Transport-controlled dissolution

The diffusion layer theory, illustrated in Fig. 15B, is the most useful and best-known model for transport-controlled dissolution. The dissolution rate here is controlled by the rate of diffusion of solute molecules across a diffusion layer of thickness h, so that kT kR in Eq. (40), which simplifies to kx = kT. With increasing distance, x, from the surface of the solid, the concentration, c, decreases from cs at x = 0 to cb at x = h. In general, c is a nonlinear function of x, and the concentration gradient dddx becomes less steep as x increases. The hyrodynamics of the dissolution process has been fully discussed by Levich [104]. In a stirred solution, the flow velocity of the liquid dissolution medium increases from zero at x = 0 to the bulk value at x = h. 

Equation (43) describes the transport-controlled dissolution rate of a solid according to the diffusion layer theory in its simplest form. The mass transfer coefficient here is given by k, = kT = Dlh. 

The viscosity (dynamic, 17, or kinematic, v) and density, p (Eq. 47), influence the dissolution rate if the dissolution is transport-controlled, but not if the dissolution is reaction-controlled. In transport-controlled dissolution, increasing 17 or v will decrease D (Eq. 53), will increase h (Eqs. 46 and 49) and will reduce J (Eqs. 51 and 52). These effects are complex. For example, if an additional solute (such as a macromolecule) is added to the dissolution medium to increase 17, it may also change p and D. The ratio of 17/p = v (Eq. 47) and D directly influence h and J in the rotating disc technique, while v directly influences the Reynolds number (and hence J) for transport-controlled dissolution in general [104]. 

Since the diffusion coefficient can be measured, the prediction of transport-controlled dissolution rates depends on a calculation of 6, which is itself a function of D, as well as of stirring rate and viscosity. A complete solution of this problem has been obtained in only one case by Levich (7 that of an ideal rotating disk under non-turbulent conditions. The derivation was made for electrode processes, but is equally applicable to dissolution, heat transfer and other heterogeneous processes. 

Figure 13.5. Transport vs surface controlled dissolution. Schematic representation of concentration in solution, C, as a function of distance from the surface of the dissolving mineral. In the lower part of the figure, the change in concentration (e.g., in a batch dissolution experiment) is given as a function of time, (a) Transport controlled dissolution. The concentration immediately adjacent to the mineral reflects the solubility equilibrium. Dissolution is then limited by the rate at which dissolved dissolution products are transported (diffusion, advection) to the bulk of the solution. Faster dissolution results from increased flow velocities or increased stirring. The supply of a reactant to the surface may also control the dissolution rate, (b) Pure surface controlled dissolution results when detachment from the mineral surface via surface reactions is so slow that concentrations adjacent to the surface build up to values essentially the same as in the surrounding bulk solution. Dissolution is not affected by increased flow velocities or stirring. A situation, intermediate between (a) and (b)—a mixed transport-surface reaction controlled kinetics—may develop.

The slowest transport-controlled dissolution/precipitation is that governed by aqueous diffusion. Diffusion rates can be estimated (cf. Bodek et al. 1988 Fetter 1988), thus we can estimate the lower limit of rates attributable to transport control. Berner (1978) suggests that the rate of diffusion-controlled dissolution R is given by... 

Two cases require consideration mass transport-controlled dissolution and dissolution at a finite rate. The former represents the simpler case in mathematical terms and we shall consider this first. 

Landau and E. Malyshev, Modeling Copper Planarization under Mass Transport Controlled Dissolution, Extended Abstract 1656, 210th Electrochemical Society Meeting, Cancun, Mexico, Oct. 29-Nov. 3, 2006. 

Under diffusion-controlled dissolution conditions (in the anodic direction) the crystal orientation has no influence on the reaction rate as only the mass transport conditions in the solution detennine the process. In other words, the material is removed unifonnly and electropolishing of the surface takes place. 

DespiC, A. R. Transport-Controlled Deposition and Dissolution of Metals 7... 

The fundamental concept of heat transport controlled moisture uptake [17] is shown in Eq. (22), where the rate of heat gained at the solid/vapor surface (W AH) is balanced exactly by the heat flow away from the surface (Q). The term All is the heat generated by unit mass of water condensed on the surface. The two most probable sources of heat generation are the heat of water condensation and the heat of dissolution. A comparison of the heat of water condensation (0.58 cal/mg water) with the heat of dissolution for a number of salts indicates that the heat of dissolution can be neglected with little error for many materials. 

The interfacial barrier theory is illustrated in Fig. 15A. Since transport does not control the dissolution rate, the solute concentration falls precipitously from the surface value, cs, to the bulk value, cb, over an infinitesimal distance. The interfacial barrier model is probably applicable when the dissolution rate is limited by a condensed film absorbed at the solid-liquid interface this gives rise to a high activation energy barrier to the surface reaction, so that kR kj. Reaction-controlled dissolution is somewhat rare for organic compounds. Examples include the dissolution of gallstones, which consist mostly of cholesterol,... 

The rate of agitation, stirring, or flow of solvent, if the dissolution is transport-controlled, but not when the dissolution is reaction-con-trolled. Increasing the agitation rate corresponds to an increased hydrodynamic flow rate and to an increased Reynolds number [104, 117] and results in a reduction in the thickness of the diffusion layer in Eqs. (43), (45), (46), (49), and (50) for transport control. Therefore, an increased agitation rate will increase the dissolution rate, if the dissolution is transport-controlled (Eqs. (41 16,49,51,52), but will have no effect if the dissolution is reaction-controlled. Turbulent flow (which occurs at Reynolds numbers exceeding 1000 to 2000 and which is a chaotic phenomenon) may cause irreproducible and/or unpredictable dissolution rates [104,117] and should therefore be avoided. 

The diffusivity, D, of the dissolved solute, if dissolution is transport-controlled (Eqs. 41-46,49,51,52). The dissolution rate of a reaction-controlled system will be independent of D. 

Where dissolution or precipitation is sufficiently rapid, the species concentration quickly approaches the equilibrium value as water migrates along the aquifer the system is said to be reaction controlled. Alternatively, given rapid enough flow, water passes along the aquifer too quickly for the species concentration to be affected significantly by chemical reaction. The system in this case is transport controlled. The relative importance of reaction and transport is described formally by the nondimensional Damkohler number, written Da. 

Among the theories proposed, essentially two main mechanisms can be distinguished these are that the rate-determining step is a transport step (e.g., a transport of a reactant or a weathering product through a layer of the surface of the mineral) or that the dissolution reaction is controlled by a surface reaction. The rate equation corresponding to a transport-controlled reaction is known as the parabolic rate law when... 

As was mentioned in the introduction to this chapter "diffusion-controlled dissolution" may occur because a thin layer either in the liquid film surrounding the mineral or on the surface of the solid phase (that is depleted in certain cations) limits transport as a consequence of this, the dissolution reaction becomes incongruent (i.e., the constituents released are characterized by stoichiometric relations different from those of the mineral. The objective of this section is to illustrate briefly, that even if the dissolution reaction of a mineral is initially incongruent, it is often a surface reaction which will eventually control the overall dissolution rate of this mineral. This has been shown by Chou and Wollast (1984). On the basis of these arguments we may conclude that in natural environments, the steady-state surface-controlled dissolution step is the main process controlling the weathering of most oxides and silicates. 

The morphology of weathered feldspar surfaces, and the nature of the clay products, contradicts the protective-surface-layer hypothesis. The presence of etch pits implies a surface-controlled reaction, rather than a diffusion (transport) controlled reaction. Furthermore, the clay coating could not be "protective" in the sense of limiting diffusion. Finally, Holdren and Berner (11) demonstrated that so-called "parabolic kinetics" of feldspar dissolution were largely due to enhanced dissolution of fine particles. None of these findings, however, addressed the question of the apparent non-stoichiometric release of alkalis, alkaline earths, silica, and aluminum. This question has been approached both directly (e.g., XPS) and indirectly (e.g., material balance from solution data). 

Sum and Skyllas-Kazacos [44] studied the deposition and dissolution of aluminum in an acidic cryolite melt. The graphite electrode was preconditioned (immersed in cryolite melt) to saturate the surface of the electrode in sodium before aluminum deposition could be observed. Current reversal chronoamperometry was used to measure the rate of aluminum dissolution in the acidic melt. Dissolution rate was mass transport controlled [45] and in the order of 0.8 10 7 and 1.8 10 7 molcm 2s 1 at 1030 °C and 980 °C respectively [44]. 

Based on the results of Berner (1978, 1983), Sparks (1988) showed that, in transport-controlled kinetics, the dissolution ions are detached very rapidly and accumulate to form a saturated solution adjacent to the surface. In surface reaction-... 

Fig. 2.3 Rate-limiting steps in mineral dissolution (a) transport-controlled, (b) surface reaction-controlled, and (c) mixed transport and surface reaction control. Concentration (C) versus distance (r) from a crystal surface for three rate-controUing processes, where is the saturation concentration and is the concentration in an infinitely diluted solution. Reprinted from Sparks DL (1988) Kinetics of soil chemical processes. Academic Press New York 210 pp. Copyright 2005 with permission of Elsevier...

Figure 1-12 Control mechanisms of mineral dissolution in aqueous solutions. Data are from Berner (1978). A straight line is drawn to separate transport control and interface reaction control although there is no theoretical basis for whether the boundary should be linear. Almost without exception, those with transport control lie above a straight line, and those with interface reaction control lie below the line. The only significant departure from the rule is the dissolution of PbS04 (cross in the figure) that lies inside the region for the interface reaction control, but is actually controlled by both interface reaction and mass transport.

There are basically three, rate-limiting mechanisms for mineral dissolution assuming a fixed degree of undersaturation. They are (1) transport of solute away from the dissolved crystal or transport-controlled kinetics. 

Dissolution occurring by a surface reaction is often slower than by transport-controlled kinetics because the latter results from more rapid surface detachment. There appears to be a good correlation between the solubility of a mineral and the rate-controlling mechanism for dissolution. Table 7.1 lists dissolution rate-controlling mechanisms for a number of substances. The less soluble minerals all dissolve by surface reaction-controlled kinetics. Silver chloride is an exception, but its dissolution... 

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