Semiempirical equations

We have considered the surface tension behavior of several types of systems, and now it is desirable to discuss in slightly more detail the very important case of aqueous mixtures. If the surface tensions of the separate pure liquids differ appreciably, as in the case of alcohol-water mixtures, then the addition of small amounts of the second component generally results in a marked decrease in surface tension from that of the pure water. The case of ethanol and water is shown in Fig. III-9c. As seen in Section III-5, this effect may be accounted for in terms of selective adsorption of the alcohol at the interface. Dilute aqueous solutions of organic substances can be treated with a semiempirical equation attributed to von Szyszkowski [89,90]... 

Equation XII-20 may be combined with various semiempirical equations. Thus if Antonow s rule applies (Eq. IV-8), one obtains... 

Another school has also developed and attempted to understand the functional dependence of adsorption on heterogeneous surfaces on the vapor pressure and temperature. Various empirical or semiempirical equations were proposed [24-26] and used later to represent experimental data and to evaluate EADF by inverting Eq. (1), which belongs to the class of linear Fredholm integrals of the first kind [27]. 

Detailed information about the conductivity of solid electrolytes can be found elsewhere.2,3,6 8,10,11 As shown in Fig. 3.1, the temperature dependence of the ionic conductivity o can, in general, be described by the semiempirical equation ... 

The Stokes-Einstein equation predicts that DfxITa is independent of the solvent however, for real solutions, it has long been known that the product of limiting interdiffusion coefficient for solutes and the solvent viscosity decreases with increasing solute molar volume [401]. Based upon a large number of experimental results, Wilke and Chang [437] proposed a semiempirical equation,... 

For this reason, process-based whole-farm simulators have been developed. The simulators comprise modules containing semiempirical equations and mechanistic models that account for soil type, crops grovm, climate, herd size, manure handling, and other factors for a variety of farms as well as modules that calculate GHG emissions, other types of... 

The evaluation of the nonelectrostatic term is usually empirical and multifarious. Based on a skillful argument, however, Volkov and coworkers [9,11,12] proposed that the nonelectrostatic term of the ion transfer energy, AGfr Cne), should be expressed by a semiempirical equation called the Uhlig equation [13], which is given by... 

Wirtz et al. obtained better agreement for the diffusion coefficients than that yielded by Eq. (2.6.6), by considering the change in viscosity in the vicinity of the diffusing particle. The semiempirical equation has the form... 

For rubberlike materials, the modulus thus defined generally shows an additional dependence on elongation (39,40,42), apparently because of increase in the non-affineness of the deformation as the elongation increases (43,44). This dependence is frequently represented (approximately) by the semiempirical equation of Mooney and Rivlin... 

Numerical solutions of the Fleitler-London, or of density functional equations, show how energies depend on separation distance, but it is more instructive to consider semiempirical equations such as the Morse potential, or especially, the very simple Rydberg equation which has been shown to apply... 

In a series of papers, Pitzer and his co-workers (4,5,6) proposed a very useful semiempirical equation for the... 

For ternary aqueous solutions of HC1 and NaCl, the following semiempirical equations are proposed to describe the activity coefficients of the electrolytes ... 

If die semiempirical equations proposed in Reference 4 are correct, then die exothermicity of die reaction RX+R1 Y —RY+R1 X would be identical in the gaseous and liquid phases. Experience shows die assumption generally results in much poorer agreement for all-solid reactions. No such semiempirical equations for enthalpy of sublimation and its estimation appear to be available, although some cancellation is expected. 

Equation of state of seawater The semiempirical equation that relates the density of seawater to its salinity, temperature, and pressure. 

Many empirical and semiempirical equations for have been published in the literature. The equations have to fulfill the boundary condition ... 

The conventional correlation for the drag on a sphere in steady motion is presented as a graph, see Fig. 5.12, called the standard drag curve , where is plotted as a function of Re. Many empirical or semiempirical equations have been proposed to approximate this curve. Some of the more popular are listed in Table 5.1. None of these correlations appears to consider all available data. 

Experimental results show that wall effects are negligible for ), up to about 0.125 for spherical-caps in low M systems. Collins suggested semiempirical equations for A simpler equation proposed by Wallis (Wl) which... 

If the excited molecule formation is mainly because of electron-cation recombination the concentration dependence of quenching of the precursor charged species may follow some form of the modified Warman-Asmus-Schuler (WAS) semiempirical equation [8], e.g. ... 

The 1997 recommendations for the OH + N02 rate constants (DeMore et al., 1997 Atkinson et al., 1997a, 1997b) may be systematically high (e.g., Donahue et al., 1997) at temperatures below 240 K. Thus, recent measurements at temperatures characteristic of the upper troposphere give rate constants that are smaller than the recommendations by 10-30% (Brown et al., 1999a Dransfield et al., 1999). In addition, 02 appears to be only about 70% as efficient a third body as N2 in the termolecular reaction. Using a modified form of the semiempirical equation for the rate constant in the falloff region (Chapter 5, Eq. (C)), which takes into account the variable collision efficiency /3,... 

CA 44, 1032(1950) (Penetrating or piercing jet theory of deton) 39) S.R. Brinkley Jr, "The Theory of Detonation Process , pp 83-8 in the "Summary Technical Report Division 8, NDRC , Vol 1 (1946). It includes Riemann formulation (pp 83-4 86) Rankine-Hugoniot condition ( 84 86) Chapman-Jouguet postulate (84) Becker semiempirical equation of state (85) Rayleigh, solution of the Riemann equation (86) and Hydro-thermodynamic theory, applications (87-8) 40) W. Loring H. 

It is interesting to compare Equation 1 with another equation, which was used to describe the decrease of Tg with decreasing molecular weight of the polymer. Based on isofree volume and viscous flow concepts, the following semiempirical equation was derived (5) ... 

The several attempts, published in the literature, to describe the kinetics of vapour phase olefin (mostly ethylene) hydration can be classified into two groups according to the basic model used. One model, for reactions catalysed by phosphoric acid supported on solids, treats the kinetics as if the process were homogeneous acid catalysis and takes into account the acid strength of the supported acid. Thus, a semiempirical equation for the initial reaction rate [288]... 

In the current statistical theory of rubber elasticity, it is suggested that the front-factor molecular forces. They have proposed a semiempirical equation of state taking into account the dependence... 

In this connection, it is very interesting that the volume and intrachain changes obtained by various experimental methods 24,29,85) [Eq. (101)] agree well with Eq. (56) following from the Tobolsky-Shen semiempirical equation of state or the related phenomenological Eq. (76). The values of y determined from the data are rather small (0.1-0.3). As has been mentioned above, according to the semiempirical approach by Tobolsky and Shen one can formally suggest that the front-factor in Eq. (28) is pressure dependent. If it is really so, then the parameter y for rubbers can be considered as an experimental coefficient similar to the coefficient of thermal expansion and compressibility 29). 

In chromatography one traditionally avoids the use of empirical parameters, such as e, and prefers Kd relative to a certain internal standard usually a low-molecular-weight substance. In this case, the distribution coefficients of macromolecules, Kd, are a function of t — tab, where tad and t are the elution times of the standard in a given mixture and in a mixture corresponding to the critical conditions. Close to the critical conditions, Kb is much more sensitive to a change in the composition of the mobile phase than to the retention times of low-molecular-weight substances, and so precision of Kd determined from t — tab will hardly be higher than that of eab calculated from the semiempirical Equation (3.16). 

Many semiempirical equations of slate with varying degree of theoretical foundations are in use. The Van der Waals equation, a two-parameter equation that gives a qualitatively correct picture of the P-V-T relations of a gas and of the gas liquid transition, is an example. 

Fundamental Property Relation. The fundamental property relation, which embodies the first and second laws of thermodynamics, can be expressed as a semiempirical equation containing physical parameters and one or more constants of integration. All of these may be adjusted to fit experimental data. The Clausius-Clapeyron equation is an example of this type of relation (1—3). 

Because the interior reference electrode is sealed, all terms in the denominator of (6.43) of this semiempirical equation are constant. It is called the Nikolskij and Eisenman equation (Morf, 1981). 

Ejector. Gas chamber pressure needs to be known in order to calculate ejector power input. A semiempirical equation was developed by Henzler (10) that related the entrainment ratio to other system variables ... 

The above mathematical treatment constitutes the creation of the form of the semiempirical equations. To actually use these equations, they must be parameterized somehow (as stressed above, Dewar used experimental data). This is analogous to the situation in molecular mechanics (Chapter 3), where a force field, defined by the form of the functions used (e.g. a quadratic function of the amount by which a bond is stretched, for the bond-stretch energy term) is constructed, and must then be parameterized by inserting specific quantities for the parameters (e.g. values for the stretching force constants of various bonds). For each kind of atom A (a maximum of) six parameters is needed ... 

The van der Waals equation is not the only semiempirical equation of state for gases containing two constants. Another equation is the Berthelot equation ... 

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