Some worked examples

Working from first principles, find the pH of a 20 cm sample of a 0.05 mol dm solution of a weak acid of p/ifa = 5.75 at 25°C. Calculate the fraction ionised, and comment on its value. How does this correlate with any assumptions made in the calculation of the pH  

This is a fairly weak acid present at a moderate concentration. 

The acid is sufficiently weak to be only slightly ionised. 

If the acid used had instead a p Ta value of 1.85 how would the reasoning be altered Find the pH, and the fraction ionised. Explain the approximations made. 

Take care The negative value is meaningless, and so only one root is taken. 

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Guedens, W. J. Yperman, J. Mullens, J. et al. Statistical Analysis of Errors A Practical Approach for an Undergraduate Ghemistry Lab, Part 1. The Goncept, /. Chem. Educ. 1993, 70, 776-779 Part 2. Some Worked Examples, /. Chem. Educ. 1993, 70, 838-841. 

One of the most basic requirements in analytical chemistry is the ability to make up solutions to the required strength, and to be able to interpret the various ways of defining concentration in solution and solids. For solution-based methods, it is vital to be able to accurately prepare known-strength solutions in order to calibrate analytical instruments. By way of background to this, we introduce some elementary chemical thermodynamics - the equilibrium constant of a reversible reaction, and the solubility and solubility product of compounds. More information, and considerably more detail, on this topic can be found in Garrels and Christ (1965), as well as many more recent geochemistry texts. We then give some worked examples to show how... 

The most common practice in cost-effectiveness analysis involving a drug is to use average wholesale price (AWP) as the measure of cost from a societal perspective. Here is a classic example. The well-regarded Gold et al. (1996) book on methods for cost-effectiveness analysis presents some worked examples of analysis done from a societal perspective. One is the study by Stinnett et al. (1996) on the cost effectiveness of dietary and pharmacologic therapy for cholesterol reduction. The authors stated that they use a societal perspective. 

However, if we had carried out the calculation as 3.66 X 8.45 X 2.11 on a calculator without rounding off the intermediate resnlt, we wonld have obtained 65.3 as the answer for E. In general, we will show the correct nnmber of significant figures in each step of the calculation. However, in some worked examples, only the final answer is rounded off to the correct number of significant fignres. The answers for all intermediate calculations will be carried to one extra figure. 

This section summarizes the classical, equilibrium, statistical mechanics of many-particle systems, where the particles are described by their positions, q, and momenta, p. The section begins with a review of the definition of entropy and a derivation of the Boltzmann distribution and discusses the effects of fluctuations about the most probable state of a system. Some worked examples are presented to illustrate the thermodynamics of the nearly ideal gas and the Gaussian probability distribution for fluctuations. 

Clearly, a textbook without some worked examples and problems that a reader can try to solve, in order to test his or her understanding of the material, would be... 

In the method chapters of this book (2-11) we explain the physical principles of each technique, but we have tried to avoid going into the theory in great detail. For the avid follower there is extension material for all chapters available on the book s website (http //www.wiley.com/go/rankin/structural), dealing with topics that are more advanced or less widely applicable than the core subjects that we present here. There is a reference to each supplementary section at the appropriate place in the printed text, as well as an index at the book s home page. We hope that this facility will also slow down the ageing process of this book, since we will endeavour to update the on-line resource with information about relevant new instrumentation or developments in methodology. At the end of each chapter we also provide references to other more formal and rigorous treatments. There are also some worked examples on the website these are also referenced in the text. 

We now turn to electronic selection rules for syimnetrical nonlinear molecules. The procedure here is to examme the structure of a molecule to detennine what synnnetry operations exist which will leave the molecular framework in an equivalent configuration. Then one looks at the various possible point groups to see what group would consist of those particular operations. The character table for that group will then pennit one to classify electronic states by symmetry and to work out the selection rules. Character tables for all relevant groups can be found in many books on spectroscopy or group theory. Ftere we will only pick one very sunple point group called 2 and look at some simple examples to illustrate the method. 

Nevertheless, they are stable to standard work-up and purification methods. The benzenesulfonyl group can be introduced using base and an aprotic solvent[3] or under phase transfer conditions[4], Table 9.2 gives some representative examples of acylation and sulfonylations. 

Much of the additional material is taken up by what 1 have called Worked examples . These are sample problems, which are mostly calculations, with answers given in some detail. There are seventeen of them scattered throughout the book in positions in the text appropriate to the theory which is required. 1 believe that these will be very useful in demonstrating to the reader how problems should be tackled. In the calculations, 1 have paid particular attention to the number of significant figures retained and to the correct use of units. 1 have stressed the importance of putting in the units in a calculation. In a typical example, for the calculation of the rotational constant B for a diatomic molecule from the equation... 

An inventor may estabHsh utiHty by providing several working examples which disclose preparation, appHcation, and even some or all of the benefits of the invention. UtiHty may also be substantiated by merely disclosing several appHcations for the invention. One method of determining the breadth or scope of an invention is to define the invention by only those elements essential to performing the intended task. This definition should then become the broadest cl aim of the patent appHcation. 

Discuss how carbon adsorption works and how it can be used in water treatment applications. Give some specific examples where this technology is used to remove specific contaminants. 

Note that these stress, strain and modulus equations are given for illustration purposes. They apply to three-point bending as shown in Fig. 2.3. Other types of bending can occur (e.g. four-point bending, cantilever, etc.) and different equations will apply. Some of these are illustrated in the Worked Examples later in this chapter and the reader is referred to Benham et al. for a greater variety of bending equations. 

Every measured quantity or component in the main equations, Eqs. (12.30) and (12.31), influence the accuracy of the final flow rate. Usually a brief description of the estimation of the confidence limits is included in each standard. The principles more or less follow those presented earlier in Treatment of Measurement Uncertainties. There are also more comprehensive error estimation procedures available.These usually include, beyond the estimation procedure itself, some basics and worked examples. 

In working Example 18.8, we have in effect assumed that the electrolyses were 100% efficient in converting electrical energy into chemical energy. In practice, this is almost never the case. Some electrical energy is wasted in side reactions at the electrodes and in the form of heat This means that the actual yield of products is less than the theoretical yield. 

This mass balance equation shows that material that is initially at radial position rin will move to radial position r for some downstream location, >0. A worked example of radial velocities and curved streamlines is given in Chapter 13, Example 13.10. 

A recursive modelling procedure will let the chemist use formulated products as a raw materials. In this paper, we illustrate this use of recursion, and provide a working example that highlights some of the complexities encountered when using this technique. 

On the negative side, some poor examples of laboratory handiwork have been seen. In most of these cases, a little extra work on the part of the handyman could have elevated the results from just passable to excellent. 

Recognition of the basic nature of condensation polymers did not advance perceptibly from the early work of Lourengo and Kraut up to 1910 or 1920, or even somewhat later. In fact, evidence of retrogression could be cited. Some further examples serve to illustrate the state of affairs prevailing at the turn of the century. 

Arbuckle has published some excellent work on microscopic studies of ice cream crystal structure and effects of hydrocolloids on this crystal structure. Some interesting examples of this work are shown in some of his papers 1,18). 

Infrared measurement of additive concentrations is a more complex analysis than initially expected, as some additives may undergo a variety of chemical reactions during processing, as shown by Reeder et al. [128] for the FTIR analysis of phosphites in polyolefins. Some further examples of IR work refer to PVC/metal stearates [129], and PE/Santonox R [68,130]. Klingbeil [131] has examined the decomposition of various organic peroxyesters (TBPB, TBPP, TBPA and TBPO) and a peroxidicarbonate (BOPD) as a function of pressure, temperature and solvent by means of quantitative FTIR using an optical high p, T reaction cell. 

EXAMPLE 18.2. A solution of HCl reacts with a solution of Na2CO, in a general chemistry lab, yielding H20, C02(g), and NaCl. Some heat and some work are also produced. State how you can recognize that heat has been produced, and identify the form of the work produced. 

In the reaction of Na2C03 with HCl, Example 18.2, some heat was produced and some work was also produced. The quantity of heat produced can be measured rather easily by the quantity of materials and the temperature rise, as will be illustrated. The quantity of work does not affect the... 

Other optical and spectroscopic techniques are also important, particularly with regard to segmental orientation. Some examples are fluorescence polarization, deuterium nuclear magnetic resonance (NMR), and polarized IR spectroscopy [4,246,251]. Also relevant here is some work indicating that microwave techniques can be used to image elastomeric materials, for example, with regard to internal damage [252,253]. 

A great variety of modifications to the steric properties, based on Ewen s work, has been carried out in order to control the stereoselectivity some representative examples are summarized in Table 9. Using modified Me2C(Cp)(Flu)MCl2 (M = Zr, Hf), high syndiospecificity has been achieved 134-136, as reported by Fina Hoechst and Mitsui Toatsu Chemicals, inc. 

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