Illustrating the Concepts

In this chapter we will illustrate some of the methods described in the previous sections. It is of course impossible to cover all types of bonding and geometries, but for highlighting the features we will look at the H2O molecule. This is small enough that we can employ the full spectrum of methods and basis sets. 

---

Before moving on to the next section, it would be very useful to work some of the Exercises and Problems. In particular. Exercises 3, 5, and 12 as well as problems 6, 8, and 11 provide insight that would help when the material of the next section is studied. The solution to Problem 11 is used throughout this section to help illustrate the concepts introduced here. 

The choice of the best method for answering this question is governed by the specific nature of the system under investigation. Few general principles exist beyond the importance of analyzing a representative sample of suitable purity. Our approach is to consider some specific examples. In view of the diversity of physical methods available and the number of copolymer combinations which exist, a few examples barely touch the subject. They will suffice to illustrate the concepts involved, however. 

A simple cooling cycle serves to illustrate the concepts. Figure 1 shows a temperature—entropy plot for an actual refrigeration cycle. Gas at state 1 enters the compressor and its pressure and temperature are increased to state 2. There is a decrease in efficiency represented by the increase in entropy from state 1 to state 2 caused by friction, heat transfer, and other losses in the compressor. From state 2 to states 3 and 4 the gas is cooled and condensed by contact with a heat sink. Losses occur here because the refrigerant temperature must always be above the heat sink temperature for heat transfer to take... 

Figure 8-5 illustrates the concept of samphng a continuous function. At integer values of the saiTmling rate. At, the value of the variable to be sampled is measured and held until the next sampling instant. To deal with sampled data systems, the z transform has been developed. The z transform of the function given in Fig. 8-5 is defined as... 

Fig. 15. A schematic model illustrating the concepts of basic structural unit, BSU, and local molecular ordering, LMO [c.g., 116].

Some stereospecific reactions are listed in Scheme 2.9. Examples of stereoselective reactions are presented in Scheme 2.10. As can be seen in Scheme 2.9, the starting materials in these stereospecific processes are stereoisomeric pairs, and the products are stereoisomeric with respect to each other. Each reaction proceeds to give a single stereoisomer without contamination by the alternative stereoisomer. The stereochemical relationships between reactants and products are determined by the reaction mechanism. Detailed discussion of the mechanisms of these reactions will be deferred until later chapters, but some comments can be made here to illustrate the concept of stereospecificity. 

There is an equivalent way of generating solutions to the electronic Schrodinger equation which conceptually is much closer to the experimentalists language, known as Valence Bond (VB) theory. We will start by illustrating the concepts for the H2 molecule, and note how it differ from MO methods. 

The simplest circuit element is the short circuit. Figure 2-66a illustrates the concept of a short circuit. A source of emf (labeled v ) produces a current that flows relatively unimpeded through the conductor resulting in a nearly zero potential drop and an infinite current. 

Polarization can be divided into activation polarization and concentration polarization , Activation polarization is an electrochemical reaction that is controlled by the reaction occurring on the metal-electrolyte interface. Figure 4-418 illustrates the concept of activation polarization where hydrogen is being reduced over a zinc surface. Hydrogen ions are adsorbed on the metal surface they pick up electrons from the metal and are reduced to atoms. The atoms combine to... 

Figure 4-419 illustrates the concept of corrosion process under concentration polarization control. Considering hydrogen evolution at the cathode, reduction rate of hydrogen ions is dependent on the rate of diffusion of hydrogen ions to the metal surface. Concentration polarization therefore is a controlling factor when reducible species are in low concentrations (e.g., dilute acids). 

Figures 54.6 and 54.7 illustrate the concept of rise and run. If one assumes that line O-A in Figure 54.6 represents a true, or target, shaft centerline, then side a of the triangle represents the amount of offset present in the actual shaft, which is referred to as the rise.

To illustrate the concept of fluorescence yield, we turn again to the K spectrum. Assume that an element is irradiated with an x-ray line energetic enough to excite the K spectrum. If the irradiation is continued, a steady state will soon be reached in which the rate at which holes are produced in the K shell (i.e., the rate at which atoms in the K state are produced) is just balanced by the combined rates of the various processes causing such holes to disappear. Let n1, n2,. . . , % be the individual rates rii at which the filling of holes leads to the production of the i lines in the K spectrum. The fluorescence yield, for this simple case is... 

We shall illustrate the concept with an example (a particle trapped in a hox), which though simple, reveals a number of important consequences of his approach. At the end of this section we shall he ready to move on to the hydrogen atom and see how the concepts apply to it and answer the remaining questions about its spectrum. 

A number of examples could be used to illustrate the concept (ref. 2) a selection are shown below. 

Figure 2 summarizes the individual steps in the proposed hybrid system showing the obtained reduction in normalized concentration (MF, UV, and NF) effluent. Figure 2 illustrates the concept of customized regeneration process according to the reuser needs. 

As little as three factors can confront the investigator with an intractable situation if he chooses to proceed classically. One way out is to use the factorial approach, which can just be visualized for three factors. An example from process optimization work will illustrate the concept. Assume that temperature, the excess concentration of a reagent, and the pH have been iden-... 

In this section, we will examine four examples that illustrate the steps, procedures, choices, and outputs involved in conducting some elementary cellular automata model simulations. The reader is advised to consult Chapter 10 to find the appropriate ways for entering parameters and making appropriate selections for each study. Following each prearranged example, some brief fiirther studies are indicated that will expand on, and fiirther illustrate, the concepts involved in the example. 

To illustrate the concept of simplicity of a vector we calculate the simplicity of two vectors f, and f2. ... 

In this formula, TBW is increased by 1 to account for the addition of the liter of 3% NaCl. Patient Encounters 4 and 5 illustrate the concepts of calculating and correcting the sodium deficit. 

A discussion of all the reasons for this phenomenon is beyond the scope of this chapter, but a simple example will illustrate the concept. Highly lipid-soluble drugs, such as pentobarbital, are preferentially distributed into adipose tissue. The result is that plasma concentrations are extremely low after distribution is complete. When the apparent volumes of distribution are calculated, they are frequently found to exceed total body volume, occasionally by a factor of 2 or more. This would be impossible if the concentration in the entire body compartment were equal to the plasma concentration. Thus, Vd is an empirically fabricated number relating the... 

---