2 Institute of Experimental Physics,University of Bremen, D-28334 Bremen, Germany

Abstract -- Dissolution and precipitation of calcium carbonate minerals in aqueous solutions with turbulent flow are controlled by a diffusion boundary layer (DBL) adjacent to the surface of the mineral, across which mass transfer is effected by molecular diffusion. To investigate the influence of such DBL to the dissolution rates of CaCO₃ a rotating disk technique was used. This technique allows an exact adjustment of the thickness of the DBL by control of the rotating speed of a circular sample of CaCO₃. Measurements of the dissolution rates in H₂O-CO₂-Ca²⁺-solutions in equilibrium with various partial pressures of CO₂ from 1· 10^-3 up to 1 atm showed a dependence of the rates R on the rotation frequency w , given by Rµ w ⁿ. The exponent n varies from 0.25 at low P_CO2 to about 0.01 at a P_CO2 of 1 atm. This reveals that the rates are not controlled by mass transport only, which would require n=0.5. The experimental data can be explained employing a theoretical model, which takes into account also the slow reaction and the chemical reactions at the surface (Dreybrodt and Buhmann, Chem. Geol. 91, 1991). Interpretation of the experimental data in view of this model reveales that conversion of CO₂ plays an important role in the control of the rates. At high P_CO2 and large thickness e >0.001cm of the DBL conversion of CO₂ occurs mainly in the DBL and therefore becomes rate limiting. This is corroborated by the observation, that upon addition of the enzyme carbonic anhydrase, which catalyzes CO₂ -conversion, the dissolution rates are enhanced by one order of magnitude. From our experimental observations we conclude that the theoretical model above enables one to predict dissolution rates with satisfactory precision. Since the precipitation rates are determined by the same mechanisms as dissolution, we infer that this model is also valid to predict precipitation rates. The predicted rates for both dissolution and precipitation can be approximated by a linear rate law R=a× (c_eq-c), where C_eq is the equilibrium concentration with respect to calcite and a a rate constant, dependent on temperature, P_CO2, DBL thickness e  and the thickness of the water sheet flowing on the mineral. Values of a are listed that can be used for a variety of geologically relevant conditions.

Dissolution and precipitation of calcite in a CO₂-H₂O system play an important role in many geological processes, such as diagenesis of calcareous deep sea sediments (Berner, 1980; Boudreau and Canfield, 1993), the formation of karst in limestone terranes (Dreybrodt, 1988; Ford and Williams, 1989; White 1988), the evolution of water chemistry in calcite depositing stream systems (Hermann and Lorah, 1987; Dreybrodt et al, 1992; Liu et al., 1995) and the global cycle of CO₂ (Archer and Maier-Reimer, 1994; Kempe, 1977). Both the precipitation and the dissolution rates in the system H₂O-CO₂-CaCO₃ are controlled by three rate determining processes: 1) The kinetics at the mineral surface, which is given by the mechanistic rate equation (called PWP equation) proposed for dissolution by Plummer et al. (1978) which has later been verified also for precipitation (Busenberg and Plummer, 1986; Reddy et al., 1981; Inskeep and Bloom, 1985). This PWP-equation reads

F=k ₁(H⁺)+k ₂(H₂CO₃^*)+k ₃-k ₄(Ca²⁺)(HCO₃^-)                     (1)

where the rate F is given in mmol cm^-2s^-1; and k ₁, k ₂, k ₃ and k ₄ are temperature dependent rate constants and the brackets denote activities in mmol cm^-3 of the corresponding species at the surface of the mineral; 2.) The slow reaction exhibits significant influence to the rates, since stochiometry requires that for each calcium-ion released from the solid one CO₂-molecule has to react to H⁺ + HCO₃^-. At large mineral surfaces and small volumes of the solution this slow reaction can be rate determining. Its kinetics has been reviewed by Kern (1960) and Usdowski (1982); 3) Mass transport of the reacting species away from and towards the mineral surface by molecular diffusion.

Taking into account these three processes Buhmann and Dreybrodt (1985 a,b, 1987) have put forward a model from which dissolution and precipitation rates can be calculated for a sheet of water of given thickness, covering the surface of the mineral. They have shown that due to CO₂-conversion the rates depend on the thickness of this water sheet. The influence of mass transport is significant, when the water is stagnant or flows laminarly on the limestone. If, however, the motion of the water becomes turbulent, molecular diffusion is enhanced by turbulent eddies, such that under otherwise unchanged conditions the rates of dissolution or precipitation, respectively, can increase by almost one order of magnitude.

Therefore Dreybrodt and Buhmann (1991) have extended it by introducing a diffusion boundary layer separating the bulk of the solution from the surface of the mineral. Mass transport through this layer proceeds by molecular diffusion, whereas in the turbulent bulk complete mixing occurs within extremely short time spans. This model predicts rates, which depend on the thickness of the diffusion boundary layer. An experimental means to create such diffusion boundary layers of well defined thickness is the rotating disk technique (Levich, 1962, Pleskov and Filinovskii, 1976).In this paper we report on measurements of dissolution rates of marble and limestone for various thickness of the boundary layer under CO₂ -pressures from 10^-3 atm up to 1 atm. These experiments are in satisfactory agreement to the rates predicted by the diffusion boundary-layer (DBL) model, and a deeper understanding of dissolution- and also precipitation rates under turbulent flow in natural systems is thus provided.

2. EXPERIMENTAL METHODS

Marble disks were cored from marble slabs of 5 mm thickness. Their diameter was 3 cm. These samples were cemented into the holder of the rotating disk apparatus. This was then mounted to the shaft of the rotating disk equipment and polished during rotation by using progressively 400 (64m m), 800 (32m m), 1200 (21m m), 2400 (10m m) and finally 400 (6m m) grid size waterproof silicon carbid paper. This procedure assures a continuous surface of the disk including its rim. The surface of the marble sample in contact with the solution was 7 cm². The marble was white and of coarse microcrystalline structure. It contained less than 1% Mg. To investigate the influence of the slow reaction carbonic anhydrase, an enzyme catalysing this reaction was introduced into the solution to concentrations of  0.1µmol  Bovine carbonic anyhdrase was purchased as lyophylized powder from Sigma and used as obtained.

Some experiments were carried out with limestone from Guilin, China. To obtain solutions in equilibrium with fixed CO₂ partial pressures (P_CO2) N₂-CO₂ gas mixtures with P_CO2 =10^-3, 5× 10^-3, 10^-2, 5× 10^-2, 10^-1 and 1 atm were bubbled through the solution.

A schematic of the experimental apparatus is shown in Fig. 1. Runs were performed in a 1.2 litre glass beaker containing 1180 ml of preset solution, which was prepared by dissolution of calcite powder to the wanted concentration under the relevant CO₂ partial pressure. The beaker was immersed in a constant temperature water bath capable of maintaining ± 0.5°C of the desired temperature. The reaction vessel was covered with Plexiglas lids with holes allowing insertion of the motorized shaft, electrodes, and a gas dispersion tube. Commercial N₂-CO₂ gas mixture with fixed P_CO2 was bubbled through a metal dispersion tube into the reaction vessel, thus providing open system conditions with respect to CO₂. The disk-shaped sample was held centred about 4 cm above the bottom of the reaction vessel on the end of a shaft. A motor drove the shaft, and gear reducers allowed the disk to rotate in the solution at speeds varying from 0 to 3500 rpm (rotations per minute). The rotating speed was held constant by a controller within 6%. The rotating disk was formed by use of epoxy from a mold into which the limestone disk and a screw was fitted, which served to fix the cast to the rotating shaft. Then using the rotating shaft as a lathe to which the cast was fixed, it was shaped into its final form, warranting that the whole set was free of asymetric centrifugal forces leading to disturbance of well defined hydrodynamic conditions. Polishing of the carbonate mineral was also performed at the rotating disk fixed to the rotating shaft of the apparatus.

Measurements of the dissolution rates were performed at a fixed rotating speed by measuring the increase in conductivity. In diluted solutions as they occur in our experiments conductivity and Ca²⁺ concentration are related linearly by

[Ca²⁺ ](mmol/l) =6.18× 10^-3× cond.(m s / cm) - 1.38× 10^-2. g =0.999

The record of a typical run with P_CO2 =1× 10^-3atm and initial [Ca²⁺ ]=3× 10^-4 mol/l is shown by Fig. 2. The straight lines give the increase in concentration at differing rotating speeds from 100 rpm up to 3500 rpm. The rates were calculated from the slopes by

                                                (2)

                        (3)

Fig. 1: Schematic of the experimental set up: 1) controller for rotating speed, 2) motor and gear, 3) rotating shaft, 4) rotating disk, 5) computer for data acquisition, 6) conductometer, 7) conductivity cell, 8) thermometer, 9) temperature sensor for 10) thermostat, 11) H₂O-CO₂-Ca²⁺ solution, 12) CO₂disperser, 13) CO₂ + N₂-supply, 14) supply tube, 15) water circulation from thermostat to waterbath cooling, 16) reactor vessel

Fig. 2: Ca²⁺-concentration versus time for a typical experiment. The straight lines show a linear increase of the concentration at constant rotating speeds. The numbers on the line denote rotating speeds listed in the inset. The corresponding dissolution rates are also given

where D is the coefficient of molecular diffusion (7.1× 10^-6cm²s^-1 at 20°C) and the kinematic viscosity of water. w is the angular velocity in S^-1. In our set-up e =5× 10^-3 cm at a rotating speed of 100 rpm, and at 3500 rpm e=8.5× 10^-4 cm. In all experiments the Reynolds number Re = r²w/u

(r radius of sample) is below 2× 10⁵, such that laminar flow can be assumed (Pleskov and Filinowskii, 1976).

                            (4)

Ca_eq²⁺ is the equilibrium concentration of gypsum (15.4 mmol/l at 20°C). Thus by measuring the dissolution rates at a given rotating speed, e   can be determined experimentally. Fig. 3 shows the result for four rotating speeds. The full line represents eqn. 3. From the good agreement between experiment and theory we conclude that the set-up works satisfactorily.

The dissolution rates measured on the marble sample at 20°C for various partial pressures P_CO2 as a function of w are illustrated by Fig. 4 in a double logarithmic plot. We have also performed experiments at a calcium concentration of 10^-3 mol/l at P_CO2 of 5× 10^-3 atm. These data are also plotted in Fig. 4. For low P_{CO2 £} 0.01atm we find

R µ Wⁿ             (5)

with exponents n between 0.13 and 0.25. For Pco_{2 ³} 0.05 atm the dissolution rates are independent on w . This behaviour shows clearly that dissolution of calcite is complex. If only mass transport by diffusion were rate limiting one would expect n=0.5 (cf. eqn. 3 and 4).

Dreybrodt and Buhmann (1991) in their diffusion boundary layer model (DBL-model) have suggested that the slow conversion of CO₂ into H⁺ + HCO₃^- might be rate limiting, especially at thicker boundary layers. To investigate this experimentally we have performed dissolution experiments adding bovine carbonic anhydrase (BCA) with a concentration of 0.1 m mol to the solution. This enzyme catalyses CO₂ -conversion thus increasing the reaction rate by about two orders of magnitude (Stryer, 1988). It has been successfully used by Dreybrodt et al. (1996 a, b) to show the rate limiting character of CO₂ -conversion for dissolution and precipitation of calcite in porous media.

Fig. 5 represents the experimental results. It depicts the dissolution rates for various P_CO2. The open squares represent the dissolution rates at a rotating speed of 100 rpm (e=5× 10^-3cm) without BCA and the full squares show the corresponding rates with BCA added. Similarly the open (full) circles depict the rates for rotating speeds of 3000 rpm (e=9× 10^-4cm) without and with BCA respectively. At low P_CO2 there is no significant change on the rates upon addition of BCA. For P_CO2 >0.01atm a significant enhancement of the rates occurs, when BCA is added. Measurements were performed at 10°C and 30°C. Table 1 lists the enhancement factors, i.e. the ratios of the rates with and without BCA under otherwise identical conditions for these two temperatures and for various P_CO2. Below P_CO2 =5× 10^-3 atm BCA has no influence on the rates.

Fig. 3: DBL-thickness e plotted versus w ^{-1/ 2}. The straight line gives the theoretical values of e by use of eqn. 3. The points mark the experimental values using a sample of gypsum and employing eqn. 4.

Fig. 4: Log of the experimental dissolution rates for various P_CO2 (listed by the symbols on the curve and by the first row in the inset) as function of the log of the rotating speed in rpm. All curves exhibit straight lines. Therefore a relation Rµ Wⁿ is valid. The corresponding values of n are listed in the second row in the inset. Open symbols refer to a calcium concentration of 3x10^-4 mol/l, closed symbols to a concentration of 1x10^-3 mol/l respectively.

Fig. 5: Dissolution rate versus P_CO2 for rotating speeds at 100 rpm (squares) and 3000 rpm (circles). The open symbols represent the rates in the solution without BCA, the full symbols those with addition of BCA (0.1m M).

Fig. 6: The geometry of the DBL-model: The DBL extends from Z =0 to Z = e . The well mixed core extends from Z =0 to Z = - d . The arrows denote fluxes of CO₃^2-, HCO₃^-, and Ca²⁺from the mineral surface. Note that F is given by the PWP-eqn. (eqn. 1) employing the activities of the species at Z = e .

Table 1. Factors of BCA   enhancement of calcite dissolution rates at differing Pco2 and rotating speeds ( 10 and 30 )

The factors are 1 for both temperatures. But at 1 atm enhancement factors are about 10. Enhancement at the higher temperature is lower than at 10°C. These results show that some change in the dissolution of calcite occurs at high P_CO2. Therefore one has to be cautious to conclude from such experiments to dissolution processes in nature, which usually occur at low P_CO2 < 5× 10^-3atm.

Fig. 6 shows the DBL of thickness e , separating the calcite surface from the turbulent bulk of thickness d . In the region of the bulk complete mixing is assumed, such that no concentration gradients can build up. This can be modelled by assuming the turbulent diffusion coefficient D_t to be higher by a factor of 10⁶ than the coefficient of molecular diffusion, which describes mass transfer in the DBL. The flux F of Ca²⁺ is given by the PWP-equation(eqn. 1) using the activities of the species at the surface of the solid phase. For stochiometrical reasons this flux must be equal to the total amount of CO² converted to H⁺ + HCO₃^- in a column with area of 1cm² covering the region from -d £ Z £ e . This is given by

                             (6)

R_CO2 is the change of the CO₂-concentration in time, resulting from two elementary reactions, occurring simultaneously.

In the following we give results, which have been obtained employing this programme. In a first step we have calculated the dissolution rates for the experiments as depicted in Fig. 4. The DBL thickness e was obtained from eqn. 3. For the experimental conditions with an area of the disk of 7cm² and a total volume of the solution of 1180 cm² the thickness d of the turbulent core is 167 cm. Fig. 7 depicts as full lines the theoretical dissolution rates as a function of e for the various CO₂ partial pressures as used in the experiment. The data points represent the experimental data from Fig. 4; but these were divided by a factor of two to fit to the theoretical curves consistently. The reason for this behaviour is not very clear. One reason may be that owing to the roughness of the surface in the order of 10^-4 cm the effective surface for dissolution is higher than the geometrical one. Another reason could be the variability of the PWP-rate constants with varying natural material. An increase of these constants by a factor of two raises the dissolution rates by the same factor (Dreybrodt and Buhmann, 1991). Rate constants of this magnitude have been observed by Compton and Daly (1984). Compton et al, (1985) have shown that surface roughness exerts significant influence to the rate constant k ₃. They have measured this constant for samples of iceland spar polished with various grid sizes. They find k ₃ = 2× 10^-10 mol cm^-2s^-1 for grit sizes of 4m as we have used, twice as large as the PWP-value. Schott et al. (1989) have reported that the dissolution rates on strained calcite are higher by about a factor of two in comparison to unstrained samples. Considering this, division of the the experimental data by two is feasible, to render them comparable to the predictions of the DBL-model using the rate constants of PWP. Fig. 7 gives evidence that the experimental data and the model predictions are in satisfactory agreement. The model also shows clearly that the rates depend on e only if the P_CO2 pressure is low, whereas almost no dependence on e is predicted for Pco_{2 ³} 0.05 atm and at e > 5× 10^-4cm.

To elicudate this behaviour we use the theoretical DBL-model to calculate the concentration profiles in the boundary layer. Fig. 8 illustrates these profiles of H⁺, HCO₃^-, CO₃^2-, and P_CO2 across the boundary layer for P_{CO2 =} 1× 10^-3atm and e =5× 10^-3cm. The boundary to the bulk is at the left hand side at Z = 0. The units for the different species are denoted at the corresponding ordinates. From the profiles one reads that H⁺ and CO₂ migrate from the bulk towards the solid phase on the right hand side whereas CO₃^2- and HCO₃^- diffuse towards the bulk. The fluxes of these species are given by the first law of Fick as

                                     (7)

where C is the concentration of the corresponding species. Therefore the fluxes can be read from the slopes of the profiles. Spatial changes of the slopes must be interpreted as changes of fluxes, which are accompanied by chemical reactions into which the corresponding species is involved.

At the calcite surface (Z = 5× 10^-3cm) due to the dissolution process there is a flux of HCO₃^- and CO₃^2- towards the bulk. In order to keep the saturation index with respect to calcite sufficiently low, CO₃^2-must react with H⁺ to form HCO₃^-. Due to mass transport by diffusion towards the bulk the flux of HCO₃^- (given by the slope of the profile) increases slightly for two reasons: Firstly CO₃^2--ions react with H⁺-ions diffusing from the bulk into the DBL and secondly close to the solid phase boundary CO₂ migrating from the bulk reacts to H⁺, which is then consumed by the reaction with CO₃^2-. Nevertheless, there is still a considerable flux of CO₃^2- into the bulk, such that most of CO₃^2- released from the solid reacts to HCO₃^- in the bulk. Therefore most of the conversion of CO₂ must also be effective there.

This, however, can no longer be valid, when the thickness of the boundary layer exceeds some critical value. If the DBL is sufficiently large the time for the species to migrate across the layer becomes so long that the chemical reactions will take place in the layer. We have therefore calculated separately the total amount of CO₂ converted in the bulk and in the DBL for various e. This is shown by Fig. 9. At a critical thickness e » 10^-2 cm the conversion rate in the layer equals that in the bulk. For smaller e the rate in the bulk by far exceeds that in the DBL. This explains, why there is a variation of the dissolution rates, with decreasing e : The rate of CO₂-conversion in the bulk is given by

                                    (8)
If d is sufficiently large only small deviations of [CO₂] from equilibrium are necessary to meet the condition F_CO2 = F_Ca. Therefore CO₂-conversion is not rate limiting. Molecular diffusion across the layer, however, still plays an important part and therefore the rates depend on e . This also explains the fact, why adding carbonic anyhdrase to the solution does not enhance the rates, as shown in Fig. 5.

We call dissolution under such hydro-chemical conditions bulk controlled. In contrast to bulk controlled dissolution at low P_CO2 the situation is quite different for high P_CO2. Fig. 10 provides the concentration profiles for P_CO2 = 0.1atm for e =5× 10^-3cm and otherwise unchanged conditions.

Note there is a steep decrease of the CO₃^2- -concentration, within 5× 10^-4 cm, which is mirrored by a steep decrease of H⁺ due to the reaction into HCO₃^-. The concentration of HCO₃^- shows a dramatic increase of its flux towards the bulk in this region. The carbonate-ions migrating towards the bulk react with the H⁺-ions diffusing towards the calcite surface and therefore the amounts of both fluxes must decrease. This is clearly seen by the slopes in the narrow region of 5× 10^-4 cm adjacent to the solid. Close to this region there is a steep increase in the flux of H⁺ towards the solid, which results from conversion of CO₂ diffusing from the bulk towards the surface of the calcite. The flux of H⁺-ions entering from the bulk is by far lower than the flux in the region of CO₂-conversion, as can be judged from the slopes. This supply of H⁺ is caused by CO₂-conversion in this region. Outside this reaction region the concentration of carbonate is practically zero, such that no carbonate is transferred into the bulk. HCO₃^-shows a constant flux towards the bulk. Fig. 11 shows the CO₂-conversion rates in the bulk and in the layer for P_CO2 = 0.1 atm as a function of e . In contrast to Fig. 9 the critical value of e =10^-3 cm is one order of magnitude less. For smaller e  the reaction is bulk controlled. For e> 1× 10^-3cm the conversion of CO₂ occurs solely in the layer. Therefore conversion of CO₂ is rate limiting and the dissolution rates become independent on e .

In this case addition of carbonic anyhdrase is effective in catalysis of CO₂-conversion and the calcite dissolution rates increase upon addition of this enzyme, as depicted in Fig. 5 for rotating speeds of 100 rpm (e =5× 10^-3cm) and also for 3000 rpm (e » 1× 10^-3 cm). The decrease of the CA enhancement factors from 10°C to 30°C (cf. table 1) can also be understood. The CO₂-conversion rate constants increase with temperature (Usdowski, 1982). Therefore at higher temperature the effect of CA is smaller than at low temperature and correspondingly the enhancement of calcite dissolution rates is also lower.

Summarizing, we have found that in the presence of a diffusion boundary layer two limits exist: At low P_CO2 (<10^-2atm) and e < 5× 10^-3cm most of the carbonate ions can diffuse across the DBL and react there to HCO₃^-, whereby the H⁺-ions necessary for this, are supplied by CO₂-conversion in the bulk. In this case of bulk control this reaction is not rate limiting and a complex interplay of the surface controlled reaction (PWP-rate-equation) and diffusion is at work.

For high P_CO2 ( >0.01 atm) CO₂-conversion takes place in a narrow region close to the solid.

Fig. 11: Calcite dissolution rate (dashed curve) and CO₂conversion rates in the bulk (· ) and in the layer () as a function of e. P_CO2 = 0.1 atm.

Table 2. Numerical values of a =a ( T, Pco2, e , d ) for calcite dissolution in the open system.

We therefore feel encouraged to list results obtained from the DBL-model for dissolution and precipitation rates for various natural occurring conditions. These can be used to give at least a first estimation, when applied to geological problems. Fig. 12 illustrates dissolution rates (a) and precipitation rates (b) for P_CO2 = 1× 10^-3atm for various thickness of e . The thickness d of the turbulent bulk is d =1 cm. For large values of e >0.005 cm rates are drastically reduced, compared to the utmost curve, which gives the maximal rates from the PWP-equation using the bulk activities of the species (e = 0). There is a significant increase in the rates for 5× 10^-3cm £ e£ 1× 10^-3 cm. Little influence of the DBL occurs, when e <2× 10^-4 cm.

All the rate curves to a reasonable extent of accuracy can be approximated by a linear relation.

R = a × (c_eq - c )                             (9)

a is a rate constant in cm s^-1 and c_eq is the thermodynamic equilibrium. It should be noted that close to equilibrium for both dissolution and precipitation, inhibition mechanisms are operative (Dreybrodt, 1988; Svensson and Dreybrodt, 1992; Dreybrodt et al., 1996 a,b). Therefore eqn. 9 is valid only for concentrations c/c_eq £ 0.9 for dissolution and c/c_eq ³ 1.2 for precipitation, respectively.

Furthermore due to CO₂ -conversion in the bulk controlled regime the thickness d of the turbulent bulk can also exert significant influence onto the rates. This is illustrated in Fig. 13, where both dissolution (a) and precipitation rates are shown for various values of d depicted on the curves. For dissolution the rates are sensitive to values of d below 1 cm. There is little change for higher values. In contrast the precipitation rates show a different behaviour. Here the limiting value of d , above which the rates stay unchanged with increasing d , is higher by a factor of 10 (d =10 cm).

To summarize such data, Table 2 lists the values of a and c_eq for dissolution rates with various P_CO2, temperatures, e , and d . Table 3 gives corresponding data for precipitation. These can serve as a help to estimate rates in natural systems to a sufficient accuracy. It should be noted here that the values in table 3 are only valid for calcium concentrations [Ca²⁺] £ 5× 10^-3 mol/l. For higher concentrations the real rates are underestimated by up to a factor of two.

We have measured by use of the rotating disk technique dissolution rates of calcite as they occur in natural waters of low P_CO2; but also for high P_CO2 > 10^-2atm up to 1 atm, which are not likely to occur in nature. The aim of these experiments was to reveal the role of the laminar diffusion boundary layer which separates the surface of the dissolving mineral from the well mixed bulk of the solution. Our results give clear evidence, that the diffusion boundary layer across which mass transport must be effected by molecular diffusion can reduce the rates significantly. Furthermore, the slow reaction plays a significant role, which so far has not been sufficiently regarded in most problems dealing with dissolution or precipitation of calcite by many researchers in this field. We have found two regions, where this reaction plays a dominant role. For P_CO2 < 10^-2 atm and thicknesses of the boundary layer below 10^-2 cm the carbonate ions released from the solid migrate across the boundary layer into the turbulent layer, where they react to HCO₃^- consuming the protons delivered by hydration of CO₂. In this bulk controlled regime the influence of CO₂ -conversion becomes evident by the dependence of the rates on the thickness of the bulk layer.

For large P_CO2 > 0.01 atm and thickness of the DBL above 10^-3 cm CO₂ -conversion takes place in a small region adjacent to the calcite surface and almost all carbonate ions released from the solid react to HCO₃^- in this narrow region. The protons needed for this reaction are delivered by CO₂ molecules, diffusing from the bulk and reacting to H⁺ + HCO₃^-close to the solid surface. In this case of layer control the rates are little dependent on the thickness of the turbulent zone, since CO₂-conversion in the DBL is rate limiting. This is corroborated by the enhancement of dissolution rates upon addition of carbonic anhydrase.

This has some important consequences. Many dissolution experiments using the rotating disk technique have employed a partial pressure P_CO2 = 1 atm and thicknesses of the DBL between 10^-3 up 1× 10^-2 cm (Herman, 1982; Herman and White, 1985). In this case rates are determined entirely by CO₂ -conversion and no valid conclusion can be drawn on the dissolution mechanisms at the surface of the mineral. This is in agreement with the observation that the dissolution rates on calcite measured by Herman (1982) compared well to the predictions of our model (Dreybrodt and Buhmann, 1991). If CO₂ -conversion controls the rates then one would expect no influence of the lithology of the limestone sample to the rates. Consequently we have measured rates with otherwise unchanged conditions for marble, for two limestone samples of Devonian origin from Guilin and a single crystal of calcite. Within the limits of error all samples exhibited identical rates. Similar results had been also obtained by Buhmann and Dreybrodt (1985 a) who showed that

during dissolution from thin stagnant films with thickness below 0.1 cm the dissolution rates are controlled by CO₂ -conversion and mass transport. Consequently different minerals (marble, limestone, and a calcite singe crystal) exhibited identical behaviour.

Fig. 12: Theoretical dissolution rates (a) and precipitation rates (b) for an open system at P_CO2 = 1× 10^-3 atm for various values e of the DBL, denoted on the curves by numbers. The corresponding e is listed in the inset.

Fig. 13: Theoretical dissolution rates (a) and precipitation rates (b) for various values d of the turbulent core (cf. Fig. 6). The values of d are related to the numbers on the curve by the corresponding insets. e = 5× 10^-3cm.

Table 3. Numerical values of a =a ( T, Pco2, e , d ) for calcite precipitation in the open system.

Often the rate equation of PWP is applied to explain field observations, by employing the activities of the corresponding species measured in the bulk water. Thus one neglects the presence of the diffusion layer and considerable overestimations of the rates are the consequence. (Liu et al, 1995). A nice example of the influence of the hydrodynamic conditions on dissolution rates in phreatic limestone caves has been given by Lauritzen (1986). He has shown that the dissolution rates in a cave consisting of a single phreatic conduit increased with increasing flow of water. At high yields of 10 m³s^-1the rates were higher by about one order of magnitude, compared to those at yields of about 1 m³s^-1.