Reaction plane

Any fast reaction can enhance mass transfer. Consider a very fast, second-order reaction between the gas-phase component A and a liquid component B. The concentration of B will quickly fall to zero in the vicinity of the freshly exposed surface and a reaction plane, within which b = Q, will gradually move away from the surface. If components A and B have similar liquid-phase diflusivities, the enhancement factor is... 

The first controversial point in this mechanism is the nature of the reaction planes where the precursor formation and the ET reaction take place. Samec assumed that the ET step occurs across an ion-free layer composed of oriented solvent molecules [1]. By contrast, Girault and Schiffrin considered a mixed solvent region where electrochemical potentials are dependent on the position of the reactants at the interface [60]. From a general perspective, the phenomenological ET rate constant can be expressed in terms of... 

The non-steady-state optical analysis introduced by Ding et al. also featured deviations from the Butler-Volmer behavior under identical conditions [43]. In this case, the large potential range accessible with these techniques allows measurements of the rate constant in the vicinity of the potential of zero charge (k j). The potential dependence of the ET rate constant normalized by as obtained from the optical analysis of the TCNQ reduction by ferrocyanide is displayed in Fig. 10(a) [43]. This dependence was analyzed in terms of the preencounter equilibrium model associated with a mixed-solvent layer type of interfacial structure [see Eqs. (14) and (16)]. The experimental results were compared to the theoretical curve obtained from Eq. (14) assuming that the potential drop between the reaction planes (A 0) is zero. The potential drop in the aqueous side was estimated by the Gouy-Chapman model. The theoretical curve underestimates the experimental trend, and the difference can be associated with the third term in Eq. (14). 

Figure 7 Stagnant film model (top) and reaction plane model (bottom) for micelle-facilitated dissolution.

Mathematical approaches used to describe micelle-facilitated dissolution include film equilibrium and reaction plane models. The film equilibrium model assumes simultaneous diffusive transport of the drug and micelle in equilibrium within a common stagnant film at the surface of the solid as shown in Figure 7. The reaction plane approach has also been applied to micelle-facilitated dissolution and has the advantage of including a convective component in the transport analysis. While both models adequately predict micelle-facilitated dissolution, the scientific community perceives the film equilibrium model to be more mathematically tractable, so this model has found greater use. 

The limitation of using such a model is the assumption that the diffusional boundary layer, as defined by the effective diffusivity, is the same for both the solute and the micelle [45], This is a good approximation when the diffusivities of all species are similar. However, if the micelle is much larger than the free solute, then the difference between the diffusional boundary layer of the two species, as defined by Eq. (24), is significant since 8 is directly proportional to the diffusion coefficient. If known, the thickness of the diffusional boundary layer for each species can be included directly in the definition of the effective diffusivity. This approach is similar to the reaction plane model which has been used to describe acid-base reactions. 

The reaction plane model with heterogeneous reactions was discussed at length for acid-base reactions in the previous section. The same modeling technique, of confining the reactions to planes, can be applied to micelle-facilitated dissolution. As with the acid-base model, one starts with a one-dimensional steady-state equation for mass transfer that includes diffusion, convection, and reaction. This equation is then applied to the individual species i, i.e., the solute, s, the micelle, m, and the drug-loaded micelle, sm, to yield... 

Two extreme cases of equation 9.2-22 or -22a or -22b arise, corresponding to gas-film control and liquid-film control, similar to those for mass transfer without chemical reaction (Section 9.2.2). The former has implications for the location of the reaction plane (at distance 8 from the interface in Figure 9.6) and the corresponding value of CB. These points are developed further in the following two examples. 

In Figure 9.6, the position of the reaction plane at distance 8 from the gas-liquid interface is shown for a particular value of cB. If cB changes (as a parameter), the position of the reaction plane changes, 8 decreasing as cB increases. This may be realized intuitively from Figure 9.6, or can be shown from equations 9.2-20, -20a, and -22a. Elimination of NA and Nb from these three equations provides a relation between 8 and cB ... 

That is, S decreases as cB increases. 8 can only decrease to zero, since the reaction plane cannot occur in the gas film (species B is nonvolatile). At this condition, the reaction plane coincides with the gas-liquid interface, and pAi, cAi, and cB (at the interface) are all zero. This corresponds to gas-film control, since species A does not penetrate the liquid film. 

Situation 4 very fast reaction, Ha> 3 and Eboundary layer. The hydrogen concentration in the bulk of the liquid falls to zero. Thus, all the catalyst in the bulk is useless. For instantaneous reactions, Ha 3, E=EX and the reaction takes place in a narrow plane located somewhere in the boundary layer the larger Ea0 the closer to the interface the reaction plane. If the limiting enhancement factor E is very high, it is said that the reaction takes place at the gas-liquid interface. Such a case is referred to as surface reaction . 

Case A Instantaneous Reaction with Respect to Mass Transfer. Since an element of liquid can contain either A or B, but not both, reaction will occur at a plane between A-containing and B-containing liquid. Also, since reactants must diffuse to this reaction plane the rate of diffusion of A and B will determine the rate, so that a change in or Cg will move the plane one way or the other (see Fig. 23.5). At steady state the flow rate of B toward the reaction zone will be b... 

Countercurrent Plug Flow of Gas and Solids. Since only one or other reactant can be present at any level in the bed, there will be a sharp reaction plane where the reactants meet. This will occur either at one end or the other of the reactor depending on which feed stream is in excess of stoichiometric. Assuming that each 100 moles of solid combine with 100 moles of gas. Figs. 26.Sa and b show what happens when we feed a little less gas than stoichiometric and a little more than stoichiometric. 

For crosscurrent flow, shown in Fig. 26.96, there will be a definite reaction plane in the solids whose angle depends solely on the stoichiometry and the relative feed rate of reactants. In practice, heat transfer characteristics may somewhat modify the angle of this plane. 

The authors believe that the reaction plane is situated at a greater distance from the electrode surface than the Helmholtz outer plane. 

Regime 5 - instantaneous reactions at an reaction plane developing inside the film For very high reaction rates and/or (very) low mass transfer rates, ozone reacts immediately at the surface of the bubbles. The reaction is no longer dependent on ozone transfer through the liquid film kL or the reaction constant kD, but rather on the specific interfacial surface area a and the gas phase concentration. Here the resistance in the gas phase may be important. For lower c(M) the reaction plane is within the liquid film and both film transfer coefficients as well as a can play a role. The enhancement factor can increase to a high value E > > 3. 

An instantaneous reaction is the fastest reaction possible and no gas is transferred into the liquid bulk. This can be utilized to determine kLa, for example with the reaction of ozone with certain fast-reacting organic compounds. The reaction develops in a reaction plane located either... 

The situation is characterized by the fact that both reactants are entirely consumed, so that cL = c(M) = 0 holds in the plane. Only in the latter case can kLa be determined. In the former case there is no transport of ozone into the liquid film, so that the mass transfer rate is only determined by kaa (Charpentier, 1981). The reaction rate depends on the mass transfer rate of ozone and pollutant to the reaction plane in the liquid film, but not on the reaction rate constant. Whether the reaction develops instantaneously in the liquid film depends on the experimental conditions, especially on the values of the applied ozone partial pressure p 03) and the initial concentration of M c(M)0. For example, the reaction tends toward instantaneous for low p(03) and high c(M)0. 

The first exponential shows the potential dependence of the rate constant upon the measured applied metal—solution potential difference (0m — 0s)- The second exponential is the double layer correction to the rate constant and accounts for the effects of both concentration and potential at the pre-reaction plane. 

It can be shown that, for the reverse reaction, the same correction applies since zr = za n as is expected since the transition state and hence the reaction plane must be the same for both forward and backward processes. 

For the validation purpose, the data from Ref. [100] are used. In this study, absorption experiments were carried out using a baffled vessel operated batch-wise with respect to liquid, and the experimental results were compared with an approximate analytical solution based on the Leveque model. The authors proposed a two-reaction-plane model and achieved a good agreement between theoretical and experimental absorption rates (see Section 9.5.4.5). 

To validate the model, the pH and concentration profiles for all species are calculated in Ref. [70], using the same parameters and conditions as those from Ref. [100]. The concentration profiles clearly demonstrate the rapid depletion of SO2 near the gas-liquid interface, and agree with the existence of two reaction planes suggested elsewhere [100],... 

The film phenomena influence the process variables along the column. The partial pressures of SO2 and CO2 are shown in Fig. 9.24. It can be seen that CO2 is absorbed in the fresh alkaline solution at the column top. Then, as the pH decreases from the top to the bottom due to SO2 absorption, the CO2 concentration increases, and the direction of the CO2 flux at the interface changes (see Fig. 9.25). CO2 desorption also occurs when the concentration in the reaction plane is higher than that at the interface, even if in the bulk it may be lower [70], The co-existence of absorption and desorption phenomena shown in Figures 9.24 and 9.25 is similar to the phenomena discussed elsewhere [104]. Further sensitivity studies regarding the effect of the buffer concentration and SO2 gas concentration can also be found [70]. 

The decrease of the concentration of the electroactive species with increasing potential has to be attributed to double layer effects. As first pointed out by Frumkin [58], in dilute solutions the electron transfer rate is affected by variations of the potential in the double layer in two ways. The potential in the outer Helmholtz plane, fa, is due to the extension of the double layer not identical to the potential in the solution (at the end of the double layer), so that the effective driving force of the reaction is DL — fa. Furthermore, the concentration of ionic reactants in the reaction plane, c, is influenced by electrostatic effects and differs from the concentration just outside the double layer, c0, by a Boltzmann term ... 

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