Some illustrative problems

Again as a test to the understanding of the theoretical description, the best procedure is to apply the arguments to an actual numerical problem. The conclusions outlined in Section 4.3 are illustrated in the worked problems below. 

00 X 10 moldm and the same approximations are made Comment on the result. 

This is a fairly weak base present at moderate concentrations. 

The self ioDisation of water can be ignored the protonation of the base will be sufficiently extensive for the base to be the main source of OH (aq). 

The base is sufficiently weak to be only slightly protonated 

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Turner, R. I., Some illustrative problems in the flow of viscoelastic non-Newtonian lubricants, ASLE Trans., 8,179 (1965). 

The first illustrative problem comes from quantum mechanics. An equation in radiation density can be set up but not solved by conventional means. We shall guess a solution, substitute it into the equation, and apply a test to see whether the guess was right. Of course it isn t on the first try, but a second guess can be made and tested to see whether it is closer to the solution than the first. An iterative routine can be set up to cany out very many guesses in a methodical way until the test indicates that the solution has been approximated within some narrow limit. 

The examples in this subsection illustrate some possible problems and solutions. 

The purpose of these 3 volumes is to present techniques of process design and to interpret the results into mechanical equipment details. There is no attempt to present theoretical developments of the design equations. The equations recommended have practically all been used in actual plant equipment design, and are considered to be the most reasonable available to the author, and still capable of being handled by both the inexperienced as well as the experienced engineer. A conscious effort has been made to offer guidelines to judgment, decisions and selections, and some of this will be found in the illustrative problems. 

This chapter contains a discussion of two intermediate level problems in chemical reactor design that indicate how the principles developed in previous chapters are applied in making preliminary design calculations for industrial scale units. The problems considered are the thermal cracking of propane in a tubular reactor and the production of phthalic anhydride in a fixed bed catalytic reactor. Space limitations preclude detailed case studies of these problems. In such studies one would systematically vary all relevant process parameters to arrive at an optimum reactor design. However, sufficient detail is provided within the illustrative problems to indicate the basic principles involved and to make it easy to extend the analysis to studies of other process variables. The conditions employed in these problems are not necessarily those used in current industrial practice, since the data are based on literature values that date back some years. 

Some physical problems, such as those involving interaction of molecules, are usually formulated as integral equations. Monte Carlo methods are especially well-suited to their solution. This section cannot give a comprehensive treatment of such methods, but their use in calculating the value of an integral will be illustrated. Suppose we wish to calculate the integral... 

In this review, we present an introduction to the theory, and exemplify the wide range of problems that can be addressed with some illustrative results from our work in the field of ah initio drug design. The problems addressed are those of activation and DNA binding of the antitumor drug cis-platin (PtCl2(NH3)2), and basic spectrometric data from a family of drugs known as psoralens. 

Where the reduction potentials of two analytes are sufficiently different a mixture may be analysed. Titanium(III), = 0-lOV may be titrated with cerium(IV) in the presence of iron(II), =0.77 V usjng methylene blue as indicator. Subsequently the total, iron plus titanium, may be determined using ferroin as indicator. The determination of iron is illustrative of some practical problems which are encountered in direct titration procedures. 

The book, divided into four sections, begins with a brief chapter describing some present problems in need of research and future trends related to polymer modification. The volume is not exhaustive but chapters were selected to illustrate specific aspects of more general areas of polymer modification. 

This chapter also allows me to cover some word-problem topics that just don t seem to fit anywhere else. You can call this the miscellaneous chapter — it contains word problems that you re likely to come across but that don t have any particular place with all the others. These problems are great for illustrating some more of the techniques that are helpful when solving math word problems. 

We now consider problems of a quantitative analysis of multiphonon transitions. Here an exact treatment seems hopeless at the present time, and to make headway at all a fair number of approximations are required. We shall give an overview of the general difficulties, discuss some (unfortunate) confusion on Born-Oppenheimer terminology, and then illustrate some quantitative problems using the adiabatic formulation (see below). The present discussion will also be used as a basis for subdividing the various papers, to be discussed in Section lOd, into various (perhaps somewhat arbitrary) categories. 

The fundamental difference between the explicit and implicit solvent models is not that one has solvent and the other does not. Rather, the difference is that the implicit model employs a homogeneous medium to represent tlie solvent where the explicit model uses atomistically represented molecules. While tlie latter choice is clearly tlie more physically realistic, the practical limitations imposed by explicit representation dictate that it is not necessarily the best choice for a given problem of interest. This section compares and contrasts the relative strengths and weaknesses of tlie two models, including some illustrative applications. 

This tutorial will guide the user through some sample problems to familiarize them with the features of CSMPlug. The following sample problems will be illustrated ... 

This chapter introduces the reader to elementary concepts of modeling, generic formulations for nonlinear and mixed integer optimization models, and provides some illustrative applications. Section 1.1 presents the definition and key elements of mathematical models and discusses the characteristics of optimization models. Section 1.2 outlines the mathematical structure of nonlinear and mixed integer optimization problems which represent the primary focus in this book. Section 1.3 illustrates applications of nonlinear and mixed integer optimization that arise in chemical process design of separation systems, batch process operations, and facility location/allocation problems of operations research. Finally, section 1.4 provides an outline of the three main parts of this book. 

This review focuses on the structural aspects and discusses several approaches to computerized structural feature analysis and automatic identification of structural similarity of organic molecules with some illustrative examples relating to the structure-activity problems. 

When the intrinsic kinetics are nonlinear, some interesting problems arise that are best discussed first with a discrete description. A possible assumption for the kinetics is that they are independent, that is, that the rate at which component 7 disappears depends only on the concentration of component I itself (this assumption is obviously correct in the first-order case). The difficulties associated with the assumption of independence are best illustrated by considering the case of parallel th order reactions, which has been analyzed by Luss and Hutchinson (1971), who write the kinetic equation for component / as -dcj/dt = kjc", 1=1, 2,. . ., N, where Cj is the (dimensional) concentration of component / at time t. The total initial concentration C(0) is , and this is certainly finite. Now consider the following special, but perfectly legitimate case. The value of C(0) is fixed, and the initial concentrations of all reactants are equal, so that C/(0) = C(0)/N. Furthermore, all the k/ s are equal to each other, k/ = k. One now obtains, for the initial rate of decrease of the overall concentration ... 

CCCII comprises ten volumes, of which the last contains only subject indexes. The first two volumes describe the development of new ligands since the 1980s, which complements Volume 2 in CCC. They also include new techniques of synthesis and characterization, with a special emphasis on the burgeoning physical techniques which are increasingly applied to the study of coordination compounds. Developments in theory, computation methods, simulation, and useful software are reported. The volumes conclude with a series of case studies, which illustrate how synthesis, spectroscopy, and other physical techniques have been successfully applied in unravelling some significant problems in coordination chemistry. 

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