theoretical prediction of single-site enthalpies of surface

PII S0016-7037(98)00262-2

Theoretical prediction of single-site enthalpies of surface protonation for oxides andsilicates in water

DIMITRI A. SVERJENSKY1,* and NITA SAHAI†

Morton K. Blaustein Department of Earth and Planetary Sciences, The Johns Hopkins University, Baltimore, Maryland 21218, USA

(Received December 8, 1997; accepted in revised form August 20, 1998)

Abstract—Surface protonation is the most fundamental adsorption process of geochemical interest. Yetremarkably little is known about protonation of mineral surfaces at temperatures greater than 25°C. Experi-mentally derived standard enthalpies of surface protonation, !Hr,1° , !Hr,2° , and !Hr,ZPC° , correspond to thereactions

"SOH ! H# " "SOH2#

"SO$ ! H# " "SOH"SO$ ! 2H# " "SOH2

#

respectively, and provide a starting point for evaluating the role of surface protonation in geochemicalprocesses at elevated temperatures. However, the experimental data for oxides do not have a theoreticalexplanation, and data are completely lacking for silicates other than SiO2. In the present study, thecombination of crystal chemical and Born solvation theory provides a theoretical basis for explaining thevariation of the enthalpies of protonation of oxides. Experimental values of !Hr,1° , !Hr,2° , and !Hr,ZPC°

consistent with the triple layer model can be expressed in terms of the inverse of the dielectric constant (1/#)and the Pauling bond strength per angstrom (s/rM-OH) of each mineral by equations such as

!Hr,ZPC% " !&r,Z'(1/#) $ (T/#)2(%#/%T)* $ B+Z(s/rM-OH) ! H+Z.

The Born solvation coefficient !&r,Z was taken from a prior analysis of surface equilibrium constants. Thecoefficients BZ+ and HZ+ were derived by regression of experimental enthalpies for rutile, &-alumina, magnetite,hematite, and silica. This approach permits widespread prediction of the enthalpies of surface protonation.Predicted standard enthalpies of surface protonation for oxides and silicates extend over the ranges (in

kcal.mole$1): !Hr,1° , $3 to $15; !Hr,2° , $0.5 to $18; !Hr,ZPC° , $4 to $33. Minerals with the largestvalues of s/rM-OH (e.g., quartz and kaolinite) are predicted to have weakly negative enthalpies and a weaktemperature dependence for their protonation equilibrium constants. Conversely, minerals with the smallestvalues of s/rM-OH (e.g., garnets and olivines) should have strong negative enthalpies and a strong temperaturedependence for their protonation equilibrium constants. Copyright © 1998 Elsevier Science Ltd

1. INTRODUCTION

Protonation reactions on the surface of oxides and silicates inwater are the most fundamental reactions controlling the reac-tivity of minerals. Although they can be represented by multi-site models (Hiemstra et al., 1989a,b; Davis and Kent, 1990;Machesky and Jacobs, 1991a,b; Walther 1997; Koretsky et al.,1998), surface protonation equilibria can also be represented byreactions between species such as"SO$,"SOH, and"SOH2#in a single-site model according to

"SOH ! H# " "SOH2! (1)

"SO$ ! H# " "SOH (2)

"SO$ ! 2H# " "SOH2! (3)

"SO$ ! "SOH2! " 2 ' SOH (4)

(Stumm et al., 1970, 1976; Schindler, 1981; Schindler and

Stumm, 1987; Davis and Kent, 1990; Dzombak and Morel,1990; Stumm and Wieland, 1990; Sverjensky and Sahai, 1996;Stumm, 1997). In the present paper, we focus on a single-sitemodel because the experimental data analyzed here refer tosuch models. The adsorption of all other aqueous cations,anions, neutral inorganic, and organic species depend on sur-face protonation reactions (James and Parks, 1982; Davis andKent, 1990). In addition, reactions such as these are thought toplay important roles in models of the kinetics of mineraldissolution (Brady and Walther, 1989; Stumm and Wieland,1990; Casey and Sposito, 1992; Lasaga et al., 1994).The experimentally measurable protonation constants corre-

sponding to Eqns. 1 and 2, K1+ and K2+ , are given by

K+1 " m"SOH2#/m"SOHaH# " a"SOH2

#/a"SOHaH#

" K110($F(0/ 2.303RT) (5)

and

K+2 " m"SOH/m"SO$aH# " a"SOH/a"SO$aH#

" K210($F(0/ 2.303RT) (6)

*Author to whom correspondence should be addressed([email protected]).†Present address: Department of Chemistry, University of Mary-

land, College Park, Maryland 20742, USA.

PergamonGeochimica et Cosmochimica Acta, Vol. 62, No. 23/24, pp. 3703–3716, 1998

Copyright © 1998 Elsevier Science LtdPrinted in the USA. All rights reserved

0016-7037/98 $19.00 # .00

3703

where the molalities of the surface species are assumed equal totheir activities (e.g., m"SOH2# - a"SOH2#). This assumption ismade because of a lack of definitive data to the contrary. It issupported by numerous analyses of surface charge as a functionof pH, ionic strength, and electrolyte type (e.g., Davis et al.,1978; James and Parks, 1982; Sahai and Sverjensky, 1997a,b).The standard states for both surface and aqueous species areassumed here to reflect hypothetical 1 molal solutions referencedto infinite dilution and a surface potential of zero at the tempera-ture and pressure of interest. In Eqns. 5 and 6, F represents theFaraday constant (96,485 Coulombs.mole$1), (0 the potential atthe surface, and K1 and K2 stand for “intrinsic” equilibriumconstants independent of solution composition (Sverjensky andSahai, 1996). It follows from Eqns. 1–6 that the protonationconstant corresponding to Eqn. 3, KZPC+ , is equal to K1+ K2+ . Theprotonation constant KZPC+ takes a special form at the zero point ofcharge. Here, m"SOH2

# - m"SO$, which implies that the surfacecharge is zero, and also that (0 - 0. Under these circumstances,KZPC+ is given by

K+ZPC " K+1 K+2 " 1/(aH#)2 " K1K2 " KZPC (7)

where KZPC refers to the equiIibrium constant for Eqn. 3. It isperhaps important to emphasize that although the numericalvalue of KZPC is obtained under a special set of conditions (i.e.,where the surface charge and (0 - 0), it is a unique value (foreach mineral) which is also valid under all conditions of solu-tion chemistry at the particular temperature and pressure ofinterest. It also follows from Eqns. 1–6 that the protonationconstant corresponding to Eqn. 4, K+n - K+2 /K+1 , is given by

K+n " K+2 /K+1 " (a"SOH)2/a"SOH2

#a"SO$ " Kn (8)

Equations 5 and 6 illustrate the well-known dependence ofsurface equilibrium constants on potential. Equations 7 and 8,however, are independent of surface potential.The enthalpies of surface protonation reactions in water are

of considerable interest in surface geochemistry for severalreasons. First, the temperature dependencies of surface proto-nation reactions are poorly known. Few experimental studies ofsurface protonation as a function of temperature have beencarried out (see below). Theoretical estimates have been madeof the temperature dependence of the zero points of charge ofoxides (Schoonen, 1994). However, the lack of enthalpies ofsurface protonation for many oxides and silicates makes itextremely difficult to evaluate the role of adsorption processesat elevated temperatures in geochemical processes such asdiagenesis in sedimentary basins and hydrothermal alterationprocesses in geothermal and ore-forming systems. Second,enthalpies of surface protonation are thought to constitute asignificant portion of experimentally derived apparent activa-tion enthalpies (Casey and Sposito, 1992; Xiao and Lasaga,1994). Numerous apparent activation enthalpies have beenobtained from empirical fits of the Arrhenius equation to thetemperature dependencies of experimentally measured dissolu-tion rates. The interpretation of these activation enthalpiesrequires knowing the enthalpies of surface protonation (Caseyand Sposito, 1992; Lasaga et al., 1994).The experimentally measurable enthalpies corresponding to

Eqns. 1 and 2, !H+r,1 and !H+r,2, can also be expected to befunctions of surface potential. The dependence can be obtained

by taking the temperature derivatives of the logarithmic formsof Eqns. 5 and 6 at constant pressure and solution composition,which yields

!H+r,1 " !Hr,10 ! F(0 $ FT(%(0/%T) (9)

!H+r,2 " !Hr,20 ! F(0 $ FT(%(0/%T) (10)

where !Hr,10 and !Hr,20 stand for the standard state enthalpies ofsurface protonation. It follows from Eqns. 9 and 10 that exper-imentally measured enthalpies of surface protonation dependnot only on the surface potential ((0), and therefore on solutioncomposition, but also on the derivative of the surface potentialwith respect to temperature (%(0/%T). In addition, it followsthat the enthalpies corresponding to Eqns. 3 and 4, !H+r,ZPC and!H+r,n, are given by

!H+r,ZPC " !H+r,1 ! !H+r,2 " !Hr,10 ! !Hr,2

0

" !Hr,ZPC0 (11)

and

!H+r,n " !H+r,2 $ !H+r,1 " !Hr,20 $ !Hr,1

0 " !Hr,n0 (12)

Enthalpies of surface protonation have been derived experi-mentally with a variety of methods. Experimental measure-ments of the temperature dependence of the zero point ofcharge, of the surface charge, or of the potential ((0) have beenreported (Berube and Bruyn, 1968a,b; Tewari and McLean,1972; Tewari and Campbell, 1976; Blesa et al., 1984; Fokkink,1987; Fokkink et al., 1987a,b; van den Vlekkert et al., 1988;Brady and Walther, 1992; Kosmulski et al., 1994; Machesky etal., 1994, 1998). Of these studies, only the potentiometrictitrations reported by Machesky et al. (1994, 1998) extendabove 95°C. Calorimetric studies have also been used to deriveenthalpies of surface protonation during titrations and fromheats of immersion (Griffiths and Furstenau, 1981; Macheskyand Anderson, 1986; Fokkink et al., 1987b; Mehr et al., 1989;de Keizer et al., 1990; Machesky and Jacobs, 1991a,b; Rud-zinski et al., 1991a,b; Casey and Cheney, 1993; Casey, 1994).The solids investigated by these methods include )-Al2O3,&-Al2O3, Co(OH)2, Co3O4, )-Fe2O3, )-FeO(OH), Fe3O4, NiO,Ni(OH)2, amorphous silica, )-SiO2, rutile, anatase, and ZrO2.Heats of immersion for more complicated silicates have beenreported (Wierer and Dobias, 1988) without deriving enthalpiesof surface protonation.Discussions of the experimentally derived enthalpies of sur-

face protonation (Blesa et al., 1990; Machesky, 1990) haveemphasized that enthalpies referring to Eqn. 3 typically rangefrom about $8 to $22 kcal.mole$1 for the solids mentionedabove. Exceptions to this range (e.g., #50 kcal.mole$1 for&-Al2O3, Akratopulu et al., 1990) have been recognized asanomalous, but there is very little understanding of why thetypical range is from about$8 to$22 kcal.mole$1. Blesa et al.(1990) have suggested a parallelism between the enthalpies ofsurface protonation and those of similar aqueous metal hydroxycomplexes in solution. But it cannot be expected that thissuggestion will hold up very well because the equilibriumconstants of surface protonation do not show a strong correla-tion with those for corresponding aqueous reactions (Sverjen-sky and Sahai, 1996). Instead, the application of crystal chem-

3704 D. A. Sverjensky and N. Sahai

ical and solvation theory shows that surface protonation andother surface equilibrium constants correlate strongly with theproperties of the solid, such as the Pauling bond strength andthe dielectric constant (Parks, 1965, 1967; James and Healy,1972a,b,c; Yoon et al., 1979; Sverjensky, 1993, 1994; Sverjen-sky and Sahai, 1996; Sahai and Sverjensky, 1997a,b). Thetheoretical basis for these correlations suggests that the prop-erties of the solids might also be useful for describing thevariation of the enthalpies of surface protonation from one solidto another. The purpose of the present paper is to develop aquantitative theoretical basis for the calculation of standardsurface protonation enthalpies that is consistent with the pub-lished experimental data and that can be used to make estima-tions for those solids that have not yet been studied experimen-tally. Intrinsic single-site surface protonation equilibriumconstants have already been represented with equations derivedfrom crystal chemical and Born solvation theory (Sverjensky,1994; Sverjensky and Sahai, 1996). Differentiating these equa-tions with respect to temperature provides a theoretical basisfor describing the enthalpies of surface protonation. The theo-retical equations will be fit to experimentally derived enthalpiesin order to establish those coefficients not already known fromthe previous analysis of surface protonation equilibrium con-stants. The resulting equations and coefficients can then used tomake predictions for minerals of geochemical interest.

2. SUMMARY OF EXPERIMENTALLY DERIVEDENTHALPIES

2.1. Enthalpies Derived from the TemperatureDependence of Surface Protonation EquilibriumConstants

A detailed comparison of the variation in enthalpies ofsurface protonation from one phase to another requires that theenthalpies be obtained in a consistent manner. However, theliterature values derived from the temperature dependence ofsurface protonation constants have not always been obtainedwith a consistent regression method (Blesa et al., 1984; Ma-chesky, 1990). In addition, even though values of !Hr,10 and!Hr,20 can be derived directly from the temperature dependenceof the surface protonation constants logK1 and logK2, the latterdepend on a surface complexation model used to evaluatesurface potential (Eqns. 5 and 6). Only values of !Hr,ZPC0 donot depend on a surface complexation model, because logKZPCis independent of surface potential (Eqn. 7).In the present study, a consistent set of enthalpies of surface

protonation were derived by least squares regression of thetemperature dependencies of experimentally derived surfaceprotonation constants from the literature. In order to do this, itwas necessary to return to the original sources of the surfaceprotonation constants. Unfortunately, in some cases these arenot tabulated and had to be obtained from the original figures inthe references. A summary of the experimentally derived sur-face protonation constants in the present study is given in Table 1.It can be seen in Table 1 that, in most instances, the exper-

imental surface protonation constants refer to a very limitedrange of temperatures. Under these circumstances, in order toderive enthalpies of surface protonation, it was necessary toassume that the heat capacities of surface protonation werezero. Enthalpies for the *th protonation reaction !Hr,*0 (* -

1,2,ZPC) were obtained by regression of the temperature de-pendence of the logarithm of the *th surface protonation con-stant, logK*,T, using the equation

log K*,T " logK*,Tr $ (!Hr,*0 / 2.303R)(1/T $ 1/Tr) (13)

where Tr represents the reference temperature 298.15 K. Theresults of these regression calculations are depicted in Figs. 1and 2 and Table 1.It can be seen in Fig. 1 that all of the experimental datasets

vary in a linear fashion with 1/T(K$1). Also, in all cases thedata cover a sufficiently large range for the determination of!Hr,*0 with Eqn. 13. However, the data for rutile and hematite(Fokkink, 1987; see also Table 1) refer to temperatures extend-ing both below and above 25°C, i.e., temperatures on both sidesof Tr in Eqn. 13. Under these circumstances, the application ofEqn. 13 yields values of !Hr0 which are the least sensitive to theassumption that the heat capacities of surface protonation arezero. It can further be seen in Fig. 1 that the two different setsof experimental data for magnetite yielded enthalpies that differby about 2 kcal.mole$1. Both magnetites were synthetic, butdifferent methods of pretreatment of solid surfaces can result indifferent measured surface properties (Machesky et al., 1998).Despite such variablity, it can also be seen in Fig. 1 that thedifference between the two magnetite enthalpies is small com-pared to the wide range of enthalpy values from about $12 to$23 kcal.mole$1 that were obtained for the solids rutile, he-matite, magnetite, and alumina.The assumption made above in using Eqn. 13 that the heat

capacities of reaction are zero is necessary unless equilibriumconstants have been measured over a wide temperature range(Sverjensky et al., 1997). For surface equilibrium constants,this has only been done in the studies of the pHZPC for rutile byMachesky et al. (1994, 1998). Under these circumstances, thetemperature dependence of the logarithm of the surface proto-nation equilibrium constant (logKZPC) can be represented bythe equation

logKZPC,T " logKZPC,Tr $ (!Hr,ZPC0 / 2.303R)

(1/T $ 1/Tr) $ (!CPr,ZPC0 / 2.303RT)(T $ Tr)

! (!CPr,ZPC0 /R).log(T/Tr)/ (14)

where !CPr,ZPC0 represents a (constant) standard heat capacityof reaction corresponding to Eqn. 3. The results of applyingEqn. 14 can be seen in Fig. 3 and Table 1.Several sets of experimental data for the temperature depen-

dence of logKZPC of rutile can be seen in Fig. 3. All the datareferring to temperatures less than 100°C in Fig. 3 have a verysimilar temperature dependence. It can also be seen in Fig. 3that the data from Fokkink (1987) and Machesky et al. (1998)agree extremely well. However, the data from Berube and deBruyn (1968a,b) and Machesky et al. (1994) are systematicallyhigher at all temperatures. All of these sets of data refer torutiles synthesized from extremely acidic HCl solutions. Thedifferences between them appear to reflect sample pretreatment(Machesky et al., 1998). In the present study, the agreementbetween the datasets from Fokkink (1987) and Machesky et al.(1998) was used as a basis for retreiving both an enthalpy anda heat capacity for surface protonation. As discussed above, thedata from Fokkink (1987) constrain the enthalpy of reaction

3705Protonation of mineral surfaces

closely ($12.6 kcal.mole$1, Fig. 1) because they extend from5 to 50°C. Using the same enthalpy, the data from Machesky etal. (1998) closely constrain a constant heat capacity of reaction(60 cal.mole$1.K $1; Eqn. 14). It can be seen in Fig. 3 that thecalculated curve of logKZPC vs. T(°C) is agree well with thedata reported by Fokkink (1987) and Machesky et al. (1998). Itcan also be seen in Fig. 3 that even though the data from Berubeand de Bruyn (1968a,b) are systematically offset from the othersets of data, and do not extend beyond 95°C, a rough idea of the

heat capacity can be obtained (90 cal.mole$1.K$1), based onthe same enthalpy equal to $12.6 kcal.mole$1.

2.2. Enthalpies Derived From Calorimetric Studies

It can be seen from Eqns. 9 and 10 that values of !Hr,1+ and!Hr,2+ obtained from calorimetric determinations will dependon surface potential and its temperature derivative. Conse-quently, calorimetric enthalpies will depend on surface charge,

Table 1. Experimental values of logKZPC, logK1 and logK2 from the literature as a function of temperature. Values of !Hr,ZPC° , !Hr,1° and !Hr,2°were obtained by regression in the present study assuming that !CP,r,ZPC° , !CP,r,1° and !CP,r,2° were zero unless otherwise noted

Phase and source of experimental data T(°C) logKZPC !Hr,ZPC° a !CPr,ZPC° b

Rutile 5 11.6Fokkink (1987; p. 22, Fig. 3.5a) 15 11.16

20 11.3825 $12.61 0.040 10.5650 10.2

Rutile 25 $9.15 0.0Berube and de Bruyn (1968; Table 1) 25 12 $12.61 90.0

50 11.3475 1195 10.7

Rutile 25 10.8 $12.61 60.0Machesky et al. (1998; p. 301) 50 10.2

100 9.4150 8.8200 8.6250 8.4295 8.6

Magnetite 25 13.8 $15.74 0.0Blesa et al. (1984; Table 1) 30 13.6

50 12.980 12.0

Magnetite 25 13.1 $17.74 0.0Tewari and McLean (1972; Fig. 4) 35 12.65

45 12.1655 11.7860 11.5980 11.090 10.8

Alumina 25 $12.27 0.0Tewari and McLean (1972; Fig. 4) 30 18.14

45 17.7060 17.3580 16.9590 16.62

Hematite 5 18.6Fokkink (1987; p. 22, Fig. 3.5b) 20 17.36

25 $23.12 0.040 16.2560 15.6

Phase and source of experimental data T(°C) logK1 !Hr,1° !CPr,1°

Magnetite 25 $6.9 0.0Blesa et al. (1984; Table 3, p. 414) 30 4.4

50 4.1580 3.7

Phase and source of experimental data T(°C) logK2 !Hr,2° !CPr,2°

Magnetite 25 $7.9 0.0Blesa et al. (1984; Table 3, p. 414) 30 9.0

50 8.780 8.2

a. kcal.mole$1. b. cal.mole$1.K$1.


ionic strength, the electrolyte used, and the pH, as well as thetemperature derivative of the surface potential (Griffiths andFurstenau, 1981; Machesky and Anderson, 1986; Fokkink etal., 1987b; Mehr et al., 1989; de Keizer et al., 1990; Macheskyand Jacobs, 1991a,b; Rudzinski et al., 1991a,b; Casey andCheney, 1993; Casey, 1994). Unfortunately, the temperaturederivative of surface potential for oxides surfaces has rarely

been evaluated (van den Vlekkert et al., 1988). The interpreta-tion of calorimetric measurements is also complicated by hys-teresis in the measurements (Mehr et al., 1989; Machesky andJacobs, 1991a,b; Casey, 1994). Virtually all of the above stud-ies report enthalpies referring to a specific electrolyte and ionicstrength or pH range. As a consequence, the results of thedifferent studies are not strictly comparable. The interpretation

Fig. 1. Experimental values of logKZPC as functions of inverse temperature. The lines represent regression of the dataconsistent with Eqn. 13 assuming that the heat capacities of surface protonation are zero.


of calorimetric experiments involving surface protonationcould be facilitated by using triple-layer or quadruple-layermodels of surface protonation. Despite possible limitations ofthese models (Sprycha, 1989a,b; Rudzinski et al., 1991a,b,1992, 1993; Charmas et al., 1995; Lumsden and Evans, 1995),they are the only surface complexation models that can, inprinciple, take into account the ionic strength dependence ofsurface charge and the dependence on specific electrolytes(Davis and Kent, 1990). In the present study, we make use ofthe triple layer analysis of heats of immersion for rutile (Rud-zinski et al., 1991a,b). The enthalpies resulting from this anal-ysis should be directly comparable to those derived from thetemperature dependencies of equilibrium constants based onthe triple layer model for Fe3O4 (Blesa et al., 1984) and&-Al2O3 (van den Vlekkert et al., 1988) discussed above.

3. THEORETICAL DEVELOPMENT

3.1. Summary of Surface Protonation Equations

Crystal chemical and solvation theory (Parks, 1965, 1967;James and Healy, 1972a,b,c; Yoon, et al., 1979; Sverjensky,

1993, 1994; Sverjensky and Sahai, 1996) permits expression ofintrinsic equilibrium constants in terms of the Pauling bondstrength per Angstrom (s/rM-OH) and the dielectric constant ofthe kth solid (#k) according to

logK1 " M1(1/#k) $ B1(s/rM-OH) ! logKii,10 (15)

logK2 " M2(1/#k) $ B2(s/rM-OH) ! logKii,20 (16)

log KZPC " MZ(1/#k) $ BZ(s/rM-OH) ! log Kii,Z0 (17)

log Kn " Mn(1/#k) $ Bn(s/rM-OH) ! log Kii,n0 (18)

where M1, M2, MZ, and Mn are given by

M1 " $ (!&r,1/ 2.303RT) (19)

M2 " $ (!&r,2/ 2.303RT) (20)

MZ " $ (!&r,Z/ 2.303RT) (21)

Mn " $ (!&r,n/ 2.303RT) (22)

and !&r,1, !&r,2, !&r,Z, and !&r,n, B1, B2, BZ, and Bn, andKii,10 , Kii,20 , Kii,Z0 , Kii,n0 , are pressure and temperature-indepen-dent coefficients that have been evaluated by analysis of sur-face protonation constants (Sverjensky and Sahai, 1996). Sep-arate sets of coefficients for the constant capacitance, thediffuse double layer, and triple layer models have been pro-vided to aid in the application of all of these models to systemsthat have not yet been experimentally studied (Sverjensky andSahai, 1996).The most general of the surface complexation models that

has been extensively applied to oxide surfaces is the triple layermodel (Davis et al., 1978; Davis and Leckie, 1978a,b; Jamesand Parks, 1982; Davis and Kent, 1990). In the present studywe use the parameterization of the triple layer model developedby Sverjensky and Sahai (1996), according to which

logK1 " 21.1158(1/#k) $ 49.2608(s/rM-OH) ! 12.9181(23)

logK2 " 21.1158(1/#k) $ 36.5688(s/rM-OH) ! 16.4551(24)

logKZPC " 42.2316(1/#k) $ 85.8296(s/rM-OH)

! 29.3732 (25)

logKn " 12.692(s/rM-OH) ! 3.537 (26)

The application of these equations to calculating and predictingvalues of the intrinsic equilibrium constants, K1, K2, KZPC, Kn,depends on the availability of dielectric constants (#k), radii andbond strength information (s/rM-OH), and on the way in whichthese quantities are used for complex, anisotropic solids. Thevalues of #k and s/rM-OH used in the present study were de-scribed in Sverjensky and Sahai (1996) and are summarizedhere in Table 2.

3.2. Equations for the Enthalpies of Surface Protonation

Taking the temperature derivatives of Eqns. 15–22, we ob-tain

Fig. 2. Experimental values of logK1 and logK2 for magentite asfunctions of inverse temperature. The lines represent regression of thedata consistent with Eqn. 13 assuming that the heat capacities ofsurface protonation are zero.


!Hr,10 " !&r,1.1/#k ! (T/(#k)

2)(%#k/%T)/ $ B+1(s/rM-OH)

! H+1 (27)

!Hr,20 " !&r,2.1/#k ! (T/(#k)

2)(%#k/%T)/ $ B+2(s/rM-OH)

! H+2 (28)

!Hr,ZPC0 " !&r,Z.1/#k ! (T/(#k)

2)(%#k/%T)/

$ B+Z(s/rM-OH) ! H+Z (29)

!Hr,n0 " !&r,n.1/#k ! (T/(#k)

2)(%#k/%T)/ $ B+n(s/rM-OH)

! H+n (30)

In taking the temperature derivatives it has been assumed thatthe Born solvation coefficients (e.g. !&r,1) are independent oftemperature. By analogy with the Born coefficients for aqueousions, e.g., +j (Shock and Helgeson, 1988), it can be expectedthat this is a reasonable assumption for enthalpies or free

energies referring to temperatures less than about 250–300°C.As a consequence, Eqns. 27–30 contain the same Born coeffi-cients as Eqns. 19–22. These Born coefficients have alreadybeen evaluated (Sverjensky and Sahai, 1996), which proves tobe a significant help in the calibration of the enthalpy equations.Equations 27–30 also contain the coefficients B+1, B+2, B+Z, B+n,and H+1, H+2, H+Z, H+n, which are related to the temperaturederivatives of the corresponding coefficients in Eqns. 15–18 bythe equations

B+1 " 2.303RT2(%B1/%T) (31)

B+2 " 2.303RT2(%B2/%T) (32)

B+z " 2.303RT2(%BZ/%T) (33)

B+n " 2.303RT2(%Bn/%T) (34)

H+1 " 2.303RT2(%logK0ii,1/%T) (35)

Fig. 3. Experimental values of logKZPC as functions of inverse temperature (K$1) for rutile from Berube and de Bruyn(1968) and from Machesky et al. (1998). The solid symbols from Machesky et al. (1998) represent the data in which theyhave expressed the most confidence. The open symbol represents a preliminary value only (Machesky et al., 1998). Inaddition, a comparison of all the experimental values of logKZPC for rutile are shown as a function of temperature (°C). Thesolid curves represent regression of the data consistent with Eqn. 14 and a constant heat capacity of surface protonation.


H+2 " 2.303RT2(%logK0ii,2/%T) (36)

H+Z " 2.303RT2(%logK0ii,Z/%T) (37)

H+n " 2.303RT2(%logK0ii,n/%T) (38)

These coefficients must be obtained by calibration of Eqns.27–30 with experimentally derived enthalpies of surface pro-tonation.The application of Eqns. 27–30 depends not only on the

availability of dielectric constants (#k), and radii and bondstrength information (s/rM-OH), the same information that wasrequired for the surface protonation equilibrium constants, butalso on knowledge of the temperature derivative of the dielectricconstant. Available data indicate that the term (T/#k2)(%#k/%T) inEqns. 27–30 is very small compared to the term 1/#k (Roberts,1950; Bosman and Havinga, 1963; Havinga and Bosman, 1965;Cygan and Lasaga, 1986). Nevertheless, it was included in thepresent study by assuming that %#k/%T was a constant equal to

Table 2. Dielectric constants and Pauling bond strengths per angstrom at 25 C and 1 bar together with calculated and predicted values of !Hr,1° ,!Hr,2° , !Hr,ZPC° , and !Hr,n° consistent with Eqns. (44), (46)–(48)

Phase #ka s/rM–OHb $!Hr,1° c $!Hr,2° c $!Hr,ZPC° c $!Hr,n° c

Kaolinite 11.8 0.2754 5.8 4.5 10.3 $1.3&-Al2O3 10.43 0.1913 10.3 11.2 21.5 0.9Rutile 120.91 0.2248 6 6.1 12.1 0.1Amor. silica 3.8073 0.3818 6.3 2.0 8.3 $4.3Goethite 11.7 0.1645 11.3 12.9 24.2 1.6Pyrolusite 10000 0.2882 2.7 1.0 3.7 $1.7Amor. Fe(OH)3 11.7 0.1645 11.3 12.9 24.2 1.6ThO2 18.9 0.1456 11.2 13.4 24.6 2.2ZrO2 22 0.1803 9.3 10.6 19.9 1.3Corundum 10.43 0.1711 11.2 12.8 24.0 1.6Hematite 25 0.1645 9.9 11.6 21.5 1.7Magnetite 20000 0.1768 8.1 9.5 17.6 1.4Quartz 4.578 0.3818 4.9 0.5 5.4 $4.4Nepheline 6.04 0.2294 10.6 10.5 21.1 $0.1Forsterite 7.26 0.1972 11.3 12.1 23.4 0.8Gibbsite 8.4 0.1716 11.9 13.5 25.4 1.6BeO 7.16 0.189 11.7 12.8 24.5 1.1MgO 9.83 0.107 14.5 17.9 32.4 3.4CaO 11.95 0.0976 14.5 18.0 32.5 3.5NiO 11.9 0.1077 14 17.3 31.3 3.3CuO 18.1 0.1686 10.2 11.7 21.9 1.5ZnO 8.49 0.1676 12.1 13.7 25.8 1.6SnO2 9 0.2177 9.4 9.7 19.1 0.3UO2 24 0.148 10.8 12.9 23.7 2.1CaTiO3 165 0.1409 10.1 12.4 22.5 2.3MgAl2O4 8.3 0.1703 12 13.6 25.6 1.6FeAl2O4 40 0.1693 9.2 10.8 20.0 1.6FeCr2O4 11.42 0.1664 11.2 12.9 24.1 1.7Zircon 11.5 0.1559 11.7 13.7 25.4 2Andalusite 6.9 0.2531 8.8 8.0 16.8 $0.8Sillimanite 11 0.2736 6.1 4.8 10.9 $1.3Kyanite 7.8 0.2402 8.9 8.5 17.4 $0.4Grossular 8.53 0.2113 9.9 10.4 20.3 0.5Almandine 4.3 0.2135 13.5 13.9 27.4 0.4Fayalite 8.76 0.1959 10.7 11.4 22.1 0.7Tephroite 9.29 0.1947 10.5 11.3 21.8 0.8Oh-Topaz 5 0.2413 11.1 10.7 21.8 $0.4Enstatite 6.78 0.2425 9.4 8.9 18.3 $0.5Jadeite 10 0.193 10.3 11.2 21.5 0.9Wollastonite 8.6 0.237 8.6 8.4 17.0 $0.2Diopside 8.02 0.2338 9.1 8.9 18.0 $0.2Hedenbergite 17.4 0.2333 7 6.9 13.9 $0.1Anthophyllite 8 0.2497 8.3 7.7 16.0 $0.6Tremolite 8 0.2402 8.8 8.4 17.2 $0.4Talc 5.8 0.2634 9.1 8.1 17.2 $1Muscovite 7.6 0.2539 8.4 7.5 15.9 $0.9Phlogopite 7.6 0.2194 10 10.2 20.2 0.2Low Albite 6.95 0.292 6.9 5.0 11.9 $1.9H-Albite 6.95 0.3533 3.9 0.3 4.2 $3.6Microcline 5.48 0.2889 8.2 6.5 14.7 $1.7Anorthite 7.14 0.2809 7.3 5.7 13.0 $1.6

a. Dielectric constants from Shannon (1993) and Olhoeft (1981). b. Values of the Pauling bond strength (s) divided by M-OH radius (Sverjensky,1994; Sverjensky and Sahai, 1996). c. kcal.mole$1.


0.001 K$1 based on data reported for forsterite (Cygan andLasaga, 1986).

4. APPLICATION TO THE DESCRIPTION OFENTHALPIES OF SURFACE PROTONATION

4.1. Parameters to be Evaluated

In Eqns. 27–30, the parameters #k and s/rM-OH are knownfrom experiment for the solids of interest (Table 2). In addition,the Born solvation coefficients (e.g. !&r,1) are known fromprior analysis of surface protonation equilibrium constants(Sverjensky, 1994; Sverjensky and Sahai, 1996). Regression ofexperimentally derived enthalpies with Eqns. 27–30 must takeinto account that only two of the enthalpies !Hr,10 , !Hr,20 ,!Hr,ZPC0 or !Hr,n0 , are independent. In the present study wechoose !Hr,20 and !Hr,ZPC0 , (Eqns. 28 and 29) for evaluation ofB+2, B+Z, H+2, and H+Z because the most data are available with thelargest ranges of the solid properties #k and s/rM-OH. Based onevaluation of these coefficients, the values of B+1, B+n, H+1, andH+n needed for calculation of !Hr,10 and !Hr,n0 in Eqns. 27 and30 can be calculated using the equations

B+1 " B+Z $ B+2 (39)

B+n " 2B+2 $ B+Z (40)

H+1 " H+Z $ H+2 (41)

H+n " 2H+2 $ H+Z (42)

which are derived from the equations developed above.

4.2. Evaluation of !Hr,ZPC0

Taking account of the known Born solvation coefficients(Sverjensky and Sahai, 1996), Eqn. 29 can be rewritten in theform

!Hr,ZPC0 $ !&r,Z.1/#k ! (T/#k2)(%#k/%T)/ " B+Z(s/rM-OH)

! H+Z (43)

Equation 43 permits experimentally derived values of !Hr,ZPC0 ,corrected for the Born solvation contribution, to be regressed interms of the parameter s/rM-OH. The experimental data used inthe present study were taken from Table 1 and from van denVlekkert et al. (1988) and are represented by the symbols inFig. 4. Calculated values of !Hr,ZPC0 (Table 2) are representedby the solid line shown in Fig. 4 and are consistent with theequation

!Hr,ZPC0 " $ 57.614.1/#k ! (T/#k2)(%#k/%T)/

! 125.208(s/rM-OH) $ 39.78 (44)

It can be seen in Fig. 4 that there is a very strong correlationbetween the experimental values of !Hr,ZPC0 (corrected forBorn solvation) and the Pauling bond strength per angstromwhich supports the application of the theoretically derived Eqn.43. The largest discrepancies between experimental and calcu-lated values are for hematite and one of the magnetite values(1.6 and 1.9 kcal.mole$1, respectively). These discrepanciesprobably indicate the overall uncertainty of enthalpy estimatesusing Eqn. 44 and similar equations (see below). The experi-

mentally derived values for alumina (Tewari and McLean,1972) and NiO (Tewari and Campbell, 1976) were not used inFig. 4 because they differ by more than 12 kcal.mole$1 fromthe calculated values and are not consistent with the experi-mental data shown in Fig. 3. In addition, the value of !Hr,ZPC0

- $20.9 kcal.mole$1 for corundum from Griffiths and Furst-enau (1981) was not included in the regression calculationsshown in Fig. 4 because it is based on an estimate of theentropy of surface protonation. The regression calculationssummarized in Fig. 4 and Table 2 instead suggest a morenegative value ($24 kcal.mole$1).

4.3. Evaluation of !Hr,20

Using a similar approach, Eqn. 28 can be rewritten in theform

!Hr,20 $ !&r,2.1/#k ! (T/#k2)(%#k/%T)/ " B+2(s/rM-OH) !H+2 (45)

Equation 45 permits experimentally derived values of !H2from Table 1 and calorimetric measurements to be corrected forBorn solvation and regressed in terms of the parameter s/rM-OH.The results are depicted in Fig. 5.It can be seen in Fig. 5 that there is a strong correlation

between the !Hr,20 corrected for Born solvation and the Paulingbond strength per angstrom. The overall equation that resultsfrom this analysis is given by

!Hr,20 " $ 28.807.1/#k ! (T/#k2)(%#k/%T)/

! 76.578(s/rM-OH) $ 23.043 (46)

The largest discrepancies between experimental and calculatedvalues are for &-alumina and magnetite values (about 1.7kcal.mole$1). The value of !Hr,20 - $9.9 kcal.mole$1 forcorundum from Griffiths and Furstenau (1981) was not in-cluded in the regression calculations shown in Fig. 5 (seeabove). Nevertheless, it is quite close to the value of $12.0

Fig. 4. Values of !H-r,ZPC corrected for Born solvation vs. thePauling bond strength per angstrom (s/rM-OH) consistent with Eqn. 43.The solid line represents a regression of the experimental data repre-sented by the solid symbols and is consistent with Eqn. 44.


kcal.mole$1 predicted in the present study (Table 2). Based onthe regression results expressed by Figs. 4 and 5 and Eqns. 44and 46, it is now possible to calculate values of !Hr,10 and !Hr,n0consistent with the above analysis of !Hr,ZPC0 and !Hr,20 .

4.4. Calculation of !Hr,10 and !Hr,n0

Using Eqns. 27, 30, and 39–42 results in the followingequations for calculating values of !Hr,10 and !Hr,n0 :

!Hr,10 " $ 28.807.1/#k $ (T/#k2)(%#k/%T)/

! 48.63(s/rM-OH) $ 16.737 (47)

!Hr,n0 " 27.95(s/rM-OH) $ 6.306 (48)

Based on the known values of #k and s/rM-OH (Table 2), Eqns.47 and 48 permit calculation of values !Hr,10 and !Hr,n0 . Theresults are given in Table 2 and are represented by the dashedlines in Figs. 6 and 7. The solid symbols in these figuresrepresent experimental values of !Hr,10 and !Hr,n0 , corrected inthe case of !Hr,10 for the calculated Born solvation term.

4.5. Prediction of !Hr,10 , !Hr,20 , !Hr,ZPC0 , and !Hr,n0

Equations 44 and 46–48 permit prediction of !Hr,10 , !Hr,20 ,!Hr,ZPC0 , and !Hr,n0 . The results are given in Table 2 and arerepresented by the solid symbols in Figs. 8–10. The values of!Hr,10 , !Hr,20 , and !Hr,ZPC0 are plotted vs. the function {1/#k $(T/#k2)(%#k/%T)} to depict the wide range of values for oxidesand silicates. Overall uncertainties associated with the esti-mates in Table 2 are difficult to assess. Uncertainties arisebecause of the application of such a simple theory to a limitedamount of experimental data. In addition, the variablity of theresults of different studies of the same mineral (e.g., magnetitein Fig. 4) suggests a significant uncertainty in the resultingpredictive equations. Overall, the uncertainties are probably of

the order of at least 1–2 kcal.mole$1 for the estimates given inTable 2.Despite these uncertainties, several features of Figs. 8–10

are noteworthy. First, oxides and silicates, as a group, show alarge range of predicted standard enthalpies of protonation. Forexample, !Hr,10 ranges from about $3 to about $15kcal.mole$1. Within this range, most oxides and silicates havenarrower distinct ranges of values. Most oxides (excludingquartz, silica, and rutile) have !Hr,10 values ranging from onlyabout $9 to $12 kcal.mole$1, whereas most silicates (againexcluding quartz) have values of !Hr,10 ranging from about $6to $11 kcal.mole$1. For values of !Hr,20 and !Hr,ZPC0 similartrends are apparent.

Fig. 5. Values of !H-r,2 corrected for Born solvation vs. the Paulingbond strength per angstrom (s/rM-OH) consistent with Eqn. 45. The solidline represents a regression of the experimental data represented by thesolid symbols and is consistent with Eqn. 46.

Fig. 6. Values of DH-r,1 corrected for Born solvation vs. the Paulingbond strength per angstrom (s/rM-OH). The dashed line representsvalues of DH-r,1 derived from the calculated values of !H°r,2 and!Hr,ZPC° consistent with Eqn. 47.

Fig. 7. Values of DHr,n° vs. the Pauling bond strength per angstrom(s/rM-OH). The dashed line represents values of DH-r,n derived from thecalculated values of !Hr,2° and !Hr,ZPC° consistent with Eqn. 48.


It can also be seen in Figs. 8–10 that the ranges of values forsilicates in each figure vary in a way that correlates roughlywith the crystal structure. Quartz and silica have enthalpies ofsurface protonation that are closest to zero (i.e., the leastnegative), followed by feldspars, and then many chain andsheet silicates together, followed by olivines. Sillimanite andkaolinite have enthalpies lying between those of albite andquartz. These trends are primarily a result of the dependence ofthe predicted enthalpies on the Pauling bond strength per ang-strom (s/rM-OH). Values of s/rM-OH are large for silicates thatcontain tetrahedrally coordinated aluminum (feldspars and sil-limanite), aluminum, and silicon without other cations (e.g.,kaolinite and sillimanite), or silicon as the only cation (e.g.,quartz and silica).An interesting feature of Fig. 10 is that it permits direct

comparison of !Hr,ZPC0 for rutile with those of other oxides. Ithas previously been suggested (Berube and Bruyn, 1968a,b;Machesky et al., 1994, 1998) that the similarities between thetemperature dependence of logKZPC for rutile and logKassoc. ofwater have a fundamental significance that permits generaliza-tion of the temperature dependence of logKZPC for rutile toother oxides. The value of !Hr,ZPC0 for rutile, about $12kcal.mole$1, is indeed similar to the !Hr,assoc.0 of water,$13.34 kcal.mole$1 (Johnson et al., 1992). However, it is clearfrom Fig. 10 that rutile is not similar to many of the otheroxides, which have predicted !Hr,ZPC0 values from about $17to $33 kcal.mole$1. Oxides have a range of values of !Hr,ZPC0

reflecting differences in the strengths of the bonding of surface

protons at the oxide-water interface, all of which depends ondifferences in the underlying crystal structures. Consequently,it appears that the similarity between !Hr,ZPC0 for rutile and for!Hr,assoc.0 water is fortuitous.Finally, it is possible to differentiate Eqns. 27–30 to obtain a

theoretical basis for the variation of !CPr,ZPC0 from one mineralto another. However, insufficient data are available to calibratethe coefficients in the resulting equations. Only in two pioneer-ing studies of the rutile surface at temperatures up to 295°C(Machesky et al., 1994, 1998) has it has been unambigouslydemonstrated that a heat capacity of surface protonation(!CPr,ZPC0 ) can be determined. Based on a value of !Hr,ZPC0 -$11.09 kcal.mole$1, Machesky et al. (1998) obtained!CPr,ZPC0 - 38 cal.mole$1K$1. This value differs from the!CPr,ZPC0 - 60 cal.mole$1K$1 obtained in the present studybecause our !Hr,ZPC0 - $12.6 kcal.mole$1 was obtained fromthe data of Fokkink (1987). Nevertheless, it can be expectedthat the heat capacities of surface protonation for other mineralswill differ substantially from that of rutile. If so, it is possiblethat the enthalpy of surface protonation contribution to appar-ent activation enthalpies may have a significant temperaturedependence.

5. CONCLUSIONS

(1) The combination of crystal chemical and Born solvationtheory provides a theoretical basis for the dependence of thestandard enthalpies of surface protonation on the Pauling bond

Fig. 8. Predicted values of !Hr,1° for the triple layer model vs. theinverse of the dielectric constant (1/#) of the solid to display the widerange of predicted enthalpies.

Fig. 9. Predicted values of !Hr,2° for the triple layer model vs. theinverse of the dielectric constant (1/#) of the solid to display the widerange of predicted enthalpies.


strength per angstrom (s/rM-OH) and the inverse of the dielectricconstant of the kth mineral (1/#k). For the *th (* - 1,2,ZPC orn) protonation reaction

!Hr,*0 " !&r,*.1/#k ! (T/#k2)(%#k/%T)/ $ B+*(s/rM-OH)

! H+* (49)

where the coefficients !&r,*, B*+ , and H*

+ are constants charac-teristic of all oxides and silicates for the triple layer model.(2) Experimentally derived enthalpies of surface protonation

are systematized by simple sets of equations for each surfaceprotonation equilibrium. The equations enable prediction of theenthalpies of surface protonation for many solids, which couldbe tested by comparison with the results of surface titrationexperiments as functions of temperature and calorimetric stud-ies. Because the predicted enthalpies are consistent with thelimited amount of experimental data currently available and thetheory developed in the present study, they should be of con-siderable use in helping to analyse the surface chemistry ofoxides and silicates. Indeed, as more data become available, theapproach described here could be used to refine the predictedenthalpy estimates given above.(3) Predicted standard enthalpies of surface protonation for

oxides and silicates extend, approximately, over the followingranges:

!Hr,10 ! $ 3 to $ 15 kcal.mole $ 1

!Hr,20 ! $ 0.5 to $ 18 kcal.mole $ 1

!Hr,ZPC0 ! $ 4 to $ 33 kcal.mole $ 1

Minerals with the largest values of the Pauling bond strengthper angstrom (e.g., quartz and silica) have the least negativeenthalpies of surface protonation. As a consequence, it can beexpected that the surface protonation equilibrium constants ofthese minerals will have the weakest temperature dependenceof all oxides and silicates. In contrast, minerals with the small-est values of the Pauling bond strength per angstrom (e.g.,garnets and olivines) have the most negative enthalpies ofsurface protonation. The surface protonation equilibrium con-stants of these minerals will have the strongest temperaturedependence of all silicates.

Acknowledgements —Discussions with W. Casey, L. Criscenti, M.Machesky, C. Koretsky, and Jim Kubicki helped the development ofthis work significantly. Reviews of the manuscript by W. Casey, M.Machesky, J. Walther, and D. Wesolowski are greatly appreciated.Financial support was provided by DuPont (through Noel Scrivner),NSF Grant EAR 9526623, and DOE Grant DE-FG02-96ER-14616.

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theoretical prediction of single-site enthalpies of surface

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