# Thermodynamic Modelling of Dolomite Behavior in Aqueous Media

Authors: Tadeusz Michałowski; Agustin G. Asuero
Journal: Journal of Thermodynamics
 (2012)

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## Abstract

The compact thermodynamic approach to the systems containing calcium, magnesium, and carbonate species is referred to dissolution of dolomite, as an example of nonequilibrium ternary salt when introduced into aqueous media. The study of dolomite is based on all attainable physicochemical knowledge, involved in expressions for equilibrium constants, where the species of the system are interrelated. The species are also involved in charge and concentration balances, considered as constraints put on a closed system, separated from the environment by diathermal walls. The inferences are gained from calculations performed with use of an iterative computer program. The simulatedquasistatic processes occurred under isothermal conditions, started at a preassumed pH0 value of the solution where dolomite was introduced, and are usually involved with formation of other solid phases. None simplifying assumptions in the calculations were made.


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## Body

## 1. Introduction

Dolomite (CaMg(CO3)2, abbr. pr1) is perceived as an unusual, metastable mineral [1, 2], and its behavior is considered as one of the most exciting topics in geology [3]. Its chemical properties should be put in context with other, most important carbonate minerals: calcite (CaCO3, pr2) and magnesite (MgCO3, pr3) [4]. The trigonal structure of calcite is composed of alternate layers of calcium and carbonate ions. The crystal structure of magnesite is the same as one for calcite, and then magnesite properties are similar to those of calcite. However, a significant difference in Pauling’s ionic radii: 99 pm (for Ca+2), 65 pm (for Mg+2), [5] causes an incompatibility of the cations in the same layer of dolomite structure. Crystal lattice of ideal dolomite (M1=184.4 g/mol, ρ1=2.899 g/cm3) consists of alternating octahedral layers of Ca+2 and Mg+2 ions, separated by layers of CO3 −2 ions [6]. Complete ordering is energetically favourable at lower temperatures. Presumably, it is the principal crystallographic constraint securing nearly ideal (Ca : Mg = 1 : 1) dolomite stoichiometry [6].Ideal dolomite, CaMg(CO3)2, is a particular case (x=0) of the compound Ca1-xMg1+x(CO3)2 [7–10], considered as a result of partial disarrangement of Ca+2 and Mg+2 in structure of real dolomite. This disarrangement is evidenced by weak or diffuse spectral lines in the reflected X-ray diffraction patterns [11]. The composition of sedimentary dolomites is not exactly stoichiometric; it ranges from Ca1.16Mg0.84(CO3)2 to Ca0.96Mg1.04(CO3)2 (i.e., -0.16<x<0.04) [1]. The mineral Ca0.5Mg1.5(CO3)2 (x=0.5) is named huntite. Sedimentary dolomites with x>0 are named calcium dolomites, and the ones with x<0 are named magnesian dolomites. The carbonate dominated by calcite (ρ2=2.71 g/cm3) is called limestone. Other carbonates: ankerite Ca(Fe, Mg, Mn)(CO3)2 and kutnohorite Ca(Mn,Mg,Fe)(CO3)2, with bivalent cations: Fe+2 and Mn+2, substituting a part of Mg+2 ions in the layered structure, enter also the dolomite group.A rock formed by the replacement of dolomite with calcite [12]
(1)CaMg(CO3)2+Ca+2=2CaCO3+Mg+2
in dedolomitisation (calcitisation) process is named dedolomite [13]. Calcium ions in this process are provided, for example, by calcium-rich water, and magnesium ions are liberated. Molar volume of dolomite (184.4/2.899=63.608 cm3/mol) is lower than two molar volumes of calcite (2·100.09/2.71=73.867 cm3/mol), and, therefore, reaction (1) causes an increase in the solid volume, equal (73.867-63.608)/63.608, that is, ca. 16% [4].Dedolomitisation is a particular case of a diagenesis [14], where an alteration of sediments into sedimentary rocks occurs. This process is accompanied by an increase in porosity, expressed by a percentage pore space. The pores form a void space or are filled with a fluid.Dolomite is thermodynamically unstable, and dedolomitisation occurs under different conditions. The dedolomitisation in alkaline media is represented by reaction [15]
(2)CaMg(CO3)2+Ca(OH)2=2CaCO3+Mg(OH)2.The reverse process (i.e., dolomitisation) occurs, for example, during evaporation of seawater. High temperatures enhance the dolomitisation process [16]. In dolomitisation, magnesium ions from seawater replace calcium ions in calcite, and dolomite is formed [17]. Dolomite growth is favoured by high Mg/Ca ratios and high carbonate contents; this fact is predictable from Le Chatelier-Brown principle [18].The main objective of this paper is to provide the way for better understanding the dolomite dissolution at different conditions [19] affected by pH, initial concentration of dolomite (C0) in the system ([pr1]t=0=C0), and concentration of CO2. The knowledge of dissolution in acidic media is essential in aspect of improved recovery of oil and gas from sedimentary basins at low temperatures [20], whereas dedolomitisation in alkaline media plays a significant role in deteriorating the concrete structure [15, 21].As will be stated in what follows, the solubility product value (Ksp1) of dolomite is a critical factor in quantitative description of its behavior in the systems where dolomite is put in contact with aqueous solutions, containing dissolved CO2 and, moreover, a strong acid (e.g., HCl) or a strong base (e.g., KOH). This paper follows the one concerning struvite [22, 23], as another representative of a group of the nonequilibrium precipitates formed by ternary salts.


## 2. Kinetics of Dolomite Dissolution and Stoichiometry

In aqueous systems, ideal dolomite can be considered as an equimolar mixture of two carbonate components: calcite and magnesite, that is,pr1=pr2+pr3. This simplifying assumption deserves some reservation, concerning relative rates of dissolution of the dolomite components; the literature provides ambiguous data in this respect, however. The results obtained according to AAS method by Lund et al. [24] exhibited stoichiometric dissolution of dolomite in relation to Ca and Mg, whereas other experiments [25] showed that pure dolomite dissolves more slowly than pure calcite.Dolomite exists in a variety of morphological forms [26]. Minerals with greater defect densities dissolve faster since their effective surface areas are greater than more perfect specimens of the same compound. The rate of surface-controlled dolomite dissolution is significantly less than one of calcite [27].Dissolution rate increases with decreasing grain size [28]. The experiments done for kinetic purposes showed that the mass loss of single dolomite crystals [29] (in response to pH and pCCO2) or one for finely dispersed dolomite particles [27] (in response to pH) was measured.The dissolution of ionic crystals is a complex process, involving some surface and transportation phenomena. Ions are transferred from the surface of the solid material to an unsaturated solution [30]. The surface phenomena depend on the morphology (microstructure) of the crystals. The rate of any dissolution process is effected by surface and transport phenomena.Kinetics of dolomite dissolution has been tested at different pH and temperatures [20]. According to a model by Busenberg and Plummer [29], the dissolution of dolomite is an effect of simultaneous action of H+, H2CO3, and H2O.The dissolution studies were usually carried out with suspensions or powdered materials employed, and the resulting concentration changes of Ca and Mg species in the bulk solution were measured [31]. For this purpose, in situ (e.g., conductometric, pH-metric, pH-static [32]) or ex situ (e.g., titrimetry, AAS [33]) methods of analysis were employed [30]. As an option, a rotating disc (RD) formed of dissolving dolomite attached at the end of rotating disc shaft was applied [34]. A loss in mass of the solid material was also measured [35].It should be noted that repeated trials to precipitate ideal dolomite under laboratory conditions at room temperature were unsuccessful [36, 37]; dolomite was precipitated at elevated temperatures (150–300°C) [38], for example, by heating calcite with Mg salt in aqueous media, at elevated CO2 pressures [39]. Dolomite is formed as a result of complex, not well-understood physicochemical phenomena [40, 41], because of the difficulties arising in preparation of stoichiometric dolomites [42]. These difficulties caused, among others, that the solubility product (Ksp1) of dolomite in water
(3)Ksp1=[Ca+2][Mg+2][CO3-2]2
measured according to different methods yielded inconsistent and unreliable results. The mechanism of dolomite formation in sedimentary environment (the so-named dolomite problem [43]) is not well understood, as hitherto [44].


## 3. Principles of Simulation of Dolomite Dissolution

### 3.1. General Remarks

Simulations are needed to check the models used. In modelling of chemical systems, different computer programs were developed. Among others, the Joint Expert Speciation System (JESS) computer program [45–48] is sometimes applied, for example, in [49]. A new approach, called Generalized Approach to Electrolytic Systems (GATES), was elaborated by Michałowski in 1992 and presented lately in some review papers [50–52] and in textbooks [53, 54]. For example, the systems with struvite [22, 23] were elaborated according to GATES.The equilibria in the system with solid carbonates are affected by total concentration (CCO2) of carbonate species, introduced by CO2 dissolved in aqueous media, and by presence of NaOH (Cb) or HCl (Ca), used to moderate pH of the solution. At  Cb=Ca, the effect is practically tantamount with absence of the related species in the solution, provided that Ca and Cb values are small; that is, the effect of ionic strength is negligible.


### 3.2. Equilibrium Constants

The set of equilibrium constants [1, 55–59] referred to the two-phase system in question is involved with solubility products (Kspi) of precipitates, (pri,i=1,…,5), expressed by (3) and
(4)[Ca+2][CO3-2]=Ksp2(pKsp2=8.48),[Mg+2][CO3-2]=Ksp3(pKsp3=7.46),[Mg+2][OH-1]2=Ksp4(pKsp4=10.74),[Ca+2][OH-1]2=Ksp5(pKsp5=5.03),
and soluble species (complexes, protonated forms):(5)[MgHCO3+1]=101.16⋅[Mg+2][HCO3-1],[MgCO3]=103.4⋅[Mg+2][CO3-2],[CaHCO3+1]=101.11⋅[Ca+2][HCO3-1],[CaCO3]=103.22⋅[Ca+2][CO3-2],[HCO3-1]=1010.33-pH⋅[CO3-2],[H2CO3]=106.38-pH⋅[HCO3-1],[MgOH+1]=102.57⋅[Mg+2][OH-1],[CaOH+1]=101.3⋅[Ca+2][OH-1],[H+1][OH-1]=Kw(pKw=14.0).One should be noted that soluble complex species:MgCO3 and CaCO3, characterized by stability constants, are different from the precipitates: pr2 = CaCO3 and pr3 = MgCO3,characterized by the solubility products (Kspi, i=2,3) values.ThepKsp1 values for dolomite reported in the literature range from ca. 16.5 to 19.35 [37, 55, 56, 60, 61], that is, within ca. 3 units. Some inferences [37] lead to the conclusion that the most probable pKsp1 value is 17.2±0.2. In this context, the value 19.35 taken in [55] seems to be excessively high. Such diversity in pKsp1 value may reflect the difficulties involved with obtaining stoichiometric dolomite.The inequalityKsp1<Ksp2·Ksp3, that is, pKsp1>pKsp2+pKsp3, valid for all Ksp1 values quoted above, expresses a kind of synergistic effect securing almost perfect arrangement of Mg+2 and Ca+2 ions in area of the corresponding planes of dolomite crystallographic structure.The values for stability constants of soluble complexes:Mg(OH)2 and Ca(OH)2, introduced for calculations in [55], are highly controversial and then were omitted in the related balances formulated below. In this context, the complexes MgOH+1 and CaOH+1 and their stability constants are well established.The dissolution of dolomite (pr1) proceeds up to the saturation of the solution against the corresponding precipitate. Provided that the solution is unsaturated against the corresponding solid phase (pri), the expression for the related solubility product (Kspi) is not valid, under such conditions. For this purpose, the expressions(6)q1=[Ca+2][Mg+2][CO3-2]2Ksp1,q2=[Ca+2][CO3-2]Ksp2,q3=[Mg+2][CO3-2]Ksp3,q4=[Mg+2][OH-1]2Ksp4,q5=[Ca+2][OH-1]2Ksp5
related to all possible precipitates (pri,i=1,…,5) were considered.


### 3.3. Formulation of the Dolomite Dissolution Model

We refer to the system obtained after introducingm1 g of dolomite into V mL of aqueous solution with dissolved CO2 (CCO2), NaOH (Cb), and/or HCl (Ca); NaOH and/or HCl are used to moderate pH value of the solution. Assuming that the volume change resulting from addition of pr1 is negligible, and denoting current (t≥0) concentration of pr1 by [pr1], [pr1]t=0=C0=(103·m1/M1)/V, we get the concentration and charge balances(7)F1(x)=[pr1]+∑Mg-C0=0,F2(x)=[pr1]+∑Ca-C0=0,F3(x)=2[pr1]+[MgHCO3+1]+[MgCO3]+[CaHCO3+1]+[CaCO3]+[H2CO3]+[HCO3-1]+[CO3-2]-(2C0+CCO2)=0,F4(x)=[H+1]-[OH-1]+Δ+2[Mg+2]+[MgOH+1]+2[Ca+2]+[CaOH+1]+[MgHCO3+1]+[CaHCO3+1]-[HCO3-1]-2[CO3-2]=0,
where(8)Δ=Cb-Ca
the expressions:(9)∑Mg=[Mg+2]+[MgOH+1]+[MgHCO3+1]+[MgCO3],(10)∑Ca=[Ca+2]+[CaOH+1]+[CaHCO3+1]+[CaCO3]
involve all soluble magnesium and calcium species; x=[x1,x2,x3,x4]⊤ is the vector of independent variables specified below. The number of variables forming the vector x is equal to the number of equations
(11)Fi(x)=0(i=1,…,4)
specified above.


### 3.4. Calculation Procedure

The relations (11) are valid, if x is chosen properly. For any other vector x′, x′≠x, at least one of the functions (11) differs from zero, and then
(12)F(x′)=∑i=14[Fi(x′)]2>0.
The calculation procedure is based on minimizing principle, quite analogous to one presented in [22, 23]. According to this approach, the sum of squares (12) is taken as the minimized function.In a dynamic process, as the dissolution ofpr1 is, a choice of ppr1=-log[pr1] as the steering variable is advised; the value of [pr1] changes during the pr1 dissolution. The next step is the choice of independent variables, xi=xi(ppr1), i=1,…,4. The xi values are involved with concentrations of some independent species. In our case, the best choice is x=[x1,x2,x3,x4]⊤=[pH,pHCO3,pMg,pCa]⊤, where pH=-log[H+1], pHCO3=-log[HCO3-1], pMg=-log[Mg+2], pCa=-log[Ca+2] were considered as independent variables. A choice of pX indices, not concentrations [X], resulted from the fact that 10-pXi>0 for any real pXi=-log[Xi] value, pXi∈ℜ. The third step is the choice of numerical values for components of the starting vector  x′=x*; if x*≠x, one can expect that F(x*)>0 (12). The x*  value is referred to particular value of the steering variable,  x*=x*(ppr1), for example, to ppr1=-logC0. The step Δppr1 is needed, and initial steps ΔpXi for pXi and lower (pXi,inf) and upper (pXi,sup) limits for expected pXi values are also required by the iterative computer programs applied in the minimization procedure.The searching ofx(ppr1) vectors, where F(x(ppr1)) is close to zero for different ppr1 values, can be made according to MATLAB iterative computer program. The searching procedure satisfies the requirements put on optimal x(ppr1) values—provided that the F(x′(ppr1)) value (12), considered as optimal one, is lower than a preassumed, sufficiently small δ-value(13)F(x′(ppr1))<δ.Then we consider that the equalityx′(ppr1)=x(ppr1) is fulfilled. It means that Fi(x(ppr1))≅0 for all i=1,…,4, that is, (7) are fulfilled simultaneously, within tolerable degree of proximity assumed for all ppr1 values taken from defined ppr1 interval. If pr1 dissolves wholly, the ppr1 covers the interval from pC0 up to the value resulting from graphical needs, that is, from the scale for ppr1 assumed to plot the related (e.g., speciation) curves. If the solubility product for pr1 is attained at defined point of the dissolution process, then upper value assumed for ppr1 is the value corresponding to this point.The iterative computer programs are usually designed for the curve-fitting procedures where the degree of fitting of a curve to experimental points is finite. In particular case, the criterion of optimization is based on differencesF(x(ppr1(N+1)))-F(x(ppr1(N))) between two successive (Nth and N+1th) approximations of the F-value, and the optimisation is terminated if the inequality(14)|F(x(ppr1(N+1)))-F(x(ppr1(N)))|<δ
is valid for any preassumed, sufficiently low δ-value, for example, δ=10-15.However, one may happen that the condition (14) can be fulfilled at local minimum different from the global minimum. It occurs when the starting values x*(ppr1) are too distant from the true value x(ppr1), where the equality F(x(ppr1))=0 is fulfilled. In this case, one should repeat the calculations for a new vector x*(ppr1), guessed at a particular ppr1 value chosen at the start for minimisation.All vectorsx=x(ppr1) obtained for different ppr1 values are the basis for plotting the speciation curves for all species (Xi) in the system. The curves are usually plotted in the logarithmic diagrams, on 2D plane, with ppr1 on the abscissa and log[Xi] on the ordinate. Other variables, for example, pH on the abscissa, can also be applied.


## 3.1. General Remarks

Simulations are needed to check the models used. In modelling of chemical systems, different computer programs were developed. Among others, the Joint Expert Speciation System (JESS) computer program [45–48] is sometimes applied, for example, in [49]. A new approach, called Generalized Approach to Electrolytic Systems (GATES), was elaborated by Michałowski in 1992 and presented lately in some review papers [50–52] and in textbooks [53, 54]. For example, the systems with struvite [22, 23] were elaborated according to GATES.The equilibria in the system with solid carbonates are affected by total concentration (CCO2) of carbonate species, introduced by CO2 dissolved in aqueous media, and by presence of NaOH (Cb) or HCl (Ca), used to moderate pH of the solution. At  Cb=Ca, the effect is practically tantamount with absence of the related species in the solution, provided that Ca and Cb values are small; that is, the effect of ionic strength is negligible.


## 3.2. Equilibrium Constants

The set of equilibrium constants [1, 55–59] referred to the two-phase system in question is involved with solubility products (Kspi) of precipitates, (pri,i=1,…,5), expressed by (3) and
(4)[Ca+2][CO3-2]=Ksp2(pKsp2=8.48),[Mg+2][CO3-2]=Ksp3(pKsp3=7.46),[Mg+2][OH-1]2=Ksp4(pKsp4=10.74),[Ca+2][OH-1]2=Ksp5(pKsp5=5.03),
and soluble species (complexes, protonated forms):(5)[MgHCO3+1]=101.16⋅[Mg+2][HCO3-1],[MgCO3]=103.4⋅[Mg+2][CO3-2],[CaHCO3+1]=101.11⋅[Ca+2][HCO3-1],[CaCO3]=103.22⋅[Ca+2][CO3-2],[HCO3-1]=1010.33-pH⋅[CO3-2],[H2CO3]=106.38-pH⋅[HCO3-1],[MgOH+1]=102.57⋅[Mg+2][OH-1],[CaOH+1]=101.3⋅[Ca+2][OH-1],[H+1][OH-1]=Kw(pKw=14.0).One should be noted that soluble complex species:MgCO3 and CaCO3, characterized by stability constants, are different from the precipitates: pr2 = CaCO3 and pr3 = MgCO3,characterized by the solubility products (Kspi, i=2,3) values.ThepKsp1 values for dolomite reported in the literature range from ca. 16.5 to 19.35 [37, 55, 56, 60, 61], that is, within ca. 3 units. Some inferences [37] lead to the conclusion that the most probable pKsp1 value is 17.2±0.2. In this context, the value 19.35 taken in [55] seems to be excessively high. Such diversity in pKsp1 value may reflect the difficulties involved with obtaining stoichiometric dolomite.The inequalityKsp1<Ksp2·Ksp3, that is, pKsp1>pKsp2+pKsp3, valid for all Ksp1 values quoted above, expresses a kind of synergistic effect securing almost perfect arrangement of Mg+2 and Ca+2 ions in area of the corresponding planes of dolomite crystallographic structure.The values for stability constants of soluble complexes:Mg(OH)2 and Ca(OH)2, introduced for calculations in [55], are highly controversial and then were omitted in the related balances formulated below. In this context, the complexes MgOH+1 and CaOH+1 and their stability constants are well established.The dissolution of dolomite (pr1) proceeds up to the saturation of the solution against the corresponding precipitate. Provided that the solution is unsaturated against the corresponding solid phase (pri), the expression for the related solubility product (Kspi) is not valid, under such conditions. For this purpose, the expressions(6)q1=[Ca+2][Mg+2][CO3-2]2Ksp1,q2=[Ca+2][CO3-2]Ksp2,q3=[Mg+2][CO3-2]Ksp3,q4=[Mg+2][OH-1]2Ksp4,q5=[Ca+2][OH-1]2Ksp5
related to all possible precipitates (pri,i=1,…,5) were considered.


## 3.3. Formulation of the Dolomite Dissolution Model

We refer to the system obtained after introducingm1 g of dolomite into V mL of aqueous solution with dissolved CO2 (CCO2), NaOH (Cb), and/or HCl (Ca); NaOH and/or HCl are used to moderate pH value of the solution. Assuming that the volume change resulting from addition of pr1 is negligible, and denoting current (t≥0) concentration of pr1 by [pr1], [pr1]t=0=C0=(103·m1/M1)/V, we get the concentration and charge balances(7)F1(x)=[pr1]+∑Mg-C0=0,F2(x)=[pr1]+∑Ca-C0=0,F3(x)=2[pr1]+[MgHCO3+1]+[MgCO3]+[CaHCO3+1]+[CaCO3]+[H2CO3]+[HCO3-1]+[CO3-2]-(2C0+CCO2)=0,F4(x)=[H+1]-[OH-1]+Δ+2[Mg+2]+[MgOH+1]+2[Ca+2]+[CaOH+1]+[MgHCO3+1]+[CaHCO3+1]-[HCO3-1]-2[CO3-2]=0,
where(8)Δ=Cb-Ca
the expressions:(9)∑Mg=[Mg+2]+[MgOH+1]+[MgHCO3+1]+[MgCO3],(10)∑Ca=[Ca+2]+[CaOH+1]+[CaHCO3+1]+[CaCO3]
involve all soluble magnesium and calcium species; x=[x1,x2,x3,x4]⊤ is the vector of independent variables specified below. The number of variables forming the vector x is equal to the number of equations
(11)Fi(x)=0(i=1,…,4)
specified above.


## 3.4. Calculation Procedure

The relations (11) are valid, if x is chosen properly. For any other vector x′, x′≠x, at least one of the functions (11) differs from zero, and then
(12)F(x′)=∑i=14[Fi(x′)]2>0.
The calculation procedure is based on minimizing principle, quite analogous to one presented in [22, 23]. According to this approach, the sum of squares (12) is taken as the minimized function.In a dynamic process, as the dissolution ofpr1 is, a choice of ppr1=-log[pr1] as the steering variable is advised; the value of [pr1] changes during the pr1 dissolution. The next step is the choice of independent variables, xi=xi(ppr1), i=1,…,4. The xi values are involved with concentrations of some independent species. In our case, the best choice is x=[x1,x2,x3,x4]⊤=[pH,pHCO3,pMg,pCa]⊤, where pH=-log[H+1], pHCO3=-log[HCO3-1], pMg=-log[Mg+2], pCa=-log[Ca+2] were considered as independent variables. A choice of pX indices, not concentrations [X], resulted from the fact that 10-pXi>0 for any real pXi=-log[Xi] value, pXi∈ℜ. The third step is the choice of numerical values for components of the starting vector  x′=x*; if x*≠x, one can expect that F(x*)>0 (12). The x*  value is referred to particular value of the steering variable,  x*=x*(ppr1), for example, to ppr1=-logC0. The step Δppr1 is needed, and initial steps ΔpXi for pXi and lower (pXi,inf) and upper (pXi,sup) limits for expected pXi values are also required by the iterative computer programs applied in the minimization procedure.The searching ofx(ppr1) vectors, where F(x(ppr1)) is close to zero for different ppr1 values, can be made according to MATLAB iterative computer program. The searching procedure satisfies the requirements put on optimal x(ppr1) values—provided that the F(x′(ppr1)) value (12), considered as optimal one, is lower than a preassumed, sufficiently small δ-value(13)F(x′(ppr1))<δ.Then we consider that the equalityx′(ppr1)=x(ppr1) is fulfilled. It means that Fi(x(ppr1))≅0 for all i=1,…,4, that is, (7) are fulfilled simultaneously, within tolerable degree of proximity assumed for all ppr1 values taken from defined ppr1 interval. If pr1 dissolves wholly, the ppr1 covers the interval from pC0 up to the value resulting from graphical needs, that is, from the scale for ppr1 assumed to plot the related (e.g., speciation) curves. If the solubility product for pr1 is attained at defined point of the dissolution process, then upper value assumed for ppr1 is the value corresponding to this point.The iterative computer programs are usually designed for the curve-fitting procedures where the degree of fitting of a curve to experimental points is finite. In particular case, the criterion of optimization is based on differencesF(x(ppr1(N+1)))-F(x(ppr1(N))) between two successive (Nth and N+1th) approximations of the F-value, and the optimisation is terminated if the inequality(14)|F(x(ppr1(N+1)))-F(x(ppr1(N)))|<δ
is valid for any preassumed, sufficiently low δ-value, for example, δ=10-15.However, one may happen that the condition (14) can be fulfilled at local minimum different from the global minimum. It occurs when the starting values x*(ppr1) are too distant from the true value x(ppr1), where the equality F(x(ppr1))=0 is fulfilled. In this case, one should repeat the calculations for a new vector x*(ppr1), guessed at a particular ppr1 value chosen at the start for minimisation.All vectorsx=x(ppr1) obtained for different ppr1 values are the basis for plotting the speciation curves for all species (Xi) in the system. The curves are usually plotted in the logarithmic diagrams, on 2D plane, with ppr1 on the abscissa and log[Xi] on the ordinate. Other variables, for example, pH on the abscissa, can also be applied.


## 4. Simulated Dedolomitisation

### 4.1. Preliminary Data

We refer first to aqueous solutions obtained before introducingpr1 into it. Any particular solution is characterised by pCCO2=-logCCO2 and pH = pH0 values, where pCCO2 equals 2, 3, 4, 5, or ∞ (the latter value refers to CCO2=0) and pH0 values cover the set of natural numbers within the interval 〈3,12〉. The pH0 value corresponds to the presence of a strong acid (HB, Ca mol/L) or strong base (MOH, Cb mol/L), see (8). The solution is then characterised by a defined pair of (pCCO2, pH0) values.After introducingpr1 into the solution, its initial (t=0) concentration in the two-phase system equals [pr1]t=0=C0 mol/L. Two values for pC0=-logC0, equal 2 and 3, were assumed. The set of parameters (pC0, pCCO2, pH0) assumed involves a multitude of different phenomena occurred in the system considered. The examples presented in the follwing concern only some particular cases.The volume change of the system, affected by addition ofpr1, can be neglected. The volume changes, involved with further dissolution and formation of precipitates, are neglected too. In order to neglect the diffusion effects, the systems were (virtually) mixed (homogenised). The dissolution has been considered as a quasistatic process, carried out under isothermal conditions.


### 4.2. Discussion on Particular Systems

At the first stage of the process that occurred afterpr1 addition, the solution is unsaturated against any particular precipitate, that is, qi<1 (i=1,…,5) in (6). At defined point, it saturates against another precipitate. As will be stated in the following, two different precipitates: pr2 = CaCO3 or pr5 = Mg(OH)2, are formed in the systems considered. Then the solution saturates against pr1 or transforms wholly to the second precipitate; in the latter case, the saturation towards pr1 is not attained. At a relatively high CCO2 value, pr1 dissolves wholly and no other precipitate is formed (qi<1). The curves expressing the relations (9) and (10) are referred to unsaturated solutions in nonequilibrium systems and termed dissolution curves. The curves expressing the relations (9) and (10), when referred to the solutions saturated against a particular precipitate, are termed the solubility curves (∑Me≤sMe, Me = Ca, Mg). The dissolution curves presented below are then terminated:(a)
by the bifurcation point, where the solubility product (Kspi) for the corresponding precipitate (pr2 or pr5) is attained (q2=1 or q5=1), or(b)
by the point, where[pr1]=0 at qi<1 (i=1,…,5), that is, pr1 dissolves wholly.The dissolution/solubility curves, obtained atpC0=2 and 3, are plotted in Figures 1(a)–1(d), for different pH0 and pCCO2 values. The curves plotted in Figures 1(c) and 1(d) refer to different sets of (pC0, pCCO2, pH0) = (3, pCCO2, pH0) values. When CCO2 exceeds distinctly the C0 value (pC0=3), pr1 dissolves wholly before the solubility product for pr2 is attained.Thelog∑Me versus pH relationships ((9), (10)) plotted for different sets of (pC0, pCCO2, pH0) values: (a) (2, pCCO2, 5); (b) (2, pCCO2, 7); (c) (3, pCCO2, 3); (d) (3, pCCO2, 6). The pCCO2 values are indicated at the corresponding curves. For further details, see text.
(a)(b)(c)(d)The plots for soluble species consisting the expressions for∑Ca and ∑Mg  ((9), (10)) and referred to different sets of (pC0, pCCO2, pH0) values are presented in Figures 2(a) and 2(b) (for pC0=2) and Figures 2(c) and 2(d) (for pC0=3).Thelog[Xi] versus pH relationships plotted for different sets of (pC0, pCCO2, pH0) values: (a) (2, ∞, 7); (b) (2, 3, 7); (c) (3, ∞, 7); (d) (3, 3, 7) for different soluble species Xi indicated at the corresponding curves: 1—Ca+2, 2—Mg+2, 3—CaOH+1, 4—MgOH+1, 5—CaHCO3 +1, 6—MgHCO3 +1, 7—CaCO3, 8—MgCO3.
(a)(b)(c)(d)As results from the course of speciation curves plotted in Figure2, the first (dissolution) step can be represented by reactions: (15a)pr1+2H2O=Ca+2+Mg+2+2HCO3-1+2OH-1,(15b)pr1+H2O=Ca+2+Mg+2+HCO3-1+CO3-2+OH-1,(15c)pr1=Ca+2+Mg+2+2CO3-2,where soluble species are formed. At the bifurcation points, the solubility product for pr2 is attained (q2=1) and then pr2 is precipitated:(16a)pr1+Ca+2=2pr2+Mg+2,(16b)pr1+H2O=pr2+Mg+2+HCO3-1+OH-1.The soluble magnesium species evolved into the solution, as the result of transformation ofpr1 into pr2, are represented by m-lines, referred to (9); each of the m-lines terminates at the point where the solubility product (Ksp1) for pr1 is attained (i.e., q1=1) and further dissolution of pr1 is stopped. As we see, the pH values corresponding to the bifurcation points are lowered with growth of CCO2 value. The growth in CCO2 causes also a significant growth in dissolution/solubility values.The points where the solubility product (pKsp1=16.54) for pr1 is attained (q1=1) at the same pair of (pC0, pCCO2) values and different pH0 values are marked and connected by lines on the corresponding plots in Figure 3. In the middle part of the resulting curves, the points are grouped together, within a small area.ThelogsMe versus pH relationships (Me = Ca, Mg) plotted at pKsp1=16.54 for pr1, ∑Me=sMe  ((9), (10)) at different (pC0, pCCO2) values: (a) (2, ∞); (b) (3, ∞); (c) (2, 4); (d) (3, 4); the numbers at the corresponding points correspond to pH0 values.
(a)(b)(c)(d)The course of the related plots is affected bypKsp1 value, assumed for solubility product of dolomite (pr1), as indicated in Figure 4 (for pC0=2) and Figure 5 (for pC0=3). The points referred to different pH0 values were omitted there, for brevity.ThelogsMe versus pH relationships (∑Me=sMe in (9), (10) at (pC0,pCCO2)=(2,5) for different pKsp1 values assumed for pr1: (a) 16.54; (b) 16.7; (c) 17.0; (d) 17.2; (e) 17.4; (f) collected effect of Ksp1 values (indicated at the corresponding curves) on logsMg value.
(a)(b)(c)(d)(e)(f)ThelogsMe versus pH relationships (∑Me=sMe in (9), (10)) at (pC0,pCCO2)=(3,5) for different pKsp1 values assumed for pr1: (a) 16.54; (b) 16.7; (c) 17.0; (d) 17.2; (e) 17.4; (f) collected effect of pKsp1 values (indicated at the corresponding curves) on logsMg value.
(a)(b)(c)(d)(e)(f)


### 4.3. Solubility Curves for Dolomite Plotted at Different PreassumedpKsp1 Values

As has been stated above, the values forpKspd attainable in the literature differ in wide range of values: from 16.54 to 19.35. Except the troubles involved with dolomite stoichiometry, these (serious) discrepancies in pKsp1 value are affected by (and resulted from) differences in solubility of calcite and magnesite. Namely, calcite constituent of dolomite dissolved more rapidly than magnesite constituent [27]. These effects, together with possible nonstoichiometry of dolomite (i.e., formation of magnesian calcite), make the system with dolomite a highly complicated one.The speciation curves for dolomite are plotted atpCCO2=5 and pC0=2 and different pKsp1 values; it results that at pKsp1 equal 16.54 and 16.7, the equilibrium solid phase is calcite. However, when pH of the solution is greater than the boundary (minimal) value, the solid phase contains two equilibrium precipitates: calcite and dolomite. However, for pKsp1=17 at lower pH values, the equilibrium solid phase is dolomite and calcite appears as the second equilibrium solid phase at pH greater than ca. 10.2. At pKsp1=19.35, calcite is not formed. The curves of log[Xi] versus pH relationships plotted for different species Xi(i.e., precipitates: pr1, pr2 and soluble complexes: Mg+2, Ca+2, MgHCO3 +1, CaHCO3 +1, MgCO3, CaCO3) are terminated at pH where the solubility product (Ksp5) for pr5 is attained. The plots of solubility curves for magnesium and calcium differ significantly at pKsp1 equal 16.54 and 16.7. At pKsp1=17, the plots bifurcate at higher pH values. At pKsp1=19.35, both plots overlap.Some thermodynamic data given above are modified, to some degree, by kinetic effects. Namely, from the data referred to dissolution of dolomite into solutions with different pH0 [27] or pH0 and p(CO2) [29] values, it results that overall effectiveness of dolomite dissolution is largely affected (limited) by dissolution of pr3 component, owing to the fact that pr2 in dolomite dissolves faster than pr3.


## 4.1. Preliminary Data

We refer first to aqueous solutions obtained before introducingpr1 into it. Any particular solution is characterised by pCCO2=-logCCO2 and pH = pH0 values, where pCCO2 equals 2, 3, 4, 5, or ∞ (the latter value refers to CCO2=0) and pH0 values cover the set of natural numbers within the interval 〈3,12〉. The pH0 value corresponds to the presence of a strong acid (HB, Ca mol/L) or strong base (MOH, Cb mol/L), see (8). The solution is then characterised by a defined pair of (pCCO2, pH0) values.After introducingpr1 into the solution, its initial (t=0) concentration in the two-phase system equals [pr1]t=0=C0 mol/L. Two values for pC0=-logC0, equal 2 and 3, were assumed. The set of parameters (pC0, pCCO2, pH0) assumed involves a multitude of different phenomena occurred in the system considered. The examples presented in the follwing concern only some particular cases.The volume change of the system, affected by addition ofpr1, can be neglected. The volume changes, involved with further dissolution and formation of precipitates, are neglected too. In order to neglect the diffusion effects, the systems were (virtually) mixed (homogenised). The dissolution has been considered as a quasistatic process, carried out under isothermal conditions.


## 4.2. Discussion on Particular Systems

At the first stage of the process that occurred afterpr1 addition, the solution is unsaturated against any particular precipitate, that is, qi<1 (i=1,…,5) in (6). At defined point, it saturates against another precipitate. As will be stated in the following, two different precipitates: pr2 = CaCO3 or pr5 = Mg(OH)2, are formed in the systems considered. Then the solution saturates against pr1 or transforms wholly to the second precipitate; in the latter case, the saturation towards pr1 is not attained. At a relatively high CCO2 value, pr1 dissolves wholly and no other precipitate is formed (qi<1). The curves expressing the relations (9) and (10) are referred to unsaturated solutions in nonequilibrium systems and termed dissolution curves. The curves expressing the relations (9) and (10), when referred to the solutions saturated against a particular precipitate, are termed the solubility curves (∑Me≤sMe, Me = Ca, Mg). The dissolution curves presented below are then terminated:(a)
by the bifurcation point, where the solubility product (Kspi) for the corresponding precipitate (pr2 or pr5) is attained (q2=1 or q5=1), or(b)
by the point, where[pr1]=0 at qi<1 (i=1,…,5), that is, pr1 dissolves wholly.The dissolution/solubility curves, obtained atpC0=2 and 3, are plotted in Figures 1(a)–1(d), for different pH0 and pCCO2 values. The curves plotted in Figures 1(c) and 1(d) refer to different sets of (pC0, pCCO2, pH0) = (3, pCCO2, pH0) values. When CCO2 exceeds distinctly the C0 value (pC0=3), pr1 dissolves wholly before the solubility product for pr2 is attained.Thelog∑Me versus pH relationships ((9), (10)) plotted for different sets of (pC0, pCCO2, pH0) values: (a) (2, pCCO2, 5); (b) (2, pCCO2, 7); (c) (3, pCCO2, 3); (d) (3, pCCO2, 6). The pCCO2 values are indicated at the corresponding curves. For further details, see text.
(a)(b)(c)(d)The plots for soluble species consisting the expressions for∑Ca and ∑Mg  ((9), (10)) and referred to different sets of (pC0, pCCO2, pH0) values are presented in Figures 2(a) and 2(b) (for pC0=2) and Figures 2(c) and 2(d) (for pC0=3).Thelog[Xi] versus pH relationships plotted for different sets of (pC0, pCCO2, pH0) values: (a) (2, ∞, 7); (b) (2, 3, 7); (c) (3, ∞, 7); (d) (3, 3, 7) for different soluble species Xi indicated at the corresponding curves: 1—Ca+2, 2—Mg+2, 3—CaOH+1, 4—MgOH+1, 5—CaHCO3 +1, 6—MgHCO3 +1, 7—CaCO3, 8—MgCO3.
(a)(b)(c)(d)As results from the course of speciation curves plotted in Figure2, the first (dissolution) step can be represented by reactions: (15a)pr1+2H2O=Ca+2+Mg+2+2HCO3-1+2OH-1,(15b)pr1+H2O=Ca+2+Mg+2+HCO3-1+CO3-2+OH-1,(15c)pr1=Ca+2+Mg+2+2CO3-2,where soluble species are formed. At the bifurcation points, the solubility product for pr2 is attained (q2=1) and then pr2 is precipitated:(16a)pr1+Ca+2=2pr2+Mg+2,(16b)pr1+H2O=pr2+Mg+2+HCO3-1+OH-1.The soluble magnesium species evolved into the solution, as the result of transformation ofpr1 into pr2, are represented by m-lines, referred to (9); each of the m-lines terminates at the point where the solubility product (Ksp1) for pr1 is attained (i.e., q1=1) and further dissolution of pr1 is stopped. As we see, the pH values corresponding to the bifurcation points are lowered with growth of CCO2 value. The growth in CCO2 causes also a significant growth in dissolution/solubility values.The points where the solubility product (pKsp1=16.54) for pr1 is attained (q1=1) at the same pair of (pC0, pCCO2) values and different pH0 values are marked and connected by lines on the corresponding plots in Figure 3. In the middle part of the resulting curves, the points are grouped together, within a small area.ThelogsMe versus pH relationships (Me = Ca, Mg) plotted at pKsp1=16.54 for pr1, ∑Me=sMe  ((9), (10)) at different (pC0, pCCO2) values: (a) (2, ∞); (b) (3, ∞); (c) (2, 4); (d) (3, 4); the numbers at the corresponding points correspond to pH0 values.
(a)(b)(c)(d)The course of the related plots is affected bypKsp1 value, assumed for solubility product of dolomite (pr1), as indicated in Figure 4 (for pC0=2) and Figure 5 (for pC0=3). The points referred to different pH0 values were omitted there, for brevity.ThelogsMe versus pH relationships (∑Me=sMe in (9), (10) at (pC0,pCCO2)=(2,5) for different pKsp1 values assumed for pr1: (a) 16.54; (b) 16.7; (c) 17.0; (d) 17.2; (e) 17.4; (f) collected effect of Ksp1 values (indicated at the corresponding curves) on logsMg value.
(a)(b)(c)(d)(e)(f)ThelogsMe versus pH relationships (∑Me=sMe in (9), (10)) at (pC0,pCCO2)=(3,5) for different pKsp1 values assumed for pr1: (a) 16.54; (b) 16.7; (c) 17.0; (d) 17.2; (e) 17.4; (f) collected effect of pKsp1 values (indicated at the corresponding curves) on logsMg value.
(a)(b)(c)(d)(e)(f)


## 4.3. Solubility Curves for Dolomite Plotted at Different PreassumedpKsp1 Values

As has been stated above, the values forpKspd attainable in the literature differ in wide range of values: from 16.54 to 19.35. Except the troubles involved with dolomite stoichiometry, these (serious) discrepancies in pKsp1 value are affected by (and resulted from) differences in solubility of calcite and magnesite. Namely, calcite constituent of dolomite dissolved more rapidly than magnesite constituent [27]. These effects, together with possible nonstoichiometry of dolomite (i.e., formation of magnesian calcite), make the system with dolomite a highly complicated one.The speciation curves for dolomite are plotted atpCCO2=5 and pC0=2 and different pKsp1 values; it results that at pKsp1 equal 16.54 and 16.7, the equilibrium solid phase is calcite. However, when pH of the solution is greater than the boundary (minimal) value, the solid phase contains two equilibrium precipitates: calcite and dolomite. However, for pKsp1=17 at lower pH values, the equilibrium solid phase is dolomite and calcite appears as the second equilibrium solid phase at pH greater than ca. 10.2. At pKsp1=19.35, calcite is not formed. The curves of log[Xi] versus pH relationships plotted for different species Xi(i.e., precipitates: pr1, pr2 and soluble complexes: Mg+2, Ca+2, MgHCO3 +1, CaHCO3 +1, MgCO3, CaCO3) are terminated at pH where the solubility product (Ksp5) for pr5 is attained. The plots of solubility curves for magnesium and calcium differ significantly at pKsp1 equal 16.54 and 16.7. At pKsp1=17, the plots bifurcate at higher pH values. At pKsp1=19.35, both plots overlap.Some thermodynamic data given above are modified, to some degree, by kinetic effects. Namely, from the data referred to dissolution of dolomite into solutions with different pH0 [27] or pH0 and p(CO2) [29] values, it results that overall effectiveness of dolomite dissolution is largely affected (limited) by dissolution of pr3 component, owing to the fact that pr2 in dolomite dissolves faster than pr3.


## 5. Final Comments

The paper provides the calculation procedure that enables some changes in speciation in the system with dissolving dolomite to be followed. In each case, the dissolution was considered as aquasistatic, isothermal process. The dissolution concept refers to the systems where solubility product of the related precipitate has not been crossed yet. The dissolution has been considered under different conditions, expressed by (a) concentration of the solid phase (C0 mol/L), (b) concentration (CCO2 mol/L) of CO2, and (c) concentration of a strong acid (Ca mol/L) or base (Cb mol/L), expressed by the value Δ=Cb-Ca. The procedure applied enables the concentrations of particular species formed at different pH of the solution to be calculated at different moments of the dolomite dissolution. This way, the dissolution (∑Me, mol/L) value was plotted. At the end of the dissolution process, ∑Me assumes its limiting value, equal to the solubility (sMe, mol/L), that is,  ∑Me=sMe. As results from the examples quoted, in some instances the solubility product of other (different from the dissolving) solid phases is crossed. Higher (relative) concentration of CO2 in the solution promotes the dissolution of dolomite.



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*Source: 723052-2012-03-14.xml*