Kinetics precipitation rate constant

The failure of conventional criteria may be due to the fact that it is not only one mixing process which can be limiting, rather for example an interplay of micromixing and mesomixing can influence the kinetic rates. Thus, by scaling up with constant micromixing times on different scales, the mesomixing times cannot be kept constant but will differ, and consequently the precipitation rates (e.g. nucleation rates) will tend to deviate with scale-up. 

In Fig. 28, the abscissa kt is the product of the reaction rate constant and the reactor residence time, which is proportional to the reciprocal of the space velocity. The parameter k co is the product of the CO inhibition parameter and inlet concentration. Since k is approximately 5 at 600°F these three curves represent c = 1, 2, and 4%. The conversion for a first-order kinetics is independent of the inlet concentration, but the conversion for the kinetics of Eq. (48) is highly dependent on inlet concentration. As the space velocity increases, kt decreases in a reciprocal manner and the conversion for a first-order reaction gradually declines. For the kinetics of Eq. (48), the conversion is 100% at low space velocities, and does not vary as the space velocity is increased until a threshold is reached with precipitous conversion decline. The conversion for the same kinetics in a stirred tank reactor is shown in Fig. 29. For the kinetics of Eq. (48), multiple solutions may be encountered when the inlet concentration is sufficiently high. Given two reactors of the same volume, and given the same kinetics and inlet concentrations, the conversions are compared in Fig. 30. The piston flow reactor has an advantage over the stirred tank... 

Other companies (e.g., Hoechst) have developed a slightly different process in which the water content is low in order to save CO feedstock. In the absence of water it turned out that the catalyst precipitates. Clearly, at low water concentrations the reduction of rhodium(III) back to rhodium(I) is much slower, but the formation of the trivalent rhodium species is reduced in the first place, because the HI content decreases with the water concentration. The water content is kept low by adding part of the methanol in the form of methyl acetate. Indeed, the shift reaction is now suppressed. Stabilization of the rhodium species and lowering of the HI content can be achieved by the addition of iodide salts. High reaction rates and low catalyst usage can be achieved at low reactor water concentration by the introduction of tertiary phosphine oxide additives.8 The kinetics of the title reaction with respect to [MeOH] change if H20 is used as a solvent instead of AcOH.9 Kinetic data for the Rh-catalyzed carbonylation of methanol have been critically analyzed. The discrepancy between the reaction rate constants is due to ignoring the effect of vapor-liquid equilibrium of the iodide promoter.10... 

Mass transfer can be described in more sophisticated ways. By taking in the previous example to represent time, the rate at which feldspar dissolves and product minerals precipitate can be set using kinetic rate laws, as discussed in Chapter 16. The model calculates the actual rates of mass transfer at each step of the reaction progress from the rate constants, as measured in laboratory experiments, and the fluid s degree of undersaturation or supersaturation. 

In kinetic reaction paths (discussed in Chapter 16), the rates at which minerals dissolve into or precipitate from the equilibrium system are set by kinetic rate laws. In this class of models, reaction progress is measured in time instead of by the nondimensional variable . According to the rate law, as would be expected, a mineral dissolves into fluids in which it is undersaturated and precipitates when supersaturated. The rate of dissolution or precipitation in the calculation depends on the variables in the rate law the reaction s rate constant, the mineraTs surface area, the degree to which the mineral is undersaturated or supersaturated in the fluid, and the activities of any catalyzing and inhibiting species. 

Do the kinetic rate constants and rate laws apply well to the system being studied Using kinetic rate laws to describe the dissolution and precipitation rates of minerals adds an element of realism to a geochemical model but can be a source of substantial error. Much of the difficulty arises because a measured rate constant reflects the dominant reaction mechanism in the experiment from which the constant was derived, even though an entirely different mechanism may dominate the reaction in nature (see Chapter 16). 

In the broadest sense, of course, no model is unique (see, for example, Oreskes et al., 1994). A geochemical modeler could conceptualize the problem differently, choose a different compilation of thermodynamic data, include more or fewer species and minerals in the calculation, or employ a different method of estimating activity coefficients. The modeler might allow a mineral to form at equilibrium with the fluid or require it to precipitate according to any of a number of published kinetic rate laws and rate constants, and so on. Since a model is a simplified version of reality that is useful as a tool (Chapter 2), it follows that there is no correct model, only a model that is most useful for a given purpose. 

Each mineral in the calculation dissolves and precipitates according to the kinetic rate law (Eqn. 26.1) used in the previous examples and the rate constants listed in Table 26.1. We take the same specific surface areas for quartz and cristobalite as we did in our calculations in Section 26.1, and assume a value of 20 000 cm2 g-1 for the amorphous silica, consistent with measurements of Leamnson el al. (1969). The procedure in react is... 

A number of factors contribute to the disparity between the predictions of kinetic theory and conditions observed in the field, as discussed in Section 16.2. In this case, we might infer the dissolution and precipitation of minerals such as opal CT (cristobalite and tridymite, Si02), smectite and other clay minerals, and zeolites help control silica concentration. The minerals may be of minor significance in the aquifer volumetrically, but their high rate constants and specific surface areas allow them to react rapidly. 

Minerals in the soil can dissolve or, if they become supersaturated, precipitate according to the kinetic rate law in the previous section (Eqn. 27.2). We take a rate constant of 4.2 x 10-18 mol cm-2 s-1 for quartz, as before, and of 30 x 10-18 mol cm-2 s-1 for potassium feldspar and 100 x 10-18 mol cm-2 s-1 for albite, from Blum and Stillings (1995). We assume a specific surface area of 1000 cm2 g-1, typical of sand-sized grains (Leamnson el al., 1969), for each of the minerals. 

The principle we have applied here is called microscopic reversibility or principle of detailed balancing. It shows that there is a link between kinetic rate constants and thermodynamic equilibrium constants. Obviously, equilibrium is not characterized by the cessation of processes at equilibrium the rates of forward and reverse microscopic processes are equal for every elementary reaction step. The microscopic reversibility (which is routinely used in homogeneous solution kinetics) applies also to heterogeneous reactions (adsorption, desorption dissolution, precipitation). 

Figure 15. Illustration of possible variations in isotopic fractionation between Fe(III),q and ferric oxide/ hydroxide precipitate (Aje(,n),q.Fenicppt) and precipitation rate. Skulan et al. (2002) noted that the kinetic AF (ni)aq-Feiricppt fractionation produced during precipitation of hematite from Fe(III), was linearly related to precipitation rate, which is shown in the dashed curve (precipitation rate plotted on log scale). The most rapid precipitation rate measured by Skulan et al. (2002) is shown in the black circle. The equilibrium Fe(III),-hematite fractionation is near zero at 98°C, and this is plotted (black square) to the left of the break in scale for precipitation rate. Also shown for comparison is the calculated Fe(III),q-ferrihydrite fractionation from the experiments of Bullen et al. (2001) (grey diamond), as discussed in the previous chapter (Chapter lOA Beard and Johnson 2004). The average oxidation-precipitation rates for the APIO experiments of Croal et al. (2004) are also noted, where the overall process is limited by the rate constant ki. As discussed in the text, if the proportion of Fe(III),q is small relative to total aqueous Fe, the rate constant for the precipitation of ferrihydrite from Fe(III), (Ai) will be higher, assuming first-order rate laws, although its value is unknown.

It is not clear whether the reaction kinetics will be significantly altered by other species present in the aqueous phase. For example, Lee et al. (1986) showed that the kinetics of the H202-S(IV) reaction in freshly collected precipitation were only 15% below those measured in laboratory pure water. On the other hand, laboratory studies by Lagrange et al. (1993) suggest that the rate constant depends both on the ionic strength and on the nature of the electrolyte and that Fe2+ catalyzes the reaction. 

Work by Voorhees and Glicksman concludes that the classical theory is correct in the limit of zero volume fraction of the coarsening phase and that both the kinetics and the size distributions are significantly dependent on the precipitate volume fraction,

diffusion-limited coarsening, given by Eq. 15.18, remains valid for all volume fractions, but the rate constant Kp is a monotonically increasing function of , as in Fig. 15.8. 

Baxendale, Evans and coworkers reported in 1946 that the polymerization of methyl methacrylate (MMA) in aqueous solution was characterized by homogeneous solution kinetics, i.e. where mutual termination of free radicals occurred, in spite of the fact that the polymer precipitated as a separate phase. Increases in the rates of polymerization upon the addition of the surfactant cetyl trimethyl ammonium bromide (CTAB) were attributed to the retardation of the rate of coagulation of particles, which was manifested in a reduction in the effective rate constant for mutual termination,... 

The permanganate ion was labeled with " Mn. The separation of Mn04 for kinetic determinations was achieved by co-precipitation with tetraphenylarsonium perrhenate, and the activity was determined with a scintillation counter. The rate constant, Arexch = 710 M s at 273 K, was obtained by fitting of the data to Eq. 65... 

Here is the forward rate constant, aj is the activity of species j in the rate-determining reaction, mj and are constants, and R and T are the gas constant and absolute temperature, respectively. The sign of the rate indicates whether the reaction goes forward or backward. The relationship of this equation to transition state theory and irreversible kinetics has been discussed in the literature (Lasaga, 1995 Alekseyev et al., 1997 Lichtner, 1998 Oelkers, 2001b). The use of this equation with = 1 is generally associated with a composite reaction in which all the elementary reactions are near equilibrium except for one step which is ratedetermining. This step must be shared by both dissolution and precipitation. 

Figure 2. Effects of different parameters on cave calcite growth rate A) precipitation rates for vmious film thicknesses (d in cm die thicker line represents average film thickness of 7.5 cm Baker and Smart, 1995) as a function of [Ca ] in the water film (T = 10°C, cavepCOz = 3x10" atm) B) precipitation rates (widi d = 0.01 cm and cave pCOt = SxlO atm) for various temperatures (in °C) C) precipitation rates for various cave pC02 levels (in 10 atm) (d = 0.005 cm, T = 10°C) D) values of the kinetic constant a as a function of temperature for different film diicknesses (d in cm)(cavepCO = 3x10 atm). All figures after B er et al. (1998).

Considering the mechanisms of sorption discussed in Section II, e.g. surface precipitation or formation of new phases involving the adsorbent and the adsorbate, the above kinetic model is not sufficient to describe isotope exchange in all relevant systems, although most experimental kinetic curves can be reproduced by proper adjustment of parameters (effective D in the solid and in the liquid film, rate constants of surface reactions) within the discussed above model. Spectroscopic studies (Section I and II) suggest that the uptake of adsorbate is often due to simultaneous formation of surface complex and surface precipitation. Single F t) curve based on the radioactivity of the solution is not sufficient to describe sorption kinetics in such systems. 

Many natural aquatic systems have a chemical composition close to saturation with respect to calcite or even dolomite. This is the case, for instance, for seawater, which is usually slightly oversaturated in the upper part of the water column and slightly undersaturated at greater depths. Under these conditions, the rates of both precipitation and dissolution contribute significantly to the overall rate of reaction. Even though the reaction paths may be very complex, there is a very direct and important link between the kinetic rate constants, according to which the rates of forward and reverse microscopic processes are equal for every elementary reaction. The fundamental aspect of this principle forms the essential aspect of the theory of irreversible thermodynamics (Frigogine, 1967). 

The reaction term R in Eq. (6.12) is determined as follows. Mn " is produced by the dissolution of solid-phase Mn oxide and is subject to reprecipitation as either an oxide or reduced phase. Because oxide reduction begins very close to the sediment-water interface, I assume that little reprecipitation as an oxide actually takes place within the deposit or that reprecipitation takes place so near to the interface that it cannot be differentiated from a boundary condition. Therefore, the Mn distribution can be considered as influenced dominantely by production and anoxic precipitation reactions over most of the sampled interval. The production term was shown in the previous section to be of the form R = Ro exp(-our) where Rq and oi are constants and x is the depth in the deposit. Precipitation reactions are commonly assumed to follow first-order or pseudo-first-order kinetics such that R = ki(C - Ceq) where /t, is a first-order rate constant and represents a depth-dependent equilibrium concentration (Holdren et al., 1975 Robbins and Callender, 1975). In LIS sediments the concentrations of many anions such as HCOs", which might precipitate with Mn, are roughly constant over the top —20 cm of sediment. This is true in particular at NWC and DEEP (Part 1). It will therefore be assumed that Ce, is constant over the depth interval of interest and that its value is the concentration to which a profile asymptotes at depth. Taken together these considerations suggest that an appropriate reaction term for Mn in the present case is... 

Here, v denotes the mean velocity of advection, and k is a rate constant of a reaction with first order kinetics. The last term in the equation R(x) is an unspecified source or sink related term which is determined by its dependence on the depth coordinate x. Instead of R(x), one might occasionally find the expression (ERj) which emphasizes that actually the sum of different rates originating from various diagenetic processes should be considered (e.g. Berner 1980). Such reactions, still rather easy to cope with in mathematics, frequently consist of adsorption and desorption, as well as radioactive decay (first-order reaction kinetics). Sometimes even solubility and precipitation reactions, albeit the illicit simplification, are concealed among these processes of sorption, and sometimes even reactions of microbial decomposition are treated as first order kinetics. 

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