4.2 Intermediate forms In essence, the argument about intermediate forms runs as follows. If whales evolved from a terrestrial ancestor through the accumulation of small differences over time, we should expect to find the fossils of a number of ‘missing links’, i.e. creatures with a mixture of terrestrial and aquatic characteristics. In fact, we might expect to find a succession of such animals, each a little bit more whale-like and a little bit less well adapted to life on land than its predecessor. To m
Acknowledgements The content acknowledged below is Proprietary (see terms and conditions) and is used under licence. Grateful acknowledgement is made to the following sources for permission to reproduce material in this unit: Figure 2(f) © National Power; Figure 3 Courtesy of IBM Corporation, Research Division, Almaden Research Center; Figure 14 ‘Fuel hoarder sentenced’ by Maurice Weaver, printed 6 April 2001, Telegraph Gro
Figures
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
3.5 Electron states and box diagrams So far, we have represented the electronic state of an atom as a collection of sub-shells. Now we turn to the states of the electrons within those sub-shells. Just as shells can be broken down into sub-shells, so sub-shells can be broken down into atomic orbitals. Each atomic orbital describes an allowed spatial distribution about the nucleus for an electron in the sub-shell. Here we shall only be concerned with their number. Consider the formula for the sub-shell electron capaci
3.3 Electronic configurations and the Periodic Table
Figure 21 has been designed for use in a particular thought experiment. The purpose of the thought experiment is to see how the electronic configuration of the atoms changes as one moves through the Periodic Table from beginning to end. We start with the hydrogen atom, which has one proton and one electron. Then we
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
Introduction This unit is an adapted extract from the course The molecular world
(S205) This unit will provide you with a detailed understanding of some of the important problems and topics that are being studied by the chemists of today, and of the ways in which associated problems might be solved by chemical methods. But to acquire this understanding you must have a good grasp of fundamental chemic
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
, then its modulus, or abso
the set of integers is
and the set of rational numbers is
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