Taylor s series

Finally, there is an interesting article" that shows how to use Taylor s series to generate shortcut methods from established theory. Examples are given for developing a criterion for replacing log mean temperature differences with average differences and for estimating the effect of temperature on reaction rate. 

The rate of reaetion (-r ) also varies with time t, sinee the eon-eentration ehanges with time. Using Taylor s series y = f(t), gives... 

Figure 2.5-1 illustrates the fact that probabilities are not precisely known but may be represented by a "bell-like" distribution the amplitude of which expresses the degree of belief. The probability that a system will fail is calculated by combining component probabilities as unions (addition) and intersection (multiplication) according to the system logic. Instead of point values for these probabilities, distributions are used which results in a distributed probabilitv of system fadure. This section discusses several methods for combining distributions, namely 1) con olution, 2i moments method, 3) Taylor s series, 4) Monte Carlo, and 5) discrete probability distributions (DPD). 

Suppose we have a function F x,y), and we carry out a Taylor s series expansion about the point (jCo,yo), thus... 

This weighting procedure for the linearized Arrhenius equation depends upon the validity of Eq. (6-7) for estimating the variance of y = In k. It will be recalled that this equation is an approximation, achieved by truncating a Taylor s series expansion at the linear term. With poor precision in the data this approximation may not be acceptable. A better estimate may be obtained by truncating after the quadratic term the result is... 

If ACp is independent of temperature, the final term in Eq. (6-16) can be neglected. Clarke and Glew expanded AH in a Taylor s series, truncating at the third derivative of ACp. and obtained Eq. (6-17). 

They point out that it is invalid to retain a higher-order (in T) term while omitting a lower-order term, for this is inconsistent with the Taylor s series expansion. Least-... 

More elaborate approaches make use of multiple substitutions to generate additional information. Consider a reaction in which substitutions can be made at two sites, X and y, so that the observed response is a function of both substituents, f x,y). Expanding this in a Taylor s series gives... 

This review includes most of the published articles from the defined area and excludes only imidazoquinolines, which were reviewed in Weissberger-Taylor s series The Chemistry of Heterocyclic Compounds (81MI1). Comprehensive Heterocyclic Chemistry II (96MI1) mentioned only some of the azoloquinolines in the first edition the authors omitted citations about this type of compounds. The trend toward interest in these compounds can be illustrated by the number of citations in Chemical Abstract as shown in Table I. Besides Chemical Abstracts Substance/Subject (Collective) Indexes, the MDL database search has been used. 

Taylor s series in terms of the normal coordinate Qi (Appendix). If the displacement is small we may ignore higher order terms in the expansion. 

An equation is said to show analytic behavior if a Taylor s series expansion about a point in the solution set of the equation converges in the neighborhood of the point. 

The functional form of U R) differs from one diatomic molecule to another. Accordingly, we wish to find a general form which can be used for all molecules. Under the assumption that the intemuclear distance R does not fluctuate very much from its equilibrium value so that U R) does not deviate greatly from its minimum value, we may expand the potential U R) in a Taylor s series about the equilibrium distance R ... 

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. 

Tacoma Narrows bridge % tangent 16 Taylor s series 32-34 tests of series convergence 35-36 thermodynamics applications 56-57, 81 first law 38-39 Jacobian notation 160-161 systems of constant composition 38 three-dimensional harmonic oscillator 125-128... 

An example of the development of a Taylor s series is provided by the expansion of the function In x around the point x = 1. The necessary derivatives become... 

An important special case of Taylor s series occurs when a = 0. Then, Eq. (34) takes the form... 

This is simply a special case of Taylor s series when h is set to zero. Exponential Series... 

For the following relationships the sign = means approximately equal to, when X is small. These equations are derived by using a Taylor s series (see Series Summation and Identities ). 

In the preceding F = fc(r, r), H = tc(r, vt)G = k(vt, v) and the normalization constant C is fixed by equating the volume integral of n to unity. For further tractability, Sano and Mozumder expand (r v) in a Taylor s series and retain the first two terms only. The validity of this procedure can be established a posteriori in a given situation. At first, the authors obtain equations for the time derivatives of the expectation values and the correlations of dynamical variables. Then, for convenience of closure and computer calculation, these are transformed into a set of six equations, which are solved numerically. The first of these computes lapse time through the relation... 

This is a non-linear relationship. Linearising using Taylor s series, as given in equation 7.24, Volume 3 ... 

The first approximation of Kasai and Oosawa (1969) has been improved by the efforts of two research groups. Zeeberg et al. (1980) employed an elegant approach with difference equations to the equilibrium exchange problem, and, through the use of a Taylor s series approximation as well as Stirling s approximation, obtained the following solution ... 

For the dilute solutions for which the osmotic coefficient is most useful, the namral logarithm in Equation (19.52) can be expanded in a Taylor s series, and terms of higher powers can be neglected. The result is... 

The variation of E2 with q is a function of the variation of the concentration of donor atoms with distance from the surface, but Melnick assumes that for small changes of q, using q = q q, E2 may be expanded in a Taylor s series in Eq, which, substituted in the above expression, yields... 

The calculation of AH° and AS° values from the pK-temperature data in each solvent mixture was performed by the nonempirical method of Clarke and Glew (26) as simplified by Bolton (27). In this method the thermodynamic parameters are considered to be continuous, well-behaved functions of temperature, and their values are expressed as perturbations of their values at some reference temperature 0 by a Taylor s series expansion. The basic equation is ... 

It is straightforward to verify that Lh provides an approximation of order 2 to L, so that vXxXx — id4) = -id6) + 0(h4). Indeed, this can be done using the expansion in Taylor s series... 

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