Approximations continuum

Aris ° was the first to address the theoretical aspects of total lumping of first-order reactions. Luss and Hutchinson later noticed that serious problems arise if one extends the continuum approach to nonlinear kinetics. Ho and Aris ° discussed the origin of the difficulties in liunping nonlinear kinetics in continuous mixtures. They proposed a single-component-identity that must be satisfied by any continuum treatment in order to overcome the difficulties. Other aspects of the mathematical and conceptual difficulties have also been examined. Krambeck addressed thermodjmamic issues. Ocone and Astarita reviewed many aspects of continuous mixtures. 

Without loss of generality, we use the rate constant A as the reactant label. The concentration of the total liunp C f) can be expressed as  

In the following we highlight some significant results obtained from the four approaches mentioned earlier. The majority of them were obtained from the continuum treatment. 

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Floris, F.M. Tomasi, J. Pascal-Ahuir, J.L., Dispersion and repulsion contributions to the solvation energy refinements to a simple computational model in the continuum approximation, J. Comp. Chem. 1991,12, 784-791... 

Therefore our caveats. Equation 47-88 and Figure 47-17 are best reserved for cases of high signal, where the continuum approximation will be valid. 

A simplified version of the argument has been given by Mould (1984).2 With the above assumption about star formation efficiency, in a continuum approximation,... 

Local exchange-correlation functionals such as generalized gradient approximations (GGA) are continuum approximations, which can, at best, average over the discontinuity. In regions where the HOMO and LUMO are significant, they provide an approximate average description [39—41] ... 

DR. P. P. SCHMIDT, (Oakland University) I m working on lithium. That s a pretty simple system, especially without its single 2s electron. And yet it s still complicated. There are several problems. One is how to model the effect of the solvent and whether, in fact, you want to throw away the modeling that has been used, namely the continuum approximation. I think the answer to that is no, you don t want to throw it away completely. But one of the notions which has been developing, and is now being tested in the thermodynamics of solvation and other problems, is what is called the semi-continuum model. In... 

In the continuum approximation of FvdM the microscopic energy can be written ... 

The validity of Equation (21) requires that the lattice must be treated as a continuum in the evaluation of the angular dependence of electric field gradients. It has also been. shown 97) that the proper factor to use in computing the electric field gradient produced by a point defect, taking into account the effect of polarization of the medium, is (2e + 3)/5 , where e is the dielectric constant of the medium. This result is based on the continuum approximation for the solid. 

As a continuum approximation, this approach should break down by the atomistic level. For islands it is presumably inappropriate for the small clusters imaged with FIM. More importantly, in many cases the stiffness may not be nearly anisotropic, as we have assumed it to be in our analysis. Then, as perhaps for Ag( 100) islands, new mechanisms may play a role. For vacancy clusters, there can be trapping in corners in systems that might seem to be cases of PD from consideration of vicinal surfaces. 

Solvent effects beyond continuum approximation can be studied. 

All of the continua discussed dius far have been isotropic in nature. An interesting question arises as to the ability of die continuum approximation to model non-isotropic media. In... 

In practice, the energy spacings Aei type (if sufficiently small) and temperature T (if sufficiently high) may allow use of classical high-T/continuum approximations to the actual quantum level distribution, by replacing the discrete quantum summation (13.78) with a classical integral. [For further discussion, see D. A. McQuarrie. Statistical Mechanics (Harper Row, New York, 1976).]... 

The classical continuum approximation (13.80) becomes questionable for light nuclei, where inertial moments are reduced and quantum rotational spacings proportionally increased, and in this case the quantum sum over angular momentum states may be substituted. Furthermore, the treatment assumes sufficient free volume for unhindered rotations, and is therefore only appropriate at the lower-density conditions of gaseous reactions. 

For the vibrational term qivib, a classical high-T continuum approximation is seldom valid, and evaluation of the discrete sum over states is therefore required over the quantum vibrational distribution. (As pointed out in Sidebar 5.13, accurate treatment of molecular vibrations is crucial for accurate assessment of entropic contributions to AGrxn.) A simple quantum mechanical model of molecular vibrations is provided by the harmonic oscillator approximation for each of the 3N — 6 normal modes of vibration of a nonlinear polyatomic molecule of N atoms (cf. Sidebar 3.8). In this case, the quantum partition function can be evaluated analytically as... 

The photoionization balance of Call in the SN1970g on 270th day is calculated in the "two level (4s and 3d) plus continuum" approximation. A lower limit for P and the solar ratio for Ca/H are adopted. 

As derived, Xq is a free energy change and so it includes entropic contributions which may arise, for example, from changes in the number of solvent orientational states before and after electron transfer occurs. In the dielectric continuum approximation, entropic effects are included through the temperature dependencies of the macroscopic dielectric quantities Dop and Ds. 

From the expression for ket in equation (55), assuming the appropriateness of the dielectric continuum approximation so the AE can be replaced by AGet0, the rate constant for electron transfer in the classical limit is given by equation (59). 

Paper [109] determined the value of Uo upon approach to the steady state from above. One-dimensional crystals were simulated of length from 8 x 103 to 2 x 104ao (ao is a lattice constant the spatial correlation in genetic pairs is neglected). The limiting values Uo = 3.5-3.6 for 500 and 700 sites in the recombination sphere (Table 7.3, third column) are close to the value 3.43 obtained in the continuum approximation by an approximate method [22] and considerably exceed the estimate 1.36 implied by the approach based on many-point densities in the linear approximation [31, 111] remember that... 

In the previous chapter, we have seen how Born s simple and successful idea of a dielectric continuum approximation for the description of solvation effects has been developed to a considerable degree of perfection. Almost all workers in this area have been trying to obtain more efficient and more precise methods for the solution of dielectric boundary conditions combined with molecular electrostatics, but the question of the validity of Born s basic assumption has rarely been discussed. This will be done in the following sections, with a surprising result. 

The mechanisms of reactions that occur in condensed phases involve the participation of solvent degrees of freedom. In some cases, such as in certain ion association reactions involving solvent-separated ion pairs, even the very existence of reactant or product states depends on the presence of the solvent. Traditionally the solvent is described in a continuum approximation by reaction-diffusion equations. Kapral s group is interested in microscopic theories that, by treating the solvent at a molecular level, allow one to investigate the origin and range of validity of conventional continuum theories and to understand in a detailed way how solvent motions influence reaction dynamics. 

The applicability of the preceding pseudocontinuum approach to convective heat transfer of gas-solid systems without heat sources depends not only on the validity of the phase continuum approximation but also on the appropriateness of the local thermal equilibrium assumption. The local thermal equilibrium may be assumed only if the particle-heating... 

It is worth pointing out that the solvation response functions for realistic models of SD chromophores resemble much more closely the TCFs that represent the low- longitudinal dielectric response of the solvent than the corresponding transverse response [18,67], This is true for the continuum approximations described in the previous section as well as for the results of MD simulations in which the solvent response is treated at the molecular level. 

Cell to Cell Transport Phenomenology. Consider a mass of cells in an external medium denoted "e" as in Fig. 1. We shall construct a continuum approximation wherein the cells are "smeared out." Such a theory will be accurate when the length scale of interest (i.e., the pattern length) is greater than a cell diameter. An important aspect of the formulation is that the equations derived by taking the continuum limit will relate effective transport rates directly to measurable properties of individual cells or pairs of cells in contact. 

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