Limiting forms

If one or more of the products is strongly adsorbed, Eq. XVIII-37 takes on another limiting form, of the type... 

The apparent activation energy is then less than the actual one for the surface reaction per se by the heat of adsorption. Most of the algebraic forms cited are complicated by having a composite denominator, itself temperature dependent, which must be allowed for in obtaining k from the experimental data. However, Eq. XVIII-47 would apply directly to the low-pressure limiting form of Eq. XVIII-38. Another limiting form of interest results if one product dominates the adsorption so that the rate law becomes... 

The complete problem with composition gradients as well as a pressure gradient, may be regarded as a "generalized Poiseuille problem", and its Solution would be valuable for comparison with the limiting form of the dusty gas model for small dust concentrations. Indeed, it is the "large diameter" counterpart of the Knudsen solution in tubes of small diameter. 

Chapter 5. ALGEBRAIC MANIPULATIONS AND LIMITING FORMS OF THE DUSTY GAS... 

Let us now turn attention to situations in which the flux equations can be replaced by simpler limiting forms. Consider first the limiting case of dilute solutions where one species, present in considerable excess, is regarded as a solvent and the remaining species as solutes. This is the simplest Limiting case, since it does not involve any examination of the relative behavior of the permeability and the bulk and Knudsen diffusion coefficients. 

The limiting form of the flux equations for large pore diameters or high pressure is best approached starting from equations (5.7) and (5.8). 

This determines the total flux at the li/nic of viscous flow. Equations (5.18 and (5.19) therefore describe the limiting form of the dusty gas model for high pressure or large pore diameters -- the limit of bulk diffusion control and viscous flow,... 

The opposite limiting form of the general flux relations, for small values of a and/or p, is most simply obtained starting from equation (5.A). At this limit it follows from equations (5.15), (5.16) and (5.17) that... 

Finally, before leaving our exploration of the dusty gas model, we must compare the large pore (or high pressure) limiting form of its flux relations with the corresponding results derived in Chapter 4 by detailed solution of the continuum equations in a long capillary. The relevant equations are (4,23) and (4,25), to be compared with the corresponding scalar forms of equations (5.23) and (5.24). Equations (4.25) and (5.24).are seen to be identical, while (4,23) and (5.23) differ only in the pressure diffusion term, which takes the form... 

It remains to check Chat our supposedly general equations (11.20) -(11.23) reduce to the previously obtained limiting forms at the extremes of bulk diffusion and Knudsen diffusion. The bulk diffusion limit is easily obtained by letting — and assuming that for all r, s,... 

The opposlCe limiting form is obtained by letting and assuming... 

Equations (11.74) and (11.75) provide the required limiting form of the flux relations. 

When bulk diffusion controls and the d Arcy permeability is large, corresponding to pores of large diameter, the flux relations for a binary mixture reduce to a limiting form given by equation (3.29) and its companion obtained by interchanging the suffixes 1 and 2, namely... 

Equation (12-31) assumes that Df is constant however, Df is rarely constant but varies with moisture content, temperature, and humidity. For long diying times, Eq. (12-31) simphfies to a limiting form of the diffusion equation as... 

A derivation for particle-phase diffusion accompanied by fluid-side mass transfer has been carried out by Rosen []. Chem. Phy.s., 18,1587 (1950) ibid., 20, 387 (1952) Jnd. Eng. Chem., 46,1590 (1954)] with a limiting form a.t N > 50 of... 

If the products of electrolysis favor other types of corrosion, electrochemical protection processes should not be applied or should be used only in a limited form. Hydrogen and OH ions are produced in cathodic protection according to Eq. (2-19). The following possible corrosion danger must be heeded ... 

Several proposals have been made to fit the borderline reactions into a well-defined mechanistic scheme. Most of these adopt one of two viewpoints either (1) borderline substrates undergo concurrent SnI and Sn2 processes, with the particular system determining which mechanism, if either, predominates or (2) all Sn reactions are related by essentially the same mechanism, which differs from case to case in the detailed disposition of electrons in the transition state. In this view pure SnI and Sn2 processes are merely the extreme limiting forms of a single mechanism, and the borderline mechanism is a merged process having some features of both. 

The goal of kinetic theory is to find the time-evolution equation for f x,v,t), with the ultimate aim being to obtain the equilibrium properties of the system through the limiting form of f x, v,t) as t cx>. 

The literature distinguishes two limiting forms of template polymerization.192 ... 

Several limiting forms result from this equation, depending on the relative values of each of the terms in the denominator. In these schemes, product binding has been ignored. If that is not the case, additional denominator terms appear. 

Thus, there are two transition states, with compositions of [TlFe54] and [TlFe4+]. Note the inverse dependence on [Fe3+] in one limiting form. This, as amplified in Rule 5, signals that Fe3+ is the product of the first step. On this basis, we write the following scheme ... 

In the Fe2+-Tl3+ reaction, the limiting form attained when [Fe2+] < A [Fe3+] showed an inverse dependence on [Fe3+], In other words, [Fe3+] was so high that the first step produced so low a [Tl2+] as to make the second step rate-controlling. Under these conditions, the first step is a rapid prior equilibrium. 

Acid catalysis. Consider the reverse of the scheme written in Eqs.(10-37)-(10-38). Derive the rate law for the reverse reaction, and discuss different limiting forms as to the type of acid catalysis demonstrated. 

The hard-core limiting forms of U(r) do not lead to physically acceptable results. We conclude that this is caused by a complete neglect of the effect of the attractive forces on the slope of the repulsive part in U(r). If the interaction energy is assumed as the sum of a Morse exponential function and the polarization energy evaluated at r = r°, the resulting transition probabilities appear useful for analyzing ion-molecule collisions. 

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