Zeroth approximation

The basic thermodynamic equation giving the change in the Gibbs free energy of any process can be written 

Krause (1969, 1970, 1971) recently refined the Scott treatment by the addition of several terms specific to blocks. The final equation can be expressed as 

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In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. 

However, one can proceed beyond this zeroth approximation, and this was done independently by Guggenheim (1935) with his quasi-chemicaT approximation for simple mixtures and by Bethe (1935) for the order-disorder solid. These two approximations, which turned out to be identical, yield some enliancement to the probability of finding like or unlike pairs, depending on the sign of and on the coordmation number z of the lattice. (For the unphysical limit of z equal to infinity, they reduce to the mean-field results.)... 

Figure A2.5.21. The heat eapaeity of an order-disorder alloy like p-brass ealeulated from various analytie treatments. Bragg-Williams (mean-field or zeroth approximation) Bethe-1 (first approximation also Guggenheim) Bethe-2 (seeond approximation) Kirkwood. Eaeh approximation makes the heat eapaeity sharper and higher, but still finite. Reprodueed from [6] Nix F C and Shoekley W 1938 Rev. Mod. Phy.s. 10 14, figure 13. Copyright (1938) by the Ameriean Physieal Soeiety.

Since exp(— IV2) < 1, this equation can be solved iteratively by using (3.28) as a zeroth approximation, whence... 

The Burnett Expansion.—The Chapman-Enskog solution of the Boltzmann equation can be most easily developed through an expansion procedure due to Burnett.15 For the distribution function of a system that is close to equilibrium, we may use as a zeroth approximation a local equilibrium distribution function given by the maxwellian form ... 

It was shown, in Eqs. (1-73), (1-74), (1-75), that a = 1, afy r> => 0, a = 0. As the zeroth approximation we shall assume that A mid /a are zero (their effects are negligibly small) if Eqs. (1-86) and (1-87) are multiplied by /a and A, respectively, we obtain the condition that og0 and -oSi are zero higher order equations would show that all the coefficients are zero. Thus, the coefficients are proportional to some power of /a (or A). The zero-order approximation to the distribution function is just the local maxwellian distribution... 

The contour lines represent points of relative density 1.0, 0.9, 0.8,..0.1 for a hydrogen atom. This figure, with the added proton 1.06 A from the atom, gives the electron distribution the hydrogen molecule-ion would have (in the zeroth approximation) if the resonance phenomenon did not occur it is to be compared with figure 6 to show the effect of resonance. 

Krause S., Microphage separation in block copol3miers Zeroth approximation including surface free energies, Macromolecules, 3, 84, 1970. 

Regular Mixtures with Complete Configurational Disorder (Zeroth Approximation)... 

The word vibrational, is in this case somewhat misleading, because enthalpic interactions are essentially static in nature and are purely static whenever the zeroth approximation is appropriate (see Della Giusta and Ottonello, 1993). Lastly, it must be noted that the excess term represents a reciprocal energy arising from the noncoplanarity of the four quadrilateral end-members (cf. section 3.8.7). The model of Davidson and Mukhopadhyay (1984) was reconsidered by Hirschmann (1991), who proposed an analogous treatment for (Ni,Mg,Fe)2Si04 mixtures. 

In solving Eq. (1) we follow an iterative process that has been refined in the last several years (Jansson, 1970 Jansson et a/., 1970 Willson, 1973 Pliva et al., 1980 Blass and Halsey, 1981). In this iterative process, the zeroth approximation to o, the ideal spectrum, is taken as i, the measured spectrum, and the iterative process can be represented as... 

The number of sites on the metal surface pa- square centimeter can be taken as equal to the number of metal atoms per square centimeter (about 1.5 X 1015). Thus, the fraction of surface covered by water molecules, 0 turns out (by inserting the appropriate numerical values into this zeroth-approximation approach) to be about 0.7. 

In the zeroth approximation, in the energy region of interest, the spectrum of states consists of one discrete state overlapping a dense but... 

Pyshnograi GV (1996) An initial approximation in the theory of viscoelasticity of linear polymers and non-linear effects. J Appl Mech Techn Phys 37(1) 123—128 Pyshnograi GV (1997) The structure approach in the theory of flow of solutions and melts of linear polymers. J Appl Mech Techn Phys 38(3) 122—130 Pyshnograi GV, Pokrovskii VN (1988) Stress dependence of stationary shear viscosity of linear polymers in the molecular field theory. Polym Sci USSR 30 2624—2629 Pyshnograi GV, Pokrovskii VN, Yanovsky YuG, Karnet YuN, Obraztsov IF (1994) Constitutive equation on non-linear viscoelastic (polymer) media in zeroth approximation by parameter of molecular theory and conclusions for shear and extension. Phys — Doklady 39(12) 889-892... 

Conditions inside pits are complicated by the aggregation of hydrous oxides and the effect of tensile stress, which increases the dissolution velocity (per unit area) at the tip over that of a plain sheet of metal. A zeroth approximation for the electrochemistry of corrosion in a pit is to take the hemispherical tip as anodic... 

A cathodically polarized air electrode (planar) has a limiting current of about 1(T4 A cm-2. In a fuel cell and the critical quantity that controls the magnitude of the current density is the thickness of the electrolyte in the meniscus of the three-phase boundaries. This varies with the shape of the meniscus and the contact angle, (d) Assume a section of the meniscus having a solution thickness of 10-5 cm and calculate the limiting current of this section, (e) In light of these zeroth approximation calculations, where do you think the maximum activity of a pore lies (Bockris)... 

Correction vectors lead to inhomogeneous linear equations of the form Ax = b that involve sparse matrices in the VB representation. An iterative small mar trix method gives rapid convergence using coordinate relaxation that resembles the Davidson algorithm for eigenvalues. We summarize the procedure here [48, 40]. We begin with a set of m orthonormal A-vectors Q,- i = 1... m and construct x the zeroth approximation to the true solution x as,... 

The specific surface areas were calculated above for two- and three-layer tobermorites, both having CaO/Si02 ratios of 1.5. Those, however, are only zeroth approximation values, because Equation 4 indicates that the two- and three-layer tobermorites have different CaO/SiOo ratios, and neither has a ratio of 1.5 the former has a lower, the latter a higher ratio. Combining the calculation of... 

In his first publications Huckel has already pointed out the possibility of the calculation of the transition frequencies from the calculated energy levels. An objection to this was, however, that in the so-called zeroth approximation (p. 279) the value of the parameters both in the valence bond and in the molecular orbital method had to be chosen quite different in order to calculate either the resonance energy or the transition frequencies in agreement with observation (Sklar, Forster)26. 

In the next approximation siJc is no longer put equal to zero, but equal to 0.25 as the best value. The solution can then be derived from that of the zeroth approximation without fresh arithmetic (Wheland). 

It appears that in a series of compounds the R.E. according to this 1st approximation method with overlap are in practically the same ratios as in the zeroth approximation (Table 24, p. 283). 

Another approximate approach to determining transport number is to use the zeroth approximation equation for ionic mobility, i.e., Eq. (4.183)... 

The starting point to elucidate the way the proton moves in solution is to consider its movement through the solvent at a steady state—constant velocity—and at a concentration so low that there is no interionic interaction (zeroth approximation). This occurs when the electric driving force ze X balances the Stokes viscous force, 6nrr v. Thus, the Stokes mobility is... 

Point-charge potential (monopole potential). By equation (45), the zeroth approximation... 

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