Interfacial transfer steady diffusion

The driving force for the transfer process was the enhanced solubility of Br2 in DCE, ca 40 times greater than that in aqueous solution. To probe the transfer processes, Br2 was recollected in the reverse step at the tip UME, by diffusion-limited reduction to Br . The transfer process was found to be controlled exclusively by diffusion in the aqueous phase, but by employing short switching times, tswitch down to 10 ms, it was possible to put a lower limit on the effective interfacial transfer rate constant of 0.5 cm s . Figure 25 shows typical forward and reverse transients from this set of experiments, presented as current (normalized with respect to the steady-state diffusion-limited current, i(oo), for the oxidation of Br ) versus the inverse square-root of time. 

FIG. 24 Steady-state diffusion-limited current for the reduction of oxygen in water at an UME approaching a water-DCE (O) and a water-NB (A) interface. The solid lines are the characteristics predicted theoretically for no interfacial kinetic barrier to transfer and for y = 1.2, Ke = 5.5 (top solid curve) or y = 0.58, Ke = 3.8 (bottom solid curve). The lower and upper dashed lines denote the current-distance characteristics for the situation where there is no interfacial transfer and where transfer occurs without limitations from diffusion in the organic phase. (Adapted from Ref. 9. Copyright 1998 American Chemical Society.)... 

We will later see when discussing the dry deposition (Chapter 4.4.1) that a similar conception is applied to explain the partial conductance the first term on the right side in Eq. (4.302) denotes the resistance of diffusion and the second the interfacial transfer. The steps that follow after interfacial transfer are the (fast) salvation and/or protolysis reactions until reaching the equilibrium, the diffusion within the droplet (until reaching steady-state concentrations or, in other words, a well-mixed droplet) and finally the aqueous phase chemical reactions. Let us first consider a pseudo-first-order reaction ... 

The fllm theory is the simplest model for interfacial mass transfer. In this case it is assumed that a stagnant fllm exists near the interface and that all resistance to the mass transfer resides in this fllm. The concentration differences occur in this film region only, whereas the rest of the bulk phase is perfectly mixed. The concentration at the depth I from the interface is equal to the bulk concentration. The mass transfer flux is thus assumed to be caused by molecular diffusion through a stagnant fllm essentially in the direction normal to the interface. It is further assumed that the interface has reached a state of thermodynamic equilibrium. The mass transfer flux across the stagnant film can thus be described as a steady diffusion flux. It can be shown that within this steady-state process the mass flux will be constant as the concentration profile is linear and independent of the diffusion coefficient. Consider a gas-liquid interface, as sketched in Fig. 5.16. The mathematical problem is to formulate and solve the diffusion flux equations determining the fluxes on both sides of the interface within the two films. The resulting concentration profiles and flux equations can be expressed as ... 

Given that, under the defined conditions, there is no interfacial kinetic barrier to transfer from phase 2 to phase 1, the concentrations immediately adjacent to each side of the interface may be considered to be in dynamic equilibrium throughout the course of a chronoamperometric measurement. For high values of Kg the target species in phase 2 is in considerable excess, so that the concentration in phase 1 at the target interface is maintained at a value close to the initial bulk value, with minimal depletion of Red in phase 2. Under these conditions, the response of the tip (Fig. 11, case (a)] is in agreement with that predicted for other SECM diffusion-controlled processes with no interfacial kinetic barrier, such as induced dissolution [12,14—16] and positive feedback [42,43]. A feature of this response is that the current rapidly attains a steady state, the value of which increases... 

The effect of increasing y is to increase the diffusion coefficient of the solute in phase 2 compared to that in phase 1. For a given value of this means that when a SECMIT measurement is made, the higher the value of y the less significant are depletion effects in phase 2 and the concentrations at the target interface are maintained closer to the initial bulk values. Consequently, as y increases, the chronoamperometric and steady-state currents increase from a lower limit, characteristic of an inert interface, to an upper limit corresponding to rapid interfacial solute transfer, with no depletion of phase 2. 

For comparable diffusion coefficients of the target solute in the two phases and nonlimiting transfer kinetics, systems characterized by different should be resolvable on the basis of transient and steady-state current responses to a value of up to 50 at practical tip-interface separations. If the diffusion coefficient in phase 2 becomes lower than that in phase 1, diffusion in phase 2 will be partly limiting at even higher values of K. On the other hand, as the value of y increases or interfacial kinetics become increasingly limiting, lower values of suffice for the constant-composition assumption for phase 2 to be valid. 

Solution of the transport problem when the process is controlled by both the interfacial electron transfer and the steady-state or linear diffusion of reactants was derived by Samec [181, 182]. These results represent the basis for the kinetic analysis, e.g., in dc polarography or convolution and potential sweep voltammetry. Under the conditions of steady-state diffusion, Eq. (60) can be transformed into a dimensionless form [181],... 

The analysis can be significantly simplified by reahzing that the rate with which the vorticity diffuses inwards, and hence establishes the fluid motion, is represented by the kinematic viscosity coeflBcient, which is of the order of 10 cmVsec and is at least one order of magnitude greater than the droplet surface regression rate. Hence quasi-steadiness for both the gas and liquid motion, with a stationary droplet surface and constant interfacial heat and mass flux, can be assumed. Once the fluid mechanical aspect of the problem is solved, the transient liquid-phase heat and mass transfer analyses, with a regressing droplet surface, can be performed. 

This section contains a simple introduction to steady state and unsteady species mole (mass) diffusion in dilute binary mixtures. First, the physical interpretations of these diffusion problems are given. Secondly, the physical problem is expressed in mathematical terms relating the concentration profiles to the diffusion fluxes. Emphasis is placed on two diffusion problems that form the basis for the interfacial mass transfer modeling concepts used in reaction engineering. 

When the adsorption/desorption kinetics are slow compared to the rate of diffusional mass transfer through the tip/substrate gap, the system responds sluggishly to depletion of the solution component of the adsorbate close to the interface and the current-time characteristics tend towards those predicted for an inert substrate. As the kinetics increase, the response to the perturbation in the interfacial equilibrium is more rapid and, at short to moderate times, the additional source of protons from the induced-desorption process increases the current compared to that for an inert surface. This occurs up to a limit where the interfacial kinetics are sufficiently fast that the adsorption/desorption process is essentially always at equilibrium on the time scale of SECM measurements. For the case shown in Figure 3 this is effectively reached when Ka = Kd= 1000. In the absence of surface diffusion, at times sufficiently long for the system to attain a true steady state, the UME currents for all kinetic cases approach the value for an inert substrate. In this situation, the adsorption/desorption process reaches a new equilibrium (governed by the local solution concentration of the target species adjacent to the substrate/solution interface) and the tip current depends only on the rate of (hindered) diffusion through solution. 

Figure 16a and b shows the effect of L on the radial dependence of the steady-state concentration and flux at the substrate/solution interface for a first-order dissolution process characterized by Ki = 10 and L = 0.1, 0.32, and 1.0. As the tip-substrate separation decreases, the effective rate of diffusion between the probe and the surface increases, forcing the crystal/so-lution interface to become more undersaturated. Conversely, as the UME is retracted from the substrate, the interfacial undersaturation approaches the saturated value, since the solution mass transfer coefficient decreases compared to the first-order dissolution rate constant. Movement of the tip electrode away from the substrate also has the effect of promoting radial diffusion, and consequently the area of the substrate probed by the UME increases. 

Thus during the steady-state stage of isothermal mass transfer the cube of average radius grows linearly with time the growth rate is a function of interfacial tension, as well as of the solubility and diffusion coefficient of dispersed substance. If an admixture that is nearly insoluble in a continuous phase is introduced into the dispersion phase, a sharp decrease in recondensation rate, as well as changes in laws describing the process, may take place. 

We now recognize k as the ratio of kf to the steady-state mass-transfer coefficient niQ = DoIrQ. When k 1, the interfacial rate constant for reduction is very small compared to the effective mass-transfer rate constant, so that diffusion imposes no limitation on the current. At the opposite limit, where k >> 1, the rate constant for interfacial electron transfer greatly exceeds the effective rate constant for mass transfer, but the interpretation of this fact depends on whether k is also large. ... 

To analyze laboratory uptake data it is necessary to determine how y depends on the other parameters of the system. Let us assume that potentially any or all of gas-phase diffusion, interfacial transport, and aqueous-phase diffusion may be influential. We assume steady-state conditions and that species A is consumed by a first-order aqueous phase reaction (12.101). At steady state the rate of transfer of species A across the gas-liquid interface, given by (12.1 IS), must be equal to that as a result of simultaneous aqueous-phase diffusion and reaction, (12.112) ... 

In the absence of convective mass transfer and chemical reaction, calculate the steady-state liquid-phase mass transfer coefficient that accounts for curvature in the interfacial region for cylindrical liquid-solid interfaces. An example is cylindrical pellets that dissolve and diffuse into a quiescent liquid that surrounds each solid pellet. The appropriate starting point is provided by equation (B) in Table 18.2-2 on page 559 in Bird et al. (1960). For one-dimensional diffusion radially outward, the mass transfer equation in cylindrical coordinates reduces to... 

A similar model that specifically considers the poison deposition in a catalyst pellet was presented by Olson [5] and Carberry and Gorring [6], Here the poison is assumed to deposit in the catalyst as a moving boundary of a poisoned shell surrounding an unpoisoned core, as in an adsorption situation. These types of models are also often used for noncatalytic heterogeneous reactions, which was discussed in detail in Chapter 4. The pseudo-steady-state assumption is made that the boundary moves rather slowly compared to the poison diffusion or reaction rates. Then, steady-state diffusion results can be used for the shell, and the total mass transfer resistance consists of the usual external interfacial, pore diffusion, and boundary chemical reaction steps in series. 

The NEQ model requires thermodynamic properties, not only for calculation of phase equilibrium but also for calculation of driving forces for mass transfer. In addition, physical properties such as surface tension, diffusion coefficients, and viscosities, for calculation of mass (and heat) transfer coefficients and interfacial areas are required. The steady-state model equations most often are solved using Newton s method or by homotopy-continuation. A review of early applications of NEQ models is available [5]. 

Thus, the interfacial mass and heat transfer fluxes are composed of a laminar, steady state diffusion term in the stagnant film and a convection term from the complete turbulent bulk phase (Taylor and Krishna, 1993),... 

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