Exponential nonuniformity

Equation (135) is the well-known Freundlich adsorption isotherm. In a number of instances this isotherm accurately describes experimental data. The interpretation of the Freundlich adsorption isotherm as resulting from exponential nonuniformity of surface is due to Zel dovich 43). 

It was later demonstrated that if the reaction mechanism corresponds to scheme (295) and the linear relation between standard Gibbs energy of adsorption and Gibbs activation energy of adsorption is obeyed [see (91)], then the kinetic (305) corresponds in general to the exponential nonuniformity of the surface with even nonuniformity included as a particular case (44). In the general case the exponent m is not equal to transfer coefficient a, but is connected with it according to (143).5... 

The results of kinetic studies conform with the notion of a reaction proceeding on a evenly or exponentially nonuniform surface in the region of medium coverages as described by the following mechanism ... 

For derivation of adsorption isotherms in the case of exponential nonuniformity, the desorption exponent q=ln b = -lna should be preferably used as an integration variable. 

Time Resolved Fluorescence Depolarization. In Equation 3, it is assumed that the polarization decays to zero as a single exponential function, which is equivalent to assuming that the molecular shape is spherical with isotropic rotational motion. Multiexponential decays arise from anisotropic rotational motion, which might indicate a nonspherical molecule, a molecule rotating in a nonuniform environment, a fluorophore bound to tbe molecule in a manner that binders its motion, or a mixture of fluorophores with different rotational rates. 

Activity-versus-time curves shown in Fig. 25 for alumina-supported Ni and Ni bimetallic catalysts show two significant facts (1) the exponential decay for each of the curves is characteristic of nonuniform pore-mouth poisoning, and (2) the rate at which activity declines varies considerably with metal loading, surface area, and composition. Because of large differences in metal surface area (i.e., sulfur capacity), catalysts cannot be compared directly unless these differences are taken into account. There are basically two ways to do this (1) for monometallic catalysts normalize time in terms of sulfur coverage or the number of H2S molecules passed over the catalysts per active metal site (161,194), and (2) for mono- or bimetallic catalysts compare values of the deactivation rate constant calculated from a poisoning model (113, 195). 

An exaggerated emphasis on heats of chemisorption has probably been harmful in the proper understanding of the role of chemisorption in surface phenomena. Thus the marked nonuniformity of all surfaces with respect to heats of chemisorption has led to rather elaborate treatments where models of surface heterogeneity (statistical distribution of energy sites) or, less successfully, specific forces of interaction between adsorbed species have been invoked to explain the non-Langmuirian adsorption isotherms. For instance the Frumkin isotherm can be obtained with a linear variation of heats of adsorption with coverage, and the Freundlich isotherm is attributed to an exponential variation of heats of adsorption. 

SO that the amount adsorbed, which is proportional to the coverage 0, is a logarithmic function of time. The exponential decrease in rate of sorption with amount sorbed can be explained quite reasonably in terms of an increase in activation energy for chemisorption with increasing coverage. This may arise from interactions between adsorbate molecules which could account for such behavior even on uniform surfaces. Much more likely, however, is the explanation suggested by Halsey that such effects arise from nonuniformity of the surface. 

Electroluminescence is observed to occur during anodization on both n- and p-type materials. The luminescence onp type is uniform on the sample surface, whereas that on n type is highly nonuniform.It occurs only when the oxide reaches a certain thickness as shown in Fig. 3.14. ° No light emission is observed below a thickness of 15 nm. For Si02 greater than 25 nm thick, the intensity of emitted light increases exponentially, the exponential factor being lOnm as shown in Fig. 3.14. 

Nonuniform temperatures, or a temperature level different from that of the surroundings, are common in operating reactors. The temperature may be varied deliberately to achieve optimum rates of reaction, or high heats of reaction and limited heat-transfer rates may cause unintended nonisothermal conditions. Reactor design is usually sensitive to small temperature changes because of the exponential effect of temperature on the rate (the Arrhenius equation). The temperature profile, or history, in a reactor is established by an energy balance such as those presented in Chap. 3 for ideal batch and flow reactors. 

Two examples follow. First, let the transverse compression vary harmonically along m then the constrictive strain rate also varies harmonically. Where is a maximum, d a Jdx is also a maximum and the two contributions to reinforce each other both vary harmonically and are in phase. Second, let the sample consist of two polymorphs of different viscosities as in Chapter 13 then a uniform constrictive strain rate leads to nonuniform transverse compression. Profiles of are exponential in such a way that where (t is less than its remote value, d a Jdx is greater than its remote value. The two contributions to e vary antipathetically and can add up to a constant sum at all points along the cylindroid axis x. ... 

The step function approach typically gives accurate results (in some cases even more accurate than the reaction front approximation). This method, however, involves more algebraic manipulations than the reaction front approximation, and it is quite often advantageous to use the reaction front approximation. The step function approach is particularly useful when the problem at hand involves several different Arrhenius exponentials. In this case, asymptotic analysis may yield nonuniform and even irrelevant results (see the discussion in [55]). 

As mentioned earlier, a very homogeneous and symmetrical target illumination is one of the keys to success in inertial confinement research. This follows from the fact that the interface between the accelerated solid state and the ablating plasma is subject to Rayleigh-Taylor instability, as a heavy material is leaning on the more tenuous plasma. The instability grows exponentially from an initial perturbation caused by nonuniformities. Better than 1% uniform compression reduces the initial perturbation to such a level that, in the final stage, the pellet is still sufficiently compressed. The quest to reach uniform ablation has led to what is known as indirect drive (Lindl 1995). The distinction between this approach and the direct drive is as follows. 

The efficiency and lifetime of the DMFC stack depend on a large number of design and operational parameters stack temperature T is among the most important. The rates of kinetic and transport processes in DMFC rise exponentially with T. Improper thermal management increases current nonuniformity over the stack volume, which may dramatically lower stack performance and lifetime. 

Bieniasz LK (2004) A fourth-order accurate, Numerov-type, three-point finite-difference discretization of electrochemical reaction-diffusion equations on nonuniform (exponentially expanding) grids in one-dimensional space geometry. J Comput Chem 25 1515-1521... 

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