicals1. Reagent Grade: Reagent grade chemical conform to the minimum standards set forth by the Reagent Chemical committee of the American Chemical Society and are used wherever possible in analytical work.2. Primary Standard Grade: Extraordinary purity is required for a primary standard. Primary standard reagent is carefully analyzed and the assay is printed on the container label. 3. Special-Purpose Reagent: chemicals that have been prepared for a specific application. Included among these are solvents for spectrophotometry and high-performance liquid chromatography.3. Special-Purpose Reagent: chemicals that have been prepared for a specific application. Included among these are solvents for spectrophotometry and high-performance liquid chromatography.2. Primary standard Grade: Extraordinary purity is required for a primary standard. Primary standard reagent is carefully analyzed and the assay is printed on the container label.

Rules for Handling Reagents and Solutions

1. Select the best grade of chemical available for analytical work.2. Replace the top of every container immediately after removal of the reagent.3. Hold the stoppers of reagent bottles between your fingers.4. Never return any excess reagent to a bottle.5. Never insert spatulas, spoons, or knives into a bottle that contains a solid chemicals.6. Keep the reagent shelf and the laboratory balance clean and neat.7. Observe regulations concerning the disposal of surplus reagents and solutions.

Cleaning and Marking Laboratory Ware

Every beaker, flask, or crucible that will contain the sample must be thoroughly cleaned before being used. The apparatus should be washed with a hot detergent solution and then rinsed, initially with tap water and finally with several small portions of deionized water. Organic solvents such as benzene or acetone may be used to remove grease films.

A chemical analysis is ordinarily performed in duplicate or triplicate. Each vessel that holds a sample must be marked so that its content can be positively identified. Flask, beaker and some crucibles have small etched areas on which semi permanent mark can be made with a pencil.

Types of Analytical Balances

An analytical balance is a weighing instrument with a maximum capacity that ranges from 1 g to a few kilograms with a precision of at least 1 part in 105 at maximum capacity.Macrobalances have a maximum capacity ranging between 160 and 200 g; measurement can be made with a standard deviation of 0.1mg.Semimicroanalytical balances have a maximum load of 10 to 30 g with a precision of 0.01mg.Microanalytical balance has a capacity of 1 to 3 g and a precision of 0.001mg.

Desiccators and Desiccants

Oven drying is the most common way of removing moisture from solids. This approach is not appropriate for substances that decompose or for those from which water is not removed at the temperature of the oven.Dried material are stored in desiccator while they cool so as to minimize the uptake of moisture. The base section of the desiccator contains a chemical drying agent (desiccants) such as anhydrous calcium chloride, calcium sulfate, magnesium perchlorate or phosphorus pentoxide.

Desiccator

Weighing by Difference

Weighing by difference is a simple method for determining a series of sample weights. First the bottle and its contents are weighed. One sample is then transferred from the bottle to a container; gentle tapping of the bottle with its top and slight rotation of the bottle control over the amount of sample removed. Following transfer, and its residual contents are weighed. The mass of the sample is the difference between the two weighings.

Weighing bottles

Simple CruciblesSimple crucibles serve only as containers. Porcelain, aluminum oxide, silica and platinum crucibles maintain constant mass and are used principally to convert a precipitate into a suitable weighing form. The solid is first collected on filter paper. The filter and contents are then transferred to a weighed crucible, and the paper is ignited.

Filtering CruciblesFiltering crucibles serve not only as containers but also as filters. A vacuum is used to hasten the filtration, a tight seal between crucible and filtering flask is accomplished with any of the several types of rubber adapters.

Filtering Crucible

Sintered-glass CruciblesSintered-glass crucibles are manufactured in fine, medium, and coarse porosities. The upper temperature limit for sintered glass crucible is ordinarily about 200oC. Filtering crucibles made entirely of quartz can tolerate substantially higher temperatures.

Filter PaperPaper is an important filtering medium. Ashless paper is manufactured from cellulose fibers that have been treated with hydrochloric and hydrofluoric acids to remove metallic impurities and silica, ammonia is then used to neutralize the acids. It is necessary to destroy the paper by ignition if the precipitate collected on it is to be weighed.

Decantation and transferring precipitate

Folding and seating a filter paper

Vacuum Filtration

Heating EquipmentMany precipitate can be weighed directly after being brought to constant mass in a low temperature drying oven. Such an oven is electrically heated and capable of maintaining a constant temperature to within 1oC. The maximum attainable temperature ranges from 140 to 260oC, depending on make and model, for many precipitate 110oC is a satisfactory drying temperature.Microwave laboratory ovens are currently appearing on the market. Where applicable, these greatly shorten drying cycles.Muffle furnace (a heavy duty electric furnace) is capable of maintaining controlled temperatures of 1100oC or higher. Long handled tongs and heat resistance gloves are needed for protection.

Volume Measurement

· Pipets

· Burets

· Volumetric flask

· Measuring cylinder

Typical pipets

Automatic pipet

Burets and volumetric flask

Reading a buret

Using pipet

Titration

Calibrating Glassware

Volumetric glassware is calibrated by measuring the mass of a liquid (water) of known density and temperature that is contained in the volumetric ware.

Laboratory Notebook

1. Record all data and observations

2. Supply each entry with a heading

3. Date each page of the notebook

4. Never attempt to erase an incorrect entry, cross it out with single horizontal line

5. Never remove a page from the notebook

6. Do not overcrowd entries

7. Keep first few pages for table of contents

Lab note book

Scales of Measurements

• (left) Carbon-fiber electrode with a 100-nanometer-diameter (100 × 10−9 meter) tip extending from glass capillary. The marker bar is 200 micrometers (200 × 10−6 meter). [From W.-H. Huang, D.-W. Pang, H. Tong, Z.-L. Wang, and J.-K. Cheng, Anal. Chem. 2001, 73, 1048.] (middle) Electrode positioned adjacent to a cell detects release of the neurotransmitter, dopamine, from the cell. A nearby, larger counterelectrode is not shown. (right) Bursts of electric current detected when dopamine is released. Insets are enlargements. [From W.-Z. Wu, W.-H. Huang, W. Wang, Z.-L. Wang, J.-K. Cheng, T. Xu, R.-Y. Zhang, Y. Chen, and J. Liu, J. Am. Chem. Soc. 2005, 127, 8914.]

The possible scale of measurements in analytical chemistry is astounding, ranging from the atomic level to the size of galaxies!

Chapter 1: Measurements1-1 SI Units

Using prefixes as multipliers

We customarily use prefixes for every third power of ten, e.g. 10-9. 10-6, 10-3 …

Converting between unitsExample: Express the energy 20 Calories in terms of ? kilojoules (kJ)

Oops! In 1999, the $125 million Mars Climate Orbiter spacecraft was lost when it entered the Martian atmosphere 100 km lower than planned. The navigation error would have been avoided if people had written their units of measurement. Engineers who built the spacecraft calculated thrust in the English unit, pounds of force. Jet Propulsion Laboratory engineers thought they were receiving the information in the metric unit, newtons. Nobody caught the error.

Why should we be concerned of units?

1-2 Chemical concentrations

A few concepts

• Solution: homogeneous mixture of two or more substances.

• Solute: a minor species in a solution.• Solvent: the major species in a solution.• Concentration: how much solute is contained in

a given volume or mass of solution or solvent.• Strong & weak electrolytes.

1-2 Chemical concentrations (Continued)

Molarity and Molality:Molarity (M) is the number of moles of a

substance per liter of solution.Molality (m) is concentration expressed as

moles of a substance per kilogram of solvent.

Problem 1-14. What is the formal concentration (expressed as mol/L = M) of NaCl when 32.0 g are dissolved in water and diluted to 0.500 L?

Answer: The molecular mass of NaCl is 58.44 g/mol The moles of sale in 32.0 g are 32.0 g /58.44(g/mol) = ……

1.3 Preparing solutions

con con dil dilM V M V

• Preparing a solution with a desired molarity

• Dilution

Problem 1-32: A bottle of concentrated aqueous sulfuric acid labeled 98.0wt % H2SO4, has a concentration of 18.0 M. HowMany millilitres of reagent should be diluted to 1.000 L to give 1.00 M H2SO4?

Solution:

STATISTICAL TESTS AND ERROR ANALYSIS

PRECISION AND ACCURACY

PRECISION – Reproducibility of the result

ACCURACY – Nearness to the “true” value

TESTING ACCURACY

TESTING PRECISION

SYSTEMATIC / DETERMINATE ERROR

• Reproducible under the same conditions in the same experiment

• Can be detected and corrected for• It is always positive or always negative

To detect a systematic error:• Use Standard Reference Materials• Run a blank sample• Use different analytical methods• Participate in “round robin” experiments

(different labs and people running the same analysis)

RANDOM / INDETERMINATE ERROR

• Uncontrolled variables in the measurement• Can be positive or negative• Cannot be corrected for• Random errors are independent of each other

Random errors can be reduced by:• Better experiments (equipment, methodology,

training of analyst)• Large number of replicate samples

Random errors show Gaussian distribution for a large number of replicates

Can be described using statistical parameters

For a large number of experimental replicates the results approach an ideal smooth curve called the GAUSSIAN or NORMAL DISTRIBUTION CURVE

Characterised by:

The mean value – x

gives the center of the distribution

The standard deviation – s

measures the width of the distribution

The mean or average, x

the sum of the measured values (xi) divided by the number of measurements (n)

n

x

x

n

1ii_

The standard deviation, s

measures how closely the data are clustered about the mean (i.e. the precision of the data)

2

ii

1n

xx

s

NOTE: The quantity “n-1” = degrees of freedom

• Variance

• Relative standard deviation

• Percent RSD / coefficient of variation

x

sRSD

Other ways of expressing the precision of the data:

Variance = s2

100x

s%RSD

POPULATION DATAFor an infinite set of data,

n → ∞ : x → µ and s → σ

population mean population std. dev.

The experiment that produces a small standard deviation is more precise .

Remember, greater precision does not imply greater accuracy.

Experimental results are commonly expressed in the form:

mean standard deviation

sx

_

The more times you measure, the more confident you are that your average value is approaching the “true” value.

The uncertainty decreases in proportion to n1/

EXAMPLE

Replicate results were obtained for the analysis of lead in blood. Calculate the mean and the standard deviation of this set of data.

Replicate [Pb] / ppb

1 752

2 756

3 752

4 751

5 760

Replicate [Pb] / ppb

1 752

2 756

3 752

4 751

5 760

n

xx i_

2i

1n

xxs

NB DON’T round a std dev. calc until the very end.

Also:

x

sRSD

100x

s%RSD

0.00500 754

3.77

0.500% 100754

3.77

Variance = s2 14.2 3.77 2

754x

3.77s 754 4 ppb Pb

The first decimal place of the standard deviation is the last significant figure of the average or mean.

Lead is readily absorbed through the gastro intestinal tract. In blood, 95% of the lead is in the red blood cells and 5% in the plasma. About 70-90% of the lead assimilated goes into the bones, then liver and kidneys. Lead readily replaces calcium in bones.

The symptoms of lead poisoning depend upon many factors, including the magnitude and duration of lead exposure (dose), chemical form (organic is more toxic than inorganic), the age of the individual (children and the unborn are more susceptible) and the overall state of health (Ca, Fe or Zn deficiency enhances the uptake of lead).

Pb – where from?• Motor vehicle emissions• Lead plumbing• Pewter• Lead-based paints• Weathering of Pb

minerals

European Community Environmental Quality Directive – 50 g/L in drinking water

World Health Organisation – recommended tolerable intake of Pb per day for an adult – 430 g

Food stuffs < 2 mg/kg Pb

Next to highways 20-950 mg/kg Pb

Near battery works 34-600 mg/kg Pb

Metal processing sites 45-2714 mg/kg Pb

CONFIDENCE INTERVALS

n

tsxμ_

The confidence interval is given by:

where t is the value of student’s t taken from the table.

The confidence interval is the expression stating that the true mean, µ, is likely to lie within a certain distance from the measured mean, x.

– Student’s t test

A ‘t’ test is used to compare sets of measurements.

Usually 95% probability is good enough.

Example:

The mercury content in fish samples were determined as follows: 1.80, 1.58, 1.64, 1.49 ppm Hg. Calculate the 50% and 90% confidence intervals for the mercury content.

n

tsx_

μ

50% confidence:

t = 0.765 for n-1 = 3

4

0.1310.7651.63 μ

05.01.63 μ

There is a 50% chance that the true mean lies between 1.58 and 1.68 ppm

Find x = 1.63

s = 0.131

n

tsx_μ

90% confidence:

t = 2.353 for n-1 = 3

4

0.1312.3531.63 μ

15.01.63 μ

There is a 90% chance that the true mean lies between 1.48 and 1.78 ppm

x = 1.63

s = 0.131

1.63

1.68

1.48

1.58

1.78

90%

50%

Confidence intervals - experimental uncertainty

1) COMPARISON OF MEANS

ns

xvalueknowntcalc

Statistical tests are giving only probabilities. They do not relieve us of the responsibility of interpreting

our results!

Comparison of a measured result with a ‘known’ (standard) value

tcalc > ttable at 95% confidence level

results are considered to be different the difference is significant!

APPLYING STUDENT’S T:

For 2 sets of data with number of measurements n1 , n2 and means x1, x2 :

Where Spooled = pooled std dev. from both sets of data

2nn

1)(ns1)(nss

21

2221

21

pooled

21

21

pooled

21calc nn

nn

s

xxt

2) COMPARISON OF REPLICATE MEASUREMENTS

tcalc > ttable at 95% confidence level difference between results is significant.

Degrees of freedom = (n1 + n2 – 2)

Compare two sets of data when one sample has been measured many times in each data set.

3) COMPARISON OF INDIVIDUAL DIFFERENCES

e.g. use two different analytical methods, A and B, to make single measurements on several different samples.

ns

dt

dcalc

tcalc > ttable at 95% confidence level difference between results is significant.

1n

)d(ds

2i

d

Where

d = the average difference between methods A and B

n = number of pairs of data

Perform t test on individual differences between results:

Compare two sets of data when many samples have been measure only once in each data set.

Example:

(di)

Are the two methods used comparable?

1n

)d(ds

2i

d

16

04.002.011.011.022.002.0s

222222

d

12.0sd

ns

dt

dcalc

60.12

0.06tcalc

2.1tcalc

ttable = 2.571 for 95% confidence

tcalc < ttable

difference between results is NOT significant.

22

21

calcs

sF

Fcalc > Ftable at 95% confidence level

the std dev.’s are considered to be different the difference is significant.

F TEST

COMPARISON OF TWO STANDARD DEVIATIONS

Q TEST FOR BAD DATA

range

gapQcalc

The range is the total spread of the data.

The gap is the difference between the “bad” point and the nearest value.

Example:

12.2 12.4 12.5 12.6 12.9

Gap

Range

If Qcalc > Qtable discarded questionable point

EXAMPLE:

The following replicate analyses were obtained when standardising a solution: 0.1067M, 0.1071M, 0.1066M and 0.1050M. One value appears suspect. Determine if it can be ascribed to accidental error at the 90% confidence interval.

Arrange in increasing order:

Q = GapRange

Analytical Figures of Merit

“Indicate a characteristic of an instrumental technique for a given analyte”

“7”Accuracy, Precision, Signal-to-Noise Ratio

Sensitivity, Limit of Detection

Linearity, Linear Dynamic Range

Accuracy

Indicates how close the measured value is to the true analytical concentration

Requires a Standard Reference Material (SRM) of other official measure

NIST: National Institute of Standards and Technology

Accuracy

Most commonly reported as percent error

│Cm - Ct│

Ct

where:

Cm = measured concentration

Ct = true concentration

x 100%

Precision

Indicates the reproducibility of repetitive measurements of equivalent samples

May be expressed as:

1. Standard Deviation (s or σ)

2. Relative Standard Deviation (RSD)

3. Confidence Limits

Precision

Standard Deviation

For an infinite number of measurements (σ)

For a finite number of measurements (s)

Standard Deviation

Note that both s and σ have the same units as the original values

How many values should be obtained?

Rule of thumb: 16

0

5

10

15

20

25

30

35

40

45

96.0 97.0 98.0 99.0 100.0 101.0 102.0 103.0 104.0

Value

Po

pu

lati

on

Total Population = 1000

0.00%

0.10%

0.20%

0.30%

0.40%

0.50%

0.60%

0.70%

0.80%

0.90%

1.00%

0 5 10 15 20 25 30 35 40 45 50

Number of Samples

Err

or

in M

ean

How far is the measured mean from the true value?

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

0 5 10 15 20 25 30 35 40 45 50

Number of Samples

Err

or

in S

td. D

ev.

How far is s from σ?

Short Cut: σ ≈ 1/5 (peak-to-peak noise)

Relative Standard Deviation

RSD = σ/mean

Where the mean may be the signal or the analyte concentration. RSD is a unit-less value, so σ must have the same units as

the mean.

RSD is often reported as %RSD, and may be used to compare different techniques.

Confidence Limits

Define an interval that encloses

the true value (Ct) with aspecified level of confidence.

1. Cm ± σ 66.7% Confidence Level

2. Cm ± 2σ 95% Confidence Level

3. Cm ± 3σ 99.0% Confidence Level

Signal to noise Ratio (S/N)

S/N = Sm/σ = 1/RSD

Notes:

1. N = noise (σ)

2. S/N is unitless

3. Always try top maximize S/N

4. S/N is used to compare instruments

5. A plot of S/N versus an instrumental parameter reaches a maximum at the optimum value for that parameter

Sensitivity

Experimental slope of a calibration curve

m = ΔS/ΔC

Sensitivity is almost always specific for one particular instrument.

0

10

20

30

40

50

60

70

80

90

100

0 2 4 6 8 10 12

Concentration (ppm)

Sig

na

l (V

)

m

LOD

LDR

Limit of Detection

The analyte concentration yielding an analytical signal equal to 3 times the standard deviation in the blank signal.

LOD = 3 x σbl / m

By definition, the LOD has just one significant figure!!

Linearity

Measure of how well the observed data follows a straight line.

SA = mC

SA = Analytical Signal

m = calibration sensitivity

Remember SA = Stot - Sbl

Linearity

Plot log(S) versus log(C)

log(SA) = log(m) + log(C)

The slope of this plot should be 1.00

A calibration curve is defined as linear if the

log-log plot has a slope in the range 0.95-1.05

Linear Dynamic Range

The concentration range over which the calibration curve is linear

Lower End → LOD

Upper End

Analyte Concentration where the observed signal falls 5% below the extrapolated line

LDR Units are

“orders of magnitude”

or

“decades”

of analyte concentration

LDR is easiest to observe on log-log plot

If linearity is poor, define an analytically useful range (AUR)

Other figures of merit may be calculated, but these 7 are sufficient.

Selectivity and Resolution may be useful in cases where more than one analyte is

determined in the same sample.

141

Chapter 2

Data Handling

142

Accuracy and Precision

Accuracy can be defined as the degree of agreement between a measured value and the true or accepted value. As the two values become closer, the measured value is said to be more accurate.

 

Precision is defined as the degree of agreement between replicate measurements of the same quantity.

143

Assuming the correct or accepted value is represented by the center of the circles below, if all values occurred within, for example, the red circles, results are precise but not accurate. If all values occurred within the yellow circle, results are both accurate and precise. If results were scattered randomly, results are neither precise nor accurate.

144

145

Example The weight of a person was measured five times

using a scale. The reported weights were 84, 83, 84, 85, and 84 kg. If the weight of a person is 76 kg weighed on a standard scale , then we know that the results obtained using the first scale is definitely not accurate.

However, the values of the weights for the five replicate measurements are very close and reproducible. Therefore measurements are precise. Therefore, a measurement could be precise but not accurate.

146

Significant Figures

At the most basic level, Analytical Chemistry relies upon experimentation; experimentation in turn requires numerical measurements. And measurements are always taken from instruments made by other workers.

Significant figures are concerned with measurements not exact countings.

147

Some information about measurements

1) Examples we will study include the metric ruler, the graduated cylinder, and the scale.

2) Because of the involvement of human beings, NO measurement is exact; some error is always involved. This means that every answer in science has some uncertainty associated with it. We might be fairly confident we have the correct answer, but we can never be 100% certain we have the EXACT correct answer.

148

3) Measurements always have two parts - a numerical part (sometimes called a factor) and a dimension (a unit). The reason for this is that we are measuring quantities - length, elapsed time, temperature, mass, etc. Not only do we have to tell how much there is, but we have to tell how much of what.

Measuring gives significance (or meaning) to each digit in the number produced. This concept of significance, of what is and what is not significant is VERY IMPORTANT.

149

The concept of significant figures (or significant digits) is important. A measurement can be defined as the comparison of the dimensions of an object to some standard.

The dimensions of an object refer to some property the object possesses. Examples include mass, length, area, density, and electrical charge. Dimensions are often called units.

150

Identifying significant digits

The following rules are helpful in identifying significant digits

1. Digits other than zero are significant.e.g., 42.1m has 3 sig figs.2. Zeroes are sometimes significant, and sometimes they are not.3. Zeroes at the beginning of a number (used just to position the decimal point) are not significant.e.g., 0.025m has 2 sig figs. In scientific notation, this can be written as 2.5*10-2m

151

4. Zeroes between nonzero digits are significant.e.g., 40.1m has 3 sig figs

5. Zeroes at the end of a number that contains a decimal point are significant.e.g., 41.0m has 3 sig figs, while 441.20m has 5. In scientific notation, these can be written respectively as4.10*101 and 4.4120*102

6. Zeroes at the end of a number that does not contain a decimal point may or may not be significant. If we wish to indicate the number of significant figures in such numbers, it is common to use the scientific notation.

152

e.g., The quantity 52800 km could be having 3, 4, or 5 sig figs—the information is insufficient for decision. If both of the zeroes are used just to position the decimal point (i.e., the number was measured with estimation ±100), the number is 5.28×104 km (3 sig figs) in scientific notation. If only one of the zeroes is used to position the decimal point (i.e., the number was measured ±10), the number is 5.280×104 km (4 sig figs). If the number is 52800±1 km , it implies 5.2800×104 km (5 sig figs).

153

Exact Numbers

Exact numbers can be considered as having an unlimited number of significant figures. This applies to defined quantities too. e.g.,

1. The rules of significant figures do not apply to (a) the count of 47 people in a hall, or (b) the equivalence: 1 inch = 2.54 centimeters.

2. In addition, the power of 10 used in scientific notation is an exact number, i.e. the number 103 is exact, but the number1000 has 1 sig fig.It actually makes a lot of sense to write numbers derived from measurements in scientific notation, since the notation clearly indicates the number of significant digits in the number.

154

Look at the following example

155

The general rule for estimation of the last digit is to record to 1/10th of the smallest division of the measuring device. So, for the common centimeter-ruler, in which the smallest division is 1mm, the estimate for the last digit can be to 0.1mm which is 1/10th, of the smallest division.

156

Suppose a student records a length of 26.2mm with a ruler (between the red marks). In his/her judgment, the length is greater than 26.1mm but less than 26.3mm, and so the best estimate is 26.2mm. The measurement can be written as 26.2±0.1 mm. The number 26.2mm contains three significant figures. Although the last digit, 2, is an estimate, it is considered to be a significant figure for the measurement.

157

How do you read it?

How do you read it?

158

159

160

161

162

Math With Significant Figures

Addition and SubtractionIn mathematical operations involving significant figures, the

answer is reported in such a way that it reflects the reliability of the least precise operation. Let's state that another way: a chain is no stronger than its weakest link. An answer is no more precise that the least precise number used to get the answer. Let's do it one more time: imagine a team race where you and your team must finish together. Who dictates the speed of the team? Of course, the slowest member of the team. Your answer cannot be MORE precise than the least precise measurement.

163

For addition and subtraction, look at the decimal portion (i.e., to the right of the decimal point) of the numbers ONLY. Here is what to do:

1) Count the number of significant figures in the decimal portion of each number in the problem. (The digits to the left of the decimal place are not used to determine the number of decimal places in the final answer.)

2) Add or subtract in the normal fashion.3) Round the answer to the LEAST number of

places in the decimal portion of any number in the problem.

164

Find the formula weight for Ag2MoO4 given the following atomic weights: Ag = 107.870, Mo = 95.94, O = 15.9994.

The number with the least number of digits after the decimal point is 95.94 which has two digits for expression of precision. Also, it is the number with the highest uncertainty. The atomic weights for Ag and O have 3 and 4 digits after the decimal point. Therefore if we calculate the formula weight we will get 375.6776. However, the answer should be reported as 375.68 ( i.e. to the same uncertainty of the least precise value.

165

Multiplication and Division

In mathematical operations involving significant figures, the answer is reported in such a way that it reflects the reliability of the least precise operation. Let's state that another way: a chain is no stronger than its weakest link. An answer is no more precise that the least precise number used to get the answer. Let's do it one more time: imagine a team race where you and your team must finish together. Who dictates the speed of the team? Of course, the slowest member of the team. Your answer cannot be MORE precise than the least precise measurement.

166

The following rule applies for multiplication and division:

The number having the least number of significant figures is called the KEY NUMBER. The LEAST number of significant figures in any number of the problem determines the number of significant figures in the answer.

This means you MUST know how to recognize significant figures in order to use this rule.

In case where two or more numbers have the same least number of significant figures, the key number is determined as the number of the lowest value regardless of decimal point.

167

2.5 x 3.42.

The answer to this problem would be 8.6 (which was rounded from the calculator reading of 8.55). Why?

2.5 is the key number which has two significant figures while 3.42 has three. Two significant figures is less precise than three, so the answer has two significant figures.

168

How many significant figures will the answer to 3.10 x 4.520 have?

You may have said two. This is too few. A common error is for the student to look at a number like 3.10 and think it has two significant figures. The zero in the hundredth's place is not recognized as significant when, in fact, it is. 3.10 is the key number which has three significant figures.

Three is the correct answer. 14.0 has three significant figures. Note that the zero in the tenth's place is considered significant. All trailing zeros in the decimal portion are considered significant.

Example

Synthetic Chemistry Analytical Chemistry

To make a new stuff To find out what is it

“white powder”

Qualitative Quantitative

to identifyThis white powder is NaCl

to detectThere is a trace admixtureof iodine in this sample

to analyzeto determine X in a sample to quantifyto measure

Run assay determinationquantitation

0.375 mg/mL caffeine

Composition, %%:Cu, 85.34(3); Zn, 11.23; Sn, 1.02(2);Fe, 0.123(2); Mn, 0.058(5); Ni, 0.229(5); …

A chocolate story: how much caffeine?

An example from Harris:

Garbage in,Garbage out!

Calibration