Graphical interpolation

For DCF calculations being done by hand, the final stage will most easily be a graphical interpolation - the alternative being a long-winded iteration process to find the actual value (which is not normally sought to better than one place of decimals). 

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Since they all necessitate a knowledge of the value of r, and of both r and either directly or indirectly, all as a function of p p°, these data are given in tabular form for reference (Table 3.2). If required, intermediate values of t may be obtained to sufficient accuracy by graphical interpolation, and the corresponding values of r can be calculated with the Kelvin formula. The values of r refer to the most commonly used model, the cylindrical pore, so that r " = r + t. The values of t are derived from the standard nitrogen isotherm for hydroxylated silica and though the values do differ... 

The 0 temperature is that value of T for which B = 0 therefore 0 can be determined from the results above by graphical interpolation. Although there is some scatter in such a graph, the best value for the temperature at which B = 0 appears to be 308.4 K, which agrees well with values determined for this system by other methods. 

Thus, log-log plots of S versus C provide an easy way to obtain the values for K (the intercept) and N (the slope of the line). The log-log plot can be used for graphic interpolation of adsorption at other concentrations, or, when values for K and N have been obtained, the amount of adsorption can be calculated from Equation 20.9. Figure 20.9 shows an example of adsorption isotherms for phenol adsorbed on Frio sandstone at two different temperatures. Note that when N = 1, Equation 20.9 simplifies to Equation 20.6 (i.e., adsorption is linear). 

One way to improve this approach it to carry out simultaneous perturbations for several, different values of AT or, equivalently, A3, and then estimate the appropriate derivative through averaging or graphical interpolation. This can be done with only small additional computational effort. 

Fig. 5.6. Calibration curve for permanganate standards. Line is a least-squares linear regression for the data. Graphical interpolation is illustrated for an unknown with an absorbance of 0.443.

The next column shows the few values of the median electron-cloud radii which can be obtained from graphical interpolation of published electron-contours based on X-rays. The agreement is fairly good, and the free-ion value for Li+ is roughly acceptable. 

Each value was the mean of individual determinations from 4-5 mice. The 50% inhibitory dose (ID50) values were determined by the method of probit-graphic interpolation for four dose levels. This assay is... 

As a test, this approximation procedure for skew boundaries was used with a nondissociating system as a model, and while the plot for the model showed a scatter of up to 10 fx/cm owing to the graphical interpolation procedures being used, no drift indicative of apparent nonideality caused by any possible deficiency in theory was observed the Qj values scattered about zero over the range of f (zj) for which they were calculated. 

Diagrams have been constructed for temperature values of 327, 350, 400, 427 and 527°C, which bracket operating conditions, and make maximum use of the tabular data of Robie, et al. at 500, 600, and 700 K. Input values at 350 and 400°C were chosen by graphical interpolation. Diagrams for 450, 475, and 500°C have not yet been constructed, because the consequences of the sharp breaks in the A G values with the phase change of sulfur at 444 C have not been... 

Repeat the trial-and-error procedure with the aid of graphic interpolation until the dew-point temperature is found. As the next estimate of the dew point, try 110°C. At this temperature, Kt = 1.756 and K2 = 0.428, as found via the procedure outlined in the previous step. Then y /Kf) + (> 2/K2) = 0.923. Since the sum is less than 1, the assumed temperature is too high. 

Also a comparison of two alternative expressions for log y given by Equations 3 and 4 is of interest. A close examination of these equations will reveal certain inherent weaknesses of the linear extrapolation method for the evaluation of E°, even when using the extended terms. First, the linear extrapolation will require a precise value of D, the dielectric constant of the solvent, which in principle is experimentally measurable. Frequently, however, it is computed by using some empirical function or by graphical interpolation. Secondly, the uncertainty of the ion-size parameter is significant and no reliable method of computation exists, nor is it directly measurable experimentally. 

A number of interpolation and extrapolation methods are commonly used, including two-point linear interpolation, graphical interpolation, and curve fitting. Which one is most appropriate depends on the nature of the relationship between x and y. 

The constants A and C can be evaluated in the usual way by solving the simultaneous equations with observed values of K and T, or by graphic interpolation. A. Smith and R. H. Lombard thus obtain. 4=—12800, and 0=0 00967, and. accordingly, the heat o dissociation at T° is —12800—0 00967cals, per mol. Hence, the heat absorbed during dissociation increases as the temp, increases, and... 

Tdcm " 44 K is the temperature at which the vapor pressure of the gaseous decomposition products equals one atmosphere, was obtained by graphical interpolation of the decomposition pressure data on FeSO (cr), reported by Greulich (4 )-... 

Although [fNT] can be taken to be proportional to the xylose concentration, there is no known experimental way to determine ka and kb explicitly. What is possible is to measure the actual yield as a function of time, xylose concentration, acidity, and temperature, for the experimental setup chosen, and to use these yield curves, together with the known pentose disappearance rate and the known furfural reslnification rate, as a graphical interpolation basis for determining the losses by the condensation reactions. Such a procedure, reported by Root, Saeman, Harris, and Neill [18], is given in an appendix chapter, but it is usually considered too complicated and too unreliable to be used for yield prognoses. 

At test termination, the concentration that kills 50% of the test organisms (LC50 value) is determined using probit analysis or graphical interpolation. Unlike in chronic toxicity tests, there is no test solution renewal, the organisms are unfed, and there is no analytical verification of the test concentrations. Furthermore, cumulative, chronic, and sublethal effects of a chemical usually are not evaluated in acute toxicity tests, although frequently behavioral changes and lesions caused by a chemical can be determined. 

The results of acute toxicity tests are reported as the LC50 and EC50 (concentration that reduces growth 50%) values and their 95% confidence intervals. Probit analysis is the most commonly used statistical method to determine LC50 values. Graphical interpolation can be used to estimate the LC50 value where the proportion of deaths versus the test concentration is plotted for each observation time. 

Graphical interpolation essentially is the plotting of the dose-response curve and reading the concentration that corresponds to the LC50 or the LC10. 

Perhaps the most widely applicable method, other than the graphical interpolation, is the moving average. The method can be used only to calculate the LC50, and there is the assumption that the dose-response curve has been correctly linearized. As with the other methods, a partial kill is required to establish a confidence interval. 

Different values reported in Table 3 are results of graphical interpolation. 

By the use of the above four methods, and also by combination of my calibrations with the numerous investigations of Kamerlingh-Onnes, I obtained the following table, the values in which were deduced, some by calculation and some by graphical interpolation they are, of course, correct only for one particular sample of platinum. 

Mnl =72000 in toluene at 37°C oMn2 = 155000 in benzene at 30°C (curves obtained by graphical interpolation).7 The horizontal lines start from an arbitrary point on one curve. The corresponding abscissae, corresponding to the intersection of one line with the two curves, have a ratio which is independent of the height of the line under consideration. 

Mnl and Mn2 respectively. By graphical interpolation, we obtained the curves which appear in Fig. 5.10. 

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