Temperature dependence fitted parameters

The combination of Eqs. (14.1)-(14.5) may be used to analyze the J t) results of entangled systems, with s treated as a temperature-dependent fitting parameter, which has the unit Da. ... 

Equations (2.23-2.25) can be used to fit the experimental data using k, and F as temperature dependent variable parameters. In addition, they are widely used to parameterize evaluated or recommended rate coefficients for combustion modelling (Chapter 3) [28]. For the reaction... 

Yu et al. [1994] carried out PALS measurements on four PS fractions (4, 9, 25, and 400 kDa, respectively) versus temperature (Figure 10.4). They evaluated the free-volume fractions on the basis of the proportionality between the free-volume fraction as probed by o-Ps and the product of the o-Ps intensity I3 and the mean cavity volume assumed spherical, as sketched previously [Eq. (10.16)]. On this basis they observed agreement with the free-volume fraction predicted as given by the lattice-hole model [Simha and Somcynsky, 1969] over a range of temperatures above Tg, the proportionality constant C being a molar mass-dependent fitting parameter. 

Figure 5.32 Calculated results for the VLc at 1 atm, activity coefficients y, excess enthalpies as a function of composition and activity coefficients at infinite dilution as/(7) for the system ethanol(l)-r -decane(2) using temperature-dependent Wilson parameters fitted simultaneously to VLE, and data A -experimental [3).

Then of course the data used should be distributed equally over the whole temperature (pressure) range Since often a lot of VLE data at atmospheric pressure are reported, perhaps some of the data have to be removed or at least a lower weighting factor for the numerous data should be used. The same is true for excess enthalpies. Most authors have measured excess enthalpies around room temperature. For fitting temperature-dependent model parameters the whole temperature range should be covered. While consistent VLE data (azeotropic data) provide the information about the composition... 

Figure 5.34 Results for acetone (1) -water (2) using recommended temperature dependent NRTL parameters fitted simultaneously to consistent VLE, h, and data.

From the data banks mentioned before, theDIPPRdatabank covers experimental pure component properties for approx. 2000 selected compounds. Additionally, recommended basic data and temperahire-dependent correlation parameters for the various pure component properties are available for aU compounds. When experimental pure component data were missing in the DIPPR data bank, predictive methods were used to calculate the required pure component data and to fit the required temperature-dependent correlation parameters. 

Another problem usually encountered with force field parameters in molecular dynamics simulations involves the implicit temperature dependence of parameters derived from experimental data. Since experimental data are usually collected at room temperature, parameters fitted to reproduce these experimental data will work best at room temperature and therefore will contain temperature effects implicitly. In molecular dynamics simulations, however, the temperature is an explicit variable, and it is assumed that the force field reproduces the structure at 0 K. The temperature effects obtained in molecular dynamics are added to properties for absolute zero. Parameters derived from data measured at higher temperatures therefore will necessarily introduce a small error. Parameters derived from ab initio calculations are free of these errors, since they are fitted to data for 0 K and do not implicitly contain temperature effects. If only experimental data are available, molecular dynamics calculations should be used to derive force field parameters.i >20 ... 

TABL 2. Fitting results of the temperature dependent influence parameter, c o(TX model with two fitting parameters (calculations by Peng-Robinson EOS)... 

It has been shown that for a correct description of the interfacial tensions of pure fluids a temperature dependent influence parameter must de used. This is especially required for more polar species like alcohols and water, while for non-polar species the temperature dependence of the influence parameter is small. In all cases the expression for the influence parameter has n optimized by fitting the computed interfacial tensions to the experimental values. It should be remarked that due to the fitting procedure the predictive value for pure species is lost. 

Establish a good set of initial parameters. In fitting rate expressions using TSR data, this can be done by fitting several (as many as is convenient) isothermal data sets with the proposed rate expression and plotting the resultant constant temperature parameters on Arrhenius plots (see ChaptCT 8). The best estimates of the Arrhenius parameters of each rate parameter obtained in this way are then used to start the iterations for an all-up fit. Notice that if the Arrhenius plots for each of the temperature dependent rate parameters are not linear, the model being used in the isothermal fitting is inadequate (see Chapter 9). 

Figure 2 Temperature dependent characteristic parameters ofP(S-b-B) during an upward and downward temperature scan, inverse of the normalized scattering peak maximum, I o/l (i o is the peak intensity at the transition), maximum position, q, and peak width, Aq, as obtained from fits of a Lorentz function to the q dependent scattering intensity. The solid lines are guides to the eyes. The data point in brackets represents the peak intensity of the sample directly q/ter the preparation. The dotted line indicates the position of the microphase separation temperature Tmst-...

Solutions in which the data can be fitted well enough to a constant value of the interaction parameter are classified as strictly regular . A better fit can be obtained by a temperature-dependent interaction parameter of the form... 

Eqn(3) allows a direct determination of LRO-parameter from resistivity measurement by using the constant A as a fit parameter. Eqn(l) is of more complicated character, where besides the SRO-parameters in the different coordination spheres there enter details of the band structure (Y,) which influence sign and magnitude of resistivity variation with degree of SRO. However, restricting to nearest neighbours and using an adequate model for the dependence of a on temperature and concentration, reliable SRO-parameters have been deduced from resistivity measurement for several solid solutions. ... 

If a data set containing k T) pairs is fitted to this equation, the values of these two parameters are obtained. They are A, the pre-exponential factor (less desirably called the frequency factor), and Ea, the Arrhenius activation energy or sometimes simply the activation energy. Both A and Ea are usually assumed to be temperature-independent in most instances, this approximation proves to be a very good one, at least over a modest temperature range. The second equation used to express the temperature dependence of a rate constant results from transition state theory (TST). Its form is... 

Here a and b are considered as fitting parameters depending on temperature. De-excitation rate constants (s < 0) are obtained from the detailed balance principle. AH fitting laws differ in the pre-exponential factor in Eq. (5.70). In the PEG model... 

It is important to note that and C2 are quantitative descriptors of the gel effect which depend only on the monomer, temperature and reaction medium. The full description of given by equation (11), requires g and g2 which are functions of the rate of initiation and extent of conversion. The kinetic parameters used in these calculations and their sources are given in Table 1. All data are in units of litres, moles and second. Figure 5 shows the temperature dependencies of and C2 and Table 2 lists these and other parameters determined by fitting the model to the data in Figures 1-4. 

A good model is consistent with physical phenomena (i.e., 01 has a physically plausible form) and reduces crresidual to experimental error using as few adjustable parameters as possible. There is a philosophical principle known as Occam s razor that is particularly appropriate to statistical data analysis when two theories can explain the data, the simpler theory is preferred. In complex reactions, particularly heterogeneous reactions, several models may fit the data equally well. As seen in Section 5.1 on the various forms of Arrhenius temperature dependence, it is usually impossible to distinguish between mechanisms based on goodness of fit. The choice of the simplest form of Arrhenius behavior (m = 0) is based on Occam s razor. 

This expression can be modified to apply directly to any of various techniques used to measure the interaction parameter, including membrane and vapor osmometry, freezing point depression, light scattering, viscometry, and inverse gas chromatography [89], A polynomial curve fit is typically used for the concentration dependence of %, while the temperature dependence can usually be fit over a limited temperature range to the form [47]... 

Ishikawa etal. proposed an approach for the determination of the ligand-field (LF) parameters of a set of isostructural lanthanide complexes. This method consists of a simultaneous fit of the temperature dependence of magnetic susceptibilities and NMR spectra for the whole isostructural series [18]. In order to avoid over-parametrization a key restriction is imposed each parameter is expressed as a linear function of the number of f electrons, n ... 

The temperature dependent T data are shown in Fig. 9. 7j values decrease from 28 ms at 21°C with increasing temperature, and show a minimum of 6.4 ms at 80° C. These results indicate the presence of the motion with a Larmor frequency of 30 MHz at this temperature. This minimum was found to be attributed to the flipping motion of a phenyl ring from the result of our other experiments discussed in later section.13 The jump rates of the flipping motion were estimated with a two-site jump model that a C-2H bond jumps between two equivalent sites separated by 180°, and that the angle made by the C-2H bond and the rotational axis is 60°. The quadrupole coupling constant of 180 kHz and the asymmetry parameter approximated to zero were used in the calculation. The calculated values for fitting with the... 

Equations of an Arrhenius type are commonly used for the temperature-dependent rate constants ki = kifiexp(—E i/RT). The kinetics of all participating reactions are still under investigation and are not unambiguously determined [6-8], The published data depend on the specific experimental conditions and the resulting kinetic parameters vary considerably with the assumed kinetic model and the applied data-fitting procedure. Fradet and Marechal [9] pointed out that some data in the literature are erroneous due to the incorrect evaluation of experiments with changing volume. 

As with the first three paraffin - water systems, only a constant parameter was required to correlate the hydrocarbon -rich phases although a temperature - dependent parameter was necessary to fit the aqueous - liquid phase data. 

J. As with the alkane - water systems, the interaction parameters for the aqueous liquid phase were found to be temperature - dependent. However, the compositions for the benzene - rich phases could not be accurately represented using any single value for the constant interaction parameter. The calculated water mole fractions in the hydrocarbon - rich phases were always greater than the experimental values as reported by Rebert and Kay (35). The final value for the constant interaction parameter was chosen to fit the three phase locus of this system. Nevertheless, the calculated three-phase critical point was about 9°C lower than the experimental value. 

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