Analytic Approximation Methods

Quasi-Stationary and Quasi-Steady-State Approximations 

The quasi-steady-state theory has been applied particularly where a condensed phase exists whose volume changes slowly with time. This is true, for example, in the sublimation of ice or the condensation of water vapor from air on liquid droplets (M3, M4). In the condensation of water vapor onto a spherical drop of radius R(t), the concentration of water vapor in the surrounding atmosphere may be approximated by the well-known spherically symmetric solution of the Laplace equation  

This can be used to compute the diffusive flux at the surface of the sphere, which determines its rate of mass increase  

Similar methods are applied by Kuczynski and Landauer (K7) to the diffusion of carbon in metallic carbides, in which a metal-carbon 

Ham further shows that the free-boundary problem, starting with a precipitate particle of negligible dimensions, is not unique, since an arbitrary spheroid will grow at constant eccentricity, its dimensions being 

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For solving this problem the analytical approximate method of integral correlations proposed by Goodman (1958) and improved by Volkov et al (1988) can be employed. This method was successfully used by Fomin et al (1994) for solving more complex problem of moving heat source within the borehole in application to melting of the paraffin deposition in the annulus. 

A further model Hamiltonian that is tailored for the treatment of non-adiabatic systems is the vibronic coupling (VC) model of Koppel et al. [65]. This provides an analytic expression for PES coupled by non-adiabatic effects, which can be fitted to ab initio calculations using only a few data points. As a result, it is a useful tool in the description of photochemical systems. It is also very useful in the development of dynamics methods, as it provides realistic global surfaces that can be used both for exact quantum wavepacket dynamics and more approximate methods. 

Equations (2.22) and (2.23) become indeterminate if ks = k. Special forms are needed for the analytical solution of a set of consecutive, first-order reactions whenever a rate constant is repeated. The derivation of the solution can be repeated for the special case or L Hospital s rule can be applied to the general solution. As a practical matter, identical rate constants are rare, except for multifunctional molecules where reactions at physically different but chemically similar sites can have the same rate constant. Polymerizations are an important example. Numerical solutions to the governing set of simultaneous ODEs have no difficulty with repeated rate constants, but such solutions can become computationally challenging when the rate constants differ greatly in magnitude. Table 2.1 provides a dramatic example of reactions that lead to stiff equations. A method for finding analytical approximations to stiff equations is described in the next section. 

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. 

After analysis by means of the streak method, a good analytical approximation for the orientation distribution is known. Orientation desmearing becomes possible. For this purpose the method described in Sect. 9.5 can be utilized. 

The partial differential equations representing material and energy balances of a reaction in a packed bed are rarely solvable by analytical means, except perhaps when the reaction is of zero or first order. Two examples of derivation of the equations and their analytical solutions are P8.0.1.01 and P8.01.02. In more complex cases analytical, approximations can be made (by "Collocation" or "Perturbation", for instance), but these usually are quite sophisticated to apply. Numerical solutions, on the other hand, are simple in concept and are readily implemented on a computer. Two such methods that are suited to nonlinear kinetics problems will be described. 

Petersson et al. had earlier proposed [23] an alternative expression E(n) = Eoo + ]CiSn+i A/(l + 1/2)6 in the context of the CBS methods developed in his group. The summation is carried out numerically in that paper, but in fact an elegant analytical approximation exists for... 

In order to describe the fluorescence radiation profile of scattering samples in total, Eqs. (8.3) and (8.4) have to be coupled. This system of differential equations is not soluble exactly, and even if simple boundary conditions are introduced the solution is possible only by numerical approximation. The most flexible procedure to overcome all analytical difficulties is to use a Monte Carlo simulation. However, this method is little elegant, gives noisy results, and allows resimulation only according to the method of trial and error which can be very time consuming, even in the age of fast computers. Therefore different steps of simplifications have been introduced that allow closed analytical approximations of sufficient accuracy for most practical purposes. In a first... 

An Approximate Method. When the third virial coefficient is sufficiently small, it frequently happens that a is roughly constant, particularly at relatively low pressures. A good example is hydrogen gas (Fig. 10.7). When this is the case, we can integrate Equation (10.51) analytically and obtain... 

Therefore, we need to find approximate methods for simultaneous reaction systems that will permit finding analytical solutions for reactants and products in simple and usable form. We use two approximations that were developed by chemists to simplify simultaneous reaction systems (1) the equilibrium step approximation and (2) the pseudo-steady-state approximation... 

In this section, we will analyze an elementary problem in quantum mechanics, the square barrier. The purpose is twofold. First, such an analysis can provide physical insight into the process, to gain a conceptual understanding. Second, analytically soluble models are indispensable for assessing the accuracy of approximate methods, such as the MBA. 

These equations, for the case of solid diffusion-controlled kinetics, are solved by arithmetic methods resulting in some analytical approximate expressions. One common and useful solution is the so-called Nernst-Plank approximation. This equation holds for the case of complete conversion of the solid phase to A-form. The complete conversion of solid phase to A-form, i.e. the complete saturation of the solid phase with the A ion, requires an excess of liquid volume, and thus w 1. Consequently, in practice, the restriction of complete conversion is equivalent to the infinite solution volume condition. The solution of the diffusion equation is... 

This is Kramers escape problem. Since no analytic solution is known for any metastable potential of the shape in fig. 40 the quest is for suitable approximation methods. This problem has received an extraordinary amount of attention from physicists, chemists and mathematicians.5 0 We describe the main features - all present already in the seminal paper by Kramers. 

It would be of considerable interest to extend the technique just presented to problems involving nonlinear equations because there are many situations in ultracentrifugation where nonideality is a dominant feature. Furthermore, it is known (4, 14) that even for two-component systems with nonideality the theory for estimating the sedimentation constant based on a diffusion-free (c = 0) approximation can lead to systematic error. Therefore, the development of an approximate procedure for nonlinear equations would be useful for further progress in analytical separation methods. 

We first follow the flow chart for the simple case of elastic scattering of structureless atoms. The number of internal states, Nc, is one, quantum scattering calculations are feasible and recommended, for even the smallest modem computer. The Numerov method has often been used for such calculations (41), but the recent method based on analytic approximations by Airy functions (2) obtains the same results with many fewer evaluations of the potential function. The WKB approximation also requires a relatively small number of function evaluations, but its accuracy is limited, whereas the piecewise analytic method (2) can obtain results to any preset, desired accuracy. 

In summary, we have shown that the kinetics of the bimolecular reaction A + B —> 0 with immobile reactants follows equation (6.1.1), even on a fractal lattice, if d is replaced by d, equation (6.1.29). Moreover, the analytical approach based on Kirkwood s superposition approximation [11, 12] may also be applied to fractal lattices and provides the correct asymptotic behaviour of the reactant concentration. Furthermore, an approximative method has been proposed, how to evaluate integrals on fractal lattices, using the polar coordinates of the embedding Euclidean space. 

Therefore, in order to obtain self-consistent results from Eq. (462), it is necessary to consider plane waves in all three directions. This is as far as an analytical approximation will go. In order to obtain solutions from the field equation (459), computational methods are required. 

Per-Olov Lowdin had a long and lasting interest in the analytical methods of quantum mechanics and my tribute to his legacy involves an application of the Wentzel-Kramers-Brillouin (WKB) asymptotic approximation method. It was the subject of a contribution(l) by Lowdin to the Solid State and Molecular Theory Group created by John C. Slater at the Massachusetts Institute of Technology. 

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