6.3.1 Refinements and difficulties In Section 6.2, we said that inter-axis repulsions vary in the order: non-bonded pair–non-bonded pair > non-bonded pair–bond pair > bond pair–bond pair There is evidence for this in the inter-bond angles in molecules. For example, in wat
4.5.2 Noble gas configurations under stress It is remarkable how many molecules and ions of the typical elements can be represented by Lewis structures in which each atom has a noble gas shell structure. Nevertheless, many exceptions exist. According to the periodic trends summarised in Section 2, the highest fluorides of boron and phosphorus are BF3 and PF5. How
4.5 More about covalent bonding So far, the valencies in Table 1 have just been numbers that we use to predict the formulae of compounds. But in the case of covalent substances they can tell us more. In particular, they can tell us how the atoms are linked together in the molecule. This information is obtained from a two-dimensional drawing of the structural form
1.2 Chemical elements Atoms of the same atomic number behave virtually identically in chemical reactions. They are therefore given the same chemical name and chemical symbol. For example, the atom of atomic number 6, which is shown in Figure 1, is a carbon atom, whose symbol is C. All materials are made of atoms, but there is a special class of substan
5.3 Stellar astrophysics If tunnelling out of nuclei is possible then so is tunnelling in! As a consequence it is possible to trigger nuclear reactions with protons of much lower energy than would be needed to climb over the full height of the Coulomb barrier. This was the principle used by J.D. Cockcroft and E.T.S. Walton in 1932 when they caused lithium-7 nuclei to split into pairs of alpha particles by bombarding them with high-energy protons. Their achievement won them the 1951 Nobel prize for physics. The same p
2.1 Overview Session 2 discusses the scattering of a particle using wave packets. We shall restrict attention to one dimension and suppose that the incident particle is initially free, described by a wave packet of the form This is a superposition of de Broglie waves, with the function
Acknowledgements The content acknowledged below is Proprietary (see terms and conditions). This content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence The author of this unit is Peter Sheldon. Grateful acknowledgement is made to the following sources for permission to reproduce material
4.4 Other Wenlock Limestone fossils Among the other fossils common in the Wenlock Limestone are brachiopods (Figure 12a and b), gastropods (Figure 12c) and bryozoans (Figure 12d). You may need to reread Section 1.3 to remind yourself about various aspects of these groups. Figure 13 (the unit image) is a reconstruction of a typical scene from a Wenlock Limestone environment. See
4.1 Trilobites As we've seen, the Cambrian explosion left the seas teeming with a huge variety of animals. In the following activity you will study some of the marine life at one particular time in the Palaeozoic Era – the middle part of the Silurian Period, 430 Ma ago. You'll look in detail at some fossils which come from a deposit in the UK called the Wenlock Limestone, famous for its many beautiful fossils. The Wenlock Limestone crops out mainly around Birmingham and the borders of Wales. Figure
4.5 What can individuals do? Whatever age they are, men, women and children can all do something to try to prevent future cardiovascular diseases in themselves or their families by eating a balanced diet (see Section 4.6), taking more exercise and modifying their lifestyles to reduce any other known risk factors. If cardiovascular diseases are pre-existi
2.10.1 Mean and standard deviation for repeated measurements In everyday terms, everybody is familiar with the word ‘average’, but in science and statistics there are actually several different kinds of average used for different purposes. In the kind of situation exemplified by Table 2, the sort to use is the mean
(or more strictly the ‘arithmetic mean’) For a set of measurements, this is de
Real functions and graphs
Sometimes the best way to understand a set of data is to sketch a simple graph. This exercise can reveal hidden trends and meanings not clear from just looking at the numbers. In this unit you will review the various approaches to sketching graphs and learn some more advanced techniques. First published on Tue, 28 Jun 2011 as
4.2 Least upper and greatest lower bounds We have seen that the set [0, 2) has no maximum element. However, [0, 2) has many upper bounds, for example, 2, 3, 3.5 and 157.1. Among all these upper bounds, the number 2 is the least upper bound because any number less than 2 is not an upper bound of [0, 2).
2.3 Inequalities involving modulus signs Now we consider inequalities involving the modulus of a real number. Recall that if a 1.1 Rational numbers The set of natural numbers is
9 When to use the calculator Despite the list of advantages given, here is a word of warning: a calculator is not a substitute for a brain! Even when you are using your calculator, you will still need to sort out what calculation to do to get the answer to a particular problem. However skilled you are at using your calculator, if you do the wrong sum, you will get the wrong answer. The phrase ‘garbage in, garbage out’ applies just as much to calculators as to computers. Your calculator is just that – a calculator!< Flexibility The calculator is very useful for ordinary arithmetic and yet it can also perform many functions commonly associated with a computer and deal with quite advanced mathematics. It is useful for both beginners and experts alike, because it has a variety of modes of operation. 8.2.5 Memory The calculator retains numbers, formulas and programs which you have stored in it, even when it is turned off. You can recall them when you need them and so save time by not having to enter the same information again. 8.2.4 Accuracy The calculator does not make mistakes in the way that human brains tend to. Human fingers do, however, make mistakes sometimes; and the calculator may not be doing what you think you have told it to do. So correcting errors and estimating the approximate size of answers are important skills in double-checking your calculator calculations. (Just as they are for checking calculations done in your head or on paper!) 8.2.2 The screen You can see the calculations that you have entered as well as the answers. This means you can easily check whether you have made any mistakes.
, then its modulus, or abso
the set of integers is
and the set of rational numbers is
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