An Adequate Approximation

A simple strategy has been developed for saturated hydrocarbons [44,96]. Assume p = 0 for any carbon-carbon bond andp = constant for any carbon-hydrogen bond. Hence 

In this form, it becomes possible to analyze the merits of Mulliken charge distributions in comparisons with physical observables. Namely, we want to learn the true value of n and the appropriate value of A for given choices of basis sets. Three approaches were followed to this end  

Comparison of adiabatic ionization potentials (IP) of normal and branched alkanes with carbon net charges, which indicates a lowering of the IPs with increasing electron population of the electron-richest bonded pair of carbon atoms in the molecule [170]. A monotonic correlation (which turns out to be linear and remarkably accurate) is possible only with atomic charges adjusted for n = 4.4083 and the corresponding p given by Eq. (5.14). 

STO-3G basis and experimental results for AS. A least-square regression led to p = 30.3+ 0.3 me and n = —4.446 0.057 [108]. Concurrently it became possible to estimate the value of the reference charge, that of the ethane carbon atom 35.1 me. Similar work for ethylenic molecules led to qc (C2H4) = 7.7 me. For comparison, calculations using a carefully optimized 4-3IG basis and configuration interaction gave 37.8 me for the carbon net charge of ethane, and 7.5 me for that of ethylene [51]. In short, A 1 with Cl wavefunctions. 

The modified population analysis [Eq. (8.4)] appears to be adequate. The correction suggested by Fig. 5.2 is now readily understood in terms of Eqs. (8.4)-(8.6) as a basis set effect. 

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The main conclusion to be drawn from these studies is that for most practical purposes the linear rate model provides an adequate approximation and the use of the more cumbersome and computationally time consuming diffusing models is generally not necessary. The Glueckauf approximation provides the required estimate of the effective mass transfer coefficient for a diffusion controlled system. More detailed analysis shows that when more than one mass transfer resistance is significant the overall rate coefficient may be estimated simply from the sum of the resistances (7) ... 

This expression can be used to describe both pore and solid diffusion so long as the driving force is expressed in terms of the appropriate concentrations. Although the driving force should be more correctly expressed in terms of chemical potentials, Eq. (16-63) provides a qualitatively and quantitatively correct representation of adsorption systems so long as the diffusivity is allowed to be a function of the adsorbate concentration. The diffusivity will be constant only for a thermodynamically ideal system, which is only an adequate approximation for a limited number of adsorption systems. 

In order to design a replacement for wood we need to look at the vibrational behaviour of soundboards in more detail. As Fig. 28.17 shows, the soundboard of a violin has quite an ornate shape and it is extremely difficult to analyse its behaviour mathematically. But an adequate approximation for our purposes is to regard the soundboard as a rectangular panel simply supported along two opposing edges and vibrating from side to side as shown in Fig. 28.18. 

In general. Equation 2-174 can be used to approximate fracture pressure gradients. To obtain an adequate approximation for fracture pressure gradients, the pore pressure gradient must be determined from well log data. ALso, the overburden stress gradient and Poisson s ratio versus depth must be known for the region. 

And finally, we may mention that a statistical analysis of the sulfur d orbital problem has been carried out with dimethyl sulfoxide as a model compound36. The results provide a clear answer to the sulfur d orbital problem, since no simultaneous reproduction of experimental geometry and an adequate approximation to the variationally optimum total energy have been possible without including d polarization functions on sulfur . 

For the vast majority of situations of interest in the design of industrial scale chemical reactors, equation 10.1.4 provides an adequate approximation to the true situation for continuous flow reactors operating at steady state. Since... 

The first terms of the power series obtained by the multipole expansion of the Coulomb intermolecular potential account for dipole-dipole interactions prevailing in systems of polar molecules. As an adequate approximation for ensembles of... 

HEM for Inclined Pipe, Discharge If a pipe leak occurs at an elevation above or below the pump or source tank the elevation change can be idealized between the starting and ending points as shown in Fig. 23-38. That is, elevation changes can be treated as an inclined pipe with a non-zero inclination factor Fj. This is an approximation to the actual piping isometrics, but is often an adequate approximation. 

An explicit equation provides an adequate approximate solution for larger values of co ... 

As the laws of dilute solution are limiting laws, they may not provide an adequate approximation at finite concentrations. For a more satisfactory treatment of solutions of finite concentrations, for which deviations from the limiting laws become appreciable, the use of new functions, the activity function and excess thermodynamic functions, is described in the following chapters. 

Nevertheless, when we include the near nuclear corrections (where the fully relativistic kinetic energy is used), the truncation of the energy functional only in the outer region up to order both in the kinetic and exchange energies turns out to be an adequate approximation. 

Fig. 4.2. Charge distribution and surface potential in a jellium model, (a) Distribution of the positive charge (a uniform background abruptly drops to zero at the boundary) and the negative charge density, determined by a self-consistent field calculation. (b) Potential energy as seen by an electron. By including all the many-body effects, including the exchange potential and the correlation potential, the classical image potential provides an adequate approximation. (After Bardeen, 1936 see Herring, 1992.)...

We shall now assume that an adequate approximation to the true ground state 0O> is given by the BO ground state, - The ground state energy, Fo, is taken to be zero. Thus,... 

While the Bingham plastic model is an adequate approximate description of foam rheology, it is by no means exact, especially at low strain rates. More detailed models attempl to relate the rheological properties of foams to the structure and behavior of the bubbles. 

Equations of an analogous type can be used for other polyurethane-forming systems, although the numerical values of the constants may be different, because Eq. (2.29) must be treated primarily as empirical. It is reasonable to expect that an adequate approximation for other systems would be n equal to 2 (which is typical for polyaddition reactions) and m also equals to 2. Such values of the constants m and n are valid, for example, for RIM-2200 (produced by Union Carbide ), which consists basically of polyester and 4.4-diphenylmethane diisocyanate.4... 

If T2 — 7 is large enough that Eq. (53) is not an adequate approximation, the next level of approximation, assuming temperature-independent heat capacities, is used ... 

The adsorption-desorption reaction in Eq. 4.3 has been applied to soils in an average sense in a spirit very similar to that of the complexation reactions for humic substances, discussed in Section 2.3.11 Although no assumption of uniformity is made, the use of Eq. 4.3 to describe adsorption or desorption processes in chemically heterogeneous porous media such as soils does entail the hypothesis that effective or average equilibrium (or rate) constants provide a useful representation of a system that in reality exhibits a broad spectrum of surface reactivity. This hypothesis will be an adequate approximation so long as this spectrum is unimodal and not too broad. If the spectrum of reactivity is instead multimodal, discrete sets of average equilibrium or rate constants—each connected with its own version of Eq. 4.3—must be invoked and if the spectrum is very broad, the sets of these parameters will blend into a continuum (cf. the affinity spectrum in Eq. 2.38). 

Clearly, the difference between eqns.(4.31) and (4.31a) is small. If we assume that genuine chromatography columns have a minimum of2500 theoretical plates, then the difference between the two equations is always less than 4%. Even for non-symmetrical ( tailing ) peaks eqn.(4.31) will almost always be an adequate approximation of eqn.(4.31a). 

A simple example will help to clarify the meaning of (44)-(48). For a self-exchange reaction of the type shown in (35) it is an adequate approximation to consider changes in Fe—O bond distances only (Sutin, 1962). In such a case (48) gives the simplified expression (49) for A, in which fc, and... 

For a not too broad Stark level an adequate approximate formula for T is obtained when one neglects the change with energy of u over the width of the level. Thus one finds from (5.51) that (Q /Q")2 assumes half of its maximum value when... 

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