Simplification linear approximation

A linear approximation of the potential is certainly too sweeping a simplification. In reality, Vr varies with the internuclear separation and usually rises considerably at short distances. This disturbs the perfect (mirror) reflection in such a way that the blue side of the spectrum (E > Ve) is amplified at the expense of the red side (E < 14).t For a general, nonlinear potential one should use Equations (6.3) and (6.4) instead of (6.6) for an accurate calculation of the spectrum. The reflection principle is well known in spectroscopy (Herzberg 1950 ch.VII Tellinghuisen 1987) the review article of Tellinghuisen (1985) provides a comprehensive list of references. For a semiclassical analysis of bound-free transition matrix elements see Child (1980, 1991 ch.5), for example. 

In case of a short mission time Tm the difference between t/s(x) and f/i(x) may be large and our conclusions will be too conservative. On the other side for long Tm the approximation Us(i) t/i(x) will be valid and our results and conclusions wiU be realistic. This simplification allows the analyst to simphfy the computing ofcostby the use of a linear approximation, as will be demonstrated later in 3.4. 

In previous sections, different simplifications and approximations have been introduced using first-order reactions involving linear models as examples. In many real situations, kinetic models corresponding to different physicochemical systems are nonlinear. As mentioned earlier, two typical scenarios where the QSS approximation can be used are gas- or liquid-phase reactions with free radicals as intermediates and catalytic or biocatalytic reactions involving catalytic surface intermediates or substrate-enzyme complexes. Within the traditional mathematical procedure for dealing with these intermediates, three steps can be distinguished ... 

For liquid-liquid systems, no such ready simplification is apparent, and experimental data is the rule, albeit the linear approximation sometimes can be used for a distribution coefficient, such as (K g), whereby at equilibrium between two phases A and B,... 

This equation may be simplifed considerably under conditions where the perturbation from Eq is very small, i.e. is very small (a) it is possible to use the linear approximation to the Butler-Volmer equation, / == Io(nFlRT)ri, (b) the exponential term may be expanded as a series, exp x = 1 + x +. .., and here and elsewhere all terms containing 77 and higher powers may be ignored, (c) the surface concentration of R remains close to Cr, and we may consider (cr) . =o c. Then... 

With a few exceptions, industrial applications of zeoUtes involve column operation in feeds containing more than two counterions. In general, therefore, the prediction of column performance involves the prediction of multicomponent equilibria and kinetics under dynamic flow conditions. Considering the complexity and diversity of these models (see Sects. 2.3 and 2.4), it is obvious that simplifications and approximations need to be made for practical coliunn modelling. For engineering pmposes, the most popular approach for coliunn modelling is the linear driving force - effective plate concept [97]. 

Chemical reaction rates may show large variations from reaction to reaction, and also with changes of temperature. It is often found that one or the other of the steps involved in the overall process offers the major resistance to its occurrence. Such a slow step controls the rate of the process. As a simplification such a rate-controlling step can be considered alone. In an alternative procedure the nonlinear relationship between rate and concentration is approximated to a linear relationship. To do this the nonlinear rate is expanded in the form of a Taylor s series and only the linear terms are retained. 

In this relationship, Vi is the initial (feed) volume of the gas. This is the case of Levenspiel s simplification where the volume of the reacting system varies linearly with conversion (Levenspiel, 1972). The last equation shows that even if we have a change in moles (sR / 0), if the conversion of the limiting reactant is veiy low, the volume of the reaction mixture could be taken as constant and eR is not involved in the solutions of the models (since eRjcA can be taken as approximately zero). 

In order to develop an intuition for the theory of flames it is helpful to be able to obtain analytical solutions to the flame equations. With such solutions, it is possible to show trends in the behavior of flame velocity and the profiles when activation energy, flame temperature, diffusion coefficients, or other parameters are varied. This is possible if one simplifies the kinetics so that an exact solution of the equation is obtained or if an approximate solution to the complete equations is determined. In recent years Boys and Corner (B4), Adams (Al), Wilde (W5), von K rman and Penner (V3), Spalding (S4), Hirschfelder (H2), de Sendagorta (Dl), and Rosen (Rl) have developed methods for approximating the solution to a single reaction flame. The approximations are usually based on the simplification of the set of two equations [(4) and (5)] into one equation by setting all of the diffusion coefficients equal to X/cpp. In this model, Xi becomes a linear function of temperature (the constant enthalpy case), and the following equation is obtained ... 

For purely intramolecular equilibria the system of differential equations (72) is already linear and the procedure described above transforms the system into the form of equation (104) exactly without any approximations. For such equilibria further simplification of equations (72) and (104) is possible by deleting all those quantities which refer to empty sets of nuclei. 

The ratio of two linear dimensions of an object is called the aspect ratio. There are a number of possible simplifications when the aspect ratio of an object or region is large (or small). For example, for the classical fin approximation, the thickness of the fin is small compared with the length, therefore the temperature will be assumed to change in the direction of the length only. 

One may surmise that the low-frequency limit, introduced while discussing the linear relaxation, would also lead to a reliable simplification in the nonlinear case since the process is governed mainly by the relaxation time xio. As we were tempted by this idea, in Ref. 67 we have supposed that the approximate expression... 

In this situation, one can achieve a further simplification by observing that it is not necessary to choose the linearly independent set 0 complex instead one can start from a real set = %, (p2,..., complete orthonormal basis for the L2 Hilbert space. The existence of such bases is well known. This means that in the approximate treatment of the eigenvalue problem for H, the relation, Eq. (2.73), is now replaced by the simpler relation... 

The calculation of the exact band structure from first principles, however, is rather complex and requires considerable simplifications. The usual and very successful method to calculate the band structure of organic charge transfer salts is a tight-binding method, called extended Hiickel approximation. In this approximation, one starts from the molecular orbitals (MO) which are approximated by linear combinations of the constituent atomic orbitals. Each MO can be occupied by two electrons with antiparallel spins. These valence electrons are assumed to be spread over the whole molecule. Usually, only the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are relevant and are, therefore, considered in most band-structure calculations [41]. 

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