Double-extrapolation method

Kinetic Light-Scattering Method. Isochronous Interpolation. When high-activity samples of endocellulase are used, the reaction proceeds so quickly that, since the measurements of scattered intensity at different angles are not performed at the same extent of reaction, the extrapolations to zero angle and the subsequent calculations are erroneous. For this reason Kratochvil et al. (27) have proposed an isochronous interpolation method, whereby the Kc/Re values are plotted against sin21 (0/2) + kft. As in the double-extrapolation method of Zimm, the value of k may be chosen arbitrarily in order to space the experimental data. 

In addition, we propose plausible reactions for the ion-goethite interactions. The intrinsic equilibrium constants describing these associations have been calculated by the double extrapolation method (19, 30). 

Figure 5. determination hij the double extrapolation method. Note that... 

Using the data presented here we have no basis for distinguishing between these alternative reactions. The possible reactions for the interactions of the ions with the goethite surface as well as the corresponding estimate of the intrinsic equilibrium constants determined by the double extrapolation method are summarized in Table I. The intrinsic acidity constants,... 

PKJ and pK (Equation 1 and 2), for aFeOOH were also determined by the double extrapolation method. A discussion of those results can be found in Davis and Leckie (19). 

PM A showed special difficulties. Several samples did not fully dissolve, and probably contained cross-linked fractions. It is remarkable that these samples originated from experiments in which the polymerization was stopped at low conversions. We did not investigate the change of molecular weight, or the fraction if cross-linked polymer, in the course of a polymerization therefore we cannot explain this effect. We observed further that Vsp/c plotted vs. concentration produced no straight lines, but more or less curved functions. This rendered the extrapolation for zero concentration more difficult. We therefore applied a double extrapolation method following Heller (21). This method is based upon the fact that the mathematical expressions... 

In the triple layer model, values for the intrinsic protonation and dissociation constants, as well as values for tlie intrinsic surface complexation constants for the background electrolyte, can be obtained from hnear, double, or electrokmetic extrapolations to zero surface charge and zero and infinite electrolyte concentration. Values of intrinsic protonation-dissociation constants and intrinsic surface complexation constants for background electrolytes obtamed for the triple layer model using the various extrapolations are compiled in Goldberg (1992). Use of graphical extrapolation methods has been criticized because the triple layer parameter values obtained are not unique (Koopal et al., 1987). 

One of the most common methods used to determine the shape-independent parameter uses the double extrapolation method proposed by Zimm (1948a). 

The choice of the available difference methods is determined by the fact that the force calculation (the evaluation of f) is the most time consuming part of the computation, involving a double summation over particles. This rules out any method of solution (such as Rimge-Kutta and extrapolation methods) that require several fimction evaluations per step. General methods requiring only one force evaluation per step are multi-value predictor-corrector methods. 

In order to estimate (5 ). and from experimental data of Rq versus c for solutions at various scattering angles 0 at constant temperature, it is convenient to use a graphical method, proposed by Zimm [28]. In this method, Kc/Rq is plotted against (sin (0/2) + kc), where k is an arbitrary constant chosen to produce a reasonable spread of data points. In Zimm s method, (S ) and A2 l are estimated from the relations of lime Kc/Rq versus sin (0/2) and lime >o c// e versus c, respectively. is obtained as the inverse of the value of the intercept of a line of lime Kc/Rq at 0 = 0° or that of a line of lime >o atc = 0. Note that the intercepts of the two lines coincide with each other. Zimm s procedure is a double-extrapolation method. 

Figure 7.11 Calculation of affinity ratios pK (a) and pQc (b) by the double extrapolation method for SiOj-NaCl. Reproduced by permission of Elsevier Science Ltd from [29]...

A graphical method, proposed by Zimm (thus tenned the Zinnn plot), can be used to perfomi this double extrapolation to detemiine the molecular weight, the radius of gyration and the second virial coefficient. An example of a Zinnn plot is shown in figure Bl.9.6 where the light scattering data from a solution of poly... 

The value obtained by Hamm et alm directly by the immersion method is strikingly different and much more positive than others reported. It is in the right direction with respect to a polycrystalline surface, even though it is an extrapolated value that does not correspond to an existing surface state. In other words, the pzc corresponds to the state of a bare surface in the double-layer region, whereas in reality at that potential the actual surface is oxidized. Thus, such a pzc realizes to some extent the concept of ideal reference state, as in the case of ions in infinitely dilute solution. 

Gouy-Chapman, Stern, and triple layer). Methods which have been used for determining thermodynamic constants from experimental data for surface hydrolysis reactions are examined critically. One method of linear extrapolation of the logarithm of the activity quotient to zero surface charge is shown to bias the values which are obtained for the intrinsic acidity constants of the diprotic surface groups. The advantages of a simple model based on monoprotic surface groups and a Stern model of the electric double layer are discussed. The model is physically plausible, and mathematically consistent with adsorption and surface potential data. 

An extension of this method has been developed for the triple layer model which allows data obtained at several values of ionic strength to be considered simultaneously (7, 13.). However, this "double extrapolation technique" involves the same sort of approximation. 

Birge-Sponer extrapolation spect A method of calculating the dissociation limit of a diatomic molecule when the convergence limit cannot be observed directly, based on the assumption that vibrational energy levels converge to a limit for a finite value of the vibrational quantum number. bir-go spon-or ik Strap-3 la-sh3n ( bis- chem a prefix indicating doubled or twice. bis ... 

A ratio used to assess the degree of cooperativity exhibited by an enzyme. It is equal to the true Ymax value (typically extrapolated from the high-substrate-concen-tration end of a double-reciprocal plot) divided by the apparent Emax value obtained from extrapolating the asymptote in the low-substrate-concentration portion of the double-reciprocal plot. For a noncooperative system, Ry will equal one positively cooperative systems will have values greater than one and negatively cooperative systems will have values less than one . This method requires good estimates of the asymptotes. 

The single and double scan methods allow estimation of the value of the characteristic parameters for both ionic and nonionic surfactants (see Table 1). The extrapolation of Eqs. 8 and 9 to SACN = 0 allows the classification of the head groups in some hydrophihcity scale. Inspection of Table 1 data indicates that the branching and isomeric structure does have quite an influence on the characteristic parameter, as it has been reported in the Hter-ature [12,27-32]. 

While these "energies" are necessarily approximate, they afford a basis for clear discrimination between sterically allowed and sterically forbidden structures. The "energy" approach also offers a means to extrapolate experimental studies (nmr, X-ray, etc.) on the conformation of small model compounds to the polynucleotide level and to test the relevance of the data in a helical complex. In addition, the method provides a starting point for a refined potential energy analysis of double helical conformation and stability. 

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