Polynomial expression

Partial Fractions Rational functions are of the type f x)/g x) where /x) and g(x) are polynomial expressions of degrees m and n respectively. If the degree of/is higher than g, perform the algebraic division—the remainder will then be at least one degree less than the denominator. Consider the following types ... 

The transfer funetion is based on equation (3.75), with the input term Z (.v) = l/.v removed. The denominator is input using the conv eommand (short for eonvolu-tion), whieh multiplies two polynomial expressions together. 

Thermal conductivities for gaseous compounds are important in unit operations involving heat transfer coefficients. Thermal conductivities can be readily computed from an empirical polynomial expression that has the following form ... 

Figure 6.12. Example of rules Gl, G2 and G3 Effect of pCo (=Pd) and of Na coverage and corresponding UWr and AO values on the rate of CO oxidation on Pt films deposited on P"-A1203 at fixed Po2=6 kPa71 Note that dr/dO(= dr/edUWR) always traces dr/dpco for negative, positive and zero (volcano peak) values. In the right figure the raw data (left) have been fitted to a polynomial expression.71 Reprinted with permission from Academic Press.

The temperature dependency of heat capacity can usually be described by a polynomial expression, e.g.,... 

In practice the implicit loop calculation can sometimes be easily avoided. This can be done, for example, by formulating an empirical polynomial expression for the saturated steam temperature as a function of the saturated steam density... 

Connell, D.W., Hawker, D.W. (1988) Use of polynomial expressions to describe the bioconcentration of hydrophobic chemicals by fish. Ecotoxicol. Environ. Safety 16, 242-257. 

Derive the polynomial expressions for the concentration dependence of electrons, metal vacancies, and anion vacancies for a compound MX showing Schottky equilibrium (Section 7.9.4). Insert values of the equilibrium constants to obtain polynomial expressions for the cases where electronic defects dominate. 

Polynomial expressions are conveniently used to represent a condensed phase which is stable in the whole temperature range of interest and which does not undergo any structural, electronic or magnetic transformations. The Gibbs energy of a compound is in the CALPHAD approach represented relative to the elements in their defined standard state at 298.15 K as a power series in terms of temperature in the form of [16] ... 

Binary solutions have been extensively studied in the last century and a whole range of different analytical models for the molar Gibbs energy of mixing have evolved in the literature. Some of these expressions are based on statistical mechanics, as we will show in Chapter 9. However, in situations where the intention is to find mathematical expressions that are easy to handle, that reproduce experimental data and that are easily incorporated in computations, polynomial expressions obviously have an advantage. 

Simple polynomial expressions constitute the most common analytical model for partial or integral thermodynamic properties of solutions ... 

The regular solution model (eq. 3.68) is symmetrical about xA = xB =0.5. In cases where the deviation from ideality is not symmetrical, the regular solution model is unable to reproduce the properties of the solutions and it is then necessary to introduce models with more than one free parameter. The most convenient polynomial expression with two parameters is termed the sub-regular solution model. 

If more than two parameters are necessary a general polynomial expression may be applied ... 

Before there can be any extrapolation there must be confidence in the model or rules being used. In practice this often has to involve an element of faith because of lack of validation data, particularly where the rule is empirical. The theory or model should be no more complex than is necessary to fit the data. The accuracy of fit to, for example, a creep curve can often be made more precise by applying ever higher order polynomial expressions, but outside the range of points these functions diverge rapidly to infinity (or minus infinity) leading to predictions that are ridiculous. 

Example According to Eq. 3.10 the complete isotopic distribution of stearic acid trichloromethylester, C19H35O2CI3, is obtained from the polynomial expression... 

Testing the Accuracy of a Calibration Spline Function. Of primary concern in calibration is the freedom from systematic errors introduced by fitting the wrong model. For judging the accuracy of the cubic spline functions, it is therefore desirable to start with a curve of known shape. Particularly difficult to adapt by ordinary polynomial expressions are... 

As a matter of fact, the volumes of NaAlSi308-KAlSi308 mixtures are not ideal and have positive excess terms. Deviations from simple Vegard s rule proportionality are mainly due to edge h Kroll and Ribbe (1983) provide three polynomial expressions relating the volumes of mixtures to the molar fractions of the potassic component Xor) valid, respectively, for monoclinic symmetry (complete disorder), triclinic symmetry of perfect order (low albite and maximum micro-cline), and the intermediate structural state ... 

Equation (5.21) assumes ternary interactions are small in comparison to those which arise from the binary terms. This may not always be the case and where evidence for higher-order interactions is evident these can be taken into account by a further term of the type Gijit = x< xj Xk Lijk, where Lijk is an excess ternary interaction parameter. There is little evidence for the need for interaction terms of any higher order than this and prediction of the thermodynamic properties of substitutional solution phases in multi-component alloys is usually based on an assessment of binary and ternary terms. Various other polynomial expressions for the excess term have been considered, see for example the reviews by Ansara (1979) and Hillert (1980). All are, however, based on predicting the properties of... 

Many systems have been found to correlate with this function but for those which do not a polynomial expression can be used instead of k2-... 

Values of the acoustic properties of water at different temperatures are given in Table 3.2 (Kaye and Laby 1986). The velocity may be described by a polynomial expansion in temperature five terms are sufficient to achieve a fit that is considerably better than the accuracy of the best available measurements (Del Grosso and Mader 1972). Thus the polynomial expression for the acoustic velocity in water is... 

However, when it comes to reactions where bonds are being made and/or broken, it is clear that, at least for the vast majority of force fields that use polynomial expressions for the bond stretching energy, the normal model is inapplicable. Nevertheless, substantial application of molecular mechanics to such TS structures has been reported, with essentially three different approaches having been adopted. 

The thermochemistry of a large number of species is available in a convenient format, either as polynomial expressions or as tabulated values. Some important sources of thermodynamic information include the following ... 

The heat capacity increases with temperature, for example, for liquid water at 20 °C the specific heat capacity is 4.182 kj kg 1 K 1 and at 100 C is 4.216k kg 1K 1 [2]. Its variation is frequently described by the polynomial expression (virial equation) ... 

Cubic and higher order polynomial expressions also arise naturally in a wide range of problems in chemistry, particularly in solubility and equilibrium problems. If we try to dissolve lead (II) chloride (PbCl2) instead of silver chloride, the solubility product expression becomes... 

The calculation of sensible heat is based on the heat capacity (at constant pressure) of the substance, CP(T), which is in units of heat (energy) per unit mass. Heat capacity information is typically in the form of coefficients for a polynomial expression... 

Figure 4-36 Volume compression of TbVQ and DyV04 calculated as a function of pressure. Solid lines are fits to a simple polynomial expression. (Reproduced with permission from Ref. 53.)...

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