Extrapolation of Data with Temperature

Extrapolation of Data with Temperature Liquid-phase excess-property data for binary systems at near-ambient temperatures appear in the literature. They provide for the extrapolation of G correlations with temperature. The key relations are Eq. (4-250), written as 

For application of Eq. (4-351) to binary systems at infinite dilution of one of the constituent species, it is divided by the product xpC2- 

The assumption here is that Cf is independent of T, making I = 0. As shown by Smith, Van Ness and Abbott [Introduction to Chemical Engineering Thermodynamics, 7th ed., p. 437, McGraw-Hill, New York (2005)1, 

Example 5 VLE at Several Temperatures For the methanol(l)/ acetone(2) system at a base temperature of Tq = 323.15 K (50 C), both VLE data 

These values allow calculation of equilibrium pressures through Eqs. (4-270) and (4-308) for comparison with the measured pressures of the data set. Values of required in Eq. (4-308) are the measured values reported with the data set. The root-mean-square (rms) value of the pressure differences is given in Table 4-7 as 0.08 kPa, thus confirming the suitability of the Margules equation for this system. Vapor-phase mole fractions were not reported hence no value can be given for rms 5 i- 

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The approximations involved in the determination of crystallinity by gas chromatography were evaluated in reference [189]. The non-ideality of the gas phase and the modification of the enthalpy of vaporization of the injected substance with temperature cause the retention diagrams at temperatures below T, to be not linear, but slightly convex against abscissa. Thus the FJ values obtained from linear extrapolation of data from temperatures higher than T, will give values lower than the true value for polymer crystallinity, especially at low temperatures. 

Although Equation (4) is conceptually correct, the application to experimental data should be undertaken cautiously, especially when an arbitrary baseline is drawn to extract the area under the DSC melting peak. The problems and inaccuracy of the calculated crystallinities associated with arbitrary baselines have been pointed out by Gray [36] and more recently by Mathot et al. [37,64—67]. The most accurate value requires one to obtain experimentally the variation of the heat capacity during melting (Cp(T)) [37]. However, heat flow (d(/) values can yield accurate crystallinities if the primary heat flow data are devoid of instrumental curvature. In addition, the temperature dependence of the heat of fusion of the pure crystalline phase (AHc) and pure amorphous phase (AHa) are required. For many polymers these data can be found via their heat capacity functions (ATHAS data bank [68]). The melt is then linearly extrapolated and its temperature dependence identified with that of AHa. The general expression of the variation of Cp with temperature is... 

In systems where the liquid phase interaction between the solute and solvent is close to ideal, then Eq. 2 can be used successfully on it s own to fit and extrapolate solubility data with respect to temperature. The technique is valuable in an industrial setting, where time pressures are always present. Solubility data points are often available without any additional effort, from initial work on the process chemistiy. The relative volume of solvent that is required to dissolve a solute at the highest process temperature in the ciystallization is often known, together with the low temperature solubility by analysis of the filtrates. If these data points fit reasonably well to the ideal solubility equation then it can be used to extrapolate the data and predict the available crystallization yield and productivity. This quickly identifies if the process will be acceptable for long term manufacture, and if further solvent selection is necessary. 

For quite a number of physically absorbed gases, Henry s law holds very well when the partial pressure of the solute is less than about 101 kPa (1 atm). For partial pressures above 101 kPa, H may be independent of the partial pressure (Fig. 14-1), but this needs to be verified for the particular system of interest. The variation of H with temperature is a strongly nonlinear function of temperature as discussed by Poling, Prausnitz, and O Connell (The Properties of Gases and Liquids, 5th ed., McGraw-Hill, New York, 2000). Consultation of this reference is recommended when temperature and pressure extrapolations of Henry s law data are needed. 

Cables specimens aged in air at temperatures between 120 and 200°C showed also a temperature shift factor (aj) that followed the Arrhenius equation with activation energy of ca. 100 kJ mof. Calculation of the remaining lifetime of cables that had been exposed to air for 25 years yielded lifetimes at ca. 40°C that were approximately half of the lifetime obtained by extrapolation of the high temperature (120-200°C) data (73). This assessment was based on 6 service cables. Gillen and coworkers (15-20) have reported similar deviations for a... 

Taylor and Smith (10] reported S (298.15 K) = 32.29 cal K mol" based on S (60 K) 6.68 cal K" mol" obtained from Debye-Einstein functions which represented their C data to only 1.8 percent from 60 to 100 K. A comparison of their extrapolated C data with those which have been measured for SrCl, BaCl, and Cal, (4] indicates that the values for SrBr decrease much more rapidly with temperature below 50 K than would be expected. We have made our own extrapolation to 0 K for... 

We adopt the more extensive data of Osborne et al. ( ), extrapolating C linearly with temperature outside the observed... 

These data contain a broad lambda type transition with a heat capacity peak at 227.5 K. Powers and Blalock (9) measured high temperature enthalpy data for KOH(cr) in both the a and B phases in a Bunsen ice calorimeter. Their enthalpy data are scattered and not precise enough to accurately define the heat capacities for the a phase. Therefore, the selected heat capacities between 298 and 516 K are estimated by graphical extrapolation of the low temperature heat capacity data. Heat capacities for the B phase are from Powers and Blalock (9). 

The value is not in full harmony with the value obtained from an extrapolation of high temperature data. Since the validity of the polarographic method has not been documented, the value of AfG° (PbSe, cr, 298.15 K) obtained from the extrapolation of the high temperature data has been selected. 

In the aqueous phase, Umland and Wallmeier [80UML/WAL] studied the reduction of selenite at the mercury electrode in the presence of Zn by polarography, see Appendix A. The solubility product of ZnSe(s) was obtained from the position of the half-wave potential of the second reduction step, HgSe(s) + Zn "" + 2e" ZnSe(s) + Hg(l), to be log (ZnSe, s, 298.15 K) = -(23.212.0). Combined with CODATA for Zn and the selected value for Se l it corresponds to AfG° (ZnSe, s, 298.15 K) = - (151.0 11.8) kJ mol. The value is in disagreement with the value obtained by extrapolation of high temperature data. Since the validity of the polarographic method has not been documented, the value of AfG°(ZnSe, a, 298.15 K) obtained from the extrapolation of the high temperature data has been selected. 

The possibility of inherent difficulties in the accurate MD simulation of slowly diffusing systems is also to be borne in mind following the finding of discrepancies between experiment and simulation observed in the nonliquid but still very relevant case of the ciystalline superionic conductor CaFj. In this system, Rahman has reported MD diffusivities of F that approach zero, on linear D scale plots of the Fig. 3 type, more rapidly than expected from the observed Arrhenius behavior of the measured diffusivities. (Rahman compared his results with extrapolations of data measured below the weak lambda transition at 1200°C, but the electrical data have validated the extrapolation to the accuracy of the comparison.) Rahman s high-temperature data have been accurately reproduced in one of the authors laboratories using a rather different form of pair potential. ... 

The present compilation of kinetic data represents the 12th evaluation prepared by the NASA Panel for Data Evaluation. The Panel was established in 1977 by the NASA Upper Atmosphere Research Program Office for the purpose of providing a critical tabulation of the latest kinetic and photochemical data for use by modelers in computer simulations of stratospheric chemistry. The recommended rate data and cross sections are based on laboratory measurements. The major use of theoretical extrapolation of data is in connection with three-body reactions, in which the required pressure or temperature dependence is sometimes unavailable from laboratory measurements, and can be estimated by use of appropriate theoretical treatment. In the case of important rate constants for which no experimental data are available, the panel may provide estimates of rate constant parameters based on analogy to similar reactions for which data are available. 

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