General principles of analytical chemistry

Introduction

Analytical chemistry encompasses the science of characterising a sample. Two broad categories can be considered. Qualitative analysis deals with the question of what elements or compounds are present. In the simplest case, this may involve only sample identification. For instance, is an unknown water sample fresh or saline? Several screening tests may rely on determining whether or not a particular substance is present or absent. This has applications in quality control during food processing where the analysis might serve to distinguish foreign bodies. Urine samples from athletes and racehorses are routinely examined for the presence of prohibited drugs. Qualitative analyses may also be used to trace the source of a sample. Pollen analysis of honey indicates the country of origin. Comparing the ‘fingerprint’ of substances in an oil slick with that in the bilge tanks of ships may enable the responsible vessel to be identified. Quantitative analysis involves determining how much of an element or molecule is present. Several types of quantitative analyses can be contemplated. These are exemplified below using the analysis of petrol (gasoline). A complete analysis would quantify the amount of each constituent in the sample. This would involve the determination of hexane, octane, toluene, tetraethyllead, etc. Such analyses are generally impossible and for environmental samples usually unnecessary. For an ultimate analysis the amount of each element is quantified.

This book is designed for those students of elementary particle physics who have no previous knowledge of quantum field theory. It assumes a knowledge of quantum mechanics and special relativity, and so could be read by beginning graduate students, and even advanced third year undergraduates in theoretical physics.

I have tried to keep the treatment as simple as the subject allows, showing most calculations in explicit detail. Reflecting current trends and beliefs, functional methods are used almost throughout the book (though there is a chapter on canonical quantisation), and several chapters are devoted to the study of gauge theories, which at present play such a crucial role in our understanding of elementary particles. While I felt it important to make contact with particle physics, I have avoided straying into particle physics proper. The book is pedagogic rather than encyclopaedic, and many topics are not treated; for example current algebra and PC AC, discrete symmetries, and supersymmetry. Important as these topics are, I felt their omission to be justifiable in an introductory text.

I acknowledge my indebtedness to many people. Professors P.W. Higgs, FRS, and J.C. Taylor, FRS, offered me much valuable advice on early drafts of some chapters, and I have benefited (though doubtless insufficiently) from their deep understanding of field theory. I was lucky to have the opportunity of attending Professor J. Wess's lectures on field theory in 1974, and I thank him and the Deutscher Akademischer Austauschdienst for making that visit to Karlsruhe possible.

The foregoing chapters have dealt with field theories, including gauge theories, and their quantisation. The stage is now almost set for applying this knowledge to particle physics. One crucial bit of scenery, however, is still missing – the idea of ‘spontaneous breaking of symmetry’. About 1960 Nambu and Goldstone realised the significance of this notion in condensed matter physics, and Nambu in particular speculated on its application to particle physics. In 1964 Higgs pointed out that the consequences of spontaneous symmetry breaking in gauge theories are very different from those in non-gauge theories. Weinberg and Salam, building on earlier work of Glashow, then applied Higgs’ ideas to an SU(2) × U(1) gauge theory, which they claimed described satisfactorily the weak and electromagnetic interactions together, in other words, in a unified way. Serious interest was shown in this theory when 't Hooft proved, in 1971, that is was renormalisable. It has met with notable experimental successes. These matters are the concern of this chapter (with the exception of renormalisation, which we deal with in the next chapter). We begin by explaining spontaneous symmetry breaking, which, when applied to field theory, is a concept that refines our notion of the vacuum.

What is the vacuum?

We begin by considering two simple physical examples. First, consider the situation illustrated in Fig. 8.1. Place a thin rod of circular cross section vertically on a table, and push down on it along its length, with a force F.

Quantum field theory

Quantum field theory has traditionally been a pursuit of particle physicists. In recent years, some condensed matter physicists have also succumbed to its charms, but the rationale adopted in this book is the traditional one: that the reason for studying field theory lies in the hope that it will shed light on the fundamental particles of matter and their interactions. Surely (the argument goes), a structure that incorporates quantum theory – which was so amazingly successful in resolving the many problems of atomic physics in the early part of this century – and field theory – the language in which was cast the equally amazing picture of reality uncovered by Faraday, Maxwell and Hertz – surely, a structure built on these twin foundations should provide some insight into the fundamental nature of matter.

And indeed it has done. Quantum electrodynamics, the first child of this marriage, predicted (to name only one of its successes) the anomalous magnetic moment of the electron correctly to six decimal places; what more could one want of a physical theory? Quantum electrodynamics was formulated in about 1950, many years after quantum mechanics. Planck's original quantum hypothesis (1901), however, was indeed that the electromagnetic field be quantised; the quanta we call photons. In the years leading up to 1925, the quantum idea was applied to the mechanics of atomic motion, and this resulted in particle-wave duality and the Schrodinger wave equation for electrons.

Environmental issues continue to be raised in the public arena. Sometimes these are not presented accurately, but only in a way designed to arouse the public and stimulate reaction. Chemistry and chemicals often comprise the target for public concern and disquiet. The only sure way to lessen the ability of pressure groups to manipulate public opinion is by educating a greater proportion of the population in environmental science. Our aims with the first edition of this book were to ensure that the fundamental principles of chemistry could be presented in an honest and straight-forward way to a wide, non-specialist audience. Hence, the text was aimed originally at undergraduate science students who were not majors in chemistry. The success of the book and feedback from users indicate that it has found a wider market than anticipated, which has included chemists requiring an introduction to the environment.

We are pleased to be able to present the second edition. We have taken the opportunity to broaden the scope of the book to satisfy this extended audience. A new section introduces the concepts of biogeochemical cycling, considering both the speciation of elements in environmental reservoirs and the exchange of material between these pools. Also included is a brief description of the role of environmental chemistry with respect to climate, dealing notably with global warming and ozone depletion.

Introduction to biogeochemical cycles

Much of the discussion of environmental processes in Chapter 6 will deal with single environmental reservoirs, such as the atmosphere, or the oceans. Many important processes involving chemicals, some of which have given rise to significant environmental problems, have involved transfer of chemicals from one reservoir or compartment to another. The transfer of chemical elements and compounds between compartments, taking into account the important influences of the biota, is termed biogeochemical cycling. Much work has been carried out to elucidate the biogeochemical cycles of the elements, but much remains to be done to describe the cycles in adequate detail.

The major transfers taking place in a biogeochemical cycle appear in Figure 5.1. An element is washed from rock and soil and enters fresh water streams and lakes in runoff, and to a lesser extent, in water flowing from ground waters into surface waters. The major part of that entering the fresh waters is carried by streamflow into the oceans. Direct coastal discharges of pollutants can also be important, especially in fully and partially enclosed seas like the North Sea. Volatile elements (e.g. mercury) and compounds are degassed from land surfaces and the sea. Less volatile materials enter the atmosphere from the land surface in wind-blown dusts (and man-made emissions) and from the sea surface in sea spray, the smaller particles of which form marine aerosol.

No-one can deny the success which quantum field theory, in the perturbative approximation, has enjoyed over the last half century. One need only mention the interpretation of quantised fields as particles, the description of scattering processes, the precise numerical agreements in quantum electrodynamics, the successful prediction of the W particle, and the beginnings of an understanding of the strong interaction through quantum chromodynamics. Yet despite these successes, the question of how to describe the basic matter fields of nature has remained unanswered – except, of course, through the introduction of quantum numbers and symmetry groups. As far as field theory goes, the matter fields are treated as point objects. Even in classical field theory these present us with unpleasant problems, in the shape of the infinite self-energy of a point charge. In the quantum theory, these divergences do not disappear; on the contrary, they appear to get worse, and despite the comparative success of renormalisation theory the feeling remains that there ought to be a more satisfactory way of doing things.

Now it turns out that non-linear classical field theories possess extended solutions, commonly known as solitons, which represent stable configurations with a well-defined energy which is nowhere singular.

‘He shut the lions' mouths, and they did not harm me, for in his sight righteousness was found in me.’

At the beginning of this work it was proposed to inquire into three questions, within the limits allowed by the subject-matter; the first two of them have been dealt with sufficiently, I believe, in the previous books. Now it remains to deal with the third, the truth of which cannot be brought to light without putting certain people to shame, and will therefore perhaps be a cause of some resentment against me. But since truth from its unchangeable throne implores us, and Solomon too, entering the forest of Proverbs, teaches us by his own example to meditate on truth and loathe wickedness; and since our authority on morals, Aristotle, urges us to destroy what touches us closely for the sake of maintaining truth; then having taken heart from the words of Daniel cited above, in which divine power is said to be a shield of the defenders of truth, and putting on ‘the breast-plate of faith’ as Paul exhorts us, afire with that burning coal which one of the seraphim took from the heavenly altar to touch Isaiah's lips, I shall enter the present arena, and, by his arm who freed us from the power of darkness with his blood, before the eyes of the world I shall cast out the wicked and the lying from the ring.