Interpolation and extrapolation

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

There are many large molecules whose mteractions we have little hope of detemiining in detail. In these cases we turn to models based on simple mathematical representations of the interaction potential with empirically detemiined parameters. Even for smaller molecules where a detailed interaction potential has been obtained by an ab initio calculation or by a numerical inversion of experimental data, it is usefid to fit the calculated points to a functional fomi which then serves as a computationally inexpensive interpolation and extrapolation tool for use in fiirtlier work such as molecular simulation studies or predictive scattering computations. There are a very large number of such models in use, and only a small sample is considered here. The most frequently used simple spherical models are described in section Al.5.5.1 and some of the more common elaborate models are discussed in section A 1.5.5.2. section Al.5.5.3 and section Al.5.5.4. 

By use of Othmer plots of reference substances, large tables of thermodynamic data can be expressed as simple correlations which are extremely accurate and easy to use. The real power of these correlations is the abiUty to interpolate and extrapolate the correlations beyond the experimental values with considerable accuracy. Mathematically stated... 

Interpolation and extrapolation performance to other operating conditions... 

Chemical kinetics deals with quantitative studies of the rates at which chemical processes occur, the factors on which these rates depend, and the molecular acts involved in reaction processes. A description of a reaction in terms of its constituent molecular acts is known as the mechanism of the reaction. Physical and organic chemists are primarily interested in chemical kinetics for the light that it sheds on molecular properties. From interpretations of macroscopic. kinetic data in terms of molecular mechanisms, they can gain insight into the nature of reacting systems, the processes by which chemical bonds are made and broken, and the structure of the resultant product. Although chemical engineers find the concept of a reaction mechanism useful in the correlation, interpolation, and extrapolation of rate data, they are more concerned with applications... 

The mathematical methods used for interpolation and extrapolation of the data obtained from accelerated tests, as described in Chapters 8 and 9, include both the mechanistic and the empirical. Arrhenius formula, based on chemical rate kinetics and relating the rate of degradation to temperature, is used very widely. Where there are sufficient data, statistical methods can be applied and probabilities and confidence limits calculated. For many applications a high level of precision is unnecessary. The practitioners of accelerated weathering are only too keen to tell you of its quirks and inaccuracies, but this obscures... 

Considering other families of similar compounds, the contributions given by Guillermet and Frisk (1992), Guillermet and Grimvall (1991) (cohesive and thermodynamic properties, atomic average volumes, etc. of nitrides, borides, etc. of transition metals) are other examples of systematic descriptions of selected groups of phases and of the use of special interpolation and extrapolation procedures to predict specific properties. 

There are thousands of references and several source books on liquid-liquid equilibrium (LLE) data for organic systems in the literature [1]. However, the available data may be incomplete or unreliable, which makes experimental work necessary. Different thermodynamic models can then be used for interpolation and extrapolation. 

It is therefore essential that thermodynamic modeling be applied to obtain a simultaneous quantitative fit to the phase diagram and thermodynamic data in order to evaluate the internal consistency of the various published data. Then a reliable framework can be established for smoothing, interpolating, and extrapolating experimental data that are costly and laborious to obtain. In Chapter 3 an associated solution model is presented. This model provides a good fit to the data for the Hg-Cd-Te system as well as for the Ga-In-Sb system, which is closer to the simpler picture of an ideal solution. 

In addition to the tabulation of new and old data on pure compounds, parameters are given for interpolating and extrapolating determined data based on the systematic way in which the properties of compounds vary within a given homologous series. 

Parameters for various empirical equations are tabulated, which permit accurate interpolation and extrapolation of the various properties within the ranges designated. 

Ep vs. 8. To draw such complete families of curves it was necessary to make rather large interpolations and extrapolations of the original data. However any errors introduced thereby will not alter the conclusions to be derived from the curves. 

There are a number of anomalies in the long form, whose exposition has caused the spilling of much ink. One of Mendeleev s greatest triumphs was his prediction of a new element ( eka-silicon ) between Si and Sn in his Table. Mendeleev had the audacity to predict some chemical properties of this new element, and his prophecies were substantially fulfilled a few years later by the isolation of germanium and a preliminary exploration of its chemistry. These predictions were made simply by interpolation between Si and Sn. Chemists, and chemistry students, have come to expect that the chemical properties within a Group follow monotonic trends properties can be predicted by interpolation and extrapolation. Experimental observations which do not fit such simple trends lead to the identification of anomalies . At one extreme, there may be a tendency to sweep such anomalies discreetly under the carpet, or even to question the validity of the data at the other extreme, strenuous efforts are made to account for anomalies by means of elaborate and sometimes fanciful theorising. 

Sound Splicing, Interpolation, and Extrapolation. Other applications of the baseline system are sound splicing, interpolation, and extrapolation. Sound splicing is sometimes used in music signal synthesis. Many instrumental sounds, for example,... 

What is the proper interpolation and extrapolation technique Experience has shown that differences in families are never quite constant, implying that linear interpolation and extrapolation is only a zeroth-order approximation. In some of the examples to follow, enough information is available to permit the use of a quadratic scheme, which may be the best approximation. For most systems, the differences between the linear and quadratic methods will be small. 

From studies of human populations exposed to certain chemicals, available data are sufficient to characterize the dose-incidence relationships for some types of cancer at high dose levels. However, as in the case of ionizing radiation, the data are not sufficient to define the dose-incidence relationships precisely for any form of cancer over a wide range of doses and dose rates. Therefore, the probability of cancer induction that may be associated with low doses of chemicals that would be of primary concern in protection of public health can be estimated only by interpolation and extrapolation of data at higher doses and dose rates, based on assumptions about the dose-incidence relationships and mechanisms of toxicity. For the few chemicals for which incidence data are available over a range of doses, the dose-incidence relationship is not inconsistent with linearity, but this result does not constitute proof of linearity. 

Alternatively, the interpolation and extrapolation steps can be combined, using the formula [75]... 

Estimating flood and pressure drop using the GPDC interpolation charts involves data interpolation and extrapolation within the framework of the generalized pressure drop correlation (GPDC) chart (Fig. 8.196 or c), This technique is expected to yield reliable estimates when appropriate data are available in the vicinity of the operating point. The reliability of the estimates will diminish with the extent to which extrapolation is required. Whenever extensive extrapolation is required, the estimates are unreliable, and the calculation is best abandoned. 

This calculation is illustrated in Fig. 5 using data on the decomposition of nitrogen pentoxide (page 64). The data of Table II are obtained by interpolation and extrapolation with this formula. 

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