Infinite value of the

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

For finite, rather than infinite, values of the dimensionless Newtonian cooling time, the stationary-state condition is given by eqn (7.21). Thus, even with the exponential approximation, both R and L involve the residence time. The correspondence between tangency and ignition or extinction still holds,... 

So far we have considered an infinite value of the gas-to-particle heat and mass transfer coefficients. One may encounter, however, an imperfect access of heat and mass by convection to the outer geometrical surface of a catalyst. Stated in other terms, the surface conditions differ from those in the bulk flow because external temperature and concentration gradients are established. In consequence, the multiple steady-state phenomena as well as oscillatory activity depend also on the Sherwood and Nusselt numbers. The magnitudes of the Nusselt and Sherwood numbers for some strongly exothermic reactions are reported in Table III (77). We may infer from this table that the range of Sh/Nu is roughly Sh/Nu (1.0, 104). 

Within the accuracy of the trapezoidal rule integration and of the graphical determination of the number of trays, the numbers 16.2 and 15.4 are substantially the same. The infinite value of the ratio of mass transfer coefficients kL/ka means that all of the resistance to mass transfer is in the gas film ... 

For n > 2, the Gegenbauer functions of the second kind are infinite at the points = 1, which correspond to 6 = 0 and 9 = it. Therefore, if there are no singularities in the physical statement of the problem, then all the constants with tildes in (2.1.5) must be zero. Moreover, for n = 0 and n = 1, the remaining constants, if nonzero, result in infinite values of the tangential velocity Vg on... 

The slope at the beginning of the stage of absorption is oblique in Figures 7.6 and 7.7. This is due to the finite coefficient of convective transfer on the surface of the rubber discs [4-6,19]. It should be recalled that a vertical tangent is associated with an infinite value of the coefficient h in... 

There is no such a thing in nature as infinitely steep potential energy walls or infinite values of the potential energy (as in the particle-in-a-box problem). This means we should treat such idealized cases as limit cases of possible continuous potential energy functions. From the Schrddinger equation = Ei/r — V r, we see... 

Fig.3. A plot of the derivative (with sign inversion) of the solid line in Fig.2., expressing the distribution of centers associated with pools of a given size, on an electron basis (thus twice the PQ/Q stoichiometry). This assumes an infinite value of the equilibrium constant. Fig.4. Computed sequences, using Kok model, and assuming an equilibrium constant of 70 between Q and pools of variable size. On an electron basis, the pool sizes are 4,8, 12,...28. Other parameters S0=0.1, Sl=0.9 (initially), 0.08 misses, and 0.03 double hits.

Ultraviolet divergences appear for infinite values of the integration moments q. For example, in Figure 5.42, the contribution of such integration is proportional to... 

The height of the desorption peak increases and the width of the peak decreases with a rise in a. At a = 2, as follows from the corresponding equations, the peak on the C-E curve should degenerate into a vertical line with an infinite value of the capacitance [3]. 

It should be said that it is too often found in scientific papers, without any relevant proofs, an infinite value of the coefficient of convection, h. This assumption would be responsible for the following abnormal process taking place over a short time, e.g., when the polymer sheet is immersed into a liquid of finite volume, at time 0, the concentration of the diffusing substance on the solid surface, being equal to that in the liquid when the partition factor is 1, decreases abruptly down to 0, and afterwards increases slightly with the concentration of the substance in the liquid. 

Solution of the Equation of Diffusion Sheet of Thickness 2L Immersed in a Liquid of Infinite Volume and Infinite Value of the Coefficient of Convection... 

The value of the concentration on the surface(s) at infinite time, or at equilibrium with the liquid is zero for a liquid of infinite volume with an infinite value of the coefficient of convection of the surface(s), when this liquid is free from diffusing substance ... 

Some profiles of concentration of the diffusing substance developed through the thickness of the sheet of thickness 2L, with -L < x < +L, are shown in Figure 1.4 with an infinite volume of liquid and an infinite value of the coefficient of convection h, showing also the symmetry with regard to the plane x = 0. 

Figure 1.8 Kinetics of transfer of the diffusing substance into (or out from) the sheet with a sheet of thickness 2L and a finite value of h, and an infinite value of the ratio of volumes defined by a. The values of R = h L/D are shown in the figure. These master curves are drawn by using the dimensionless numbers M/M and D-tHJ for the co-ordinates.

An infinite value of the coefficient of convection h can be obtained from the kinetics of release of the additive expressed in terms of time, at the beginning of the process, from the shape of the curve, but a vertical tangent is far from being easy to determine precisely, even by expanding the scale of time. 

It is necessary when reading the instructions of the model to find the equation upon which it is based. For example, a model built only by using Equation 1.34 obtained with an infinite value of the coefficient of convection is a very poor model the accuracy of the value determined for the diffusivity can be appreciated by comparing the kinetics curves obtained either by calculation with the model or by plotting the experimental data. Moreover, the kinetics curves would be expressed preferably either in terms of time or by using the square root of time, and be drawn with an expanded scale of time. 

Diffusion with an Infinite Value of the Coefficient of Convection... 

For an infinite value of the coefficient of convection and low values of time expressed by the ratio M/M < 4, the series in Equation (1.38) vanishes, leading to a simple relationship, as shown already in (1.39), applied to the length a instead of L ... 

This infinite value of the flux is associated with the vertical tangent at the origin of time of the kinetics of transfer of the diffusing substance. 

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