Modified additivity rule

The modified additivity rule introduced by Deutsch et al. (1997) attempted to account for the effects of molecular bonding by introducing empirically determined weighting factors that depend on the atomic orbital radii and the electron occupation numbers of the various atomic orbitals. A detailed comparison with existing molecular ionization cross section data for molecules of the form AB suggested the following explicit form of the ionization cross section of... 

The above-described modified additivity rule can be extended (Deutsch et al., 1998a Deutsch et al., 1998b) to the calculation of electron impact ionization cross sections for molecules of the form A ByC, and ApB C,D. The corresponding expressions in their simplest form (factors of n that cancel have been omitted for simplicity) are... 

Fig. 7. Absolute total single SiH4 ionization cross section as a function of electron energy. The various symbols refer to the following data triangles (A), Basner et a/., (1997a) circles ( ), Chatham et al. (1984) squares ( ), Krishnakumar and Srivastava (1995) diamonds (A), Haaland (1990) the dashed line represents a calculation using the modified additivity rule (see text for details), and the dotted line refers to the BEB calculation of Kim and coworkers (Ali et al., 1997).

The total and selected partial electron impact ionization cross sections of TMS are shown in Fig. 10. Also shown in Fig. 10 are the calculated total single TMS ionization cross sections from the modified additivity rule and from the BEB model of Kim and coworkers (Ali et al., 1997). There is reasonably good agreement between the two calculated cross sections and between the calculated cross sections and the measured cross section of Basner et al. (1996) (at least for impact energies below about 80 eV). The cross section of McGinnis etal. (1995) is considerably smaller than the two calculated cross sections and the measured cross section of Basner et al. (1996). 

Fig. 10. Absolute ionization cross sections of TMS as a function of electron energy. The squares ( ) and circles ( ) refer to the total ionization cross sections of McGinnis et at. (1995) and Basner el al. (1996), respectively. Also shown are the calculated cross section of Kim and coworkers (Ali et al, 1997) (dotted line) and a calculated cross section using the modified additivity rule (dashed line). The partial cross section for the most abundant fragment ion (m/z = 73) from Basner et al. (1996) is indicated by the triangles (A).

Fig. 12. Absolute ionization cross section of HMDSO as a liinction of electron energy. The squares ( ) denote the total ionization cross section of Basner et at. (1999) the open circles (O) refer to the data of Seefeldt et al. (1985). A calculated cross section based on the modified additivity rule is shown as the diamonds ( ). Also shown are two partial ionization cross section for the fiagment mjz = 147 (full circles, ) and the CHj ion at m/z = 15 (triangles, ).

The solubilities of carbon dioxide in aqueous solutions of seven binary and three ternary mixed salts chosen from eight kinds of electrolytes were measured at 25°C and 1 atm partial pressure of carbon dioxide by the saturation method. The experimental results were not correlated easily by the modified Setschenow equation, but they were correlated very well by the empirical two-parameter equation. The parameters in the equation for the binary and ternary solutions could be estimated by assuming an additive rule for the parameters of the component salt systems. This method, therefore, is useful for predicting the solubility of carbon dioxide in aqueous mixed-salt solutions. 

It should be noted that the Doi and Ohta theory predicts oifly an enhancement of viscosity, the so called emulsion-hke behavior that results in positive deviation from the log-additivity rule, PDB. However, the theory does not have a mechanism that may generate an opposite behavior that may result in a negative deviation from the log-additivity rule, NDB. The latter deviation has been reported for the viscosity vs. concentration dependencies of PET/PA-66 blends [Utracki et ah, 1982]. The NDB deviation was introduced into the viscosity-concentration dependence of immiscible polymer blends in the form of interlayer slip caused by steady-state shearing at large strains that modify the morphology [Utracki, 1991]. 

If the effects of mass transfer resistance and axial dispersion were linearly additive, as they are for a linear system, then the breakthrough curve plotted in terms of the modified time parameter t would be independent of 6. That this is approximately, though not exactly, true may be seen from Figure 8.22i. However, the deviation from the linear addition rule becomes important only when S is large and the isotherm is highly nonlinear (/3 < 0.5). Even for a highly nonlinear isotherm (= 0.33) the linear addition principle evidently provides a useful approximation except in the extreme case of low mass transfer resistance and large axial dispersion (5 > 5). 

One commenter su ested that cleaning verification be required on other horizontal surfaces within the work area, in addition to windowsills and uncarpeted floors. EPA agrees with this commenter becanse the Dnst Stndy demonstrated that, in nearly all cases, the cleaning verification step resulted in lower dust lead levels and, in most cases, the verification step was needed to achieve cleannp of all of the leaded dust deposited on the floors by the renovation. EPA is also concerned about the possible contamination of surfaces that are used to prepare, serve, and consume meals. EPA expects that movable surfaces, such as tables and desks, will be moved from the work area before work begins. Therefore, EPA has modified the rule to require cleaning verification on all countertops in the work area. 

Modifying the rules of addition for reciprocals of energy values of subsystems with reference to complex structures, the formula for calculating P, -parameter of a complex structure is obtained ... 

Figure 60 illustrates two-step creep and recovery data at different stresses for a reinforced polymer along with the predictions finm the Schapery creep formulation and those obtained from simply applying a modified form of the Boltzmann superposition principle. Without going into the details of the procedures of obtaining all the parameters, it is clear that the model captures much of the observed nonlinear response, while the modified Boltzmann rule does not. (Note that the modified Boltzmann rule simply assumes additivity of responses, but without the linearity assumptions.) Figure 61 shows the creep and recovery data... 

If we consider A and A3 as components, then the phase rule Eq.(3.12) allows up to four coexisting phases. However, we have an additional equilibrium constraint imposed by Eq. (3.69), reducing the degrees of freedom by one and the maximum number of coexisting phases to three. The modified phase rule therefore is... 

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