The coordination chemistry of weathering: III. A generalization on the dissolution rates of minerals

o Cacmochtmica Ano Vd. 52,pp. 1%9-1981 Copy*t 6 1988Pcrprmon Rar pk. Printed in U.S.A.

Geochimrca

0016-7037/88/53.00

+ .slo

The coordination chemistry of weathering: III. A generalization on the dissolution rates of minerals ERICH WIELAND, BERNHARD WEHRLI and WERNER STUMM Swiss

Federal Institute for Water Resources and Water Pollution Control (EAWAG), Swiss Federal Institute of Technology (ETH), Zurich, Switzerland (Received August 14, 1987; accepted in revisedform April 27, 1988)

Abstract-A general rate law on the surface-controlled dissolution of oxides and silicates is discussed. Combining concepts of surface coordination chemistry with established models of lattice statistics and activated complex theory we propose a general rate law for the acid- and ligand-promoted dissolution of minerals R = kx,P,S where R is the proton- or ligand-promoted dissolution rate [molesm -* s-l), k stands for the appropriate rate constant [se’], x. denotes the mole fraction of dissolution active sites [ -1, P, represents the probability to find a specific site in the coordinative arrangement of the precursor complex [ -1 and S is the surface concentration of sites [ mol m-*1. Surface complexes (surface chelates and metal proton complexes) are precursors in the rate-limiting detachment of a central metal ion from the surface into the solution. We develop a mechanistic model, which clarifies the pHdependence of dissolution rates. First appropriate surface protonation isotherms are derived. The surface protonation equilibria of different minerals at constant ionic strength match a single Freundlich isotherm with a slope of cu. 0.2. This result explainsthe fnquently reportedfractionalpHdependence of dissolution rates. Based on the Bragg-Williams approximation a lattice statistical procedure is outlined which permits the calculation of the probability P, of the precursor complex. The consistency of reported experimental dissolution rates and activation energy is tested and, as a consequence, different possibilities to correlate reaction rates (kinetic rate constants [s-l] are not reported) or apparent activation energies with thermodynamic data in the form of free linear energy relations are explored. We postulate that the site energy (Madelung energy) of the most stable lattice constituent (generally a metal cation at a surface site) or the ion formation energy of the solid-characteristic of the free energy needed to break the essential bonds in the lattice-are suitable free energy parameters to be correlated with the dissolution rates (log RH at pH 5) =MOH, =MOH;

GLOSSARY OF SYMBOLS A a C CH,

CL,

cj

m n p,

Preexponential factor in Arrhenius equation (26) Activity Integral capacitance of a flat double layer [Farads m-‘1 Surface concentration of protons, adsorbed ligands and precursor complexes respectively

R RH,

[molesm-*1 E.,

4,

EM

/

F GO Ho h i i

=M, =Mj, =ML

Arrhenius activation energy, ion formation energy and Madelung energy respectively [J moles-‘] “Apparent” activation energy as determined from the temperature dependence of reaction rates: Em = E. + E,, where E, accounts for the temperature dependence of preequilibria. Decay frequency of an activated complex / = kB. T/h (s-‘1. Faraday constant [Coulombsmole-‘] Standard free energy [J moles-‘] Standard enthalpy [J mole-‘] Planck’s constant [J *s] Rate order based on =MOH ; Subscript: Denotes precunor complex. Number of nearest neighbor protons forming a precursor complex/rate order based on =MOH; Equilibrium constant of reaction i Rate constant of proton and ligand-promoted dissolution reqectively Boltzmann constant [J * K-‘] Surface metal sites, precursor configuration of surface metal site, and surface chelates, respectively

RL

s so T w x.,

XH,

XL

ZPC I( (T ‘Ti]

t[il

Uncharged and protonated surface hydroxo Broup Exponent of a Freundlich isotherm Fractional rate order based on [H ‘1 Probability to find a specific site in the geometric and coordinative arrangement of the precursor complex [ -1 Molar gas constant [J K-’ mol-‘] Rate of proton and ligand-promoted dissolution respectively [moles m -’ h -‘I Surface density of total crystallographic sites [molesm-‘1 Standard entropy [J K-’ mol-‘1 Temperature [K] Interaction energy (Frumkin isotherm parameter) [J mol-‘1 Molar fraction of dissolution-active (step, kink) sites, of protonated sites and of sites occupied by the adsorbed ligand L respectively [ -1 Subscript: denotes the zero point of charge Transmission coefficient Surface charge density [moles m -2] Activity coefficient [ m 2 mol-‘1 Surface concentration of species i [molesm-*] Dissolvedbulk concentration of species i [M] Superscript: denotes activated state INTRODUCIION

CHEMICAL

WEATHERING, i.e., the dissolution of minerals by the action of water and its solutes, is an important feature of the global hydrogeochemical cycle of elements. Surface processes rather than transport processes are the rate-controlling

steps in the dissolution 1969

of most slightly soluble oxides and

1970

E. Wieland, B. Wehrliand W. Stumm

silicates. In the two previous papers ( FURRERand STUMM, 1986; ZINDERet al., 1986) we have illustrated that most of the dissolution reactions are critically dependent on the coordinative interactions taking place on these surfaces, above all on the interaction of surface oxo- and hydroxo groups with H+, OH- (surface protonation) and with suitable ligands (anions and weak acids). The mechanisms were described by the attachment of protons or ligands to the reaction sites prior to the detachment of the metal species from the latter. The detachment step is rate-determining; thus the reaction rate was shown to depend on the concentrations of the surface species. We have extended our studies to other minerals, especially kaolinite and muscovite. In connection with these studies ( WIELANDand SNMM, in preparation), we deem it desirable to attempt to develop further the ideas given by RJRRERand STUMM (1986) into a more unified theory on the kinetics of dissolution and to investigate whether linear free energy relationships (LFER) permit a general characterization of the dissolution rates of different minerals and to test such a theory for consistency with geological and laboratory experimental evidence. The kinetic arguments presented here are drawn from the work of LASAGA( 198 l), AAGAARD and HELGESON( 1982), MORGAN and STONE (1985) and the experimental developments on understanding weathering processe sby LAGACHE (1965), WOLLAST (1967), HELGESON(1971), LUCE ef al. (1972), PA&S (19831, BUSENBERGand CLEMENCY(1976), PETROVIC (1976), GRANDSTAFF (1986), HOLDREN and BERNER(1979), HELGE~ONel al. (1978, 1984) CHOU and WOLLA!_X(1985), MURPHY and HELGE~ON (1987), and BLUM and LASAGA(1988). As in the preceding papers, the dissolution theory is based on the surface complex formation model (S~HINDLER et al., 1976; STUMM et al.. 1976, 1980, 1987; SFQSITO, 1983; SCHINDLERand STUMM, 1987) which considers that the surfaces of hydrous oxides and silicates contain hydroxyl surface groups with well defined coordination properties for interaction with H +, OH -, metal ions, anions and weak acids (ligands). This paper is organized along the following lines: we first derive a simplified, generalized rate law for the dissolution of minerals, compatible with the general framework of activated-state theory. We show that the dissolution rate of a mineral, R, can be interpreted as R = kCj where C, is the concentration (activity) of the precursor of the activated complex. The concentration C/can be expressed as the product of the surface concentration of active sites, S [ mol me21 and Pi, a factor that accounts for both the surface coverage of dissolution promoting reactants (protons or ligands) and their coordinative geometric arrangement. Then a simple lattice statistical argument is derived in order to relate Pj to the mole fraction of protonated sites, xn. Finally, we explore various possible approaches to apply LFER to characterize the dissolution rates of various minerals. A GENERAL RATE LAW FOR THE SURFACECONTROLLED DI!!SOLUTION OF MINERALS

When a mineral dissolves, the coordinative partners of the crystal constituents change upon dissolution. The central

metal ions of the mineral surface will exchange their ligands, going into solution, e.g., the 02- incipiently coordinated to Al ions in an aluminum oxide layer will be replaced by HzO, OH- or other ligands when the Al ions are transferred into solution. The key to understanding the rate of the overall reaction is an understanding of the factors that affect the elementary steps. The sequence of the various reactions may be quite involved, but the following steps are obvious: i) mass transport of solutes (H’, OH-, ligands) to the mineral surface; ii) surface attachment of the solutes (adsorption or surface complex formation); iii) various surface chemical reactions (inter-lattice transfers etc.); iv) detachment of reactants from the surface; and v) mass transport of reactants into the bulk of the solution. In the surfacecontrolled dissolution rate of minerals, a reaction step at the surface is rate-determining. In the case of steady state conditions with respect to the surface concentration of the precursor, the dissolution kinetics follow a zero order rate law: R = constant [ mol me2 h-l]. To fix ideas, a simple reaction scheme can be proposed in terms of a consecutive reaction; for example, in a ligand-promoted disco lution reaction: =M + L(aq) “+ =ML

(la)

=ML 2 ML(aq)

(lb)

a fast adsorption or surface complex formation step is followed by a slow and thus rate-determining detachment of the metal ligand complex from the surface lattice. The dissolution rate can be written ( FURRER and STUMM, 1986)as

=d[Ml

R

L

T*G

1 = kL(

=ML} = k,.K,,,,{=M)[L]

(2)

where RL = the dissolution rate [ mol mm2h-’ 1,. kL - a rate constant [h-l]; m = concentration of suspension [g I-‘]; a = specific surface [m2 g-l]; { =M } , { =ML) are concentrations (or activities, respectively) for the concentrations of the free surface sites (metal centers), and the surface sites occupied by L (metal ligand complex) [ mol m-*1. The rate law (2) has been confirmed experimentally for a variety of bidentate (surface chelate forming) ligands ( FURRERand STUMM,1986; ZINDER efal., 1986). ActivafedComplex Theory (ACT). Reaction (la) and (I b) for the surface-controlled dissolution can be represented by the general scheme K*

A * A* L products. The “mononuclear” decomposition can be interpreted as the ratedetermining elementary step. The mean features of ACT are summarized in Fig. 1. Accordingly, the ligand-promoted reaction (Eqn. 2) could be written in such a way that the detachment corresponds to the decomposition of an activated surface complex. =ML :

=ML’

=ML’ L ML(aq).

(3a)

(3h)

The surface metal ligand complex, as a suitable site for detachment, is thus the precursor of the activated complex and is in local equilibrium with it. The dissolution rate R [ mol m -2 h-‘1 most generally is proportional to the concentration (activity) of the precursor species C, [ mol m-‘] R = kc,.

(4)

1971

Coordination chemistry of weathering: 111

Reaction

cootdlnate

-

Elementary reactions: KS A +=

A+

-Lb

$

3

Rate

products

*

L.

c

yj

7

-

exp

(1)

(- &

.*

) ;

C, a x.P,S.

kB*T

f = FF_

RT

[S-'1

The dissolution rate, as given in a most generalizing way in F.qn. (4). can now be expressed in a somewhat more detailed but still comprehensive way as R,, = k,,x.P,S (iii) (iv)

- yj

f) k = f exp (- AG

(5b)

(ii)

and rate constant:

if I( = 1 and y*

We assume in the following that dissolution promoting reactants adsorb on all sites with the same probability. If the precursor consists of a single site with an adsorbed bide&ate ligand L then P, = xL, where xL is the mole fraction of sites occupied by a ligand. Notice that in a lattice statistical model the empirical mole fractions are interpreted as probabilities for the microscopic sites to be “empty” or occupied by an adsorbed species. We will discuss in more detail below, that the formation of an activated complex in acid promoted dissolution involves several protons adsorbed to nearest neighbor OH-groups of a surface metal center. On a lattice with N such nearest neighbors the probability for a precursor with j excess protons is related to a product of N individual probabilities. These are represented by the chance of an OH-group to be protonated or uIVhrgex$ given by xH and (1 - xH) respectively. (See Eqn. (21) below for a lattice with N = 4.) Obviously there are surface sites with widely differing activation energies; sites at surface defects, steps, kinks, pits, etc. are more active than others. In a simplifying way, this can be taken care of by multiplying Swith a mole fraction x, which gives the ratio of active sites to total (active and less active) sites

;

R

= f exp

=

k Cj [in01 111” s-l]

(bs’)) . exp (_ w’*) RT It

(v)

(vi)

FIG. 1. Activated complex theory for the surface-controlled dissolution of a mineral far from equilibrium. (Modified from MORGAN and STONE, 1985.)

Equation (4) is the same as the mte law given more specifically in Eqn. (2). because the precursor is =ML. The role of surface species. The reaction scheme ( 1,2) can be extended to solutes other than ligands that affect the dissolution rate, e.g., in the proton-promoted dissolution L is replaced by H+. The binding of protons onto the surface is usually very fast ( ASTUMIAN et al., 1981). Although we can readily determine the extent of surface protonation, it is now more difficult to measure analytically the concentration (activity) of the precursor species, Cj. It is thus necessary to seek ways to express Cj in terms of the experimentally accessible surface protonation because empirical rate laws suggest that two or more surface protons participate in the ratelimiting step (FIJRRER and STUMM, 1986). Therefore, a specific coordinative arrangement between surface protons and their nearest neighbor metal centers is assumed to form the precursor complex. A simplified, plausible relationship implies that C’jis proportional density of surface sites, S, [moles m-‘1 and (2) to the probability P, to find a site in the coordinative arrangement of the (I) to the

precursor complex:

C, a P,.S.

(54

(6)

where S, the density of metal centers in the lattice surface, is readily measurable and in principle accessible from crystallographic considerations. The first three terms on the right hand side of Eqn. (6) need some additional interpretation: x,; the densityof dissolutionactivesites.The surface sites are characterized by different activation energies concerning their tendency to dissolve. The most active sites are most likely localized on exposed surface positions such as kinks or defect sites; they are characterized by a smaller Madelung or lattice stabilization energy. Consider a simplified example: A dissolution reaction occurs chiefly at steps, which are separated by terraces with a mean depth of 50 sites. In this case x. = ‘/M.Natural samples may show very complex morphologies with an entire spectrum of site energies. At first sight it may seem somewhat arbitrary to account for the dissolution active sites in terms of a simple mole fraction x,. But one needs to realize that the detachment at active sites and at less active sites ,Ulc, active sites -c Me-aq (7a) less active sites y

Me. aq

(7b)

are parallel reactions in which the faster reaction (7a) is dominant and rate-determining. The mole fraction x. depends primarily on many morphological properties of the mineral (roughness, porosity, crystallographic plane, surFace defects, etc.). It can also depend on the way the minerals have become ground and on particle size ( HOLDREN and SPEYER, 1985). However, BLUM and LASAGA (1987) estimate that the dislocation density ofquartz seldom exceeds lo9 cme2. This small surface concentration of structural defects will not noticeably influence dissolution rates. In well crystallized samples like this the dissolution process itself will produce surface irregularities. The same mechanism may apply for silicates, where preferential dissolution of more nzactive cations will increase the mole fraction x. of silica sites in an activated configuration. The dissolution under steady state condition-as explained before (FIJRRER and STUMM, 1986)-requires that active sites are constantly regenerated dunng the dissolution reaction and that the surface to volume ratio remains constant during the time period of consideration. Furthermore, x. can be influenced by solutes that become adsorbed and block active dissolution sites, e.g.. organic substances, phosphates, but also metal ions such as V02+ on b-A1203(WIELAND, 1988), or Al( III) on albite (CHOU and WOLLAST, 1985). Interestingly, the

1972

E. Wieland, B. Wehrh and W. Stumm

most dissolution-active sites such as kinks arc also more disposed to adsorb reactive solutes. Modifying x. from the solution side is most important also in corrosion control. The dissolution rate of “passive” oxides covering metals often controls the dissolution rate of the underlying metal. Pj; theprobabilityofprecursorarrangement.Mostgenerally, a solute that becomes bound to a surface site may make this surface site (the central metal unit) more or less detachable. The increasing rate of dissolution of most minerals with increasing [H +] in solution is caused by the protonation of the surface (in excess relative to the point of zero proton charge). But P, is not simply equal to the molar fraction xn = Cn/S; the geometric coordinative arrangement of the hound surplus protons is of particular importance. The protons may be assumed to be relatively mobile in the surface layer; i.e., they can be shifted for example from OH groups to neighboring oxygen bridges (like in tautomerism ) . In a simplifying way, a random distribution of surplus protons in the surface layer may be assumed. A sufficient weakening of the critical metal-oxygen bond occurs if a sufficient number, j, of oxide or hydroxide groups neighboring the surface central metal ion-in case of the surface of &AlzO, j = 3 ( FIIRRER and STUMM, 1986)-have hecome protonated. Thus, we need a quantitative relationship between P, and xn. This will he derived in the subsequent section. k,; the dissociationof specific bonds.According to Eqn. (6) kH does not depend on the surface characteristics (morphology, surface protonation) of oxides and silicates. The specific rate constant kH rather represents the configurational (AS*) and energetic (AH’) changes between the activated complex At and the precursor A ( Eqn. (vi) on Fig. I). Actually, the kinetic parameters A*, AH” and AC” for surface processes cannot be derived from statistical mechanics. As a conceptual approach it is suitable to postulate that the rate constant kHmainly is a measure of the metal-oxygen bond broken in the rate-determining step of a detachment reaction. This is in line with previous studies (SCHOTT ef al.,198 1; SCHO-~Tand BERNER, 1983) explaining the incongruent loss of Mg, Ca and Fe relative to Si during the weathering process of various silicates by the difference in the Madelung energies of the cations. Hence, Ap could be related to the enthalpy change during the dissociation of a metal-oxygen bond in a lattice. Although the energies of bond dissociation of metaloxygen bonds in gaseous diatomic molecules are experimentally determined, suitable energies of metal-oxygen bonds in lattice structures are not reported. In a subsequent chapter we therefore establish linear free-energy relationships in order to correlate the kinetic terms (rate constant kH or activation energy E.) with suitable thermodynamic parameters approximating the enthalpy of activation Afl. SURFACE

A thermodynamic approach

Different models of the acid base equilibria at the mineral water surface have been proposed. Critical assessments were recently presented by SPOSITO(1984), HAYES and LECKIE (1987) and !~CHINDLERand STUMM (1987). Although these models of surface complexation are based on differing assumptions regarding electric double layer properties and molecular structure, they are able to fit the experimental data equally well ( WE~TALL and HOHL, 1980). According to the constant-capacitance model of SCHINDLERand STUMM(1987), the protonation of surface OH-groups (=MOH) may be characterized by the dissociation constant K&: =MOH; e,

=

The lower the pH, the faster minerals and oxides dissolve. Many authors have shown that the empirical rate law

(8)

where RH and kHare the experimental dissolution rate [mOl me2 h-r] and the rate constant, respectively, exhibits a frac-

tional rate order n. Frequently values in the range 0 -z n < 0.5 are found. Similar observations have been published for alkaline media where negative fractional exponents n have been established (see Table 1). Only for hydroxides like gibbsite (PULFER et al., 1984) and for very high proton concentrations (CORNELL et al., 1976) have values of n as high as n = 1 been reported. The kinetic treatment that is developed here for the proton-promoted dissolution may therefore be applied accordingly to the effects of alkaline media on mineral dissolution. Various interpretations have been given for these findings: HELG~SON et al. (1984) postulated the formation of an ac-

2 = MOH + H+

(9)

{=MoH}‘[H’lexp(-F*c/CRT)

(10)

{ =MOH;}

= K& (int) exp( -F2a/CR7J

(11)

where F, R and T represent the Faraday constant, the molar gas constant and the temperature mspectively. SFOSITO(1983) has shown that Eqn. (I I) can be derived from a general lattice statistical mean field model, which is mathematically equivalent to the Bragg-Wis approximation. The two model parameten, the “intrinsic” constant K:, (int) at zero surface charge and the integral capacitance of the flat electric double layer C [Faradsm -‘I can be determined from

Table 1

PROTONATTON

RH = ku[H+]”

tivated complex of the type (K,Na)Al( 0H)a.4Si30t4. CHOU and WOLLAST (1985 ) attributed the fractional rate law to the nonideal behavior of the surface, which could be calculated using the regular solution theory. A similar explanation has been given by GBAUER and STUMM (1982). In the following, we first derive two suitable isotherms to describe surface protonation data and then we will outline a more mechanistic picture of the acid-promoted dissolution process based on lattice statistics.

Some fractjonal rate orders of the H+ and OH--promoted dissolution

Mineral a-Al,% Be0

Acid

Base

n

b) i

a)

cl cl

"N4 "N4

0.41 0.17

3 2

a-FeGOH

d)

e)

0.33 0.46

Fe(OH1, Feldspars Albite Nepheline

fl g) h) i)

"N03 tic1 HClO,

3

Cr(OH),

Silica

kl

a)

b) cl d) e) f) g) h) I) k)

0.48

Buffer

0.4 0.45 - 0.2

NaOH

- 0.46

IKl

Fractional rate order based on ["+I Rate order based on (=MOH,'] or (=MO-] FURRER and STUW! (19861 ZINDER, FURRER and ST&IN (1986) SEO. FURUICHI OkAMOTO and SAT0 (19751 FURUICHI, SAT0 and OWTO (1969) HELGESON, MURPHY and AAGAARD (1984) CHOU and YOLLAST (1985) TOLE. LASAGA, PANTANO and WHITE (1986) WIRM and GIESKES (1979)

2

Coordination chemistry of weathering: III titration curves. In absence of other specifically adsorbing species the surface charge may be approximated by ,J=

{=MOH:}.

(12)

Solving Eqn. (10) for [H+] and using Eqn. (12) yields the well-known

F~mkin-Fow~er-Guggenheim {FFG) isotherm (FRUMKIN, I925; FOWLERand GU~ENHEIM, 1939): [H+] =

K:, (int)aS_

{ =MOH$ }

ApH = pHzpc - pH

(13) If we define the mole fraction xH and the interaction energy w [J

mol-'1 : xH = { =MOH:}/S

(14)

w = FZ.S/2C

(17)

is plotted as a master variable. The index “ZPC” denotes the zero point of charge ( STUMM and MORGAN,1981). This concept is applied in Fig. 2b to seven different oxides and one Latex sample. In the range ApH > 1 a straight line fits all the experimental data within a factor of 2. Therefore a suitable geochemical approximation can be given by a F~un~ich master isothe~ of the form:

{ =MOHfl

exp(F*-t=MOI-GlIC’ZW.

~=MOH:J

1973

= KF.[H+]m iH+lgn,

(18)

with the parameters p& = -6.5 1 and m = 0.2 1. The specific value for the exponent m of the different oxides is contained in Table 2. Equation (18) corresponds to a tangent or secant to the more precise F~mkin curve (Eqn. 13). The relation is useful to transform dissolution rate laws of tractional order n, based upon solution pH (Eqn. 8), into the appropriate expression for a surface-controlled reaction. The substitution of [H+] by { =MOH: } in relation (8) yields:

(15)

R+j= k~~([H+]~~/K~“).{

=MOH:j”‘”

(19)

then Eqn. (13) takes the more familiar form: =&*{=MOHt+j’. [H+] = Ir,, (int) -e

exp(Zw=xn/RT). H

(16)

In the two previous papers of this series ( FIJRRER and STUMM, al., 1986) it has been shown that, within experimental, error integer values (i = n/m) of the reaction order result, if the rate law is formulated in terms of protonated surface sites. The 1986; ZINDERet

In Fig. 2a, tbe FFG-equation (Eqn. 13) is applied to the protonation equilibria at different oxide surfaces. The parameters K:, (int) and C were determined from published titration curves using a typical c~s~l~p~c site density S = 1.67 - 10-s mol me2 (~Ke~ndjng to 10 sites nmm2) and the linear regression method as described by SCHINDLER and STIJMM(1987). The isotherm parameters are summarized in Table 2. As Fig. 2a indicates, the Frumkin equation (Eqn. 13) is able to represent different protonation data over a wide range of pH. Generally, the surface protonation increases only slightly with decreasing pH. This nonideal (or “non-~ngmui~an”) behavior is due to electrostatic interactions at the oxide water interface, which leads to an inhibition of the proton uptake at the positively charged surface. ANDERSONet al. (1976) and, recently, HARDING and HEALY (1985) and FOKKINK(1987) have shown that the same protonation curve describes various particle surfaces if the difference

log

(20)

{=MOH,+l

reaction orders of the acid-promoted dissolution have been determined as i = 3 in the case of 6-A120,and cu-FeOOHand i = 2 for Be0 (Table I). Lattice statistics as a mechanistic model The question remains bow second and third-order rate laws in proton&d surface sites { =MOH: 1 can be interpreted on the basis of a molecular mechanism. FURRER and STUMM(1986) presented evidence that tire rate-limiting step of the dissolution process is the detachment of an activated complex surrounded by i (i = 2 or 3) protons,

log ~=MOH*+~

-4

~9 Beryllium oxide -5

* Goethite

IJ Beryllium oxide

Q Aluminum oxide

l

+ Zirconium oxide

-5

A Anatase

Goethite

Y Hematite

* Aluminum oxide + Zirconium oxide A Anatase

-6

-6 6

6

4

PH FIG. 2. Surface p~tonation isotherms. Dots represent experimental data from titration curves at ionic strength f = 0.1 (Hematite, i = 0.2). References are indicated in Table 2. The concentration of protonated sites f =MOH$) is given in moles m-‘. BET surface data were used to calculate the surface concentrations. a) Frumkin isotherms (Eqn. 13). b) Surface concentration as a function of pHzpc - pH = ApH (E qn. 18). The adsorption isotherm at ApH 5 1 can be interpreted as a Freundlich master isotherm (some of the data points for Be0 are off scale),

1974

E. Wieland, B. Wehrli and W. Stumm

Table 2

Protonation isotherm parameters of different oxides

a) Surface

p"zpc

d)

e)

C

c)

w

m

nF/cm2

kJ/mol

b) pK& tint)

TiOs,

fl

6.25

4.92

79

99

0.19

ZrOs s-Al24 a-FeOOH a-FqOs

g) h) i) k)

6.4 8.7 7.28 8.67

4.72 7.32 6.03 7.47

148 115 167 94

51 68 47 a3

0.13 0.16 0.16

Fe,&, Be0 Latex

g) I) ml

5.63 8.71 6.45

151 134 113

52 58 69

0.085 0.14

6.8 10.2 8.0

(4 - j)=uncharged OH groups is given by the Bernoulli scheme (RNETTI, 1974): 4! j!(4 -j)!

Pj = 7

..&.(l

The first factor on the right-hand side represents the number of possible geometric arrangements of j protons on four neighbor sites. The exclamation mark stands for the factorial (4! = 4 - 3 - 2.1). Obviously, c Pj=

1 and

e)

f)

:I i)

:/ m)

(22)

j-0

(23)

iz

cl d)

pH of the zero point of charge Intrinsic protonation equilibrium constant (Eq. lla) Integral double layer capacitance Interaction energy parameter of the FFG isotherm (Eq. 16). The high site's density used in these calculations yields high values of w (Eq.15). Freundlich slope for ApH > 2 (Eq. 181 this studv REGAZZONI-,BLESA and MAROTO (1983) KUMMERT and STUMM (1980) SIGG and STUMM (1981) FOKKINK (1987) FURRER and STUMM (1986). m was calculated for ApH > 4 HARDING and HEALY (1985)

- XHy-‘.

The first summation is equivalent to the statement that each metal site belongs to one of the five possible configurations shown in Fig. 3, whereas Eqn. (23) reflects the fact that each surface proton belongs to four different metal sites. The maximum proton density that results from titration experiments is typically 10 times smaller than the crystallographic site density S (Table 3). As a consequence the mole fraction xn becomes xH Q 1 and Eqn. (2 1) simplifies for real systems to Pj =

4

0 i

X-ii

(24)

Lattice statistics provide a simple framework to estimate probabilities (molecular fractions) of such geometric arrangements at surface sites. Different two-dimensional lattice statistical models have been derived which account for interactions between adsorbed species: The exact solution of the Ising model (BAXTER, 1982) requires an evolved mathematical treatment but considers the effect of the interaction energy on the two-dimensional distribution. The quasi chemical approximation (HILL, 1960) allows only for interaction in pairs between the nearest neighboring species, whereas the Bragg- Williams approximation (HILL, 1960) starts from the basic assumption that the species are distributed at random. In this case the interaction energy does not influence the configurational entropy of the system. Here we choose the Bragg-Williams approximation because the FFG-isotherm (Eqn. 13 and Fig. 2a) was consistently derived using this regular solution model (FOWLER and GUGGENHEIM, 1939). Recent experimental evidence indicates that protons at interfaces are extremely mobile ( PRATS et al., 1986), which could result in fast random motion among surface sites. The Bragg-Williamsapproximation allows the calculation of macroscopic properties such as free energiesand activity coefficientsfrom lattice statistical partition functions. Here we are interested only in the direct calculation of mole fractions of precursor complexes. To derive the appropriate equations we need only to define the geometry of the lattice and to exploit the basic assumption of the Bragg-Williams model: Random distribution of species. The chessboard geometry is a convenient way to represent two surface species (metal centers = M, and OH-groups with a 1: I stoichiometry (see Fig. 3). If the hydroxyl species are allowed to be randomly protonated, we may distinguish 5 types of metal sites Mj, according to the number of nearest neighbor protons (j = 0, 1,2, 3,4). In the absence of other species, the probability of finding a hydroxyl group in the protonated or uncharged state is equal to xu or (1 - xn), respectively. The probability Pj to find a metal site surrounded by j-protonated and

FIG. 3. The chessboard model of oxide interfaces. a) Lattice statistical arrangement of metal sites M and surface OH-groups. b) Within a binary mixture of protonated (black) and uncharged (hatched) OH groups five types of metal centers Mj exist according to the number of nearest neighbour protons (j = 0, 1,2,3,4).

1975

Coordination chemistry of weathering: III

for j > 2, where 4 represents the first term at the right-hand side 0 of Eqn. (21). Equation (24) can easily be related to empirically determined rate laws (Eqn. 20) assuming that the rate orderj (j = 2 or 3) corresponds to a dissolution mechanism which involves a surface complex =Mj with j (2 or 3) nearest neighbor protons. The combination of Eqns. (6) and (24) yields

RH = kL’.

4 0j

.XJH.X,.S.

(25)

Two factors in Eqn. (25 ) are readily accessible from experimental data: Frumkin or Freundlich isotherms may be used to transform measured pH values into the mol fraction of protonated hydroxyl functions, xH. The total site density S may be obtained from crystallographic lattice parameters. Experimental methods to estimate the fraction of active sites x. are missing. Therefore, only the product kh’. x. is accessible from empirical rate constants. ACTIVATION ENERGY OF THE RATE-DETERMINING STEP

The rate constant kH as given in Eqn. (6) may be defined in terms of the Activated Complex Theory (ACT) (Fig. 1). The entropy term in Eqn. (vi) (Fig. 1), AS’*, is an indicator for the change of a precursor structure towards the formation of the activated complex. ASo* changes according to the configurational changes of a precursor being activated. During the monomolecular desorption of gaseous molecules from a surface, for example, the entropic states of precursor and activated complex are similar and, thus ASot is small or zero ( GLASSTONE etal.,194 1). The mononuclear detachment of a single metal center from the surface of a mineral is assumed to be comparable to the monomolecular desorption reaction mentioned. Thus Eqn. (vi) (Fig. 1) simplifies to kH=A-exp

% (

)

(26)

where A ( = f exp( ASO*/R) is approximately constant for dissolution processes. The enthalpy of activation, AH@, may be related to the Arrhenius activation energy E,, ( HELGESON et al., 1984) by E. = AH” + RT.

The activation energy of weathering processes usually is >30 kJ mol-’ and hence, RT is small in comparison to AHo* (RT = 2.5 kJ mol-’ at T = 25°C). It follows from Eqn. (27) E, m AHo*.

(28)

It is important, however, to realize that the comparison given in Eqn. (28) and the interpretation of Arrhenius parameters in terms of ACT is strictly limited to elementary reaction steps. Complex reaction sequences need to be analyzed in terms of possible mechanisms and ratedetermining steps. Note that dissolution processes are complex kinetic systems of surface complexation, ion exchange and detachment processes. Hence the activation energy of a dissolution process (Table 4) obtained from the temperature dependence of the overall reaction typically is an apparent activation energy ( EaPP) . However, Eappmay be used to derive an estimate for E,, the activation energy of the rate-determining elementary step supposing the temperature dependence of preceding reaction steps forming the precursor (pre-equilibrium of the surface with protons or ligands, ion exchange reactions) are negligible or known. We denote E wp = E, + E,

(2%

where Epppis the apparent activation energy obtained from the temperature dependence of the overall rate (Arrhenius plot), E, is an enthalpy quantity resulting from the temperature dependence of the preceding steps and E, is the Arrhenius activation energy of the rate-limiting detachment step. WOOD and WALTHER (1983 ) have found similar Arrhenius curves for different materials over a large temperature range (E, = 55 kJ mol-‘), but as Table 4 shows this information may not be extended to low temperatures. Assuming the validity of the given approximations and by combining Eqns. (6)) (26)) (28 ) and (29) the proton-promoted dissolution rate RH (referring to the rate-determining step) may be expressed as

(27) =-

E SPP - & + log A + log Cj 2.303 RT (30)

Table 3

Maximum surface protonation XH from titration curves

al

Oxide

'H max

Ti$" b-Al,% cz-FeOOH

a)

xH

=

0.05 0.10 0.09

Sb' 2.0 10-s > 2.1 10-s 2.8 10-S

[=MOH,+} / S;

Authors BOEHM and HERRMANN (1967) PERI (1965) YATES (1975)

the maximum surface proton concentra-

tion is taken from experimental titration curves. References see Table 2. =

concentration of surface sites in moles mm2 estimated from the crystallographic structure.

b)

S

c)

Anatase.

where, ideally, Cj ( = XaPjS) and A are constants for various minerals. In the following, we first test Eqn. (30) comparing the experimental dissolution rates RH with the related activation energies EBppof the various minerals reported in previous studies. Secondly, linear free-energy relationships (LFER) are established to estimate the dissolution rate RH (or activation energy E., respectively) of oxides and silicates using well defined thermodynamic parameters. Data selection. Table 4 lists the available data of dissolution rates and apparent activation energies as well as thermodynamic and structural data for 14 oxide and silicate minerals. Data selection was restricted to iron-free minerals (exceptions:

1976

E. Wieland, B. Wehrli and W. Stumm Table 4

Literaturadata on dlssolution rates,thcrmodyna~lc Infonsation, and activationenergy of sraaaselecteddnerals.

Mineral

TASS, [kJ/mol 1

i261

WART2

257

- 4.01

Reference [kJ?wI] 156000

,mi;m$

[k$%]

- 10.60

71

--v_________ *4

- 12.20

RIMSTIDT and BARNES

(1980)

VAN LKER et al. (1960)

~SCOVITE

- 178

* 517

- 339

135800

- 9.03

~OLINITE

- 149

- 432

- 283

134000

- 8.96

*63.75t 3

this

&lJMINUM

- 988

-2070

-1090

126000

- 8.59

*56.6&14.1

FURRER and STWM

LIN and CLEMENCY

OXIDE (a-Al,O,)

(1981)

work (1986)

this work

ERCON

AUGfTE -

128000

- 8.65

53.5

: 7.94

78

TOLE (19851

SCHOTT and WERNER

(1985)

SIEGEL and PFANNKUCH

ESTATITE

- 537

- 764

- 227

135000

- 8.18

50

ALLBITE -

-

- 239

- 174

144000

- 8.20

38

65.6

SCHOTT et al. (1981)

EUSENBERG and CLEBtENCY (19761

- 7.84

-

K-FELOSPARS

13.7

- 157

- 143

143000

- 8.21

CHOU and NOLLAST 119851

EUSENBERG and CLEMENCY

38

HELGESON et al.

- 478

AHORTHITE -

- 937

- 459

133000

- 8.32

(1984)

(1976)

(1984)

EUSENEERG and CLEMENCY

35

11976)

SCHOTT and PETIT (19871

FORSTERITIC

- 977

- 1320

- 342

125000

- 8.36

38

GRANOSTAFF

(1981)

- 495

- 631

- 136

134000

- 6.32

38

SCHOTT et al. (1981)

KIVINE

EOPSlDE

SCHOTT and PETIT (1987)

8&YERllE

- 372

- 772

- 400

112000

- 8.27

~RYLLIUM

-1160

-1670

- 516

114oOO

- 7.69

50 t17

PULFER et al.

(1984)

FURRER and STUWl (1986)

OXIDE _______--_---_--.-_4_____I_

Sol

$01

_____----__f_--________

%l

:

_-

standard free energy change, enthalpy change and entropy change of the proton-Promoted of a mineral (standard mineral cell containing 24 ol;ygen atomst (ROEIE, HEMIN~AY

and FISHER, 1978).

cell containing

Ei

molal energy of formation of a mineral from the gaseous ions (standard mineral

109 $.,

experinantal dissolution rate of a sir&era1 at

E aw

apparentactivationenergyof the overall weathering reaction of a mineral at pH 5 (*

The experimental

dissolution

24 oxygenatoms) (KELLER, 1954)

PH 5 (*

pH

4)

PH 4).

values log RN and Ea are taken from literature given in the reference list.

forsteritic olivine-18.8% FeO, augite-27% Fe203) where surface-controlled dissolution rates at pH = 5 were reported. This pH was selected because many data on dissolution rates are reported for this pH, which is particularly representative of soil-water systems. Furthermore, for many minerals the dissoIution rates have been shown to be only slightly pH-

dependent in the near neutral pH range. At pH = 5, the dissolution reaction may be inferred either to occur congruently or at least it may be assumed that the formation of a new phase occurs in a subsequent reaction step that is of little consequence to the kinetics of the dissolution step (KELLER, 1978).

Coordination chemistry of weathering: III

In order to make the thermodynamic parameters for different minerals comparable, they are “normalized”-following the suggestion by KELLER (1954)-by considering a standard mineral cell containing 24 oxygen atoms with the following stoichiometry of the dissolution reaction:

M(1),, W2)m2- - *M(i),,,, SLO,(OH),

+ a H+

+ b Hz0 + = ml M(1)‘; + m2 M(2)‘;. + mi M(i)‘i+

+

c

HISO

-+ dHz0

(31)

where a, 6, c, d, e, mi, p, q represent the appropriate stoichiometric coefficients. Consistency of kinetic data The measured dissolution rates, log RH, are plotted (Fig. 4) as a function of the apparent activation energy, EDpp(obtained from the temperature dependence of the dissolution). As expected, the dissolution rate for many minerals shows a tendency to increase with decreasing Ewp. The regression line drawn (defined above all by the experimental dissolution rates of quartz, b-Alz03, bayerite, zircon, enstatite and diop side) is characterized by the slope -A log R,IAE,, = (2.3RT)-‘, as described by Eqn. (30). Although we have to account for some minerals that do not fit this regression, we consider the correlation reasonably consistent, especially if one takes into account that the data reported are from different research groups. Furthermore, the data imply a constancy ofA - C, in Eqn. (30) and that E, is either negligible or constant. Although a larger number of data were needed to draw definite conclusions, we infer that E.,,p is plausibly a good approximation for E,, and in turn for AHo* or AGw of the “activation” of the precursor to the activated complex (Fig. 1).

l oxide

-11 _i

-7_g=-10 r

E 0 E

-

-9 _*

I2

0

20

40

Eapp

60

00

100

[k J mol-‘]

FIG. 4. Experimental values of the rates RH and activation energies E, for the dissolution of oxides and silicates at pH 5 (data from references listed in Table 4). The abbreviations used in the diagram refer to the letters underlined in Table 4.

1977

It is interesting to see whether one can arrive at reasonable explanations for those minerals that are off the regression. pH values are not given for the dissolution of augite; apparent activation energies obtained at lower pH values are typically higher ( SCHO-~-Tand PETIT, 1987). In case of olivine, the minerals were not subject to laboratory crushing prior to the dissolution experiments; thus xp (Eqn. 6) may be smalier in these preparations than those encountered in crushed samples. [This would be in line with the observation (PA&S, 1983) that the dissolution rates observed in the field are smaller than those measured in the laboratory.] All feldspars recorded in Table 4 have apparent activation energies that are about 10 kJ mol-’ smaller than those on the regression line. Perhaps this could be accounted for by the fact that comparable preequilibration steps, e.g., the replacement of Na+ by H’ in albite (CHOU and WOLLAST, 1984) and Ca”, K’ by H+ in anorthite and K-feldspars, precede the dissolution reaction. This would imply that AH (= E, in Eqn. 29) for these reactions were typically of the order of 10 kJ mol-‘.

Can dissolutionrates be predicted by linear free energy relations (LFER) ? We are searching for a measure of the energetic-s of forming the activated complex from the precursor ( AGo* or AH’*, Fig. 1). This essentially kinetic parameter cannot be derived from statistical mechanics. But the reasonably consistent relationship found between log RH and Ea,,,, motivates us to inquire whether simple relations between AHOS and thermodynamic parameters such as the enthalpy of the dissolution reaction (Eqn. 3 I), AH:,, or the ion formation energy Ei exist. In Fig. 5a and 5b, the LFERs between the dissolution rate (log RH) and, respectively, AH:, of the proton-promoted overall dissolution reaction and Ei, the ion formation energy are plotted. It is obvious from these plots that only the reactivity of the oxides correlates well with thermodynamic parameters. While silicate data represent a wide range in AHO,,, Ei and site energy, their dissolution rates are quite close to the value for A&03. It is obvious from both ofthese plots that silicates are more reactive than oxides. It is not surprising that AH:,, a thermodynamic parameter for the enthalpy change between reactant and products, is not a good measure for AH’*, the energetics of the formation of the activated complex from the precursor. Ei is the sum of energy contributions resulting from the attraction of each cation for its surroundings; it is computed by adding the energies of the bonds between the constituent cations and oxygen of a mineral starting with the elements in the gaseous ionic state ( KELLER, 1954). The lattice energy appears to be a valuable parameter to estimate the dissolution rate of oxides of A-type (hard acid) cations. However, for silicates this type of LFER does not apply; their lattice energies, the way they are calculated, comprise as a collective parameter, the stability of all cations and anions, whereas plausibly the surface-controlled dissolution

E. Wieland, 8. Wehrli and W. Stumm

f

eDt

d

*iQOU

eDf lb

site energy

15

i

[k J mol”]

“y.. *oxide / IW .5, a) LFBR between the dissolution rate Rn of oxides and silicates of a m&r& and the e&a&y change A@$&, of the co~spondi~~ proton-~orno~ overali reaction; b) LFER between the dissohmon rate RHof oxides and siiicatcs of a mineral and its ion forrn~~~n’~ergy 4; cf LFER between the ~~~~0~ rates & of oxides and &c&es and the site energy I& of the most stable cation (except alkali met&) or Si, respectively,

0 silicate

l oi

fUOO

I

I

I

100

120

140

Ei

[kJ

1

mol-‘f

rate depends solely on the lattice stability of that type of cation which is removed in the rate-determining step from the lattice. Structural effects in mineral dissolution We may thus, in a simplifying way, postulate that the dissolution rate should be proportional to the site energy of the most stable lattice cation, i.e., the bond of the most stable cation with respect to increasing site energy in the mineral lattice (excluded Si4+ in all minerals except SiOz), The main contribution to the forces stabilizing a cation in a particular site of the oxygen sphere of oxides and silicates is given by the electrostatic Madelung energy. The Madelung or site energy, EM, is defined as the energy that would be required to separate a particular ion (point charge) from its equilibrium position in a crystal structure to infinite distance. The ranges of site energies of common ions in silicates are given in Table 5 (Down, 1980; data from RAYMOND, 1971, an$ OHASHI, 1976 ) . Figure 5c depicts plots of dissolution rate, log RH , as

a function of the site energy, EM of the most stable cation (Ca, Mg, Be, Al, Zi) and Si in the minerals of Table 4. With oxides, the correlation between rate (or AZfOt) and EM is plausible. A satisfactory correlation is also found for the layer silicates kaolinite, muscovite and the Ca-Mg silicates, diopside and forsterite. But here again, some marked excep tions have to be accounted for: in the case of Z&O, we are uncertain about EM (see Table 5); the binding of Zr-0 in ZrSi04 is possibly weaker than in ZrOz. In the case of enstatite, a different mechanism than that postulated might occur. In the case of feldspars, equilibria that precede the dissolution may change the Madelung energy. Possibly the dissolution of feldspars reelects the dissolution of the Al-free silicon framework (LOU and WOLLAX, 1985 ). CONCLUSION 1. The dissolution of minerals is a consequence of chemical, surface chemical and physical reaction steps. If the dissolution kinetics is controlled by surface processes, the trans-

Coordination chemistry of weathering: III Site Energy(Made1 ung)

Table 5

EM [ kJ/mol] Na+

112911293971 4305 -

Kt ca2+ u92+ Al3 + IV

10450

Als+ VI

10868

*0d+

7400

*zr+

13300

1338 1338 4807 5058

17556 - 19646

Si4+ d-

5016 -

6270

OH-

1170 -

1317

E,,,:

ranges

of Madelung or site

energies,

respectively

cation

Mz+

or an anion

common

minerals

*estimated

(OOUTY,

of a A'-

in

1980);

values (Wieland,1988).

1979

that Eapp is often a good approximation of Ea. Most suitable free energy parameters to be correlated with dissolution rates are thus, the ion formation energy Ei or the site energy, the Madelung energy, EM, for the most stable lattice constituent. 5. In order to make further progress towards a generalization of dissolution rates of different minerals, and to test the consistency of the concepts with geological evidence, more data, especially measured dissolution rates of a great variety of minerals (in solutions whose pH and other variables are well defined) and their temperature dependence are needed ( BLUMand LASAGA,1988 ) . The determination of useful sets of data should include the careful determination of surface protonation and adsorption equilibria in order to calculate empirical rate laws in terms of precursor concentrations Cj.

Acknowledgements-This research has been supported by the Swiss National Foundation. Valuable suggestions were received from James I. Drever, Gerhard Furrer, Jacques Schott and two anonymous reviewers. Editorial handling: T. Pates

port of the reactants and of the products can be neglected, and constant (time-invariant) dissolution rates per unit surface area are observed as long as the system is far from dissolution equilibrium. In such systems kinetic equations should be formulated in terms of surface concentrations. Experimental studies should therefore establish the relevant adsorption equilibria. 2. Surface complexation models (STUMM et al., 1976; SCHINDLER and STUMM, 1987; SPOSITO, 1984) such as the constant capacitance model will prove to be essential tools for the quantitative description of the reactive surface species in mineral dissolution. We have shown that the equations of the constant capacitance model correspond to the well-known Frumkin and Freundlich isotherms. The adsorption of species is responsible for fractional rate orders if bulk concentrations are used to formulate rate laws. The inclusion of electrostatics in dissolution kinetics adds little complexity. At constant ionic strength the electric double layer produces very similar effects on different materials as Fig. 2b demonstrates. 3. Complex dissolution mechanisms which involve several surface species are best described by lattice statistical models. This method allows the calculation of precursor concentrations as a function of the surface density of dissolution promoting species such as excess protons. 4. The activated complex theory provides a model to bridge the gap between thermodynamic information (surface coordination and lattice or site energy) and kinetic information. The rate constant k is related to the free energy of conversion ( AGot or AH’*) of a suitable surface complex (precursor) to an activated surface complex (Fig. 1); this energy can be compared according to the Arrhenius theory with the activation energy E. of the rate-determining step. The fact that for many minerals a consistent relationship was found between the apparent activation energy, EaW, (from the temperature dependence of the overall dissolution rate) and the dissolution rate (log R) lends support to the inference

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