Equivalent linear

In reahty the chemistry of breakpoint chlorination is much more complex and has been modeled by computer (21). Conversion of NH/ to monochloramine is rapid and causes an essentially linear increase in CAC with chlorine dosage. Further addition of chlorine results in formation of unstable dichloramine which decomposes to N2 thereby causing a reduction in CAC (22). At breakpoint, the process is essentially complete, and further addition of chlorine causes an equivalent linear increase in free available chlorine. Small concentrations of combined chlorine remaining beyond breakpoint are due primarily to organic chloramines. Breakpoint occurs slightly above the theoretical C1 N ratio (1.75 vs 1.5) because of competitive oxidation of NH/ to nitrate ion. Organic matter consumes chlorine and its oxidation also increases the breakpoint chlorine demand. Cyanuric acid does not interfere with breakpoint chlorination (23). 

Column type Exclusion limit (polystyrene equivalent) Linear operating range (polystyrene equivalent)... 

The presence of a large number of chain-ends in the fully synthesized dendrimer molecules makes them highly soluble and also readily miscible, for example with other dendrimer solutions. The solubility is controlled by the nature of the end-groups, so that dendrimers with hydrophilic groups, such as hydroxyl or carboxylic acid, at the ends of the branches are soluble in polar solvents, whereas dendrimers with hydrophobic end-groups are soluble in non-polar solvents. The density of the end-groups at the surface of the dendrimer molecule means that they have proportionately more influence on the solubility than in linear polymers. Hence a dendritic polyester has been shown to be more soluble in tetrahydrofuran than an equivalent linear polyester. 

The first column of Table 14.3 gives the response notation (or, equivalently, the factor combination). The next eight columns list the eight factor effects of the model the three main effects (A, B, and C), the three two-factor interactions (AB, AC, and BC), the single three-factor interaction (ABC), and the single offset term (MEAN, analogous to PJ in the equivalent linear model). 

Nickel carbonyl is by far the best-known of these compounds. All the data support a tetrahedral structure with four equivalent, linear Ni—C—0 arrangements. The Raman (3, 62, 61) and infrared spectra (8, 53, 135, 136) have been investigated and various assignments of the fundamental frequencies have been put forward on the basis of Td symmetry (53, 136, 208). Several approximate coordinate analyses have been made using these assignments, but the spectra of isotopically substituted species have not been measured, so exact evaluations of force constants have not yet proved possible. The Ni—C and C—0 stretching force constants are particularly important and the values obtained by workers using various approximations are included in Table I. 

An equivalent linear structure of CoP-CO was found to be a second-order saddle point (N-x = 2) when the 6-31G(d) basis set... 

How many times does a given treatment appear with each of the other treatments Graph the design with blocks on the vertical axis and treatments on the horizontal axis. If the usual model for this design is ytj = ju, + y, + iy + (y, write an equivalent linear model using yS s, and write the corresponding X matrix. 

Appendix 2 on bifurcation describes a linearization process for nonlinear systems of DEs at a steady state. Linearization forms a locally equivalent linear DE system y (t) = Ay(t) at a steady state. The eigenvalues of the linearized system matrix A determine the dynamic stability or instability of the particular steady state from their lay with respect to the imaginary axis in C. 

The advantage of an MCR synthesis over an equivalent linear synthesis is clear, in that the size of a linear derived library is a function of the number of steps and individual inputs. For example, the library from a three-step synthesis with 10 inputs at each step would have a size of 103 compounds, whereas that from a three-... 

Then, the nonconvex Eq. (9) can be replaced by the following set of equivalent linear inequalities [4,5]... 

Because thermodynamic and kinetic criteria are not usually equivalent, linear free energy relationships correlating reaction rates with thermodynamic variables should be used cautiously.23... 

The difference between the mean-square dimension of a polymer with long chain branching and an equivalent linear polymer with the same number of segments (Figure 12-46) can be expressed in terms of a parameter, g, equal to the ratio of the respective radii of gyration31 (Equation 12-64) ... 

Figure 8-21. Equivalent linear and multistep gradient profiles.

Extrapolation of the linear experimental laboratory results is less than accurate unless the effects of areal displacement and gravity are taken into account. A recent study by Chilton (1987) embodies computer support for predicting two-dimensional and three-dimensional production curves for oil recovery with mobility-controlled carbon dioxide. Figure 1 shows the three-dimensional results and indicates a 26% increase in oil production when the mobility of the gas is decreased tenfold by appropriate means. In contrast, the equivalent linear model predicted only 5.4% increase in oil production with the same decrease in mobility. 

Figure 8.5. Catalytic cycle with intermediate X2 as macs (shown encircled), cut to give equivalent linear pathway starting and ending with macs X2.

Rules 7.17 and 7.18 regarding necessary conditions for negative reaction orders must also be reformulated. They are best expressed with reference to the equivalent linear pathway ... 

For a reaction order to be negative, the respective participant (reactant, product, or silent partner) must be a product in a reversible step. In the equivalent linear pathway, this step may not be the last nor be preceded by an irreversible one. 

For a reactant or silent partner, the step sequence of the equivalent linear pathway must be such that the step in which the species in question is a product must precede that or those in which it is a reactant. 

The later a reactant enters the equivalent linear pathway, the lower is its reaction order. 

If significant fractions of the catalyst material may be bound in the form of reaction intermediates, the rules for reaction orders in noncatalytic simple pathways no longer apply. However, if one of the cycle members—the free catalyst or an intermediate—is a macs (most abundant catalyst-containing species, containing practically all of the catalyst material), the rules for noncatalytic pathways can be adapted The rate equation and reaction orders for the cycle are the same as for an imaginary equivalent linear pathway that starts and ends with macs. A cycle member that contains only an insignificant fraction of catalyst material is a lacs (low-abundance catalyst containing species), and the denominator terms it contributes can be dropped. 

As an example of the first case, an annular flow channel was calibrated using the mass-transfer limited oxidation of H2 on platinized Alloy 600 in 0.1 m NaOH as a function of the rotational velocity of a flow-activating impeller [24]. By assuming, a priori, the form of the mass-transfer correlation (corresponding to that for ambient temperature), it was possible to calculate the equivalent linear flow velocity as shown in Fig. 5. These data were then used to establish correlations between the Sherwood and Stanton numbers and the Reynolds number corresponding to the temperature of interest (Fig. 6). We note that at the highest temperature (250 °C), and for the highest... 

Fig. 5. Dependence of equivalent linear flow velocity in the annular channel on temperature and flowactivating impeller rotational velocity [24], Reproduced from Corrosion J. 44, 186 (1988) by permission of the Editor.

Besides using a lot of mobile phase, analyses run at these high flow rates are not compatible with an MS detector without some sort of split to divert flow. For many LC analyses, flow splitters are a fact of life when using MS detection. However, Fig. 2B illustrates that separation of the chromatographic test sample can be further scaled to a 2.1 -mm i.d. column, now running at an equivalent linear velocity (1.0 mL/min). Under the conditions used in... 

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