Interpolation or Extrapolation

After identification of the elements that contain feet of particle trajectories the old time step values of F at the feet are found by interpolating (or extrapolating for boundary nodes) its old time step nodal values. In the example shown in Figure 3.6 the old time value of Fat the foot of the trajectory passing through A is found by interpolating its old nodal values within element (e). 

When structure-property relationships are mentioned in the current literature, it usually implies a quantitative mathematical relationship. Such relationships are most often derived by using curve-fitting software to find the linear combination of molecular properties that best predicts the property for a set of known compounds. This prediction equation can be used for either the interpolation or extrapolation of test set results. Interpolation is usually more accurate than extrapolation. 

Normal Boiling Temperature The normal boiling temperature (point) is the temperature at which the vapor pressure equals exac tly 101,325 Pa (1 atmosphere). Caution shomd be taken in using values from older references, where the temperature may be reported for the prevaihng pressure (0.95-0.97 atm) rather than at 1 atmosphere. If at least two values of vapor pressure very close to 1 atmosphere are available, the normal boihng point can be interpolated or extrapolated on a plot of logP vs. l/T. Tme section on vapor pressure discusses this in more detail. 

NOTE pj p f Pc) of 1/2, -A, and 14 may be used to cover the ranges 0.2—0.4, 0.4—0.6, 0.6-0.7, and 0.7—0.8, respectively, with a maximum error in .q of 5 percent at pL = 6.5 m atm, less at lower pLs. Linear interpolation reduces the error generally to less than 1 percent. Linear interpolation or extrapolation on T introduces an error generally below 2 percent, less than the accuracy of the original data. 

Depth of localized corrosion should be reported for the actual test period and not interpolated or extrapolated to an annual rate. The rate of initiation or propagation of pits is seldom uniform. The size, shape, and distribution or pits should oe noted. A distinction should be made between those occurring underneath the supporting devices (concentration cells) and those on the surfaces that were freely exposed to the test solution. An excellent discussion of pitting corrosion has been pubhshed [Corro.sion, 25t (January 1950)]. 

We can make two different uses of the activation parameters AH and A5 (or, equivalently, E and A). One of these uses is a very practical one, namely, the use of the Arrhenius equation as a guide for interpolation or extrapolation of rate constants. For this purpose, rate data are sometimes stored in the form of the Arrhenius equation. For example, the data of Table 6-1 may be represented (see Table 6-2) as... 

Mathematical Models. As noted previously, a mathematical model must be fitted to the predicted results shown In each factorial table generated by each scientist. Ideally, each scientist selects and fits an appropriate model based upon theoretical constraints and physical principles. In some cases, however, appropriate models are unknown to the scientists. This Is likely to occur for experiments Involving multifactor, multidisciplinary systems. When this occurs, various standard models have been used to describe the predicted results shown In the factorial tables. For example, for effects associated with lognormal distributions a multiplicative model has been found useful. As a default model, the team statistician can fit a polynomial model using standard least square techniques. Although of limited use for Interpolation or extrapolation, a polynomial model can serve to Identify certain problems Involving the relationships among the factors as Implied by the values shown In the factorial tables. 

Interpolation or extrapolation from equation for correlating temperature relationships, e.g., the Clausius-... 

Supporting points of the storage profiles Interpolated or extrapolated amounts of products... 

This rule holds reasonably well when C or t varies within a narrow range for acute exposure to a gaseous compound (Rinehart and Hatch, 1964) and for chronic exposure to an inert particle (Henderson et al., 1991). Excursion of C or / beyond these limits will cause the assumption Ct = K to be incorrect (Adams et al., 1950, 1952 Sidorenko and Pinigin, 1976 Andersen et al., 1979 Uemitsu et al., 1985). For example, an animal may be exposed to 1000 ppm of diethyl ether for 420 min or 1400 ppm for 300 min without incurring any anesthesia. However, exposure to 420,000 ppm for lmin will surely cause anesthesia or even death of the animal. Furthermore, toxicokinetic study of fiver enzymes affected by inhalation of carbon tetrachloride (Uemitsu et al., 1985), which has a saturable metabolism in rats, showed that Ct = K does not correctly reflect the toxicity value of this compound. Therefore, the limitations of Haber s rule must be recognized when it is used in interpolation or extrapolation of inhalation toxicity data. 

The calculation of compressibility factors of gaseous ethanol can be made with equation 2.18, because the second virial coefficient (B) is available at different temperatures [20] and the saturation vapor pressures can be interpolated or extrapolated from the experimental data (figure 2.4). One obtains Z = 0.991 at... 

The use of any model more complex than Eq. (1) has frequently been criticized as an attempt to read too much into a set of kinetic data. The implicit results of such a misapplication would be a proliferation of meaningless constants this would thus lead to unnecessary effort in data fitting and interpretation, or to misleading interpolation or extrapolation. 

Figure 8.5 shows that can be estimated most precisely when the third experiment is located at x,3 = 0. This is reasonable, for the contribution of the third experiment at x, = 0 to the variance associated with involves no interpolation or extrapolation of a model if the third experiment is carried out at X = 0, then any discrepancy between y,3 and the true intercept must be due to purely experimental uncertainty only. As the third experiment is moved away from x, = 0, does increase, but not drastically the two stationary experiments remain positioned near X, = 0 and provide reasonably good estimates of hg by themselves. 

Data are available at other states requiring different thermodynamic correlations for interpolation or extrapolation. 

From Abdel-Alim and Hamielec (Al), Brabston and Keller (BIO), Hamielec et al (H6), LeClair (L5), LeClair and Hamielec (L7), and LeClair et al. (L9). Interpolated or extrapolated. 

The Eyring approach has the advantage that the pseudothermodynamic activation parameters can be readily related to the true thermodynamic quantities that govern the equilibrium of the reaction. The Arrhenius equation, on the other hand, is easier to use for simple interpolations or extrapolations of rate data. 

Rule 14. Potential parameters may be estimated by linear interpolation or extrapolation between homologous members of a chemical series of compounds. 

The results of this trial-and-error calculation do not have a characteristic curve such as the one given previously in Figure 12-5. However, the relationship between Sxjf and pf is essentially linear so that linear interpolation or extrapolation can be used with good effect to obtain succeeding trial values of pf. 

Marcus5 8 taught us that the most appropriate and useful kinetic measure of chemical reactivity is the intrinsic barrier (AG ) rather than the actual barrier (AG ), or the intrinsic rate constant (kQ) rather than the actual rate constant (k) of a reaction. These terms refer to the barrier (rate constant) in the absence of a thermodynamic driving force (AG° = 0) and can either be determined by interpolation or extrapolation of kinetic data or by applying the Marcus equation.5 8 For example, for solution phase proton transfers from a carbon acid activated by a ji-acceptor (Y) to a buffer base, Equation (1), k0 may be determined from Br A ns ted-type plots of logki or... 

From this curve it is possible to interpolate or extrapolate for additional values that have not been measured. Interpolation is using the curve to read values that lie between data points. 

Thus, this book is concerned with predictions. These are usually based on correlations of known information, with interpolation or extrapolation, as required. Reid and Sherwood (1958) distinguished correlations of three different types ... 

After two iterations have been completed, interpolation or extrapolation is used to bring AF.bl = 0 based on the last two values of AE to speed the calculation. The results of the iteration are ... 

Reactor purchased costs are as follows with straight-line interpolation or extrapolation to 30,000 gal acceptable ... 

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