Limiting Cases of the General Equation

Consider the case of low overpotential, often referred to as the micropolarization re on or the linear current-potential region. The exponential terms in Eq. (5.26) can be linearized, with the use of only the first two terms in the Taylor expansion of the exponential term 

This approximation is valid for x 1. It yields the following linear relationship between the current density and the overpotential 

Note that we have derived here Eq. (5.6), which was written intuitively in Section 5.1.1. It is also interesting that the symmetry factor, p, disappeared from Eq. (5.26) in the process of linearization. Thus, the rate of reaction a close to equilibrium does not depend on the detailed shape of the energy barrier for activation, (which determines the value of P). It does, however, depend on the magnitude of the energy of activation, which manifests itself in the value of jo. 

It is interesting to note that in Section 5.1.1 we estimated the linear region to be approximately 5 mV, whereas here we find it to be 20 mV for p = 0.5. This discrepancy arises because in Section 5.1.1 we linearized the function exp(x ), while here we linearize the difference between two exponential terms. For P = 0.5 this is given by sinh( x). Thus, Eq. (5.26) can be written as 

We turn our attention now to the case of high overpotential. One of the two exponential terms in Eq. (5.26) becomes negligible with respect to the other. For a large anodic overpotential one has 

---

Limiting cases of the general convective-diffusion equation are often helpful. If the time dependence is ignored, i.e., 5C/0t = 0 (for example, at low bulk protein concentration, at long times, and/or when the rate of adsorption is much greater than the transport to the surface), then we have... 

We have shown that the Nernst-Planck equation is only a limiting case of the generalized Maxwell-Stefan equations. Nevertheless, many ionic systems of interest are dilute and the Nernst-Planck equation is widely used. 

The performance equations of ideal reactors, which are well known to any reactor engineer, are just the limiting cases of these general mass conservation equations. Possible simplifications of these equations are discussed later in the chapter after discussing all the governing equations and their dimensionless forms. 

Other adsorption isotherms have been used in the literature to analyze experimental data. However, it can usually be shown that they are limiting forms of the general isotherm derived here (equation (10.8.38)). For example, in early work the importance of the term in the diffuse layer potential (zj f ([) ) was not recognized. By using equation (10.8.7) for this contribution, the exact dependence of this term on Qni and Qad is obtained. Equation (10.8.7) can be simplified in limiting cases and the form of the isotherm without an explicit dependence on ([) obtained. 

Most biological processes can be characterized by three types of rate equations, two of which are limiting cases of the third more general rate equation. The first of these is the first order rate equation represented by Equation 1. In this rate process, k represents the first order rate constant and C... 

Except for the limiting case of the irreversible isotherm discussed above the prediction of the temperature and concentration profiles requires the simultaneous solution of the coupled differential heat and mass balance equations which describe the system. The earliest general numerical solutions for a nonisothermal adsorption column appear to have been given almost simultaneously by Carter and by Meyer and Weber. These studies all deal with binary adiabatic or near adiabatic systems with a small concentration of an adsorbable species in an inert carrier. Except for a difference in the form of the equilibrium relationship and the inclusion of intraparticle heat conduction and finite heat loss from the column wall in the work of Meyer and Weber, the mathematical models are similar. In both studies the predictive value of the mathematical model was confirmed by comparing experimental nonisothermal temperature and concentration breakthrough curves with the theoretical curves calculated from the model using the experimental equilibrium... 

As one can note, Eq. (11) is strongly nonlinear due to the presence of the exponential terms, which makes its analytical solution in the general case impractical. However, important limiting forms of the PB equation can be formulated susceptible for analytical handling. 

Wlien describing the interactions between two charged flat plates in an electrolyte solution, equation (C2.6.6) cannot be solved analytically, so in the general case a numerical solution will have to be used. Several equations are available, however, to describe the behaviour in a number of limiting cases (see [41] for a detailed discussion). Here we present two limiting cases for the interactions between two charged spheres, surrounded by their counterions and added electrolyte, which will be referred to in further sections. This pair interaction is always repulsive in the theory discussed here. 

The general equation can be further reduced to the case of infinite dilution limit, a binary mixmre, ionic solutions, and so on. These equations are supplemented by closure relations such as the Percus-Yevick (PY) and hypernetted chain (HNC) approximations. 

Let us now turn to diffusion in the general case, without worrying about the exact mechanism or the rates of diffusion of the various species. As an example to illustrate how we would analyze a diffusion-limited solid state reaction, we use the general equation describing formation of a compound with spinel (cubic) structure and stoichiometry ... 

We will develop an analytical formulation of the statement in Equation (8.2). This will be done for surface flame spread on solids, but it can be used more generally [1], As with the ignition of solids, it will be useful to consider the limiting cases of thermally thin and thermally thick solids. In practice, these solutions will be adequate for first-order approximations. However, the model will not consider any effects due to... 

The use of the excess ligand condition, equation (57), spares the need to consider the continuity equation (52) for the ligand. Then, two limiting cases of kinetic behaviour are particularly simple the inert case and the fully labile case. As we will see, these cases can be treated with the expressions (for transient and steady-state biouptake) developed in Section 2, and they provide clear boundaries for the general kinetic case, which will be considered in Section 3.4. 

After these technical preliminaries, we may consider the solution of the Liouville equation (10). However, we shall not discuss the most general situation but we shall limit ourselves to the special case where ... 

The expressions of the Sections 1.5 and 1.6 are general and apply to any solution of the Schrodinger equation. In the special case of a Morse potential, the radial integrals in Eq. (1.34) can be evaluated, with some approximations, in closed form. The approximation consists in replacing the lower limit of integration by -oo. This approximation is similar to that used in Section 1.3 when obtaining the wave functions. Thus... 

It has been mentioned above that with nonuniform catalyst surfaces, as they mostly occur in practice, the above equations give merely an upper limit of the velocity although in such catalysts the true surface would be expected to be greater than the geometrical one. In nearly all the cases where a reaction has been measured on different catalysts, it has been found that fco is by no means a universal constant as would have to be expected from the simple theory. There is a regular connection between fco and the activation energy, of the general form ... 

---