Approximation Order

Comprehensive reviews of additive practices are available in the Hterature (18,21—23) and in extensive patent coverage. The common types of additives are discussed in approximate order of the frequency of their use. 

Other preparative methods for alkoxysilanes, in approximate order of declining utOity, are given by the following equations (34—40) ... 

Resonance effects are the primary influence on orientation and reactivity in electrophilic substitution. The common activating groups in electrophilic aromatic substitution, in approximate order of decreasing effectiveness, are —NR2, —NHR, —NH2, —OH, —OR, —NO, —NHCOR, —OCOR, alkyls, —F, —Cl, —Br, —1, aryls, —CH2COOH, and —CH=CH—COOH. Activating groups are ortho- and para-directing. Mixtures of ortho- and para-isomers are frequently produced the exact proportions are usually a function of steric effects and reaction conditions. 

Dipolarophiles utilized in these cycloadditions leading to five-membered heterocycles contain either double or triple bonds between two carbon atoms, a carbon atom and a heteroatom, or two heteroatoms. These are shown in Scheme 9 listed in approximate order of decreasing activity from left to right. Small rings containing a double bond (either C=C or C=N) are also effective dipolarophiles, but these result in six- and seven-membered ring systems. 

Carbon steel and alloy combinations appear in Table 11-12 Alloys in chemical- and petrochemical-plant sei vice in approximate order of use are stainless-steel series 300, nickel. Monel, copper alloy, aluminum, Inconel, stainless-steel series 400, and other alloys. In petroleum-refinery sei vice the frequency order shifts, with copper alloy (for water-cooled units) in first place and low-alloy steel in second place. In some segments of the petroleum industiy copper alloy, stainless series 400, low-alloy steel, and aluminum are becoming the most commonly used alloys. 

Because alterations to equipment design can be cumbersome and expensive, a more economical approach may be to change the metallurgy of affected components. Metals used in typical cooling water environments vary in their resistance to erosion-corrosion. Listed in approximate order of increasing resistance to erosion-corrosion, these are copper, brass, aluminum brass, cupronickel, steel, low-chromium steel, stainless steel, and titanium. 

Galvanic corrosion in typical industrial cooling water systems is the net result of the interplay of these factors. Some factors may accelerate the corrosion process others may retard it. In their approximate order of importance, the more influential factors are discussed below. 

Graded Adsorbents and Solvents. Materials used in columns for adsorption chromatography are grouped in Table 12 in an approximate order of effectiveness. Other adsorbents sometimes used include barium carbonate, calcium sulfate, calcium phosphate, charcoal (usually mixed with Kieselguhr or other form of diatomaceous earth, for example, the filter aid Celite) and cellulose. The alumina can be prepared in several grades of activity (see below). 

Wilkinson s method for the estimation of the reaction order is illustrated for first-order (left) and second-order (right) kinetic data. The first-order reaction is the decomposition of diacetone alcohol (Table 2-3 and Fig. 2-4) data to about 50 percent reaction are displayed. The slope gives an approximate order of 1.2. The second-order data (Fig. 2-2) give a precise fit to Eq. (2-59) and an order of two exactly. 

Major Suspension Reactor Problems. The critical problems in defining or optimizing a suspension reactor can be divided into three areas. In approximate order of importance, these are ... 

In comparing nucleophiles whose attacking atom is in the same row of the periodic table, nucleophilicity is approximately in order of basicity, though basicity is thermodynamically controlled and nucleophilicity is kinetically controlled. So an approximate order of nucleophilicity is NH2 >RO > OH > R2NH > ArO > NH3 > pyridine > F > H2O > CIO4, and another is R3C > R2N > RO > F (see Table 8.1). This type of correlation works best when the structures of the nucleophiles being compared are similar, as with a set of substituted phenoxides. Within such a series, linear relationships can often be established between nucleophilic rates and pAT values. 

Table 10.10 lists some leaving groups in approximate order of ability to leave. The order of leaving-group ability is about the same for SnI and Sn2 reactions. 

TABLE 10.10 Leaving Groups Listed in Approximate Order of Decreasing Ability to Leave ... 

Some radicals (e.g., triphenylmethyl) are so unreactive that they abstract hydrogens very poorly if at all. Table 14.3 lists some common free radicals in approximate order of reactivity. ... 

The groups are listed in approximate order of ease of reduction. See Ref. 498... 

All these immobilization techniques run the risk of altering activity compared with the native enzyme. Improved activity is occasionally reported, but this is the exception. The immobilization techniques listed above are in approximate order of loss in activity. Physical entrapment normally causes no change. Adsorption will distort the shape of the molecule compared with the native... 

We note in passing that a key role of the parameters cr, and ctj just as the parameter cr in the previous example is connected not only with the approximation order, but also with stability of the appropriate difference scheme. This important property will appear in subsequent discussions in Chapter 5, Section 1. 

The error of approximation on a grid. So far we have considered the local difference approximation meaning the approximation at a point. Just in this sense we spoke about the order of approximation in the preceding section. Usually some estimates of the difference approximation order on the whole grid are needed in various constructions. 

We now should raise the question how does the accuracy order of a scheme depend on the approximation order on a solution Because the error — solves problem (38) with the right-hand side V ft (and... 

A rigorous definition of stability of a difference scheme will be formulated in the next section. The improvement of the approximation order for a difference scheme on a solution of a differential equation will be of great importance since the scientists wish the order to be as high as possible. 

It is evident that the approximation order remains unchanged and A is identical with operator (15) up to the terms 0( h j ). 

Obviously, scheme (2) is of first-approximation order with respect to r and h, since its residual behaves as follows ... 

The solution can uniquely be found from the above conditions. This scheme has the approximation order 0 a — 0.5)r + -[ h" ). 

Power law expressions are useful as long as the approximate orders of reactant concentration are constant over a particular concentration course. A change in the order of the reaction corresponds to a change in the surface concentration of a particular reactant. A low reaction order usually implies a high surface concentration, a low reaction order, and a low surface reaction of the corresponding adsorbed intermediates. In order to deduce (Eq. (1.17b)) the rate of surface carbon hydrogenation, the power law of Eq. (1.18) has been used. 

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