Empirical fitted constants

Equations 16 and 19 are iixlicative of the versatility of the form of the isotherm relation, equation 13, insofar as the latter provides for description of solute complexation with either (or both) of the solvent components of binary stationary phases. In addition, each of what might otherwise appear simply to be empirical fitting constants can be precisely interpreted (equations 17 and... 

Here, Cj and are empirically fitted constants that will vary somewhat for different oils. Reasonable values are 0.008 K and 0.18, respectively. 

Infrared spectra are straightforward to predict theoretically, demanding development of a force field (FF) to determine frequencies and dipole derivatives for intensities. These parameters were initially obtained using empirically fitted force constants and simple models for transition dipoles (Krimm and Bandekar, 1986 Torii and Tasumi, 1996). 

In transforming the independent variables alone, it is assumed that the dependent variable already has all the properties desired of it. For example, if the /s are normally and independently distributed with constant variance, at least approximately, then any transformations such as described in Section VI,B,1 would be unnecessary. Under such assumptions, Box and Tidwell (B17) have shown how to transform the independent variables to reduce a fitted linear function to its simplest form. For example, a function that has been empirically fitted by... 

The critical input parameters are then (1) the grain size, which should be known for each case, (2) the Aci temperature which is calculated from thermodynamics, (3) the effective diffusion activation energy, Qea, and (4) the empirical constants aj for each element. Qea and aj were determined by empirically fitting curves derived using Eq. (11.12) to experimentally observed TTT curves, and the final formula for calculating r was given as... 

The rate constants for the simple alkanes can be empirically fit by assigning rate constants to the primary and secondary C-H groups, along with sub-... 

Mehran and Tanji (1974) mentioned several problems with the above approach, including (1) assuming first-order kinetics, (2) empirical fitting of rate coefficients, (3) rate coefficients that are constant, implying that concentrations of substrates are not limiting, and (4) no consideration for environmental factors such as temperature, soil-water content, and aeration. 

The reference temperature (T ) is chosen differently depending on whether an empirical data fit or a "universal" WLF fit are carried out. If the data are fit empirically, T, should be taken as the value that gives the best fit of the data. The values of constants Cj and C2 are then calculated after superposition. This is done by shifting the curves and must be performed prior to using the nomograph program. The empirically determined constants may then be substituted in the WLF equation and a nomograph of the desired viscoelastic functions plotted. 

Under elevated pressures, the rocksalt stmcture transforms into the CsCl structure. Changes in lattice constants and measurements of other physical properties have provided much quantitative information for empirical fits to equations of state. Modem theoretical tools are used for obtaining a deeper understanding of charge being transferred back from the anion towards the cation in the aUcah hahdes. However, as can be seen from calculations of their cohesive properties, even nowadays there are problems to be solved for such simple stmctures. The data for the halide ions, in particular, are quite useful and may be transferred to other halide systems, and give good predictive values for more complex systems. 

Figure 9 Bond distance versus force constant data for intra-row bonding in the row of the periodic table containing the second transition series (n = 5, Rb to Xe) (a) data fit to empirical force constant-bond distance functions (see text) (b) same data fit to the exponential decay function that we have proposed. Identity of the points is given in Ref. 42...

The difference between the extended Debye-Hiickel equation and the Pitzer equations has to do with how much of the nonideahty of electrostatic interactions is incorporated into mass action expressions and how much into the activity coefficient expression. It is important to remember that the expression for activity coefficients is inexorably bound up with equilibrium constants and they must be consistent with each other in a chemical model. Ion-parr interactions can be quantified in two ways, explicitly through stability constants (lA method) or implicitly through empirical fits with activity coefficient parameters (Pitzer method). Both approaches can be successful with enough effort to achieve consistency. At the present, the Pitzer method works much better for brines, and the lA method works better for... 

Sverdrup (1990) and Sverdrup and Warfvinge (1995) developed Ae PROFILE model to calculate mineral weathering rates by means of a geochemical mass-balance procedure. This model differs from the others in that it uses dissolution constants, which were, for the most part, determined in the laboratory. Empirical fitting parameters, such as surface area of mineral exposed, are used to adjust the model to the real system being described. The model appears to work satisfactorily in many catchments if the fitting parameters are chosen judiciously. This requires a considerable amount of knowledge... 

Another simpler systematic potential was derived for chlorine, using the two atomic sites for the distributed multipoles and dispersion coefficients. The anisotropic atom-atom repulsion potential was derived from the overlap model, with the proportionality constant and one major anisotropic coefficient being adjusted by empirical fitting to the crystal structure. This empirical adjustment appeared to effectively absorb the missing contributions, including the many-body effects, because the potential was able to reproduce a wide range of properties of the solid and liquid from Monte Carlo simulations. 

Table 4.2. Comparison of rotational constants in MHz observed and calculated from empirical fit to vibrational potential function for oxetanone-3ls)...

The numerical results for the flow characteristics jj1- and j have been presented in Fig. 3.29 as functions of the EPR dimensionless density A. The calculations were also performed for the empirical turbulence constant vro = 0.03U h. The shear velocity [/, (A) and the displacement height d(A) turned out to increase, but t/,(A) and Z0(A) decrease with increase in the EPR density. It is possible to vary vro in order to fit these theoretical profiles to experimental data, thus determining the empirical constant. 

For the analysis of the spectra of trans-(CH)x, valence-force-field calculations have been performed, the force constant being empirically fitted so as to reproduce the observed vibrational frequencies (Inagaki et al., 1975 Schiigerl and Kuzmany, 1981). The energy gradient scheme based on the ab initio crystal orbital method has also been attempted for the study of the vibrational structure of trans- and c -(CH)x (Teramae et al., 1984). The calculated result of the vibrational frequencies has been found to depend upon the quality of the basis set. It has been demonstrated that, e.g., for the 4-31G basis, a uniform scaling of the force constants, multiplying by 0.8, is required to adjust the vibrational frequencies of the trans-(CH), (Table III). 

As has already been indicated, any separation involves only the part of the isotherm which lies at or below the feed-concentration level. An empirical fit, based upon the assumption of a constant separation factor, can often be made to this part. For this purpose, Eq. (7) can be rewritten in the form... 

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