A Textbook of Analytical Chemistry

By Y. Anjaneyulu and K. Chandrasekhar

Analytical Chemistry plays a vital role in many disciplines like Chemistry, Physics, Biology and industrial process. Scientists, engineers and technicians in all fields of Chemistry, Biotechnology, Water Resources, Engineering and Medicine often need information from chemical analysis. In addition, after the analysis is done, interpretation of the data is complicated and plays a major role. The present book is for providing training to analytical chemists in various modern analytical methods, which is a major task for Universities, Industrial organizations and government laboratories because of the rapid growth in new techniques, and necessity of analyzing multi-complex samples in short time.
Salient Features
* Many of the topics are presented with thorough background of related chemical principle
* Wide range of new analytical techniques that are useful in modern analytical chemistry are presented in this book, so that necessary confidence can be acquired to obtain high quality analytical data
* Basic principles of Gravimetric, Volumetric Analysis with latest developments and Statistical Treatment of Analytical data are provided
* Atomic absorption, inductively couple plasma spectroscopy and flame photometry are presented with details of design, fabrication and operational variables and their applications are presented with good examples
* Electroanalytical method are given in detail and discussed with their specific applications

• Language: English
• Publisher: BSP BOOKS
• Release date: Nov 5, 2019
• ISBN: 9789388305754

Book preview

Anjaneyulu.

Chapter 1

Introduction to Analytical Chemistry

Introduction

Analytical chemistry deals with quantitative analysis of composition of substances and complex materials in various matrices, by measuring a physical or chemical property of a distinctive constituent of the component or components of interest. Analytical methods are classified according to the property of the analyte measured.

1.1 History of Development of Analytical Chemistry

From the vast number of organic compounds now known to man, many have been applied in the diverse fields of medicine, science and technology with highly beneficial results. As organic chemistry developed in the 19th century, the recognition of functional group reactivity led to a more intelligent use of organic compounds, among which was their use as chemical reagents in analytical chemical processes. The work of F. Feiglhas provided ample evidence, not only of the inestimable value of organic compounds in chemical analysis, but also of the possibilities for synthesizing new and more effective reagents.

The very early history of chemical analysis has been admirably summarised by F. Szabadvary in his comprehensive History of Analytical Chemistry (1966). Therein, he describes the wet reaction which Pliny used to establish whether a sample of copper sulphate had been adulterated with iron sulphate or not, using papyrus soaked in an extract of gall nuts from the Oak, Quercus infectoria. Pliny described the colour reaction that occurred and thus originated the technique of spot-testing with an organic reagent (gallotannic acid). If this is accepted as a valid example, then Pliny’ s test is all the more remarkable in that a period of some 1600 years had to elapse before similar analytically-based reactions begin to be described in the writings of scientists and philosophers in the 16th and 17th centuries. Otto Tachenius, a Pharmacist and physician from Westphalia, described in some detail (1666) the reactions of the aqueous extract of gall nuts with solutions of several metal salts, and thus may be credited with the publication of the first systematic examination of an organic reagent for metal ions. Such primitive beginnings necessarily involve natural substances, principally those of vegetable origin. Extracts of plant, particularly the flowers, provided solutions of colouring matter for use in the dyeing of clothes. It is not surprising that the colour changes of such solutions in the presence of acids and alkalis attracted the interest of the medieval chemists and pharmacists.

Robert Boyle provided the best example of an early chemist employing colour reactions as diagnostic tests for various chemical substances and the use of plant and animal extracts, such as syrup of violets, extracts of cornflower, cochineal and litmus, served to introduce these reagents to a wider chemical fraternity. His use of an infusion of Lignum nephriticum (a species of pterocarpus) as a test for acids and alkalis is of added interest in that the sky-blue fluorescent colour of the solution is discharged by the addition of acid, thus anticipating the introduction of fluorescent indicators by some 250 years. Boyle’s Syrup of violets became a recognized test for acids and alkalis, and led to its adoption as the first acid-base indicator in what subsequently developed into titrimetric analysis. Technical analysis and industrial chemistry were developed rapidly in the 18thcentury and most of the new chemistry on those days was analytical chemistry.

In 1767, William Lewis published a short treatise on the quality of American potashes in which he described one of the earliest examples of an exact titrimetric process, of a sample of potash with acid and detected the end point using certain vegetable juices, or on paper stained with them. Ten years later, in 1777, Carl Friedrich Wenzel described the chemical use of litmus, curcuma and brazilin in titrimetric processes and from that time the use of such substances became commonplace in acidimetry and alkalimetry.

The first book in English to treat exclusively the subject of volumetric analysis, the first editions of which appeared in 1863, was by Francis Sutton. Even in the second editions, published in 1871, Sutton described only three indicators for use in acid-base titrations: litmus, cochineal and turmeric paper. The first synthetic organic compounds to be used as an acid-base indicator appears to be phenolphthalein, proposed by E. Lluck in 1877, following A. Baeyer’s synthesis of this and other phthaleins in 1871. This application was closely followed (1878) by W.von Miller’s and other’s observation that several azo dyes, in particular methyl orange, are unaffected by carbonic acid and are thus ideal indicators for the titration of carbonates with mineral acids (technical analysis of soda). These synthetic organic compounds thus constitute early applications as reagents in chemical analysis.

Later on, analytical chemistry assumed the supporting role of an inseparable tool in advancing in depth knowledge in any scientific field. Investigations in virtually all of the physical and biological sciences are obliged to make use of analytical data in the course of their work. Analytical techniques became indispensable in the biochemist’s study of living matter and its metabolic processes. The physician relies heavily upon the results of the analysis of body fluids in making his diagnoses. The classification of a mineral is incomplete without knowledge of its chemical composition. Analytical techniques were employed in identifying the products of high-energy bombardments by physicists.

Thus, every experimental investigation of today’s chemistry relies to a greater extent upon the result of analytical measurements. A thorough background in analytical chemistry is thus a vital necessity for all those who aspire to be chemists, regardless of their field of specialisation.

In general, the types of problems encountered in analytical chemistry are:

(a)   Qualitative analysis of elements, ions, atomic groups, and functional groups in mixtures of substances.

(b)   Qualitative analysis of a single chemical species or of each in mixtures of species

(c)   Quantitative analysis for elements, ions, atomic groups, and functional groups in mixtures of substances

(d)   Quantitative analysis of a single chemical species or mixtures of species throughout the sample or at its surface

Many paths are available to achieve the information sought, and many individual techniques may be incorporated into the route leading to a completed analysis. An important part of analytical chemistry is choosing the optimum pathway, a choice which is simplified only through the assimilation of knowledge and experience. Thus, in solving analytical problems an analytical chemist is often required to design or repair electronic systems, arrange optical systems, design instruments, interpret spectra and other instrumental data, perform classical analysis with simple chemicals and solutions, develop and evaluate new procedures or modify old ones, separate simple and complex mixtures, purify samples, write computer programmes and statistically evaluate the reliability of the data.

1.2 What is Analytical Chemistry?

A scientific discipline that develops and applies methods, instruments and strategies to obtain information on the composition and nature of matter in space and time.

Scientific and technological developments are moving at such a pace that during a scientist’s working lifetime of about 40 years there are bound to be many new major advances. Meaning, the job of an analytical chemist is constantly having to adapt to new developments. Instrument and computer manufacturers are producing analytical machines that are ever-increasing in power and scope. Analytical chemists now nearing the end of their working careers, all the major instrumental methods have been introduced since they completed their formal education. Analytical chemistry has found its way into such fields as veterinary sciences, archaeology, farming, and space travel (Fig 1.1).

To sum up analytical chemistry the best word to describe it is boundless

Today analytical chemists are expected to be a vital link in an extraordinary number of diverse fields e.g. the concentration of oxygen and of carbon dioxide are determined in millions of blood samples every day and used to diagnose and treat illness, to quantitative analysis of plants and the soil in which they grow.

The intellectual contributions of analytical chemists to the understanding of nature, to the chemical industries and to the sciences of health and the environment are exciting, rewarding, progressive, vital and growing.

The modern principles and applications of analytical chemistry in today’s world are as important and as relevant to our existence as any other field

1.3 The Process of Analytical Chemistry

A   Define the Question. What information is needed? How accurately must, the answer be known? What legal constraints apply to the methods?

B   Select the best available method for the analysis. This is the most difficult, step and requires the most sound background on various analytical techniques. Analytical chemists tend to be generalists.

C   Sampling. The sample to be analysed must be representative of the bulk of material of interest. It must not be contaminated or altered prior to analysis. Sample must, be measured accurately.

D   Sample Preparation. Most samples do not come ready for analysis. The process of getting them ready for a measurement is, in many respects, more important than the measurement, itself. Without accurate preparation, the measurement is useless.

E   Separation. This may not always be required, but often interferences must be removed prior to analysis.Most separations involve an equilibration between phases, eg.Solid-Liquid (precipitation), Gas-Liquid (distillation, GC), Liquid-Liquid (extraction,HPLC). Modern chromatographic methods often allow the separation of as many as 1000 components in a mixture.

F   Measurement-Analysis. This is the step most often associated with Analytical Chemistry. However, as you will see, it is only one of many steps in the process. It is a step often done by specialists when the specific measurement involves complex instrumentation.Robotic systems are also becoming important here.

1.4 Methods of Detecting Analytes

1.   Physical means

•   Mass

•   Colour

•   Refractive index

•   Thermal conductivity

2.   With electromagnetic radiation (Spectroscopy)

•   Absorption

•   Emission

•   Scattering

3.   By an electric charge

•   Electrochemistry

•   Mass spectrometry

As you can see, there are a limited number of ways to detect an analyte. However, in each of the above general categories there are a large multitude of specific analytical techniques.

1.5 Applications of Analytical Chemistry

1.5.1 Helping to save lives

Laboratories in hospitals contain considerable numbers of analytical personnel. Their duties include analysing blood and urine that, will help doctors in their diagnosis of a patient’s illness.

The progress of an individual disease can be tracked by monitoring the concentration of key components in blood or urine.

The effectiveness of a course of drugs can be assessed by analysing for an active ingredient or its metabolites (Compounds into which the body converts the drug in order to try and remove it.)

Drugs are not effective at low levels but are harmful at very high concentrations so a patient has to be checked regularly to make sure that the levels of administered drugs in the blood is correct and is still being effective in combating the ailment. This whole situation generates massive work loads for analytical chemists who work in hospital laboratories as the number of samples being taken and the number of patients being treated with drugs at any one time is very large. This particular application is also stressful as the results of these tests have to be generated quickly but also with a high level accuracy and the analytical chemists also have to stand up to testing questions from doctors trying to ascertain a patient’s ailment.

1.5.2 Consumer and environment protection

The quality of materials we use every day such as food and drink is controlled by GOVT. A public analyst checks these products regularly to ensure that they conform to the legal requirements. Their work is very important and the highest standards are required. To qualify to be an analyst of this type special qualifications have to be obtained, the Mastership of Chemical Analysis. Central & Local Governments have large numbers of scientists working for them including a large number of analytical chemists.

The scientific workers working in the Govt establishments (like CSIR) not only analyse foods, drinks, medicines & pesticides but provide an invaluable service to Customs & Excise by examining beers, wines, spirits, oils, petrol, tobacco products and a variety of contraband including drugs.

They also monitor the level of toxic and hazardous chemicals in the work place such as the level of asbestos fibers, mercury vapour or lead dust. These are all potentially very harmful substances. Scientific civil servants may be in the situation that they have to measure small amounts of potentially hazardous chemicals as well as dealing with the work of an extremely large number of analyses.

The checking of drinking water is a process that has to be carefully and constantly monitored. Water authorities make great use of analytical chemists and their skills to keep a check on the quality of water, not just in terms of materials that should be removed but chemicals that should be added e.g. Cl and F. A close watch can also be kept on the contents of surface waters and what is discharged in a sewerage system and in particular industrial plants that discharge effluents. The sudden arrival of a high concentration of a transition metal at sewage works can kill the bacteria used in the water purification process. Sludge is produced and used as fertiliser, the concentration of which must be known to ensure that toxic metals don’t get into the food chain. As with drugs, there is a concentration range within which the action of the metal is beneficial, perhaps even desirable. Organisms may suffer at metal concentrations above, causing a toxic effect, and below, causing a deficiency, of the beneficial level.

1.5.3 Farming support

Today with every passing hour the efficiency of farming is increasing drastically. The levels of nutrients in soil are monitored which gives the farmers information which they require to manage their farms and how much fertiliser or pesticide to add to the soil. A number of laboratories provide this service to the farmers through soil scientists, agricultural analytical chemists and others. Levels of potentially harmful materials are monitored not only in the soil but also in the harvested crops and livestock.

It is very important that the levels of these pesticides, fertilisers, weedkillers etc. are applied to crops and soils that could pass along the food chain. The Ministry of Agriculture, Fisheries & Foods has laboratories which monitor residual concentrations of some components of animal foodstuffs or veterinary drug residues in the meat supplied to the consumer.

1.5.4 Crime prevention and detection

Helping to catch criminals

Analytical Chemists are very important in helping the police to solve crimes and bring the culprits to justice.

Although there is no substitute for the logical human mind, the days of people like Sherlock Holmes have long since gone.

These days Analytical Chemists work closely with the police to help them to piece together crime scenes. In many cases Forensic Scientist, as they are called in this Held, gather evidence from contact made by criminals on items at the crime scene.

Microscopic examination and measurement of chemical components is the basis for this aspect of Analytical Chemistry and it is this factor of making so much from such small samples which makes this area of police work so fruitful in the detection and combating of crime.

This particular application of analytical chemistry is seen by some as being a very untrustworthy form of detection but time after time they come up triumphant and silence those who cast doubts over their techniques.

1.5.5 Maintain fair play

Blowing the whistle on the cheats

Since the dawn of time people have been competing against each other just to be given the accolade of being The Best. But in the pursuit of perfection some of these people have turned to artificial means to enhance their personal skills. Old wive’s remedies and myths were used for years and years to aid in the training regime and performance of these athletes.

But today sport is no longer a form of recreation, it is really a form of employment with top athletes earning massive bank balances. A reflection of this is that the use of performance enhancing drugs (P.E.D.’s) has been encountered. Those athletes desperate to achieve fame (and in many cases fortune.) are prepared to take drugs which can have unpleasant side-effects just in the hope of being the victor.

Analytical Chemistry is now used widely in the Held of sport to monitor the levels of P.E.D.’s in competing athletes. At major championships the winners are subjected to dope tests to try and keep the sport clear of artificial enhancements. The analysts are also used heavily through the season where sports people are subjected to random tests. These tests done are looking at the level of certain drugs in the person system. Often it is minute amounts which are being detected and so the analysts have to be very accurate as to the level of the chemical in the sample taken. Someone found guilty of taking P.E.D.’s can be subjected to a fine, suspension, or in radical cases banned lifetime from the sport.

In recent times there have been many well-documented instances of sports persons being caught with P.E.D.’s in the system.

This can lead to some grey areas in the whole system. This is because there is a strict list of compounds which sports people are prohibited to have in their bodies. Make note-prohibited in their bodies not prohibited to take. Some of these prohibited substances are found in quite legitimate products and some athletes have to go before committees to explain how/why the banned substance was in the sample they gave.

It is in this type of situation that the role of the analytical chemist is stretched in opposing directions. The committee wants to ensure that only un-assisted athletes are rewarded for their achievements so want the tests to pick out minute concentrations of the compounds that they have banned. The athlete may been clean but somehow shows a positive dope test and looks back to an independent analytical chemist to prove his innocence.

1.6 Skills of an Analytical Chemist

Analytical Chemists are employed in various fields but the common core to all these applications is that to a high degree of accuracy the presence or concentration of a particular compound needs to be determined. Without this high degree of accuracy the results obtained would not be very useful.

So in the carrying out his/her duties an analytical chemist relies upon a few basic skills which are the key to their results being useful in accordance with their brief.

One of the most important skills that is needed by an analytical chemist is to be accurate in the observations that are made in the course of an investigation. An assay is a good way to demonstrate this skill. Take a look at the series of flasks shown above being used in a titration. Flask A shows a clear solution before a reagent from the burette is added. When some reagent is added the situation, as shown in B, is seen where the solution will temporarily colour but disappear back to A on stirring. The end point of the titration is when the flask is like C, the first sign of a permanent colour. If an analytical chemist took situation D to be the end point of the titration then problems would occur. In D too much reagent has been added and the end point has been passed. Any calculations or deductions made as a result of D would be of no use as it is highly inaccurate. The situation in D could be worse, but in a field where minute quantities are being analysed such over calculations could have disastrous consequences leading to medicines having toxic levels of certain constituents etc.

Observational skills are also important when it comes to the reading of apparatus. This is not only important in the field of analytical chemistry but scientists in general have to develop the skill of being accurate in all measurements that they make. The level of accuracy in any determination starts right from the first stage. Without the accurate reading of scales on equipments, as above, there would be a lower level of accuracy in the final reports produced. So although the scale above is split into 5 subdivisions between each large mark an analytical chemist must develop the skill of reading an ‘invisible’ set of subdivisions to give a reading to the nearest 0.1 or better such as 0.05. So we have looked at the importance of an analytical chemist showing good observational skills in analysis. Another skill to be developed is selecting the right equipment for a measurement or process. If measuring the mass of a sample supplied to him/her, any analyst would be immediately dismissed if they use A. This is far too inaccurate in determining the physical quantity required. Those who use B get another range but actually the use of an analytical balance, such as G is the best selection to accurate values. G will measure to 4 or more decimal places the mass of something. The point to bear in mind here is that a measurement stated to a large degree of accuracy can be rounded if less accuracy is required but you cannot obtain a more accurate result from a reading which has been under-analysed and measured.

An analytical chemist has often a very heavy workload and can thus not afford to waste time when constantly working to a deadline. This means that they have acquired skills at recognizing exactly when measurements of items have to be very accurate or whether some rough approximation is sufficient. So there is a skill that has to be demonstrated as to which of the pieces of equipment above has to be employed in the measurement of analyte for any one reaction.

Here is another example of when the skill of an analytical chemist can be put to use. Both of the above arc 250ml flask but one would be suitable for diluting a solution in an analytical process while the other will be useful for preparing standard solutions.

Not all the skills needed by an analytical chemist involve actual hands-on the reaction. In these modern days there are a lot of very expensive machines available to chemists which allow certain processes to be carried out under very strict conditions, ultimately making the analysis far more accurate. Above is shown an example of such a piece of equipment. It is a spectrophotometer which you can see by the various readouts included can carry out not just one but many different jobs. Here the skills employed are more in the preparation of the samples supplied to these machine and then in the operation of such machinery. These machines can give very accurate results but in many cases it is still down to the operator to take the readouts and with his/her knowledge present the findings to those who have commissioned the testing.

1.7 Analytical Chemistry and Society

There are many other things which play a part in the role of being an analytical chemist other than analysing the chemical and/or the composition of a material sample.

In the pharmaceutical industry an analytical chemist could be given the job of finding out the nature and composition of a competitor’s products. The research laboratories will be making new materials that will have to be analysed, and the levels of toxic materials in effluent that a factory discharges into the atmosphere or into water has to be monitored to protect society in general.

There are many other aspects in which analytical chemists are involved. The illustration in page 3 shows the numerous interactions that an analytical chemist can have with other professional individuals and groups.

Analytical chemistry is the science of making quantitative measurements. In practice, quantifying analytes in a complex sample becomes an exercise in problem solving. To be effective and efficient, analysing samples requires expertise in:

1.   The chemistry that can occur in a sample

2.   Analysis and sample handling methods for a wide variety of problems (the tools-of-the-trade)

3.   Proper data analysis and record keeping

To meet these needs, Analytical Chemistry courses usually emphasize equilibrium, spectroscopic and electrochemical analysis, separations, and statistics.

Analytical chemistry requires a broad background knowledge of chemical and physical concepts. With a fundamental understanding of analytical methods, a scientist faced with a difficult analytical problem can apply the most appropriate teclmique(s). A fundamental understanding also makes it easier to identify when a particular problem cannot be solved by traditional methods, and gives an analyst the knowledge that is needed to develop creative approaches or new analytical methods.

1.8 Terms to Know

Qualitative Analysis-reveals the chemical identity of the species in the sample.

Quantitative Analysis-establishes the relative amount of one or more of the species (analytes) in numerical terms.

Analyte-the components of a sample t hat are to be determined.

Assay-the process of determining how much of a given sample is in the material indicated by its name.

Replicate Samples-portions of a sample of approximately the same size that are carried through an analytical procedure at the same time and in the same way.

Interference-a species that causes an error by making the quantity being measured either larger or smaller.

Sample Matrix-all of the components making up the sample containing the analyte.

1.9 Questions

1.   What is analytical chemistry?

2.   Write briefly about the history of development of analytical chemistry.

3.   Describe briefly the process of analytical chemistry.

4.   What are the different methods of detecting analytes?

5.   What are the applications of analytical chemistry?

6.   What are the skills required for an analytical chemist?

7.   State briefly the applications of analytical chemistry in crime prevention and detection.

Chapter 2

Statistical Treatment of Analytical Data

2.1 Review of Fundamental Concepts

2.1.1 Moles

Chemists knows that atoms and molecules react in definite proportions. But one cannot conveniently count the number of atoms or molecules.

But since the chemist has determined their relative masses, their reaction can be described on the basis of the relative masses of atoms and molecules reacting, instead of the number of atoms and molecules reacting.

For example, in the reaction

We know that one silver ion will combine with one chloride ion.

We know further that 107.870 mass units of silver will combine with 35.453 mass units of chlorine.

To simplify calculations, the chemist has operationally defined the mole as the atomic, molecular or formula weight of a substance expressed in grams.

Since a mole of any substance contains the same number of atoms or molecules as a mole of any other substance, atoms will react in the same mole ratio as their atom ratio in the reaction.

In the above example, one silver ion reacts with one chloride ion, and so each mole of silver ion reacts with one mole of chloride ion. Each 107.87 gm of silver reacts with 35.453 gm of chloride.

Example 2.1 Calculate the number of grams in one mole of CaSC>4. H2O

Solution One mole is the formula weight expressed in grams. The formula weight is

The number of moles of a substance is calculated from

where formula weight represents the atomic or molecular weight of the substance. Thus,

Since we usually work with millimole quantities, a more convenient form of equation 2.1 is

Conversely, we can calculate the grams of material from the number of moles:

Again, we usually work with millimole quantities, and so

Example 2.2 Calculate the number of moles in 500 mg of Na2 WO4 (Sodium tungstate)

Solution

Example 2.3 How many milligrams are in 0.250 m mol of Fe2O3 (ferric oxide)?

Solution

2.1.2 Concentration of solutions

Concentration of stoichiometric reactions are usually expressed in terms of molarity, normality and formality. Gas concentrations are usually expressed by their pressures or partial pressures.

Molarity is the unit, of concentration expressed as the number of moles of solute dissolved per litre of solution. Thus,

For example, 1.000 M solution of KC1 can be prepared by dissolving 1.000 mole of KC1 (74.55 grains) in an amount of water or some other solvent, to get, exactly 1.000 litre of solution. Since KC1 is completely dissociated, the solution will be exactly 1.000 Al in K+ and 1.000 M in Cl"-ion, if water is used as a solvent. Molar concentration of KC1 is therefore zero. In case KC1 were partially dissociated, its molar concentration would be determined by the amount of dissociation taking place.

Example 2.4 A solution is prepared by dissolving 1.26 g AgNO3in a 250 ml volumetric flask and diluting to vol. Calculate the molarity of the silver nitrate solution. How many millimoles of AgNO3 were dissolved?

Solution

Example 2.5 How many grams per millilitre of NaCl are contained in a 0.250-M solution?

Solution

Example 2.6 How many grams of Na2SO4 should be weighed out to prepare 500 ml of a 0.100 M Solution?

Solution

Example 2.7 Calculate the concentration of potassium ion in grams/litre after mixing 100 ml of 0.250 M KC1 and 200 ml of 0.100 Μ K2 SO4.

Solution

Formality (F) may be defined as the number of gram formula weight (GFW) of the solute dissolved per litre of solution.

For KC1 solution, the concentration can be expressed as 1.000 F KC1. The concentration can also be expressed as 1.000 FK+ and 1.000 FC1" as the KC1 is completely ionised.

In acetic acid solution, there are three species present. These are H3 O+, C2 H3 O^.

However, when 0.1000 M of HC2H3O2IS dissolved in 1.000 litre of solution, a 0.1000F HC2H3O2 solution is obtained.

Normality (N) may be defined as the number of gram equivalent weights of solute per litre of solution.

Unlike molarity and formality, normality varies according to the reaction in which the solute participates. Thus exact calculation of gram equivalent weight can be carried out after determining the specific changes in chemical identity of t he solute in the course or reaction.

Example 2.8 Calculate the equivalent weights of the following substances (a) NH3 (b) H2 C2 O4 (in reaction with NaOH) (c) KMnO4 (Mn is reduced to Mn+2 ) (d) Na3 PO4 (in precipitating Ag3 PO4 )

Solution

(a)   

(b)   

(c)   The Mn undergoes a 5-electron change, from valence +7 to +2.

(d)   Each mole of phosphate ion reacts with three equivalents of Ag+ (3Ag+ + Nas PO4 Ag3 PO4 + 3 Na+ )

Example 2.9 Calculate the normality of the solutions containing the following (a) 5.300 g/litre Na2 CO3 (when the CO3 ² reacts with 2 protons) (b) 5.267 g/litre K2Cr2C2the Cr is reduced to Cr+3) (c) 2.68 g/litre PESCUused as a precipitating agent for Ba+2

Solution

(a) CO3 ² reacts with 2H+ to H2CO3.

(b) Each Cr+6 is reduced to Cr+3, a total change of 6e~/molecule of Κ^Ο^Ογ.

(c) Each SO2-⁴ reacts with one Ba+² (which contains 2 equiv/mole)

2.1.3 Expression of Analytical Results

Results of an analysis may be reported in many ways, and the beginning analytical chemist should be familiar with some of the common expressions, and units of measure employed. Results will nearly always be report cd as concentration, on either a weight or a volume basis: The quantity of analyte per unit weight or per volume of sample. The units used for the analyte will vary.

Solid samples

Calculations for solid samples are based on weight. The most common way of expressing the results of macro determinations is to give the weight of analyte as a percent of the weight of sample (weight/weight basis). The general formula for calculating percent on a weight /weight basis, which is the same as parts per 100, then is

Trace concentrations arc usually given in smaller units, such as parts per thousand (ppt %), parts per million (ppm), or parts per billion (ppb). These are calculated in a manner similar to parts per hundred (%).

Table 2.1 Common units for expressing trace concentrations

Example 2.10 A 2.6 g sample of plant tissue was analysed and found to contain 3.6 μ g of zinc. What is the concentration of zinc in the plant in ppm? In ppb?

Solution

3.6μ g/2.6g = 1.4 μ g/g =1.4 ppm

3.6 x 10³ ng/2.6 g = 1.4 x 10³ ng/g = 1400 ppb

One ppm is equal to 1000 ppb. One ppb is equal to 10 ⁷%

Example 2.11 25.00 mL of solution are found to contain 3.45 x 10 x moles of fluoride ion.

What is ppb F for this solution?

Solution We need ug F/Litres to get parts per billion. First calculate Amount A.

Then divide by Amount B. 0.6554 ug F/0.02500 litres = 26.2 ug/L, or 26.2 ppb.

Example 2.12 Prepare 200 ml of 6 M ammonia solution from the 15.4 M reagent. This is often called a dilution problem, but it can be solved from the basic definition of Molarity.

Solution We need 200 ml of 6 M solution. We are told the volume; we need to calculate the amount of ammonia needed. (6 moles ammonia/litre) x 0.20 litres = 1.20 moles of ammonia. The source of the ammonia will be the 15.4 M solution. Again using the units:

= 77.9 mL of 15.4 M ammonia reagent needed. Add 77.9 rnL of 15.4 M ammonia reagent to 100 mL of water (for safety), then dilute to 200 mL total volume.

Liquid Samples

Results for liquid samples may be reported on a weight/weight basis, as above, or they may be reported on a weight/volume basis. The latter is probably more common, at least in the clinical laboratory. The calculations are similar to those above. Percent on a weight/volume basis is equal to g of analyte per 100 ml of sample, while mg% is equal to mg of analyte per 100 ml of sample.

2.2 Significant Figures

The minimum number of digits required to express a value in scientific notation without loss of accuracy. Zeros arc significant when they occur:

1.   in the middle of a number

2.   at the end of the number on the right hand side of the decimal point

The last (farthest to the right) significant figure in a measured quantity always has some associated uncertainty.

Addition and Subtraction

Rules for rounding off numbers

1.   Addition and subtraction: express all numbers with the same exponent and align all numbers with respect to the decimal point. Round off the answer according to the number of decimal places in the number with the fewest decimal places. Round up except when exactly halfway-then round to nearest digit.

2.   Multiplication and division: limited to the number of digits contained in the number with the fewest significant figures.

Example 2.13 E xpress each answer with the correct number of significant digits

(a)   1.021 + 2.69 = 3.711

(b)   12.3 — 1.63 = 10.67

(c)   4.34 x 9.2 = 39.928

(d)   0.0602/(2.133 x 10⁴ ) = 2.84903 x 10 -6

Solution

Logarithms and Antilogarithms

If a is the base 10 logarithm of n (a = log n), then n = 10'

•   a = log n

•   10a = 10(logn) = n

•   n = antilog a

Natural logarithms (In) are based on the number e (=2.718281 ) instead of 10

•   b = ln n

•   eb = elnn=n

A logarithm is composed of a mantissa and a character The number of digits in mantissa of log x = number of significant figures in x. The character corresponds to the exponent of the number when written in scientific notation

Example 2.14 339 can be written in scientific notation as 3.39 x 10². It has three significant figures so the mantissa of the logarithm of 339 should have three significant figures and the character of the logarithm of 339 will be 2.

Solution log 339 = 2.530199698 —> 2.530

In converting a logarithm to its antilogarithm, the number of significant figures in the antilogarithm should equal the number of digits in the mantissa

antilog (6.142) = 1O⁶.⁴¹² = 1.386755829 x 10⁶ —> 1.39 x 10⁶

2.3 Limitations of Analytical Methods

The function of the analyst is to obtain a result as near to the true value as possible by the correct application of the analytical procedure employed. The level of confidence that the analyst may enjoy in his result will be very small unless he has knowledge of the accuracy and precision of the method used as well as being aware of the source of errors which may be introduced. Quantitative analysis is not simply a case of taking a sample, carrying out a single determination and then claiming that the value obtained is irrefutable.

2.3.1 Precision and Accuracy

Precision is a measure of the reproducibility of a result. Accuracy refers to how close a measured value is to the true value

Absolute and Relative Uncertainty

Absolute uncertainty is an expression of the margin of uncertainty associated with a measurement. Absolute uncertainty has the same units as the measurement.

Relative uncertainty is an expression comparing the size of the absolute uncertainty to the size of its associated measurement. It is a dimensionless quotient.

Relative Uncertainty = (absolute uncertainty/magnitude of measurement) % Relative Uncertainty = Relative Uncertainty x 100

Propagation of Uncertainty

Addition and Subtraction

For addition and subtraction, use absolute uncertainty in addition or subtraction: e2 = [e12 +e22 +...en2⁰.⁵

where

•   eg = absolute uncertainty of calculated value

•   ei = absolute uncertainty for individual value

•   n = number of experimental values

2.3.2 Types of Errors

The precision of a measurement is readily determined by comparing data from carefully replicated experiments. Unfortunately, an estimate of the accuracy is not, as easy to obtain. To determine the accuracy, we have to know the true value, which is usually what we are seeking in the analysis.

Results can be precise without, being accurate and accurate without, being precise.

Systematic Error

Systematic Error (also called determinate error) is a consistent error that can be detected and corrected. Ways to detect systematic error:

1.   Analyse samples of known composition. Your method should reproduce the known answer.

2.   Analyse blank samples containing none of the analyte being sought,. If you observe a nonzero result,, your method responds to more than you intended.

3.   Use different, analytical methods to measure the same quantity. If the results do not agree, there is an error associated with one (or more) of the methods.

4.   Samples of the same material can be analysed by different people in different laboratories (using the same or different methods). Disagreements beyond the expected random error indicate systematic errors.

Systematic Error

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. Note that, bias affects all of the data in a set in the same way and that it bears a sign.

In general, a systematic error in a series of replicate measurements causes all the results to be too high or too low. An example of a systematic error is the unsuspected loss of a volatile analysis while heating a sample.

Sources of systematic Errors

There are three types of systematic errors: (1) Instrumental errors are caused by non ideal instrument behavior, by faulty calibration, or by use under inappropriate conditions. (2) Method error arise from non ideal chemical or physical behavior of analytical systems.(3) Personal errors results from the carelessness, inattention, or personal limitations of he experimenter.

Instrument Errors

All measuring devices are potential source of systematic errors. For example, pipets, burets, and volumetric flasks may Hold or deliver volumes slightly different from those indicated by their graduations. These differences arise from using glassware at a temperature that differs significantly from the calibration temperature, from distortions in container walls due to heating while drying, from errors in the original calibration, or from contaminates on the inner surface of the containers. Calibration eliminates most systematic errors of this type.

Electronic instruments are subject to instrumental systematic errors. These can have many sources. For example, errors may emerge as the voltage of a battery operated power supply decreases with use. Errors can also occur if instruments are not calibrated frequently or calibrated incorrectly. The experimenter may also use an instrument under conditions in which errors are larger. For example, a pH meter used in strongly acidic media is prone to an acid error, as discussed in chapter 20. Temperature changes cause variation in many electronic components, which can lead to drifts and errors. Some instruments are susceptible to noise induced from the alternating current (ac) power lines, and this noise may influence precision and accuracy. In many cases, errors of these types are detectable and correctable.

Random Error

Random error (also called indeterminate error) is due to the limitations of the physical measurement and cannot be eliminated. A better experiment may reduce the magnitude of the random error but cannot eliminate it, entirely.

Indeterminate (or random) errors.

1.   Usually are related to insufficiently controlled variations in experimental conditions.

2.   Affect precision, but not accuracy.

3.   Cannot be eliminated, but can be treated (statistically).

4.   Are related to the small, random errors in an experiment that combine to give an overall error.

5.   For example, 4 random errors exist in an experiment: Up U2, U3, and U4.

(a)   The four errors follow a Gaussian (normal error curve) distribution, randomly combining in all possible combinations.

(b)   As the number of errors gets bigger, the better the fit to the normal distribution.

(c)   Empirically, replicate data from most quantitative analytical experiments behave just like indeterminate errors => They follow a normal distribution.

(d)   Note that all the uncertainties (random errors) need not be the same size.

(e)   Sources of the uncertainty could include judgment of water levels, pipette angles and drainage times, balance performance (e.g., vibrations, drafts, etc.), interpretations of endpoints, etc.

6.   Sample data: 50 replicate pipettings with a 25-mL pipette.

7.   Because indeterminate error generally follows a normal distribution, calculations can be based upon characteristics of the normal distribution curve.

Another type of error is gross error. Gross errors differ from indeterminate and determinate errors. They usually occur only occasionally, are often larger, and many cause a results to be either high or low. They are often the product of human errors.

For example, if part of a precipitate is lost before weighing, analytical results will be low. Touching a weighing bottle with your fingers after its empty mass is determined will cause a high mass reading for a solid weighed in the contaminated bottle. Gross errors lead to outliers, results that appear to differ markedly form all other data in a set of replicate measurements. Various statistical tests can be performed to determine if a result is an outlier.

Method Errors

The non ideal chemical or physical behavior of the reagents and reactions on which an analysis is based often introduces systematic methods errors. Such sources of non ideality include the slowness of some reactions, the incompleteness of others, the instability of the some species, the non specificity of most reagents, and the possible occurrence of side reactions that interfere with the measurement process. For example, a common method error in volumetric analysis results from the small excess of reagent required to cause an indicator to undergo the color change that signals completion of the reaction. The accuracy of such an analysis is thus limited by the very phenomenon that makes the titration possible.

The analytical method used involves the decomposition of the organic samples in hot concentrated sulfuric acid, which converts the nitrogen in the samples to ammonium sulfate. Often a catalyst, such as mercuric oxide or selenium or copper salt, is added to speed the decomposition. The amount of ammonia in the ammonium sulfate is then determined in the measurement step. Experiments have shown that compounds containing a pyridine ring, such as nicotinic acid, are incompletely decomposed by the sulfuric acid. With such compounds, potassium sulfate is used to raise the boiling temperature. Samples containing N-O or N-N linkage must be pretreated or subjected to reducing conditions. Without these precautions, low results are obtained. It is highly likely the negative error, (x3-xt) and (x4 —xt).

Errors inherent in a method are often difficult to detect and are thus the most serious of the three typed of systematic error.

Personal Errors

Many measurements require personal judgments. Examples include estimating the position of a pointer between two scales divisions, the color of a solution at the end point in a titration, or the level of liquid with respect to a graduation in a pipet or buret. Judgments of this type are often subject to systematic, unidirectional errors. For example, one person may read a pointer consistently high, another may be slightly slow in activating a timer, and a third may be less sensitive to color changes. An analyst who is insensitive to color changes tends to use excess reagent in a volumetric analysis. Analytical procedures should be always be adjusted so that any known physical limitations of the analyst cause negligibly small errors.

A universal source of personal error is prejudice, or bias. Most of us, no matter how honest, have a natural tendency to estimate scale reading in a directional that improves the precision in a set of results. Alternatively, we may have a preconceived notion of the true value for the measurement. We then subconsciously cause the results to fall close to this value. Number bias is another source of personal error that varies considerably from person to person. The most frequent number bias encountered in estimating the position of a needle on a scale involves a preference for the digits 0 and 5. Also common is a prejudice favoring small digits over large and even numbers over odd.

2.3.3 The Effect of Systematic Errors on Analytical Results

Systematic errors may be either constant or proportional, 'the magnitude of a constant error stays essentially the same as the size of the quantity measured is varied. With constant errors, the absolute error is constant with sample size, but the relative error varies when sample size is changed. Proportional errors increase or decrease according to t he size of the sample taken for analysis. With proportional errors the absolute error varies wit h sample size, but the relative error stays constant with changing samples size.

Constant Errors

The effect of a constant error becomes more serious as the size of the quantity measured decreases. The effect of solubility losses on the results of a gravimetric analysis. The excess of reagent required to bring about a color change during a titration is another example of constant error. The volume, usually small, remains the same regardless of the total volume of reagent required for titration. Again, the relative error from this source becomes more serious as the total volume decreases. One way of reducing the effect of constant error is to increase the saiiiplc size until the error is tolerable.

Example 2.15 Suppose that 0.50 mg of precipitate is lost as a result of being washed with 200mL of wash liquid. If the precipitate weighs 500 mg, t he relative error due to solubility loss is -(0.50/500) x 100% = -0.1%. Loss of the same quantity from 50mg of precipitate results in a relative error of-0.1%.

Proportional Errors

A common cause of proportional errors is the presence of interfei itig contaminants in the sample. For example, a widely used method for determining copper is based on the reaction of copper (II) iron with potassium iodide to give iodine. The quantity of iodine is then measured and is proportional to the copper. Iron (III), if present, also liberates iodine from potassium iodide. Unless steps are taken to prevent this interference, high results are observed for the percentage of copper because the iodine produced will be a measure of t he copper (II) and iron (III) in the samples. The size of this error is fixed by the fraction of iron cont amination, which is independent of the size sample taken. If the sample is doubled, for example, t he amount of iodine liberated by both the copper and the iron contaminant is also doubled. 'Thus, the magnitude of the reported percentage of copper is independent of sample size.

Example 2.16 Find the absolute error

Solution The answer is 3.06 + / — 0.04 keep same number of significant figures in answer.

If necessary, you can determine the relative uncertainty at the end of the calculation

%r.u. = (0.04/3.06) x 100 = 1.31% 1 %

Multiplication and Division

For multiplication and division, use percent relative uncertainty

% eg = percent relative uncertainty for calculated value

% ei = percent relative uncertainty for individual value

n = number of experimental values

Example 2.17 Find the relative uncertainty [1.76(+/ —0.03) x 1.89(+/ —0.02)]/0.59(+/ —0.02) = 5.6(+/—?)

Solution First convert absolute uncertainty to percent relative uncertainty

You should retain extra insignificant figures throughout the calculation and round at the end. When using a calculator, don’t worry about, significant figures until the final answer. If necessary, you can determine the absolute uncertainty at, the end of the calculation

Exponents and Logarithms

Use relative uncertainty in calculations involving exponents and logarithms

If y = log x ey 0.43429(ex/x)

Example 2.18 If x = 5.23(+/ — 0.06), what, is the error in y if y = logx?

Solution y = log5.23(+/ — 0.06) = 0.719(+/—?)

First convert absolute uncertainty to relative uncertainty (0.06/5.23) = 0.011 ey = 0.43429 (0.011/5.23) = 0.0010 y = 0.719(+/-0.001) or y = 0.719(+/-0.1%)

If y = 10x

ey = (2.3026ex )y

The Gaussian Distribution

Statistics gives us tools to accept, conclusions that have a high probability of being correct and to reject conclusions that do not.

We say that the variation in experimental data is normally distributed when replicate measurements exhibit, a bell-shaped (Gaussian) distribution

Properties of the Gaussian or Normal Error Curve

1.   Maximum frequency at zero indeterminate error

2.   Positive and negative errors occur at the same frequency (curve is symmetric)

3.   Exponential decrease in frequency as the magnitude of the error increases.

4.   The formula for the normal distribution is

5.   where y = the frequency of a result, x σ = the population standard deviation μ = the population mean.

6.   For data following a normal distribution, the function is completely defined by the mean and standard deviation.

7.   If you assume that the population mean is zero, the standard deviation is 1, and graph the function on a x,y-plot, a bell-shaped curve is produced with.

(a)   68.3% of the area between +/-1 standard deviation unit of the mean.

(b)   95.5% of the area between +/-2 standard deviation units of the mean. c. 99.7% of the area between +/-3 standard deviation units of the mean.

8.   The probability of measuring a value X in a certain range is proportional to the area of the curve under that range.

9.   The total area under the curve must be unity (1) since the probability of making all measurements of X is one (1).

10.   Note that the width or spreading out of the curve is driven by the standard deviation.

(a)   Large standard deviation = wide bell curve.

(b)   Small standard deviation = narrow bell curve.

11.   It is common to normalize the curve by expressing the deviation from the mean in terms of standard deviation units using Z, where Z is.

12.   Since Z is the deviation from the mean expressed in standard deviation units, ALL normal error curve plots will be the same shape in these Z units.

13.   Normal distribution curves re-plotted in units of Z are said to be normalized.

14.   General properties of a normal error curve (a normalized Gaussian distribution of errors):

(a)   The mean (or average) is the central point of maximum frequency (i.e., the top of the bell curve).

(b)   The curve is symmetric on both sides of the mean (i.e., 50% per side).

(c)   The is an exponential decrease in result (X) frequency as you move away from the mean.

(d)   Areas under the curve correspond to known frequencies.

(e)   Percentages (e.g., 68.3%, 95.5%, 99.7%, etc.) apply to single measurements in the set.

(f)   If time and expense permit, you need to perform more than 20 replicates when possible to be sure that the sample mean and standard deviation are sufficiently close to the population mean and standard deviation.

15.   If time and expense are prohibitive, data sets can be pooled from a series of duplicate experiments.

where V = the number of observations in a given set ΛΑ = the number of pooled sets

2.4 Elementary Statistics Relevant to Analytical Chemistry

Definition of Terms:

1.Mean (average):

(a)   Of a sample (i.e., if N, the number of measurements, is < 20). = (sum of all measure-ments)/(number of measurements)

(b)   Of a population (i.e., if V > 20). = population mean.

2.Median is equal to middle result when the data are arranged by size.

(a)   If the data is an odd numbered set, the median is the middle value.

(b)   If the data is an even numbered set, the median is the average of the middle two values.

Example 2.19 An odd-numbered set,: 2.9, 2.6, 2.4, 2.3, 2.2

Solution

2.9

2.6

2.4

2.3

2.2

Sum = 12.4

X = 12.4/5 = 2.5 Median = 2.4

Example 2.20 An even-numbered set: 0.1000, 0.0902, 0.0886, 0.0884

X = 0.3672/4 = 0.0918 Median == (0.0902+0.0886)/2 = 0.0894

3.Standard deviation: The standard deviation, s, measures how closely the data are clustered about the mean. The smaller the standard deviation, the more closely the data are clustered about the mean.

Of a sample (N ≤ 20),

where Xi represents an individual measurement and Xis the sample average.

For example, given the following data set:

For an infinite set, of data, Ό’ the mean is designated by Ό’ (the population mean) and the standard deviation is designated by (the population standard deviation).

Of a population (N > 20),

where μ is the population mean

The degrees of freedom of a system are given by the quantity η — 1.

Example 2.21 For the following data set, calculate the mean and standard deviation. Replicate measurement s from the Calibration of a 10 mL pipette.

Solution

Mean = [(9.990 + 9.993 + 9.973 + 9.980 + 9.982)/5] = 9.984

Standard Deviation (s·) = {[(9.990 - 9.984)² + (9.993 - 9.984)² + (9.973 - 9.984)² + (9.980 -9.984)² + (9.982 - 9.984)²]/(5 - 1)} = 8 x 10-3

Standard Deviation and Probability

Percentage of Observations in Gaussian Distribution

The more times you measure a quantity, the more confident you can be that the average value of your measurements is close to the true population mean. The uncertainty decreases in proportion to l/(n)⁰.⁵, where n is the number of measurements. The standard deviation (or error) of the mean = s/(n)⁰.⁵

4.Variance (s² )

5.Relative standard deviation (RSD) and coefficient of variation (CV).

(a)   RSD = s/X

(b)   RSD can be expressed as a percentage, parts per thousand, etc.

1) CV(%) = (s/X) x 100% => CV is RSD expressed as %

2) RSD and CV usually give a clear picture of the data quality.

3) Large RSD or CV implies poor quality.

6. Student’s t

Confidence Intervals

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

Confidence interval: = x ± (ts/(n)°·⁵ )

where s is the measured standard deviation, n is the number of observations, and t is the Student’s test.

Example 2.22 A chemist obtained the following data for the alcohol content in a sample of blood: percent ethanol = 0.084, 0.089, and 0.079. Calculate the 95% confidence limit for the mean.

Solution average = (0.084 + 0.089 + 0.079)/3 = 0.084

From Table, t at the 95% confidence level with t wo degrees of freedom is 4.303. So the 95% confidence interval = 0.084 ± (4.303)(0.005)/(3)°·⁵ = 0.084 ± 0.012

Comparison of Means with Student’s t

We arbitrarily adopt the 95% confidence level as a conservative standard.

We will say that two results do not differ from each other unless there is a > 95% chance that our conclusion is correct.

The preceding statement is based on the concept, of a null hypothesis. The null hypothesis assumes that the two values being compared are, in fact, the same.

Thus, we can use the t test (for example) as a measurement of whether the null hypothesis is valid or not.

7. Comparing a Measured Result to a Known Value

Example 2.23 A new procedure for the rapid analysis of sulphur in kerosene was tested by analysis of a sample which was known from its method of preparation to contain 0.123% Sulphur. The results obtained were: %S 0.112, 0.118, 0.115, and 0.119. Is this new method a valid procedure for determining sulphur in kerosene?

Solution To answer this question, we must see if the known value falls within the 95% confidence interval of the data from the new procedure.

average = 0.116; s = 0.0033

95% confidence interval = 0.116 ± (3.182)(0.0033)/2

= 0.116 ± 0.005

= 0.111 to 0.121 which does not contain the known value

Because the new method has a j5% probability of being correct, we can conclude that this method will not be a valid procedure for determining sulphur in kerosene

8. Q Test for Bad Data

Sometimes one datum appears to be inconsistent with the remaining data. When this happens, you are faced with the decision of whether to retain or discard the questionable data point. The Q Test allows you to make that decision: Q = (gap/range) where gap = difference between the questionable point and the nearest value and range = total spread of data

If Q(observed) > Q(tabulated), the questionable data point should be discarded.

Example 2.24 Can the value of 216 be rejected from the following set of results? Data: 192, 216, 202, 195 and 204

Solution Q = (gap/range)

gap = 216-204 = 12

range = 216-192 = 24

Q = (12/24) = 0.50 Q(tabulated) = 0.64

Q(observed) is less than Q(tabulated) (0.50 < 0.64) so the data point cannot be rejected.

9. F Test

The F test provides a simple method for comparing the precision of two sets of identical measurements.

where is the standard deviation of method 1 and s2