Simple application

Here we consider a number of simple examples involving gases. Most of the following applications are based on assuming that the gases are ideal. This means that pressure, P, volume, V, and temperature, T, are related via 

Remark 2 Equation (2.7) is simple but nonetheless important, because it is used frequently throughout this text as a first and often satisfactory approximation. 

But this does not look like much progress. With some foresight we compute the following differential 

Inserting Eq. (2.19) into Eq. (2.12) completes the proof of the above statement. Remark Integrating Eq.(2.19) using Eq.(2.7) we immediately obtain 

This means that if an ideal gas is compressed (expanded) isothermally, i.e. V Vo (V Vo), its entropy is decreased (increased). 

The expansion of the natural gas is a spontaneous process and thus work must have been lost. According to the Gouy-Stodola relation, this lost work is related to the entropy production of the process. 

The expansion has been assumed to be adiabatic, and thus the entropy generated equals the entropy increase of the gas, AS, as the entropy change of the environment, AS0, can be set to zero because the process is adiabatic. The amount of lost work can now be calculated from the entropy values Sj and S2 of 1 mol of methane at the initial and final conditions, respectively. However, this requires knowledge not only of the final pressure P2, which is known, but also of the final temperature T2, which is unknown. Here, the first law helps us out. Applying Equation 2.39 and substituting zero for Win and Qout, we find AH = 0 or H2 = Hv From the IUPAC data series number 16, dealing with methane [1], we find that the molar enthalpy and entropy at initial conditions are, respectively, 

the lost work can be calculated and we find Wlost = T0(S2 - S,) = 3324J/mol. Note that this calculation did not require the application of the concept of exergy. 

we wish to calculate which fraction this lost work is of the work originally available in the gas. The chemical exergy of the gas, assumed to be methane, is significant, 831.65 kj/mol, but it should be excluded from the calculation because no chemistry is involved in the expansion step. The work available in the gas at initial and final conditions can be calculated from Equation 6.11  

The fraction of nonchemical work available in the gas that has been lost in the expansion process can now be calculated from W /Exj = 3.325/13.501 = 0.246. If we had included the chemical exergy of the gas, this number would have been reduced to 0.00393, but as the expansion step is strictly nonchemical, this result is meaningless. Of course, the calculation of Wlost itself would not be affected as the chemical exergy would have to be included in both Exj and Ex2 and would drop out. 

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For two Bom-Oppenlieimer surfaces (the ground state and a single electronic excited state), the total photodissociation cross section for the system to absorb a photon of energy ai, given that it is initially at a state x) with energy can be shown, by simple application of second-order perturbation theory, to be [89]... 

The cross relation has proven valuable to estimate ET rates of interest from data tliat might be more readily available for individual reaction partners. Simple application of tire cross-relation is, of course, limited if tire electronic coupling interactions associated with tire self exchange processes are drastically different from tliose for tire cross reaction. This is a particular concern in protein/protein ET reactions where tire coupling may vary drastically as a function of docking geometry. 

Potassium and sodium borohydride show greater selectivity in action than lithium aluminium hydride thus ketones or aldehydes may be reduced to alcohols whilst the cyano, nitro, amido and carbalkoxy groups remain unaffected. Furthermore, the reagent may be used in aqueous or aqueous-alcoholic solution. One simple application of its use will be described, viz., the reduction of m-nitrobenzaldehyde to m-nitrobenzyl alcohol ... 

A simple application of the reaction may bo mentioned. Refluxing of (I) with 48 per cent, hydrobromic aeid and glacial acetic acid leads to hydrolysis and decarboxylation and the production of a mixture of the yl tctone yr-di-phonylbutyrolaotone (II) and the isomeric unsaturated acid yY-dlphenyl-vinylacotic acid (III) reduction by the Clemmonsen method or catalytically... 

Two simple applications may be mentioned. With cyciohexene (I) 3-bromo-cycZohexene (II) is obtained in good yield the latter upon dehydrobromlnation with quinoline affords an 80-90 per cent, yield of 1 3-cycfohexadiene (III) ... 

An account of the general technique of chromatographic adsorption has been given in Section 11,46. The simple applications to be described are —... 

Clearly, the factors determining Ko are far more complex than is indicated by a simple application of the Kozeny-Carman equation, and when possible, filter design should be based on experimental determinations made under conditions approximating those expected in the planned installation. 

The essence of surge protection is determining when and how much to open or close the recycle valve. In the simple application of Figure 6-45, the antisurge controller (UIC) will position the recycle valve using a control algorithm based on AP, AP, and data on the location of the surge limit. 

Table 3 compares packings as an aid to initial selection. For simple applications use Pall Rings for studies as a tried and true packing. This will give conservative results when compared w ith more recent random packings. 

S. S. H. Naqvi, S. M. Caspar, K. C. Hickman, and J. R, McNeil. A Simple Technique for Linewidth Measurement of Gratings on Photomasks. Proc. SPIE. 1261, 495, 1990. K.P. Bishop, S.M. Caspar, L.M. Milner, S.S.H. Naqvi, and J.R, McNeil, rasterization using Scatterometry. Proc. SPIE. 1545, 64, 1991. These papers discusses a simple application of scattering from surhices that are intentionally patterned. 

Operational sequence diagrams are flcw-charting techniques that represent any sequence of control movements and information collection activities that are executed in order to perform a task. Various activities in the diagram are represented with a symbolic notation, supported where necessary by a text description. For the majority of simple applications, OSDs assume a linear flow drawn from top to bottom with a limited degree of branching and looping. The symbols used are usually tailored to fit the type of task being studied and its level of analysis. 

Some of the solvent assignments seem capricious, such as CCU in ARP, CF3COOH in AD, and CHCI3 and CftHjNHi in MISC. From one point of view, it is a triumph that a purely statistical treatment, devoid of chemical experience or intuition, can generate a scheme more or less in accord with chemical concepts. However, it does not seem to be superior to the simple application of chemical ideas. 

The remarkable thing is that the HF model is so reliable for the calculation of very many molecular properties, as 1 will discuss in Chapters 16 and 17. But for many simple applications, a more advanced treatment of electron correlation is essential and in any case there are very many examples of spectroscopic states that caimot be represented as a single Slater determinant (and so cannot be treated using the standard HF model). In addition, the HF model can only treat the lowest-energy state of any given symmetry. 

An important feature of the typical system is its modularity. The most powerful EMCS installations have all three components, but often only a single module is necessai y for simple applications, such as controlling a single air-conditioning unit, or for most applications in the residential sector. Thus local and control modules arc capable of stand-alone operation without higher-level components. The functions,... 

The phenomenon of salt hydrolysis may be regarded as a simple application of the general Bronsted-Lowry equation... 

The set of metallic radii and the equation permit the reasonably satisfactory discussion of observed interatomic distances in many intermetallic compounds. In some crystals, however, the simple application of the radii and the equation leads to disagreement with observation. 

A simple application of the flow-model suggests why bioapatite carbonate 8 C values should directly follow the 8 C values for the diet as a whole. Experimentally, this observation seems to be generally confirmed (Ambrose and Norr 1993 Tieszen and Fagre 1993). In terms of the flow-model, it is... 

The above equations gave reasonably reliable M value of SBS. Another approach to modeling the elastic behavior of SBS triblock copolymer has been developed [202]. The first one, the simple model, is obtained by a modification of classical rubber elasticity theory to account for the filler effect of the domain. The major objection was the simple application of mbber elasticity theory to block copolymers without considering the effect of the domain on the distribution function of the mbber matrix chain. In the derivation of classical equation of rabber elasticity, it is assumed that the chain has Gaussian distribution function. The use of this distribution function considers that aU spaces are accessible to a given chain. However, that is not the case of TPEs because the domain also takes up space in block copolymers. 

As a very simple application of the approach presented in sect. 2, we confine our attention to a model system consisting of 2Ne independent charged particles... 

As a simple application of the variation method to determine the ground-state energy, we consider a particle in a one-dimensional box. The Schrodinger equation for this system and its exact solution are presented in Section 2.5. The ground-state eigenfunction is shown in Figure 2.2 and is observed to have no nodes and to vanish at x = 0 and x = a. As a trial function 0 we select 0 = x(a — x), 0 X a... 

These considerations are borne out by explicit model calculations (see Fig. 16), which give results close to those predicted by simple application of the Verwey-Niessen... 

R. Kubo, Statistical-mechanical theory of irreversible processes. 1. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Japan 12, 570 (1957) R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29, 255 (1966). 

Several examples of the application of quantum mechanics to relatively simple problems have been presented in earlier chapters. In these cases it was possible to find solutions to the Schrtidinger wave equation. Unfortunately, there are few others. In virtually all problems of interest in physics and chemistry, there is no hope of finding analytical solutions, so it is essential to develop approximate methods. The two most important of them are certainly perturbation theory and the variation method. The basic mathematics of these two approaches will be presented here, along with some simple applications. 

In recent years much attention has been paid to the use of explosives for the shaping and working of metals. Figure 14.12 illustrates a simple application of the process for forming a dished end of a vessel. A flat metal blank is placed over a suitable mould and the space between them evacuated. Above the blank is water and in this a suitable explosive charge is fired. The metal takes the form of the mould with little or no spring-back and usually does not require further treatment. The process is particularly... 

We will illustrate the operation of the GA through a simple application— determination of the lowest energy arrangement of a small group of dipoles. Suppose ten identical dipoles are spaced at equal intervals along a straight line (Figure 5.7). 

The next thing to notice is that the widely spaced features are very approximately equally spaced. This suggests that the g- and -matrix principal axes are non-coincident. You might think that a simple application of the above equations would suffice for a complete analysis. It is not quite so simple, and a nonlinear least-squares program is required.1,6 Table 7.4 shows the fitted parameters. 

In similar manner, spikes in the image from cosmic rays are extinguished by simple application of the median filter with a small submatrix size (3 or 5). 

Given a series of tests with the new material, the average yield x would be compared with po- If x < Po, the new supplier would be dismissed. If x > Po, the question would be Is it sufficiently greater in the light of its corresponding reliability, i.e., beyond a reasonable doubt If the confidence interval for p included po, the answer would be no, but if it did not include po, the answer would be yes. In this simple application, the formal test of hypothesis would result in the same conclusion as that derived from the confidence interval. However, the utility of tests of hypothesis lies in their generality, whereas confidence intervals are restricted to a few special cases. 

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