Rigorous expressions

To derive Eq. (6.34) for a system of point charges in slab geometry, we proceed in a fashion analogous to the one employed for bulk systems in Section 6.2.1 and Appendix F.l. In other words, we divide the original chaige density related to Eq. (6.33) 

However, the potential (r ) related to p (r ) differs from its bulk counterpart [see Eq. (6.14)] because the basic simulation cell of the current slab system is repeated in only two (of the three) spatial dimensions. Nevertheless, we can still apply our basic strategy detailed in Appendix F.1.1.2 to find the explicit expression for (r ). 

We start by expanding the potential in Fourier space according to 

Akz = 27r/sz. This is consistent with viewing the system in slab geometry as three-dimensional with a basic cell becoming infinitely large in the 2-direetion (i.c., s — oo) such that AA — 0. 

we consider the behavior of the function /(A , 2 — Zj,a) for small wavenumbers k. To this end we perform a Taylor expansion of both the 

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Although PVT equations of state are based on data for pure fluids, they are frequently appHed to mixtures. 7h.e virial equations are unique in that rigorous expressions are known for the composition dependence of the virial coefficients. Statistical mechanics provide exact mixing rules which show that the nxh. virial coefficient of a mixture is nxh. degree in the mole fractions ... 

An important question is whether one can rigorously express such an average without referring explicitly to the solvent degrees of freedom. In other words. Is it possible to avoid explicit reference to the solvent in the mathematical description of the molecular system and still obtain rigorously correct properties The answer to this question is yes. A reduced probability distribution P(X) that depends only on the solute configuration can be defined as... 

This chapter takes up three aspects of kinetics relating to reaction schemes with intermediates. In the first, several schemes for reactions that proceed by two or more steps are presented, with the initial emphasis being on those whose differential rate equations can be solved exactly. This solution gives mathematically rigorous expressions for the concentration-time dependences. The second situation consists of the group referred to before, in which an approximate solution—the steady-state or some other—is valid within acceptable limits. The third and most general situation is the one in which the family of simultaneous differential rate equations for a complex, multistep reaction... 

Equation (7) is a rigorous expression relating the instantaneous vapor and liquid compositions with respect to monomer 1. However, the liquid composition (X ) needs to be related to the polymerization conversion in order to complete the model. 

Although allowing for axial variations in v with a constant wall concentration c yields Eq. (20-59), as a more rigorous expression applicable to higher concentrations [Trettin et al., Chem. Eng. Comm., 4, 507 (1980)] the form of Eq. (20-58) is convenient in analyzing a variety of complex behavior. 

The solid-liquid equilibrium for NaCl is rigorously expressed by ... 

A slightly more rigorous expression for o may be obtained by keeping an additional term in the Taylor expansion for the variance of the ratio,... 

In the case of dipolar produced line width, a rigorous expression for the second moment (Equation 17) of the resonance absorption line allows one to obtain quantitative structural information (Section II,B and C). The... 

Equation (95) is obtained from the virial expansion of the equation of state for rigid spheres for higher densities the rigid-sphere equation of state obtained from the radial distribution function by Kirkwood, Maun, and Alder has to be used (K10, Hll, p. 649). When Eq. (95) is substituted in Eqs. (92), (93), and (94) one then obtains the rigorous expressions for the coefficients of viscosity, thermal conductivity, and selfdiffusion of a gas composed of rigid spheres. 

Solution of the Boltzmann equation gives the velocity distribution function throughout the gas as it evolves through time, for example, due to velocity, temperature, or concentration gradients. A practical solution to the Boltzmann equation was found by Enskog [114], which is discussed in the next section. This approach is used to calculate rigorous expressions for gas transport coefficients. 

Then we divide the interval of integration over / by two above-mentioned intervals. We retain rigorous expression in exponential and power functions involved in the integrand in Eqs. (70a) and (70b), but in other co-factors we set / = 0 and g = 0 in the first and second intervals, respectively. The spectral function then is approximated by a sum... 

In this section, microdisc electrodes will be discussed since the disc is the most important geometry for microelectrodes (see Sect. 2.7). Note that discs are not uniformly accessible electrodes so the mass flux is not the same at different points of the electrode surface. For non-reversible processes, the applied potential controls the rate constant but not the surface concentrations, since these are defined by the local balance of electron transfer rates and mass transport rates at each point of the surface. This local balance is characteristic of a particular electrode geometry and will evolve along the voltammetric response. For this reason, it is difficult (if not impossible) to find analytical rigorous expressions for the current analogous to that presented above for spherical electrodes. To deal with this complex situation, different numerical or semi-analytical approaches have been followed [19-25]. The expression most employed for analyzing stationary responses at disc microelectrodes was derived by Oldham [20], and takes the following form when equal diffusion coefficients are assumed ... 

It can be deduced easily that the rigorous expression of the current is... 

Any of the above definitions of supersaturation can be used over a moderate range of system conditions, but as outlined in the following paragraph, the only rigorous expression is given by Eq. (10). 

The simplest rigorous expression for the barrier, based on Fig. 6.1a, is given by Equation 6.1,... 

Using a lattice model, Ruckenstein and Li derived a more rigorous expression for the steric interaction between two parallel plates.20... 

Equation 55 is a rigorous expression for the number of overall transfer units for equimolar counterdiffusion, in distillation columns, for instance. 

Thus, a more rigorous expression for the relaxation force is... 

Within the frame of MB-RSPT through the fourth order, a rigorous expression for the contribution from doubly excited configurations... 

A more rigorous expression is derived by noting that at equilibrium, partial fugacities of each component are the same in each phase, that is... 

Equations 12.16a and 12.16b are the rigorous expressions of components contributed from the trans population in the AHB approximation. They remain valid for any pump intensity in the absence of reverse cis-trans photo-isomerization. In practice, however, low-intensity pumping is the most interesting and important case, because the largest anisotropy in molecular distribution can be obtained far from excitation saturation, and at the low pump intensity, the photodegradation of chromophores is considerably reduced. 

The substitution of eqs 8—10 into eq 4, and considering infinite dilution of the solute (component 2), 5delds the following rigorous expressions for the derivative (9 In... 

Eq. (2) does not contain any adjustable parameter and can be used to predict the gas solubility in mixed solvents in terms of the solubilities in the individual solvents (1 and 3) and their molar volumes. Eq. (2) provided a very good agreement [9] with the experimental gas solubilities in binary aqueous solutions of nonelectrolytes a somewhat modified form correlated well the gas solubilities in aqueous salt solutions [17]. The authors also derived the following rigorous expression for the Henry constant in a binary solvent mixture [9] (Appendix A for the details of the derivation) ... 

The Kirkwood-Buff theory of solutions for ternary mixtures was used to analyze the gas solubility in a mixed binary solvent composed of a high molecular weight and a low molecular weight cosolvent, such as the aqueous solutions of water soluble polymers. A rigorous expression for the composition derivatives of the gas activity coefficient in ternary solution was used to derive the composition dependence of the Henry constant under isobaric and isothermal conditions. The obtained expressions as well as the well-known Kri-chevsky equation were tested for the solubilities of Ar, CH4, C2H6 and C3H8 in the aqueous solutions of PPG-... 

In this paper, the fluctuation theory of solutions was applied to the solubility of drugs in aqueous mixed solvents. A rigorous expression for the composition derivative of the activity coefficient of a solute in a ternary solution (Ruckenstein and Shulgin, 2001) was used to derive an equation for the activity coefficient of a solute at infinite dilution in an ideal mixed solvent and an expression for the solubility of a poorly soluble solid in an ideal mixed solvent (Eq. (23)). This simple equation can predict the solubility in terms of those in the individual constituents of the mixed solvent and... 

As in a previous paper [Int. J. Pharm. 258 (2003) 193-201], the Kirkwood-Buff theory of solutions was employed to calculate the solubility of a solid in mixed solvents. Whereas in the former paper the binary solvent was assumed ideal, in the present one it was considered nonideal. A rigorous expression for the activity coefficient of a solute at infinite dilution in a mixed solvent [Int. J. Pharm. 258 (2003) 193-201] was used to obtain an equation for the solubility of a poorly soluble solid in a nonideal mixed solvent in terms of the solubilities of the solute in the individual solvents, the molar volumes of those solvents, and the activity coefficients of the components of the mixed solvent. 

The following rigorous expression for the activity coefficient (k2°°) of a solid solute (the designation of the components in this paper is as follows the solid solute is component 2, the water is component 3 and the... 

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