Square-root model

Equation 2.18 effectively incorporates the retardation effects into the mobility determination for high concentration solutions. As an example, for aqueous solution at room temperature T = 298K), using D = 78.56 and t] = 0.008948, the variation of the mobility of the positive ion with concentration in 1,1 valency electrolytes of HCl, KNO3, and NaCl are plotted in Figure 2.5 according to Equation 2.18. The variation of the transference numbers of the cations with the concentration are also plotted to discern its effect on the mobility of each ion. As observed, the square root model represents the reduction of the mobility of each ion with increasing concentration, where the reduction appear to be mostly dependent on A. ... 

To evaluate the penetration of carbonation, the square-root model can be considered (Section 5.2.1). In most cases it is reasonable to assume that the conditions of exposure of the structure wiU not change in time and thus the carbonation coefficient K can be calculated from the present carbonation depth K = It... 

The most straightforward models to implement are normal models, followed by square root models and then lognormal models. The process that is used will have an impact on the distribution of future interest rates predicted by the model. A generalised distribution is given in Figure 3.2. 

An intriguing example of a structurally very simple self-replicating systems utilizes amidinium carboxylate salt bridges to enhance the catalytic condensation of an aniline structure with a benzaldehyde derivative (Figure IS). " Several structures with various substituents in para-position to the recognition sites have been investigated. For instance, autocatalysis was established in the formation of imine 14 from aniline 12 and aldehyde 13. Adding preformed template to the reaction mixture increased the initial rate of product formation, and the obtained experimental data was in accordance with the square root model. 

Barnes and Hunter [290] have measured the evaporation resistance across octadecanol monolayers as a function of temperature to test the appropriateness of several models. The experimental results agreed with three theories the energy barrier theory, the density fluctuation theory, and the accessible area theory. A plot of the resistance times the square root of the temperature against the area per molecule should collapse the data for all temperatures and pressures as shown in Fig. IV-25. A similar temperature study on octadecylurea monolayers showed agreement with only the accessible area model [291]. 

Fig. IV-25. The evaporation resistance multiplied by the square root of temperature versus area per molecule for monolayers of octadecanol on water illustrating agreement with the accessible area model. (From Ref. 290.)...

P(r,i) is the pairwise potential, which, depending upon the model, can be considered tc include electrostatic and repulsive contributions. The second term is a function of th< electron density, and varies with the square root, in keeping with the second-momen approximation. The electron density for an afom includes contributions from the neigh bouring atoms as follows ... 

Note that both the penetration and the surface-renewal theories predict a square-root dependency on D. Also, it should be recognized that values of the surface-renewal rate s generally are not available, which presents the same problems as do 6 and t in the film and penetration models. 

Given that, from the penetration theory for mass transfer across an interface, the instantaneous rale ol mass transfer is inversely proportional to the square root of the time of exposure, obtain a relationship between exposure lime in the Higbie mode and surface renewal rate in the Danckwerts model which will give the same average mass transfer rate. The age distribution function and average mass transfer rate from the Danckwerts theory must be deri ved from first principles. 

More complicated rate expressions are possible. For example, the denominator may be squared or square roots can be inserted here and there based on theoretical considerations. The denominator may include a term k/[I] to account for compounds that are nominally inert and do not appear in Equation (7.1) but that occupy active sites on the catalyst and thus retard the rate. The forward and reverse rate constants will be functions of temperature and are usually modeled using an Arrhenius form. The more complex kinetic models have enough adjustable parameters to fit a stampede of elephants. Careful analysis is needed to avoid being crushed underfoot. 

This means that the precision of the prediction decreases with the square root of time. This describes the random walk model. A drift can be easily built into such a model by the addition of some constant drift function at each successive time period. 

In vitro dissolution was virtually complete after 6-8 hr. Since the plot of cumulative drug release versus time is hyperbolic, the authors attempted to fit the data to the Higuchi matrix dissolution model (116,117), which predicts a linear correlation between cumulative drug release and the square root of time. Linearity occurred only between 20 and 70% release. 

The nonlinear character of log has not often been discussed previously. Nevertheless, Jorgensen and Duffy [26] argued the need for a nonlinear contribution to their log S regression, which is a product of H-bond donor capacity and the square root of H-bond acceptor capacity divided by the surface area. Indeed, for the example above their QikProp method partially reflects for this nonlinearity by predichng a much smaller solubility increase for the indole to benzimidazole mutation (0.45 versus 1.82 [39, 40]). Abraham and Le [41] introduced a similar nonlinearity in the form of a product of H -bond donor and H -bond acceptor capacity while all logarithmic partition coefficients are linear regressions with respect to their solvation parameters. Nevertheless, Abraham s model fails to reflect the test case described above. It yields changes of 1.8(1.5) and 1.7(1.7) [42] for the mutations described above. 

In this simplified model, it is assumed that liquid may leave the plate, either by flow over the weir Ln(weir) or by weepage Ln(weep)- Both these effects can be described by simple hydraulic relations, in which the flow is proportional to the square root of the available hydrostatic liquid head. The weir flow depends on the liquid head above the weir and hence... 

NFS spectra of the molecular glass former ferrocene/dibutylphthalate (FC/DBP) recorded at 170 and 202 K are shown in Fig. 9.12a [31]. It is clear that the pattern of the dynamical beats changes drastically within this relatively narrow temperature range. The analysis of these and other NFS spectra between 100 and 200 K provides/factors, the temperature dependence of which is shown in Fig. 9.12b [31]. Up to about 150 K,/(T) follows the high-temperature approximation of the Debye model (straight line within the log scale in Fig. 9.12b), yielding a Debye tempera-ture 6x) = 41 K. For higher temperatures, a square-root term / v/(r, - T)/T,... 

Fig. 9.12 (a) NFS spectra of FC/DBP with quantum beat and dynamical beat pattern, (b) Temperature-dependent /-factor. The solid line is a fit using the Debye model with 0D = 41 K below 150 K. Above, a square-root term / - V(Tc - T)/Tc was added to account for the drastic decrease of /. At Tc = 202 K the glass-to-liquid transition occurs. (Taken Ifom [31])... 

The penetration front was measured from the images in Figure 3.4.9 and plotted against the square root of time in Figure 3.4.10. The plot indicates that this relationship is linear and its slope is a measure of the sorptivity [28]. This type of experiment, coupled with gravimetric measurements, allow for the modeling of... 

A critical comparison between experiment and theory is hindered by the range of experimental values reported in the literature for each molecule. This reflects the difficulty in the measurement of absolute ionization cross sections and justifies attempts to develop reliable semiempirical methods, such as the polarizability equation, for estimating the molecular ionization cross sections which have not been measured or for which only single values have been reported. The polarizability model predicts a linear relationship between the ionization cross section and the square root of the ratio of the volume polarizability to the ionization potential. Plots of this function against experimental values for ionization cross sections for atoms are shown in Figure 7 and for molecules in Figure 8. The equations determined... 

In the early days of water radiolysis, it was empirically established in several instances that the reduction of molecular yield by a scavenger was proportional to the cube root of its concentration (Mahlman and Sworski, 1967). Despite attempts by the Russian school to derive the so-called cube root law from the diffusion model (Byakov, 1963 Nichiporov and Byakov, 1975), more rigorous treatments failed to obtain that (Kuppermann, 1961 Mozumder, 1977). In fact, it has been shown that in the limit of small concentration, the reduction of molecular yield by a scavenger should be given by a square root law in the orthodox... 

---