Equation modifications

Figure 4-1. Effect of UNIQUAC equation modification for an alcohol-hydrocarbon system.

Levey AS, Bosch JP, Lewis JB, et al. A more accurate method to estimate glomerular filtration rate from serum creatinine A new prediction equation. Modification of Diet in Renal Disease Study Group. Ann Intern Med 1999 130 461 470. 

The time method of lines (continuous-space discrete-time) technique is a hybrid computer method for solving partial differential equations. However, in its standard form, the method gives poor results when calculating transient responses for hyperbolic equations. Modifications to the technique, such as the method of decomposition (12), the method of directional differences (13), and the method of characteristics (14) have been used to correct this problem on a hybrid computer. To make a comparison with the distance method of lines and the method of characteristics results, the technique was used by us in its standard form on a digital computer. 

Gaseous anaesthesia in mice is an equilibrium process between the anaesthetic gas and the phase in which the gas exerts its effect (the biophase). As such, it should be amenable to treatment by the Clausius-Clapeyron equation. Modification of this equation is required when the distribution of a series of gases is to... 

Of the two-parameter cubic equations, modifications of the Redlich-Kwong equation are among the most popular. In this chapter we consider only one cobic equal en of state Soave s" version of the Redlich-Kwoeg equation its performance is typical of modem cubic eqanlions. The Suave-Redlich-Kwong equation incorporates (he following prescription for purameter 0 ... 

The Benedict-Webb-Rubin equation may be generalized in a manner similar to that used for the Beattie-Bridgman equation. Modification then leads to... 

Using the above equation we can determine the values of equilibrium constants when reaching the certain output of nanoparticles, sizes of nanoparticles and shapes of nanoparticles formed with the appropriate equation modification. The internal cavity sizes or nanoreactor reaction zone and its geometry significantly influence the sizes and shapes of nanostructures. 

Dieterici s equation A modification of van der Waals equation, in which account is taken of the pressure gradient at the boundary of the gas. It is written... 

In 1972, Soave published a method of calculating fugacities based on a modification of the Redlich and Kwong equation of state which completely changed the customary habits and became the industry standard. In spite of numerous attempts to improve it, the original method is the most widespread. For hydrocarbon mixtures, its accuracy is remarkable. For a mixture, the equation of state is ... 

A modification of the foregoing procedure is to suspend the plate so that it is partly immersed and to determine from the dry and immersed weights the meniscus weight. The procedure is especially useful in the study of surface adsorption or of monolayers, where a change in surface tension is to be measured. This application is discussed in some detail by Gaines [57]. Equation 11-28 also applies to a wire or fiber [58]. 

The modification of the surface force apparatus (see Fig. VI-4) to measure viscosities between crossed mica cylinders has alleviated concerns about surface roughness. In dynamic mode, a slow, small-amplitude periodic oscillation was imposed on one of the cylinders such that the separation x varied by approximately 10% or less. In the limit of low shear rates, a simple equation defines the viscosity as a function of separation... 

Still another situation is that of a supersaturated or supercooled solution, and straightforward modifications can be made in the preceding equations. Thus in Eq. IX-2, x now denotes the ratio of the actual solute activity to that of the saturated solution. In the case of a nonelectrolyte, x - S/Sq, where S denotes the concentration. Equation IX-13 now contains AH, the molar heat of solution. 

The importance of the solid-liquid interface in a host of applications has led to extensive study over the past 50 years. Certainly, the study of the solid-liquid interface is no easier than that of the solid-gas interface, and all the complexities noted in Section VIM are present. The surface structural and spectroscopic techniques presented in Chapter VIII are not generally applicable to liquids (note, however. Ref. 1). There is, perforce, some retreat to phenomenology, empirical rules, and semiempirical models. The central importance of the Young equation is evident even in its modification to treat surface heterogeneity or roughness. ... 

Using appropriate data from Table II-9, calculate the water-mercury interfacial tension using the simple Girifalco and Good equation and then using Fowkes modification of it. 

The very considerable success of the BET equation stimulated various investigators to consider modifications of it that would correct certain approximations and give a better fit to type II isotherms. Thus if it is assumed that multilayer formation is limited to n layers, perhaps because of the opposing walls of a capillary being involved, one... 

The ideal gas law equation of state thus leads to a linear or Henry s law isotherm. A natural modification adds a co-area term ... 

The only modification of equation (Al.6.90) for spontaneous Raman scattering is the multiplication by the density of states of the cavity, equation (Al.6.24). leading to a prefactor of the fonn cojCOg. ... 

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

Surface electron charge density can be described in tenus of the work fiinction and the surface dipole moment can be calculated from it ( equatiou (Bl.26.30) and equation (B1.26.31)). Likewise, changes in the chemical or physical state of the surface, such as adsorption or geometric reconstruction, can be observed through a work-fimction modification. For studies related to cathodes, the work fiinction may be the most important surface parameter to be detenuined [52]. 

This theoretical foundation of electron diffraction as given by Wierl [26] and modification of this equation partly following suggestions made by Soltzberg and Wilkins [27] gives Eq. (24), with the atomic property p for the atoms i and j, a reciprocal distance s, and the distance tu between the atoms i and j. 

The equations of motion are integrated using a modified velocity Verlet algorithm. The modification is required because the force depends upon the velocity the extra step involves... 

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