Maths for Chemists

Graham Doggett, Martin Cockett

• Publisher: Royal Society of Chemistry
• Year: 2015
• Print ISBN: 9781849733595
• eBook ISBN: 9781782624950
• Edition: 1
• Topic: Mathematics
• Subtopic: Higher Education
• Index: Mathematics

The two volumes of Maths for Chemists provide an excellent resource for all undergraduate chemistry students but are particularly focussed on the needs of students who may not have studied mathematics beyond GCSE level (or equivalent). The texts are introductory in nature and adopt a sympathetic approach for students who need support and understanding in working with the diverse mathematical tools required in a typical chemistry degree course. The early chapters of Maths for Chemists Volume I: Numbers, Functions and Calculus provide a succinct introduction to the important mathematical skills of algebraic manipulation, trigonometry, numbers, functions, units and the general grammar of maths. Later chapters build on these basic mathematical principles as a foundation for the development of differential and integral calculus. In spite of the introductory nature of this volume, some of the more important mathematical tools required in quantum chemistry are deliberately included, through a gradual introduction to, and development of, the concept of the eigenvalue problem. Ideal for the needs of undergraduate chemistry students, Tutorial Chemistry Texts is a major series consisting of short, single topic or modular texts concentrating on the fundamental areas of chemistry taught in undergraduate science courses. Each book provides a concise account of the basic principles underlying a given subject, embodying an independent-learning philosophy and including worked examples.

1

Numbers and Algebra

Numbers of one kind or another permeate all branches of chemistry (and science generally), simply because any measuring device we use to record a characteristic of a system can only yield a number as output. For example, we might measure or determine the:

• Weight of a sample.

• Intensity or frequency of light absorption of a solution.

• Vibration frequency for the HCl molecule.

• Relative molecular mass of a carbohydrate molecule.

Or we might:

• Confirm the identity of an organic species by measuring its boiling point.

• Measure, or deduce, the equilibrium constant of a reversible reaction.

• Wish to count the number of isomeric hydrocarbon species with the formula C₄H_10.

In some of these examples, we also need to:

• Specify units.

• Estimate the error in the measured property.

Clearly then, the manner in which we interact with the world around us leads us quite naturally to use numbers to interpret our experiences.

In many situations, we routinely handle very large and very small numbers, so disparate in size that it is difficult to have an intuitive feel for order of magnitude. For example:

• The number of coulombs (the basic unit of electrical charge) associated with a single electron is approximately 0.000 000 000 000 000 000 160 2177.

Decimal numbers are commonly written with a space between every group of three digits after the decimal point (sometimes omitted if there are only four such digits).

• The equilibrium constant for the electrochemical process

is of the order of 1 followed by 4343 zeros.¹ In chemical terms, we have no problem with this answer, as it indicates that the equilibrium is totally towards the right side (which means that the aluminium electrode will be completely consumed and the gold electrode untouched).

These two widely different examples, of a type commonly experienced in chemistry, illustrate why it is so important to feel at ease using numbers of all types and sizes. A familiarity and confidence with numbers is of such fundamental importance in solving quantitative chemical problems that we devote the first two chapters of this book to underpinning these foundations. Our main objective is to supply the necessary tools for constructing models to help in interpreting numerical data, as well as in achieving an understanding of the significance of such data.

Aims:

In this introductory chapter, we provide the necessary tools for working with numbers and algebraic symbols, as a necessary prelude to understanding functions and their properties – a key topic of mathematics that impinges directly on all areas of chemistry. By the end of the chapter you should be able to:

• Understand the different types of numbers and the rules for their combination.

• Work with the scientific notation for dealing with very large and very small numbers.

• Work with numerical and algebraic expressions.

• Simplify algebraic expressions by eliminating common factors.

• Combine rational expressions by using a common denominator.

• Treat units as algebraic entities.

1.1 Real Numbers

1.1.1 Integers

One of the earliest skills we learn from childhood is the concept of counting: at first we learn to deal with natural numbers (positive, whole numbers), including zero, but we tend to ignore the concept of negative numbers, because they are not generally used to count objects. However, we soon run into difficulties when we have to subtract two numbers, as this process sometimes yields a negative result. The concept of a negative counting number applied to an object can lead us into all sorts of trouble, although it does allow us to account for the notion of debt (you owe me 2 apples is the equivalent of saying “I own −2 apples”). We therefore extend natural numbers to a wider category of number called integers, which consist of all positive and negative whole numbers, as well as zero:

Counting numbers have been in use for a very long time, but the recognition of zero as a numeral originated in India over two millennia ago, and only became widely accepted in the West with the advent of the printed book in the 13th century.

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Table of contents

• Cover image
• Title page
• Copyright
• Preface
• Contents
• 1 Numbers and Algebra
• 2 Functions and Equations: Their Form and Use
• 3 Limits
• 4 Differentiation
• 5 Differentials
• 6 Integration
• 7 Differential Equations
• 8 Power Series
• 10 Working with Arrays I: Determinants
• 11 Working with Arrays II: Matrices and Matrix Algebra
• 12 Vectors
• 13 Simple Statistics and Error Analysis
• Worked Answers to Problems
• Glossary
• Subject Index