4.6 Summary of Section 4 The chemical formulae of many substances can be understood by arguing that their atoms attain noble gas structures by chemical combination. In ionic compounds, this is achieved by the transfer of electrons from one atom to another; in molecular substances, it happens through the sharing of electron pairs in covalent bonds. But in both cases, bonds between atoms consist of shared pairs of electrons. In covalent compounds the sharing is fairl
4.5.4 Resonance structures Gaseous oxygen occurs as O2 molecules. But ultraviolet light or an electric discharge converts some of the oxygen to ozone (Box 6). This has the molecular formula O3. Many people know that gaseous ozone in the stratosphere protects us from harmful sola 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 1.2.1 Isotopes All atoms of the same element have identical atomic numbers, and are chemically similar, but they may not be identical in other ways. Figure 2f shows copper. All copper atoms have atomic number 29: all their nuclei contain 29 protons. But they also contain uncharged particles called neutrons. In natural copper, the a 6 Summary Scattering is a process in which incident particles interact with a target and are changed in nature, number, speed or direction of motion as a result. Tunnelling is a quantum phenomenon in which particles that are incident on a classically impenetrable barrier are able to pass through the barrier and e 2.8 Descriptive statistics Scientists collect many different types of information, but sets of data may be very loosely classified into two different types. In the first type, so-called ‘repeated measurement’, an individual quantity is measured a number of times. An astronomer wanting to determine the light output of a star would take many measurements on a number of different nights to even out the effects of the various possible fluctuations in the atmosphere that are a cause of stars ‘twinkling’. In the seco Acknowledgements The content acknowledged below is Proprietary (see terms and conditions) and is used under licence. All materials included in this unit are derived from content originated at the Open University. 1 Developing modelling skills The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook. Click 'View document' to open the workbook (PDF, 0.2 MB). Acknowledgements The content acknowledged below is Proprietary (see terms and conditions) and is used under licence. Wade_In_Tulsa, photos All other materials included in thi Acknowledgements The content acknowledged below is Proprietary (see terms and conditions) and is used under licence. All materials included in this unit are derived from content originated at the Open University. 4.4 Further exercises In this exercise, take
4.3 Least Upper Bound Property In the examples just given, it was straightforward to guess the values of sup E and inf E. Sometimes, however, this is not the case. For example, if
In such cases, it i 1.1 Rational numbers The set of natural numbers is
6.2 Getting the feel of big and small numbers Very small and very large numbers can be difficult to comprehend. Nothing in our everyday experience helps us to get a good feel for them. For example numbers such as 1099 are so big that if Figure 1 was drawn to scale, you would be dealing with enormous distances. How big is big? First express 1 000 000 000 in scientific notation as 109. Next, to find out how many times bigger 1099 is, use your calculator to divide 1099 by 109 Introduction This unit explores reasons for studying mathematics, practical applications of mathematical ideas and aims to help you to recognize mathematics when you come across it. It introduces the you to the graphics calculator, and takes you through a series of exercises from the Calculator Book, Tapping into Mathematics With the TI-83 Graphics Calculator. The unit ends by asking you to reflect on the process of studying mathematics.
In order to complete this unit you will need Learning outcomes By the end of this unit you should be able to: Section 1: Sets use set notation; determine whether two given sets are equal and whether one given set is a subset of another; find the union, intersection and difference of two given sets. Section 2: Functions determine the image of a given function; determine whether a given function is one-one Learning outcomes After studying this unit you should: be able to perform basic algebraic manipulation with complex numbers; understand the geometric interpretation of complex numbers; know methods of finding the nth roots of complex numbers and the solutions of simple polynomial equations. 2.4.1 Try some yourself 1 Write the following as a number to a single power: (a) 26 ÷ 22 (b) 1010 ÷ 107 (c) 78 ÷ 74 Introduction This unit is an adapted extract from the course Mathematical methods and models
(MST209) This unit lays the foundations of Newtonian mechanics and in particular the procedure for solving dynamics problems. The prerequisite skills needed for this unit are the ability to solve first- and second-order differential equations, a knowledge of vectors, and an understanding of the concept of a
Box 6: Ozone is blue
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Exercise 29
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then it can be shown that E is bounded above by 3, but it is not so easy to guess the least upper bound of E.
the set of integers is
and the set of rational numbers is
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