Diffusion, equimolar

Diffusivity measures the tendency for a concentration gradient to dissipate to form a molar flux. The proportionality constant between the flux and the potential is called the diffusivity and is expressed in m /s. If a binary mixture of components A and B is considered, the molar flux of component A with respect to a reference plane through which the exchange is equimolar, is expressed as a function of the diffusivity and of the concentration gradient with respect to aji axis Ox perpendicular to the reference plane by the fpllqvving relatipn 6 /... 

Condition (iv). For equimolar counter diffusion in a binary mixture, the... 

Equimolar Counterdiffusion in Binary Cases. If the flux of A is balanced by an equal flux of B in the opposite direction (frequently encountered in binary distillation columns), there is no net flow through the film and like is directly given by Fick s law. In an ideal gas, where the diffusivity can be shown to be independent of concentration, integration of Fick s law leads to a linear concentration profile through the film and to the following expression where (P/RT)y is substituted for... 

Multicomponent Diffusion. In multicomponent systems, the binary diffusion coefficient has to be replaced by an effective or mean diffusivity Although its rigorous computation from the binary coefficients is difficult, it may be estimated by one of several methods (27—29). Any degree of counterdiffusion, including the two special cases "equimolar counterdiffusion" and "no counterdiffusion" treated above, may arise in multicomponent gas absorption. The influence of bulk flow of material through the films is corrected for by the film factor concept (28). It is based on a slightly different form of equation 13 ... 

Equimolar Counterdiffusion. Just as unidirectional diffusion through stagnant films represents the situation in an ideally simple gas absorption process, equimolar counterdiffusion prevails as another special case in ideal distillation columns. In this case, the total molar flows and are constant, and the mass balance is given by equation 35. As shown eadier, noj/g factors have to be included in the derivation and the height of the packing is... 

General Situation. Both unidirectional diffusion through stagnant media and equimolar diffusion are idealizations that ate usually violated in real processes. In gas absorption, slight solvent evaporation may provide some counterdiffusion, and in distillation counterdiffusion may not be equimolar for a number of reasons. This is especially tme for multicomponent operation. 

Xm are not. For unimolecular diffusion through stagnant gas = 1), and reduce to T and X and and reduce to and equation 64 then becomes equation 34. For equimolar counterdiffusion = 0, and the variables reduce tojy, x, G, and F, respectively, and equation 64 becomes equation 35. Using the film factor concept and rate equation 28, the tower height may be computed by... 

In dilute systems the logarithmic-mean insoluble-gas and nonvolatile-hquid concentrations approach unity, and Eq. (5-261) reduces to the dilute-system formula. For equimolar counter diffusion (e.g., binary distillation), these log-mean factors should be omitted. See Eq. (5-189). 

Effective diffusivities for these ions in equimolar concentration ratio and with various inert electrolytes, have been determined by several methods (see Table III). The mobility products obtained from capillary cell (stagnant diffusion) and rotating-disk measurements are in fairly good agreement. 

The current peaks observed when fast potential ramps are applied appear similar to the one shown in Fig. 8. The peak currents (triangles in Fig. 10) can be satisfactorily interpreted in terms of a pure-diffusion model (S10) in equimolar ferri-ferrocyanide solution,... 

The combined diffusivity is, of course, defined as the ratio of the molar flux to the concentration gradient, irrespective of the mechanism of transport. The above equation was derived by separate groups working independently (8-10). It is important to recognize that the molar fluxes (Ni) are defined with respect to a fixed catalyst pellet rather than to a plane of no net transport. Only when there is equimolar counterdiffusion, do the two types of flux definitions become equivalent. For a more detailed discussion of this point, the interested readers should consult Bird, Stewart, and Lightfoot (11). When there is equimolal counterdiffusion NB = —NA and... 

In this equation the entire exterior surface of the catalyst is assumed to be uniformly accessible. Because equimolar counterdiffusion takes place for stoichiometry of the form of equation 12.4.18, there is no net molar transport normal to the surface. Hence there is no convective transport contribution to equation 12.4.21. Let us now consider two limiting conditions for steady-state operation. First, suppose that the intrinsic reaction as modified by intraparticle diffusion effects is extremely rapid. In this case PA ES will approach zero, and equation 12.4.21 indicates that the observed rate per unit mass of catalyst becomes... 

Alkenes are scavengers that are able to differentiate between carbenes (cycloaddition) and carbocations (electrophilic addition). The reactions of phenyl-carbene (117) with equimolar mixtures of methanol and alkenes afforded phenylcyclopropanes (120) and benzyl methyl ether (121) as the major products (Scheme 24).51 Electrophilic addition of the benzyl cation (118) to alkenes, leading to 122 and 123 by way of 119, was a minor route (ca. 6%). Isobutene and enol ethers gave similar results. The overall contribution of 118 must be more than 6% as (part of) the ether 121 also originates from 118. Alcohols and enol ethers react with diarylcarbenium ions at about the same rates (ca. 109 M-1 s-1), somewhat faster than alkenes (ca. 108 M-1 s-1).52 By extrapolation, diffusion-controlled rates and indiscriminate reactions are expected for the free (solvated) benzyl cation (118). In support of this notion, the product distributions in Scheme 24 only respond slightly to the nature of the n bond (alkene vs. enol ether). The formation of free benzyl cations from phenylcarbene and methanol is thus estimated to be in the range of 10-15%. However, the major route to the benzyl ether 121, whether by ion-pair collapse or by way of an ylide, cannot be identified. 

First, consider the gradient of cA. Since A is consumed by reaction inside the particle, there is a spontaneous tendency for A to move from the bulk gas (cAg) to the interior of the particle, first by mass transfer to the exterior surface (cAj) across a supposed film, and then by some mode of diffusion (Section 8.5.3) through the pore structure of the particle. If the surface reaction is irreversible, all A that enters the particle is reacted within the particle and none leaves the particle as A instead, there is a counterdiffusion of product (for simplicity, we normally assume equimolar counterdiffusion). The concentration, cA,at any point is the gas-phase concentration at that point, and not the surface concentration. 

Often, to simplify the measurement of n, the current recorded on the process under examination is compared with that given, under the same experimental conditions, by an equimolar amount of a species whose n is known. This is really a good and simple method, but it requires that the compared species possess a comparable diffusion coefficient D (i.e. in first instance, the species used for comparison must possess a molecular weight roughly similar to that of the species under examination). 

In a mixture of two gases A and B provided that the total pressure, and hence the total molar concentration is constant, d[A]/dx and d[B]/dx are equal and opposite and A and B tend to diffuse in opposite directions. In equimolar counter-diffusion, the two components diffuse at equal and opposite rates =- ba- Equation (108) can then be integrated... 

For small values of [A], this reduces to eqn. (110) which was derived from equimolar counter-diffusion and thus, at low concentration, the effect of bulk flow can be neglected. 

For convenience, let the flux of A within the ash layer be expressed by Fick s law for equimolar counterdiffusion, though other forms of this diffusion equation will give the same result. Then, noting that both (2a dCJdr are positive, we have... 

In the specific instance of A diffusing through a catalytic pore and reacting at the end of the pore to form gaseous component B, equimolar counterdiffusion can be assumed, and an effective transition region diffusivity, D a is independent of concentration and can be calculated from the Knudsen and binary diffusion coefficients ... 

This simplified diffnsivify is sometimes used for diffusion in porous cafalysfs even when equimolar counterdiffusion is nol occurring. This greafly simplifies fhe equations. When no reactions are occurring, the diffusivity is a function of concentration (in terms of mole fraction, xa) ... 

Combiue your iuformatiou to estimate the trausitioual diffusivity for uitrogeu,. You can use au average couceutratiou for uitrogeu iu your calculatiou. How does it compare to the two diffusivities you calculated iudividually How does it compare to the transitional diffusivity if there were equimolar counterdiffusion—that is reaction at the end of the pore ... 

It should be noted that D g is a property of the mixture of A and B, and is defined with reference to the mixture and not to the fixed coordinates. Except in the case of equimolar counter-diffusion of A and B, the diffusion of A would result in the movement of the mixture as a whole. However, in the usual case where the concentration of A is small, the value of ) g is practically equal to the value defined with reference to the fixed coordinates. 

It is easy to define the number of transfer units for equimolar counter-diffusion (distillation) and for dilute systems, when the mole fraction of the solute is so small that the... 

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