Thermodynamic properties of H4SiO4 in the ideal gas ...

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Geochimica et Cosmochimica Acta 75 (2011) 3853–3865 www.elsevier.com/locate/gca

Thermodynamic properties of H4SiO4 in the ideal gas state as evaluated from experimental data Andrey V. Plyasunov ⇑ Institute of Experimental Mineralogy, Russian Academy of Sciences, Chernogolovka, Moscow Region 142432, Russia Received 7 December 2010; accepted in revised form 14 April 2011; available online 21 April 2011

Abstract Solid phases of silicon dioxide react with water vapor with the formation of hydroxides and oxyhydroxides of silica. Recent transpiration and mass-spectrometric studies convincingly demonstrate that H4SiO4 is the predominant form of silica in vapor phase at water pressure in excess of 102 MPa. Available literature transpiration and solubility data for the reactions of solid SiO2 phases and low-density water, extending from 424 to 1661 K, are employed for the determination of Df G0 , Df H 0 and S0 of H4SiO4 in the ideal gas state at 298.15 K, 0.1 MPa. In total, there are 102 data points from seven literature sources. The resulting values of the thermodynamic functions of H4SiO4(g) are: Df G0 = 1238.51 ± 3.0 kJ mol1, Df H 0 = 1340.68 ± 3.5 kJ mol1 and S0 = 347.78 ± 6.2 J K1 mol1. These values agree quantitatively with one set of ab initio calculations. The relatively large uncertainties are mainly due to conflicting C 0p data for H4SiO4(g) from various sources, and new determinations of C 0p would be helpful. The thermodynamic properties of this species, H4SiO4(g), are necessary for realistic modeling of silica transport in a low-density water phase. Applications of this analysis may include the processes of silicates condensation in the primordial solar nebula, the precipitation of silica in steam-rich geothermal systems and the corrosion of SiO2-containing alloys and ceramics in moist environments. Ó 2011 Elsevier Ltd. All rights reserved.

1. INTRODUCTION Thermodynamic properties of gaseous silicon hydroxides are necessary for analyzing the vapor phase silica transport in the earth crust and the condensation phenomena in the primordial solar nebula (Hashimoto, 1992). Other potential applications involve the problem of silica precipitation in steam-rich geothermal systems and recent concerns with the corrosion or deposition of SiO2-containing alloys and ceramics in high-temperature moist environments (Allendorf et al., 1995; Jacobson et al., 2005; Singh and Vora, 2005). Although there is a general agreement among researchers since the 1950s that the main form of silica in water is H4SiO4, decisive experimental evidence of that for silica in steam appeared only recently. Two experimental transpiration studies (i.e. studies of the partial pressures of Si over ⇑ Tel.: +7 915 328 2674.

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solid phases in the presence of steam) of the reaction of water vapor with cristobalite (Hashimoto, 1992; Jacobson et al., 2005) agree that at 1100–1650 K and water pressures close to 0.1 MPa the vapor phase concentration of Si is determined by the reaction SiO2 ðsÞ þ 2H2 OðgÞ ¼ H4 SiO4 ðgÞ:

ð1Þ

The stoichiometry of the predominant reaction was confirmed by plotting the logarithm of the partial pressures of Si-containing species (calculated from analytical concentrations of Si in the condensate of the vapor phase by assuming the ideal gas law) versus the logarithm of the water vapor pressure and observing a slope close to 2. At much lower water pressures or higher temperatures, i.e. at lower values of the chemical potential of water, another reaction, most probably SiO2 ðsÞ þ H2 OðgÞ ¼ SiOðOHÞ2 ðgÞ;

ð2Þ

appears to contribute to the material balance of Si in vapor. However, available experimental (Hildebrand and Lau,

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A.V. Plyasunov / Geochimica et Cosmochimica Acta 75 (2011) 3853–3865

1994; Jacobson et al., 2005) and ab initio (Allendorf et al., 1995) results for the reaction (2) are in rather poor agreement, with the scatter of about 80 kJ mol1 in the resulting value of the enthalpy of formation, DfH0, of SiO(OH)2(g), see Table 5 of Jacobson et al. (2005). Many other gaseous species may form in the Si–O–H system (Allendorf et al., 1995; Hildebrand and Lau, 1994), however, only in small amounts. Mass spectrometric data on the volatile species formed from SiO2(s) and water at 1473–1773 K and water vapor pressure between 0.018 and 0.094 MPa (Opila et al., 1997) also confirm that H4SiO4 is the primary reaction product, with only a small amount of SiO(OH)+, believed to be an ionized fragment of SiO(OH)2, detected. The contributions of SiO2(g) and other silicon oxides are negligible in the presence of water vapor. It may be appropriate to note that at high temperatures/ pressures and high silica concentrations, especially in the neighborhood of the upper critical end point of the SiO2– H2O system, the speciation of silica becomes much more complicated, with the extensive formation of dimers and higher polymers, see Zotov and Keppler (2000, 2002), Newton and Manning (2003, 2008), Gerya et al. (2005). However, the current work is concerned with the dilute solutions of silica, mainly in the low-density water, where the contributions of species other than H4SiO4 are expected to be negligible. 2. THERMODYNAMIC PROPERTIES OF H4SIO4(G) 2.1. Review of literature values First of all, it is necessary to emphasize that the current work is concerned with the properties of gaseous H4SiO4 in the ideal gas (standard) state. According to IUPAC (Mills et al., 1993), the standard state for the gaseous species is defined as follows: “the standard state for a gaseous substance, whether pure or in a gaseous mixture, is the (hypothetical) state of the pure substance B in the gaseous state at the standard pressure P ¼ P and exhibiting ideal gas behaviour”. This definition is used for the standard state of all gaseous species considered. In this work the standard pressure P = 0.1 MPa.

The standard state thermodynamic properties of H4SiO4(g) were obtained by Allendorf and coworkers (1995) from a combination of ab initio calculations and empirical corrections. Revised data, corrected for hindered rotors, can be downloaded from the publicly available GasPhase Database of Sandia National Laboratories (see http://www.sandia.gov/HiTempThermo/). Results are shown in Table 1. The values of the ideal gas heat capacity of H4SiO4(g) over the temperature range 250–2000 K were approximated by a polynomial, see Table 1. Rutz and Bockhorn (2005) also employed ab initio and density functional theory calculations for the determination of the thermodynamic properties of H4SiO4(g). Note that the DfH0 values calculated using various levels of theory differ up to 25 kJ mol1. The agreement of the theoretical DfH0(Tr) and S0(Tr) values recommended by Allendorf et al. (1995) and Rutz and Bockhorn (2005) is only within 9 kJ mol1 and 9 J K1 mol1, respectively. Here and below Tr = 298.15 K. The molar heat capacities, C 0p , of H4SiO4(g), tabulated in these works, differ within 2–8% over the temperature range 300–1500 K. The uncertainties of C 0p data result in large differences when the high-temperature thermodynamic functions are extrapolated to 298.15 K, see below. For the data treatment the values of C 0p from Allendorf et al. (1995) were accepted, as these results are presented in greater detail and cover wider temperature ranges. However, the currently existing uncertainties in C 0p values are taken into account when estimating errors in DfH0(Tr) and S0(Tr) values of H4SiO4(g) obtained in this work, see below. Hashimoto (1992) made a careful transpiration study of the reaction of b-cristobalite and water vapor between 1373 and 1773 K at atmospheric pressure, created by the oxygen–water mixture, and partial pressures of water ranging from 0.01 to 0.06 MPa. From the temperature dependence of the equilibrium constant he determined by the “secondlaw method” (i.e. assuming that over a small temperature range Dr C 0p ¼ 0) for the reaction (1) DrH0(1600 K) = 56.7 ± 1.7 kJ mol1 and DrS0(1600 K) = 66.2 ± 1.0 J K1 mol1. By combining these results with the known thermodynamic properties of cristobalite (Robie and Hemingway, 1995), water in the ideal gas state (Cox et al., 1989; Robie and Hemingway, 1995), and C 0p for H4SiO4(g) from Allendorf et al. (1995), see Table 1, the values of DrH0(Tr) =

Table 1 Values of the standard state DfH0 and S0 of H4SiO4(g) at Tr = 298.15 K and P ¼ 0:1MPa from various sources. References

DfH0 kJ mol1

S0 J K1 mol1

Allendorf et al. (1995) Rutz and Bockhorn (2005)

1342.2 ± 13.8 1333.11

342.54 333.93

Comments

Ab initio calculations This value is selected by the authors from ab initio calculations. Note that DfH0 values calculated using various levels of theory differ up to 25 kJmol1 Jacobson et al. (2005) 1351.3 ± 1.7 Third-law treatment of transpiration data, accepting S0(298) and C 0p from Allendorf et al. (1995) Hashimoto (1992) 1339.88 348.3 Recalculated from author’s values at 1600 K, see text Jacobson et al. (2005) 1341.29 347.5 Recalculated from authors’ values at 1200 K, see text This work 1340.68 ± 3.5 347.78 ± 6.2 This work, based on a simultaneous treatment of 102 transpiration and vapor solubility data for solid silicon dioxide phases, see text C 0p /Ra = a0 + a1 102 T1 + a2 105 T2 + a3 108 T3 + a4 1011 T4 + a5 1015 T5 a0 = 2.87914; a1 = 5.89126; a2 = 9.47715; a3 = 7.84564; a4 = 3.15382; a5 = 4.89073 (250–2000 K) a

Our fit of C 0p data tabulated in Allendorf et al. (1995) and http://www.sandia.gov/HiTempThermo/.

Thermodynamics of H4SiO4 in the ideal gas state

52.17 kJ mol1 and DrS0(Tr) = 72.77 J K1 mol1 were obtained, resulting in DfH0(Tr) and S0(Tr) for H4SiO4(g), given in Table 1. Jacobson et al. (2005) also presented results of a transpiration study of the reaction of b-cristobalite and water vapor between 1098 and 1375 K. The total pressure, close to 0.1 MPa, was created by water–argon mixtures with mole fractions of water from 0.05 to 0.6. Their results, treated by the “second-law method” are consistent with the following values of the enthalpy and the entropy changes for the reaction (1) at 1200 K: DrH0(1200 K) = 54.6 ± 2.7 J mol1 and DrS0(1200 K) = 67.5 ± 2.1 J K1 mol1. As it was done in the case of Hashimoto (1992) data, results are recalculated to 298.15 K: DrH0(Tr) = 54.6 kJ mol1 and DrS0(Tr) = 73.57 J K1 mol1, with DfH0(Tr) and S0(Tr) for H4SiO4(g) given in Table 1. It should be noted that for their final recommendation of DfH0(Tr) Jacobson et al. (2005) selected results of the “third-law method” calculations, i.e. using for each experimental point the relation X X S 0 ðT Þ ðH 0 ðT Þ H 0 ðT r ÞÞ; Dr H 0 ðT r Þ ¼ RT ln K þ T i

i

ð3Þ where summations cover all reaction’s participants, with the values of S0(Tr) and C 0p for H4SiO4(g) accepted from Allendorf et al. (1995). According to Eq. (3), the standard enthalpy of reaction at Tr = 298.15 K can be calculated from each experimental point, providing that the thermal functions (S 0 ðT r Þ,C 0p ) are known with high accuracy for reaction’s participants. It is expected that the calculated with Eq. (3) values of Dr H 0 ðT r Þ are approximately constant, with a small scatter due to random experimental errors. However, the obtained by this method values of Dr H 0 ðT r Þ, taken from Tables 1 and 3 in the Jacobson et al. (2005) work, show a strong and systematic change, up to 10 kJ mol1, correlating with the temperature of experiments, as seen in Fig. 1. The most probable explanation of that “drift” is that the selected value of S 0 ðT r Þ for

3855

H4SiO4(g) is inconsistent with high-temperature experimental data. The observation of these discrepancies was a main reason to revise the thermodynamic properties of gaseous H4SiO4 by taking into account all available relevant experimental data. 2.2. Experimental data on reaction of water vapor with solid silicon dioxide phases The recent transpiration studies of reaction of water vapor with cristobalite at very high temperatures were discussed above. Other works are concerned with the solubility of amorphous silica or quartz in low-density water steam (Straub and Grabowski, 1945; the same data are presented by Straub, 1946; Morey and Hesselgesser, 1951; Wendlandt and Glemser, 1963; Heitmann, 1964, the abridged set of data was published in Heitmann, 1965; Martynova et al., 1975). All these studies used a dynamic method, when water vapor (or water diluted with nitrogen in experiments of Wendlandt and Glemser, 1963) was passed through solid phases. The concentration of silica in the samples was determined by colorimetry (a “molybden blue” method) or gravimetry. The establishment of equilibrium was usually confirmed by checking that the silica content is independent of steam flow rates. An experimental determination of the solubility of solids in the vapor phase is a challenging task. An obvious problem is the kinetics of establishing equilibrium at low temperatures. A specific difficulty is the mechanical carry-over of small drops of liquids into the vapor phase. As the solubility of a solid in the liquid phase is much higher than in the gas phase, the net result may be an elevated concentration of a solute in the condensate of the vapor phase compared with the truly equilibrium conditions. In a typical industrial setup the extent of the mechanical carry-over is 0.01–0.05% in the balance of steam (Styrikovich and Martynova, 1963), and it was proven very difficult to lower it below 103% even in the laboratory flow apparatus (Straub, 1946). Consequently, data at lower temperatures, where the contribution of the mechanical carry-over to the material balance of a solute in the vapor phase may be dominant, often are not reliable. 2.3. Selection of data sets for further treatment Before proceeding to the treatment of experimental data, a selection of most reliable values should be done. Previously, an analysis of literature values was done by Martynova et al. (1975) when discussing the T–P–X coordinates of boundary lines of the phase diagram of the SiO2– H2O system. Their own (Martynova et al., 1972) study of the partition of silica between the vapor and liquid phases of water allowed the determination of vapor–liquid distribution constant, KD: KD ¼

Fig. 1. A strong and systematic temperature drift of the Dr H 0 ðT r Þ values for the reaction SiO2(c) + 2H2O(g) = H4SiO4(g) at the standard pressure P = 0.1 MPa, calculated by Jacobson et al. (2005) from experimental data for the reaction of b-cristobalite and water.

Y ; X

ð4Þ

where Y and X stand for the mole fraction of silica in the vapor and liquid phases of water, respectively. Values of KD, coupled with known solubilities of amorphous silica

3856


or quartz in liquid water, allow conclusions on the quality of data in the vapor phase. This analysis found the results of Straub and Grabowski (1945) to be most reliable. Therefore, all these data were accepted.1 Martynova et al. (1975) concluded that Heitmann’s (1964, 1965) solubility data in water vapor were not reliable at low temperatures. For example, results at 433 K are at least 10 times too high for amorphous silica and up to 80 times too high for quartz. In addition, the low-temperature vapor phase solubility data for amorphous silica and quartz are intrinsically inconsistent, because the vapor–liquid distribution constants for dissolved silica are very different when calculated from quartz and amorphous silica solubility values. Martynova et al. (1975) also pointed out that the bending of Heitmann’s experimental isobaric solubility data at low pressures, when approaching the saturation temperature, cannot be easily explained (this effect is absent in Straub and Grabowski (1945) data). On the other hand, Heitmann’s data at higher pressures, when extrapolated to the saturation line, appear to be consistent with the results of Martynova and coworkers. Based on these observations, we excluded from further consideration Heitmann’s data on solubility of amorphous silica obtained at temperatures below 550 K and data on solubility of quartz at T < 600 K and all data at pressures below 0.6 MPa. The rest of experimental points were given a lower weight of 0.5. Note that Harvey and Bellows (1997) in their review also gave low weights to Heitmann’s data, since they concluded that “the reproducibility of Heitmann’s measurements is probably not better than approximately a factor of 2”. The upper temperature range of accepted data on solubility of amorphous silica in water has been set at 780 K, based on observations (Morey and Hesselgesser, 1951; Heitmann, 1965) that the transformation of amorphous silica to cristobalite or quartz is slow, and it is possible to obtain reproducible data on solubility of amorphous silica in low-pressure water even at 773 K. We also accepted data of Morey and Hesselgesser (1951) at 673 and 773 K, and Wendlandt and Glemser (1963) at 673 K. At 773 K results of Morey and Hesselgesser (1951) and Heitmann (1964, 1965) of quartz solubility are comparable, however, values of Wendlandt and Glemser (1963) are up to 3–4 times lower, see Fig. 2, therefore, the latter data at this temperature were excluded. Note that in the work of Wendlandt and Glemser (1963) experimental data are presented only graphically, however, the review of Harvey and Bellows (1997) gives the corresponding numerical data taken from the dissertation of Wendlandt H.G. “Reactionen zwischen Oxiden und Wasser bei hoheren Temperaturen und verschiedenen Dichten”, Universita¨t Go¨ttingen

1 Note that in the corresponding paragraph of Martynova et al. (1975) there is a reference to Morey and Hesselgesser (1951), stating that those data “for amorphous silica at pressures 35– 5.7 kgf/cm2 at temperatures 260–343 °C are consistent with data of this work along the water saturation line”. However, it is an obvious misprint, because Morey and Hesselgesser did not report measurements below 400 °C, whereas data at 260–343 °C and various pressures are plotted in Fig. 6 of Straub and Grabowski (1945).

Fig. 2. Comparison of experimental data on solubility (Y, mole fraction of SiO2) of quartz in water at 773 K as a function of water density, calculated from the equation of state of Wagner and Pruß (2002). (See above-mentioned references for further information.)

(1963). Three data points of Martynova et al. (1975) on the solubility of amorphous silica in saturated water vapor of density below 0.15 kg m3 were also accepted. An important question is the selection of the upper bound of water density for the data on solubility of solid phases of silica in steam. As discussed later, in Section 2.5, it was decided to use only data at water densities below 15 kg m3. Both sets of ln K0 data from transpiration studies of Hashimoto (1992) and Jacobson et al. (2005) were accepted. Note that in Hashimoto’s paper data are presented only graphically, however, numerical values of his ln K0 at 1375–1661 K are tabulated by Jacobson et al. (2005). The final set of the accepted data is presented in Table 2. 2.4. Thermodynamic properties of solid silicon dioxide phases: quartz, cristobalite, amorphous silica Experimental data refer to vapor phase solubilities or transpiration studies of three solid SiO2 phases: amorphous silica (AS), quartz (Q) and cristobalite (CR). The caloric properties, the entropy S0(Tr) and the heat capacity as a function of temperature, are known for these phases (Robie and Hemingway, 1995; Richet et al., 1982). The enthalpies of formation of the phases are tied up to the recommended by CODATA value of the enthalpy of formation of aquartz at 298.15 K (Cox et al., 1989), and are intrinsically consistent within a few hundreds of J mol1 (uncertainties of the phase transitions among SiO2 polymorphs, see Richet et al., 1982).2 One of the checks of the consistency of 2 Thermodynamic properties of amorphous silica, given in Richet et al. (1982), refer to a silica glass with very low OH content. The water content in amorphous silica, used for measurements quoted in this work, may vary. However, it was repeatedly shown (see, for example, Gunnarsson and Arno´rsson, 2000) that the solubility results did not depend on the water content of silica gel employed.


3857

Table 2 Data sets used in the determination of DfG0 and S0 of H4SiO4(g) at 298.15 K. Reference

Solid Phasea

T range, K

P range, MPa

Number of points

bJ mol1 D

c J mol1 jDj

Straub and Grabowski (1945) Morey and Hesselgesser (1951) Wendlandt and Glemser (1963)d Heitmann (1964)e Heitmann (1964)e Martynova et al. (1975) Hashimoto (1992)f Jacobson et al. (2005)

AS Q Q AS Q AS CR CR

533.2–616.5 673.2773.2 673.2 572.2–776.2 624.2–865.2 424.25–496.06 1375–1661 1074–1375

0.3452.76 3.45–6.89 2.128 0.88–4.90 0.88–4.90 0.49–2.45 0.1 0.1

32 3 1 21 6 3 10 26

161 4322 3497 782 310 45 65 211

579 4322 3497 871 907 417 182 946

a b c d e f

AS, Q, CR designates amorphous silica, quartz, cristobalite, respectively. is the average difference between experimental and calculated G0 ðH4 SiO4 ðgÞÞ. D T ;P is the average of the absolute values of the difference between experimental and calculated G0 ðH4 SiO4 ðgÞÞ. jDj T ;P The numerical values are quoted from the review of Harvey and Bellows (1997). Weight of data points is taken equal to 0.5. Primary data are not reported in the paper, however, most data points are tabulated in Jacobson et al. (2005).

thermodynamic data for silica phases is by means of calculations of the relative solubilities of these phases in water and comparison with numerous and consistent experimental data. Such a check was made by Gunnarsson and Arno´rsson (2000), who found that data for quartz and cristobalite were internally consistent, but the Gibbs energy of formation of amorphous silica appeared to be systematically too positive up to 0.65–1.36 kJ mol1 over the temperature range 273–623 K. Although these discrepancies are well within the expected uncertainty of the Gibbs energy of formation for amorphous silica (±2.1 kJ mol1, see Robie and Hemingway, 1995), the use of tabulated thermodynamic functions for AS may lead to the systematic bias during the data treatment. Therefore, we used the experimental results for the relative solubilities of Q and AS at 273–573 K and saturated water vapor to determine the values of S 0 ðT r ; P Þ and Df G0 ðT r ; P Þ of amorphous silica, which provide the best agreement with aqueous solubility data, see the Appendix A. Note that obtained in this way values of the entropy and the Gibbs energy of formation of AS, see Table A1, are comfortably within the uncertainty brackets given by Robie and Hemingway (1995). The accepted values of the thermodynamic properties of solid silicon dioxide phases are given in Table A1. 2.5. Data treatment for the determination of DfG0 and S0 of H4SiO4(g) at 298.15 K, 0.1 MPa Accepting for gaseous H2O and H4SiO4 the standard state of ideal gas at the standard pressure P = 0.1 MPa one writes for the Gibbs energies of these species, GT ;P :

P /H4 SiO4 Y P

ð5Þ

P /1 H4 SiO4 Y P

ð7Þ

and GT ;P ðH2 OðgÞÞ G0T ðH2 OðgÞÞ þ RT ln

P /H2 0 ; P

ð8Þ

where superscripts 1 and * refer to infinite dilution and pure substance, respectively. Accepting for the solid SiO2 the standard state of pure substance in the solid state at the standard pressure P = 0.1 MPa one writes: Z P GT ;P ðSiO2 ðsÞÞ ¼ G0T ðSiO2 ðsÞÞ þ V ðSiO2 ðsÞÞdP P

G0T ðSiO2 ðsÞÞ

þ V ðSiO2 ðsÞÞðP P Þ

ð9Þ

Although the contribution of the last term is small in the low-pressure range used, it was nevertheless accounted for, with the expansion of solid phases calculated using the equation and parameters from Holland and Powell (1998), see Table A1: pffiffiffiffi pffiffiffiffiffiffiffiffi V 1;T ¼ V 1298 ð1 þ a0 ðT 298Þ 20a0 ð T 298ÞÞ: ð10Þ

Dr GðT ; P Þ ¼ 0 ¼ GT ;P ðH4 SiO4 ðgÞÞ 2GT ;P ðH2 OðgÞÞ GT ;P ðSiO2 ðsÞÞ;

ð11Þ

and the corresponding statement for ln K 0 of the reaction (1) is given by

and GT ;P ðH2 OðgÞÞ ¼ G0T ðH2 OðgÞÞ þ RT ln fH2 0 P /H2 0 ð1 Y Þ ¼ G0T ðH2 OðgÞÞ þ RT ln ; P

GT ;P ðH4 SiO4 ðgÞÞ G0T ðH4 SiO4 ðgÞÞ þ RT ln

For the reaction (1) at equilibrium

GT ;P ðH4 SiO4 ðgÞÞ ¼ G0T ðH4 SiO4 ðgÞÞ þ RT ln fH4 SiO4 ¼ G0T ðH4 SiO4 ðgÞÞ þ RT ln

where fi and /i stand for the fugacity and the fugacity coefficient of a species i; Y is the vapor phase mole fraction of H4SiO4; the superscript 0 designates the standard-state property at P = P . At very low solubilities of silica (the maximum Y = 1.31 105), when 1–Y 1, the following simplifications are valid:

ð6Þ

ln K 0 ¼ ln

Y /1 H4 SiO4 P

P

ð/H2 0 Þ2

V ðSiO2 ðsÞÞ ðP P Þ RT

ð12Þ

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The values of the fugacity coefficient of water, /H2 0 , are accurately known over very wide T and P ranges (Wagner and Pruß, 2002). If the partial molar fugacity coefficient of dissolved silica at infinite dilution in water, /1 H4 SiO4 , were known, then the values of the equilibrium constant of reaction (1) could be calculated at any pressures. However, /1 H4 SiO4 is not known. One of the possible strategies would be to use only data at very low water densities q1 , say, less than 2% of the critical density of water, i.e. at q1 < 6 kg m3, were deviations of /1 H4 SiO4 from 1 are expected to be very small. However, there are very few data at such low densities and due to exceedingly low solubilities they are often not reliable. At higher densities the values of /1 H4 SiO4 are expected to be less than 1, reflecting strong attractive interactions between molecules of H4SiO4 and H2O, both of which are able to form hydrogen bonds. The method proposed to estimate values of /1 H4 SiO4 is described below. The discussion uses the water density instead of more customary in earth sciences pressure because the intermolecular interactions are functions of distances. In this sense, the pressure is a secondary variable. In addition, there was much discussion in the geochemical literature, devoted to the solubility in the near-critical ranges, about advantages of using density, not pressure, when interpolating and extrapolating data. At low to moderate densities, up to approximately 1/3 of the critical density of a mixture, the deviations from ideality are accurately described by the virial equation of state, Þ PV truncated at the second virial coefficient: RT ¼ 1 þ BðT , V where V is the volume of the mixture and B(T) is the second virial coefficient of the mixture. The rigorous statements for the composition dependence of B(T) and the fugacity coefficients of the mixture’s components are given in the thermodynamic textbooks (see, for example, Prausnitz et al., 1999). For the particular case of interest for the current work – very dilute solutions of H4SiO4 (component 2) in practically pure water (component 1) – the following equations are given for /1 and /1 2 (Prausnitz et al., 1999): ðT ÞP P ln /1 ¼ B11RT and ln /1 2 ¼ ð2B12 ðT Þ B11 ðT ÞÞ RT , where B11(T) is the well-known (Harvey and Lemmon, 2004; Wagner and Pruß, 2002) second virial coefficient of water and B12(T) is the second cross virial coefficient for interactions between molecules of H2O and H4SiO4. The evaluation of B12(T) at various temperatures is planned as a separate project; however, preliminary results for the H2O–H3BO3 and H2O–H4SiO4 systems suggest that the following relation may connect B12(T) and B11(T) in water– hydroxide mixtures: B12(T) n B11(T), where n is close (within 1) to the number of OH groups in the molecule of a hydroxide. If one accepts the approximation 7B11 ðT ÞP B12(T) 4B11(T), then ln /1 ¼ 7ln /1 , with H4 SiO4 RT the expected error within 30–40%. Although this estimate is considered more realistic than the ideal mixing model, it is still important not to use it at high water densities as this approximation, if significantly in error, may introduce bias in the data treatment. Therefore, we chose the water density 15 kg m3 (5% of the critical density of water) as a cutoff density value for acceptance of solubility data for the determination of the standard state properties of

gaseous H4SiO4. For all accepted data points /1 > 0.90 and /1 H4 SiO4 > 0.48. It was decided to perform a simultaneous treatment of data to determine the values of the Gibbs energy of formation, Df G0 , and the entropy, S0, of H4SiO4(g) at Tr = 298.15 K and P = 0.1 MPa. First, for each accepted experimental point the value of ln K0 was calculated according to Eq. (12), with /1 and /1 H4 SiO4 evaluated as described above. The contribution of the last term of Eq. (12) was neglected for transpiration data (always less than 0.0004 in ln units), but it was accounted for the solubility data, where it adds up to 0.021 ln units. Second, the values of ln K 0 were converted to the Gibbs energy change, Dr G0 , for the reaction (1) at the ideal gas standard state pressure of 0.1 MPa according to: Dr G0 ðT Þ ¼ RT ln K 0 ¼ G0T ðH4 SiO4 ðgÞÞ 2G0T ðH2 OðgÞÞ G0T ðSiO2 ðsÞÞ

ð13Þ

The value of Dr G0 is the stoichiometric sum of the Gibbs energies, G0T , of the reaction (1) participants at the standard state pressure of 0.1 MPa, which are given by G0T ¼ Df G0T r S 0T r ðT T r Þ þ

Z

T

Tr

C 0p dT T

Z

T

Tr

C 0p dT : T ð14Þ

Third, for each experimental point the values of the auxiliary function F were calculated as follows: F ¼ Dr G0 þ 2G0T ðH2 OðgÞÞ þ G0T ðSiO2 ðsÞÞ Z T Z T 0 C p ðH4 SiO4 ðgÞÞ dT : C 0p ðH4 SiO4 ðgÞÞdT þ T T TR Tr ð15Þ It follows that F ¼ Df G0T r ðH4 SiO4 ðgÞÞ S 0T r ðH4 SiO4 ðgÞÞ ðT T r Þ:

ð16Þ

In total, 102 values of F over the temperature range 424– 1661 K were collected. The weighted least-squares fit of all data resulted in the following values at 298.15 K for H4SiO4(g): Df G0 = 1238.51 ± 0.51 kJ mol1; S0 = 347.78 ± 0.72 J K1 mol1, with uncertainties given as 2r. How robust are these values? First, we performed the fit of the same data set in the “ideal mixture of ideal gases” approximation, i.e. for /1 = 1 and /1 H4 SiO4 = 1, with the following results: Df G0 = 1239.74 ± 0.53 kJ mol1; S0 = 346.65 ± 0.75 J K1 mol1. Therefore, the model employed for fugacity coefficients changes Df G0 by 1.23 kJ mol1 and S0 by 1.13 J K1 mol1 compared with the “ideal” approximation. Second, if the weight of points is assumed to be proportional to T or T1, then the value of Df G0 changes by less than 0.3 kJ mol1 and the value of S0 by less than 0.3 J K1 mol1. The exclusion of any data set has a similar effect. The biggest effect is due to the difference in C 0p values of H4SiO4(g) as given in Allendorf et al. (1995) and Rutz and Bockhorn (2005). Values of C 0p from Rutz and Bockhorn (2005) at 300 K are 2.2% lower than the ones from Allendorf et al. (1995), but systematically higher at


T = 400 K and above, with the difference increasing from 0.9% at 400 K to 8.1% at 1500 K. If the C 0p results of Rutz and Bockhorn (2005) are employed for the treatment of data at T < 1500 K, which is the upper temperature bound of Rutz and Bockhorn (2005) calculations, then the values of Df G0 and S0 of H4SiO4(g) at 298.15 K change by 1.9 kJ mol1 and by 6.1 J K1 mol1, respectively. Taking into account these variations (and the uncertainties of Df H 0 and S0 of solid phases), one estimates the total uncertainty of Df G0 ± 3.0 kJ mol1 and of S0 ± 6.2 J K1 mol1. These values of Df G0 and S0 of H4SiO4(g) correspond to Df H 0 = 1340.68 ± 3.5 kJ mol1, with necessary values of S0 for H2(g), O2(g), Si(c) taken from Cox et al. (1989). The final results are given in Tables 1 and Table A1. The relatively large uncertainties arise due to conflicting C 0p data for H4SiO4(g) from various sources; new determinations of C 0p would be helpful to resolve the discrepancies and decrease the uncertainties of thermodynamic functions of formation and the entropy for this substance. The value of Df H 0 is consistent within the stated uncertainties with the results calculated from data of Hashimoto (1992) and Jacobson et al. (2005) by the “second-law method”, and it is in agreement with ab initio calculations of Allendorf and coworkers (1995), but not with the ab initio value of Rutz and Bockhorn (2005), see Table 1. The value of Df H 0 is 11 kJ mol1 more positive compared with the value, recommended by Jacobson et al. (2005) on the basis of the “third-law method”. The value of S0 is close to the “second-law” values of S0 due to Hashimoto (1992) and Jacobson et al. (2005), and it is 5.3 and 13.9 J K1 mol1 larger than the ab initio values of Allendorf et al. (1995) and Rutz and Bockhorn (2005), respectively. Table 3 gives the calculated values of the thermodynamic properties of H4SiO4(g) over the temperature range 298.15–2000 K.

3859

Over the temperature range of availability of data the values of the auxiliary function F change by about 430 kJ mol1, so, this function is not convenient for a graphical presentation. To make the scatter of results from various sources more visible, we recalculated available experimental data to the values of the standard Gibbs energy of solution, Dsol G0 , of quartz (Q) according to the reaction SiO2 ðquartzÞ þ 2H2 OðgÞ ¼ H4 SiO4 ðgÞ:

ð17Þ

The following relation connects F and Dsol G0 : Z T Dsol G0 ¼ F þ C 0p ðH4 SiO4 ðgÞÞdT T

Z

Tr T

Tr

C 0p ðH4 SiO4 ðgÞÞ dT T

2G0T ðH2 OðgÞÞ G0T ðQðcÞÞ;

ð18Þ

0

Fig. 3 shows the values of Dsol G calculated from various data sets (symbols). The solid line is calculated using the values of the thermodynamic functions of H4SiO4(g) obtained in this study, while the dotted line corresponds to the thermodynamic functions of H4SiO4(g) from Jacobson et al. (2005). 3. APPLICATIONS FOR TRANSPORT OF SILICA IN LOW-DENSITY STEAM H4SiO4 is the dominant form of silica in low-density water at all temperatures of interest for geochemistry. In order to illustrate this point, mole fractions (Y) of H4SiO4(g), SiO2(g) and SiO(g) in water vapor of density 10 kg m3 in equilibrium with quartz were calculated over the temperature range 500–1700 K, as shown in Fig. 4. All calculations are done in the approximation of the ideal mixing of ideal

Table 3 Thermodynamic properties of H4SiO4(g) at 298.15–2000 K. ðG0T H 0298 Þ T 1 1

T

C 0p

S 0T

K

J K1 mol1

J K1 mol1

JK

298.15 Uncertainty 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

115.25

347.78 ±6.2 348.494 383.746 413.627 439.402 461.962 481.977 499.964 516.324 531.365 545.320 558.364 570.623 582.187 593.123 603.484 613.318 622.682 631.648

347.78 ±6.2 347.782 352.496 361.811 372.645 383.827 394.869 405.563 415.834 425.662 435.059 444.047 452.655 460.909 468.834 476.452 483.785 490.851 497.668

115.56 129.25 138.28 144.24 148.33 151.40 154.03 156.56 159.11 161.70 164.24 166.58 168.61 170.25 171.51 172.59 173.86 175.95

mol

H 0T H 0298

G0T

kJ mol1

kJ mol1

0.000

1238.51 ±3.0 1239.154 1275.818 1315.725 1358.407 1403.498 1450.714 1499.827 1550.653 1603.047 1656.890 1712.081 1768.537 1826.182 1884.953 1944.788 2005.632 2067.436 2130.155

0.213 12.500 25.908 40.054 54.695 69.687 84.961 100.490 116.273 132.313 148.611 165.154 181.917 198.864 215.954 233.160 250.479 267.959

3860


however, the current errors in the thermodynamic properties of this compound in the gas phase (Jacobson et al., 2005) amount to the uncertainty in its concentrations up to 4 orders of magnitude. 4. CONCLUSIONS

Fig. 3. Values of Dsol G0 at Pr = 0.1 MPa for the reaction SiO2(quartz) + 2H2O(g) = H4SiO4(g), calculated using values of the auxiliary function F from various sets of data (symbols), or using the thermodynamic functions of H4SiO4(g) recommended in this work (solid line) or by Jacobson et al., 2005 (dotted line).

Fig. 4. Calculated mole fractions Y of H4SiO4(g), SiO2(g) and SiO(g) in water vapor of density 10 kg m3 at various temperatures in equilibrium with quartz.

The recent transpiration and mass-spectrometric studies (Hashimoto, 1992; Jacobson et al., 2005; Opila et al., 1997) demonstrate that H4SiO4 is the predominant form of silica in low-density water vapor at pressures in excess of 102 MPa, where components other than SiO2 and H2O are not important. Available literature data on the reactions of solid silica phases and low-density water, extending from 424 to 1661 K, in all 102 points, are employed for the determination of Df G0 , Df H 0 and S0 of H4SiO4 in the state of ideal gas at 298.15 K. The resulting values are: Df G0 = 1238.51 ± 3.0 kJ mol1, Df H 0 = 1340.68 ± 3.5 kJ mol1 and S0 = 347.78 ± 6.2 J K1 mol1, with the necessary auxiliary thermodynamic data for water in the ideal gas state and solid phases taken from Cox et al. (1989) and Robie and Hemingway (1995). The values of C 0p for H4SiO4(g) are accepted from ab initio calculations (Allendorf et al., 1995). The obtained result for Df H 0 of H4SiO4(g) agrees with one set of ab initio calculations (Allendorf et al., 1995), but not with another (Rutz and Bockhorn, 2005), and is consistent within the stated uncertainties with the values calculated from data of Hashimoto (1992) and Jacobson et al. (2005) by the “second-law method”. The value of S0 is close to the “second-law” values from data of Hashimoto (1992) and Jacobson et al. (2005), and it is 5.3 and 13.9 J K1 mol1 larger than the ab initio values of Allendorf et al. (1995) and Rutz and Bockhorn (2005), respectively. The thermodynamic properties of this species, H4SiO4, in the ideal gas state are necessary for realistic modeling of silica transport in a low-density water phase. Applications of this analysis may include the processes of silicates condensation in the primordial solar nebula, the precipitation of silica in steam-rich geothermal systems and the corrosion of SiO2-containing alloys and ceramics in moist environments. ACKNOWLEDGMENTS

gases, and the mole fractions Y are calculated as ratios of partial pressures of H4SiO4(g), SiO2(g) and SiO(g) to the partial pressure of water. The species SiO(g) is formed according to the reaction SiO2(s) + H2(g) = SiO(g) + H2O(g), with the values of hydrogen fugacity f(H2(g) = P(H2(g)) taken equal to the ones, corresponding to the equilibrium thermal dissociation of water H2O(g) = H2(g) + 0.5O2(g) without excesses of either H2 or O2. The thermodynamic properties of SiO2(g) and SiO(g) are taken from JANAF tables (Chase, 1998). The concentrations of H4SiO4(g) are larger than those of SiO2(g) and SiO(g) by many orders of magnitude (note that the concentration of SiO(g) depends on the hydrogen pressure and will be much larger at high P(H2(g))). The form SiO(OH)2(g) is expected to occupy the space between H4SiO4(g) and SiO2(g),

This work has benefited from comments and suggestions of GCA Associate Editor D.A. Sverjensky and three anonymous reviewers.

APPENDIX A After decades of controversy, the problem of equilibrium values of quartz solubility in liquid water at low temperatures appears to be finally resolved. Solubility results, obtained during the 1960s–1970s (Fournier, 1960; Morey et al., 1962; Mackenzie and Gees, 1971), all around 5– 7 ppm at 298.15 K, were much lower than the values, based on thermodynamic extrapolations of accurate high-temperature quartz solubility data, which all converted to the va-


lue of 10.8 ppm at 298.15 K (Van Lier et al., 1960; Siever, 1962). In 1997, Rimstidt (1997) presented results of careful experiments on quartz solubility, which lasted at room temperatures up to 13.5 years, i.e. much longer than the previous ones. Rimstidt’s results give at 298.15 K the solubility value of 11.0 ± 1.1 ppm, in a perfect agreement with the value of 10.8 ppm obtained by the extrapolation of accurate high-temperature quartz solubility data. It follows that earlier literature results might be unequilibrated due to insufficient exposition times (up to 3 years). (Note that some geochemists still think that the question of the equilibrium solubility of quartz in liquid water at 298.15 K is not yet resolved. The arguments used refer to the correlating equations of Walther and Helgeson (1977) and Manning (1994), which both give the value close to 104 m (6 ppm). However, it should be noted that the model of Walther and Helgeson (1977) requires an unusually strong, “electrolyte-like” temperature dependence of the partial molar heat capacity, C 1 p;2 , of SiO2(aq), which changes from 100 J K1 mol1 at 298 K to +20 J K1 mol1 at 373 K. For comparison, for another inorganic hydroxide, H3BO3, the change of C 1 p;2 between 298 and 373 K is only around 30 J K1 mol1 (Hnedkovsky et al., 1995), and the equation of Rimstidt (1997), see below, is consistent 0 with C 1 p;2 ðSiO2 ðaqÞ C p ðquartzÞ, i.e. changes for about 1 1 10 J K mol over the temperature range 298–373 K. This difference in C 1 p;2 corresponds to the difference in the Gibbs energy increment over the same temperature range of up to 2.0–2.5 kJ mol1, i.e. up to 0.4 log10 units for an equilibrium constant. The arguments about the expected temperature change of C 1 p;2 of a neutral species at ambient temperatures support Rimstidt’s value for the equilibrium quartz solubility). Rimstidt (1997) proposed an equation describing the most reliable quartz solubility data for the temperature range 273–573 K at saturated water pressure P 1 : log10 m ¼

1107:12ð10:77Þ 0:0254ð0:0247Þ T

ðA1Þ

This equation allows calculating the Gibbs energy change, Dr G0 , for the reaction SiOðquartzÞ þ 2H2 OðlÞ ¼ H4 SiO4 ðaqÞ 0

ðA2Þ

as Dr G ¼ RT ln m. At 298.15 K, 0.1 MPa, Dr G0 21.34 ± 0.24 kJ mol1. Combining this value with the Gibbs energy of formation of quartz and liquid water (Cox et al., 1989), one obtains for H4SiO4(aq) the value of Df G0 ¼1309.23 ± 1.03 kJ mol1. Now the solubility of amorphous silica in liquid water at 298.15 K can be calculated, assuming that the same species, H4SiO4(aq), determines the solubility of the amorphous phase in water. Accepting for amorphous silica the value of Df G0 ¼ 849.28 ± 2.12 kJ mol1 from Robie and Hemingway (1995), one obtains Dr G0 ¼ RT ln m ¼14.33 ± 2.13 kJ mol1, which corresponds to the solubility of amorphous silica at 298.15 K in liquid water of 3.09 103 mol kg1, or 185 ppm of silica. This value is much higher than all available literature data (Alexander et al., 1954; Krauskopf, 1956; Greenberg and Price, 1957; Siever, 1962; Morey et al., 1964; Marshall, 1980; Marshall and Warakomski, 1980;

3861

Dandurand and Schott, 1987; Gallup, 1998; Hamrouni and Dhahbi, 2001), which indicate the solubility within 110–140 ppm, with most values clustering around 120 ppm. Note that the calculated solubility value is very uncertain, as the range between 79 and 438 ppm agrees with the error of 2.13 kJ mol1 in the Gibbs energy of amorphous silica. One of the possible explanations of this discrepancy could be a significant contribution of the silica dimer H6Si2O7 to the material balance of silica in equilibrium with a more soluble amorphous phase. The dimer (as well as higher polymers) is expected to contribute significantly to the total concentration of silica at high temperatures/high silica concentrations, especially in the neighborhood of the upper critical end point, see Zotov and Keppler (2000, 2002), Newton and Manning (2002, 2003, 2008), Gerya et al. (2005). However, evidence of the existence and stability of such a dimer at low temperatures/low concentrations of silica is inconclusive and its large concentration is ruled out by most investigators, see Gunnarsson and Arno´rsson (2000). To illustrate this point, we made the following simple calculations: Zotov and Keppler (2002) gave values of DrH = 12.6 kJ mol1 and DrS = 40.7 J K1 mol1 for the silica dimerization reaction at P = 500 MPa. These data result in the values of the equilibrium constant for the dimerization reaction K dim ¼ X dimer =ðX monomer Þ2 of 0.94 at 298 K and 3.3 at 573 K. The calculated fraction of the dimer in the material balance of silica in equilibrium with amorphous silica is less than 0.001% at 298 K and less than 0.5% at 573 K. These estimates involve a rather long extrapolation and refer to P = 500 MPa (note that the pressure dependence of this reaction has opposite signs in (Zotov and Keppler, 2002) and (Newton and Manning, 2002) works), however, they confirm that there are not convincing reasons to expect a significant contribution of species other than monomeric H4SiO4 to the solubility of amorphous silica at temperatures to 573 K. The systematic difference between the Gibbs energy of H4SiO4(aq), calculated from quartz and amorphous silica solubilities, was noted and discussed by Gunnarsson and Arno´rsson (2000). These authors found that the difference varies from 650 to 1360 J mol1 and persists at all temperatures they considered (273–623 K). Gunnarsson and Arno´rsson (2000) came to the conclusion that the most probable cause for the discrepancy could be the uncertainty in the enthalpy of solution of solid phases in solvents, which is used to derive the enthalpy of formation of amorphous silica. The task is to determine the values of DfH0, DfG0, and S0 at 298.15 K, 0.1 MPa for amorphous silica, which provide a consistency between the reliably established solubilities of quartz and amorphous silica in liquid water at 273–573 K at saturated water pressure P 1 , using the thermodynamic properties of quartz recommended by CODATA (Cox et al., 1989) as an anchor. The following procedure was employed to achieve this goal: one writes an analog of Eq. (A2) for the case of amorphous silica and then obtains a relation: RT ln

mðQÞ ¼ G0 ðQÞ þ G0 ðASÞ mðASÞ

ðA3Þ

3862


where Q designates quartz and AS designates amorphous silica. In essence, (A3) is the equation for the reaction

Q = AS. Considering G0 of solid phases as a function of temperature and pressure, one writes:

Table A1 Thermodynamic properties of substances (at P = 0.1 MPa) used in calculations. Results of this work are shown in bold. Substance

DfH0(298.15 K) 1

kJ mol

S0(298.15 K) JK

1

1

mol

DfG0(298.15 K) 1

kJ mol

Quartz 910.7 ± 1.0a 41.46 ± 0.20a 856.29 ± 1.0b Cristobalite 908.4 ± 2.1c 43.4 ± 0.1c 854.57 ± 2.1c Amorphous silica 902.65 ± 1.01e 48.21 ± 0.47e 850.25 ± 1.0e Amorphous silica 901.6 ± 2.1c 48.5 ± 1.0c 849.28 ± 2.1b H4SiO4(g) 1342.25 ± 3.3e 346.65 ± 6.1e 1239.74 ± 2.8e Si(c) 0 18.81 ± 0.08a 0 O2(g) 0 205.152 ± 0.005a 0 H2(g) 0 130.680 ± 0.003a 0 241.826 ± 0.040a 188.835 ± 0.010a 237.14 ± 0.04b H2O(g) H2O(l) 285.83 ± 0.04a 69.95 ± 0.03a 228.58 ± 0.04b C 0p (Quartz, 298–844 K)c = 81.145 + 1.828 102 T1.81 105 T2–6.985 102 T0.5 + 5.406 106 T2

V (298.15 K) 1

a0 104

cm mol

K1

22.69c 25.74c 27.27c 27.27c

0.65d 0.81d 0.81f

3

C 0p (Quartz, 844–1700 K) = 57.96 + 9.330 103 T + 1.835 106 T2 DtrH(a–b, 844 K)c = 0.625 kJ mol1 C 0p (Cristobalite, 298–523 K)c = 4160 + 2.548 T6.286 107 T2 + 7.168 104 T0.5 C 0p (Cristobalite, 523–1800 K)c = 72.75 + 1.300 103 T4.132 106 T2 DtrH(a–b, 523 K)c = 1.39 kJ mol1 C 0p (Amorphous silica, 298–1700 K)c = 74.64–7.259 103 T3.114 106 T2 + 5.570 106 T2 C 0p /R (H4SiO4(g), 250–2000 K)e = 2.87914 + 5.89126 102 T9.47715 105 T2 + 7.84564 108 T33.15382 1011 T4 + 4.89073 1015 T5 C 0p (H2O(g), 298–2500 K)c = 27.057 + 1.7584 102 T + 2.7696 105 T2–27.656 T0.5–2.5097 106 T2 a b c d e f g

Cox et al. (1989). Calculated from DfH0 and S0 using entropies of elements given in the Table. Robie and Hemingway (1995) A parameter of Eq. (10) taken from Holland and Powell (1998). Obtained in this work, see text. A parameter of Eq. (10) assumed to be equal to that for cristobalite. Allendorf et al. (1995); see also http://www.sandia.gov/HiTempThermo/.

Fig. A1. Experimental results on amorphous silica solubility (molality) in water at saturated water pressure at 273–573 K. Symbols: (1) Hamrouni and Dhahbi (2001); (2) Gunnarsson and Arno´rsson (2000); (3) Gallup (1998); (4) Dandurand and Schott (1987); (5) Chen and Marshall (1982); (6) Marshall and Warakomski (1980); (7) Marshall (1980); (8) Fournier and Rowe (1977); (9) Martynova et al. (1975); (10) Morey et al. (1964); (11) Greenberg and Price (1957); (12) Kitahara (1960b); (13) Siever (1962); (14) Krauskopf (1956); (15) Alexander et al. (1954); (16) Hitchen (1935).


3863

Fig. A2. Residual of the data points used to calculate the values of the function Z. The symbols are the same as in Fig. A1.

Table A2 Values of the enthalpy of transition Q(quartz) = AS(amorphous silica), DH, at Tr = 298.15 K based on the literature data on enthalpy of solution of these phases in various solvents at T = Ts, as quoted from Richet et al. (1982). Referencesa

Ts

Mulert (1912) 293 Wietzel (1921) 298 Wietzel (1921) 298 Troitzsch (1935) 350 Roth and Troitzsch (1949) 350 Humprey and King (1952) 298 Kracek et al. (1953) 298 Hummel and Schwiete (1959) 300 Takahashi and Yoshio (1970) 298 Wise et al. (1963) 298 Holm et al. (1967) 970 Navrotsky et al. (1980) 985 Mean value and uncertainty (as 2r) a b c

DH(Ts) kJ mol1

(H(Q, Ts)–Q, Tr))kJ mol1

(H(AS, Ts)–H(AS, Tr)) kJ mol1

DH(Tr)b kJ mol1

9.24 ± 1.26 9.46 ± 0.63 9.46 12.68 ± 0.30 12.47 ± 0.42 8.87 ± 0.46 9.12 ± 0.25 9.50 ± 0.84 9.62 7.45 ± 2.72 6.07 ± 0.84 6.99 ± 0.33

-0.21 0 0 2.44 2.44 0 0 0.08 0 0 41.25 42.30

0.22 0 0 2.23 2.23 0 0 0.07 0 0 43.52 44.56

9.23 ± 1.26 9.46 ± 0.63 9.46 12.47 ± 0.30c 12.26 ± 0.42c 8.87 ± 0.46 9.12 ± 0.25 9.49 ± 0.84 9.62 ± 0.63 7.45 ± 2.72 8.34 ± 0.84 9.25 ± 0.33 9.03 ± 0.42

Complete references see in Richet et al. (1982). Calculated as DH(Tr) = DH(Ts) + (H(Q, Ts)–H(Q, Tr))) + (H(AS, Ts)–H(AS, Tr)). Excluded from calculating the mean value as an outlier.

RT ln

mðQÞ ¼ ½Df G0 ðT r ; QÞ Df G0 ðT r ; ASÞ mðASÞ ½S 0 ðT r ; QÞ S 0 ðT r ; ASÞ ðT T r Þ Z T Z T 0 0 C p ðQ; T Þ C p ðAS; T Þ þ Tr "TZr # Z T C 0 ðQÞ T C 0 ðASÞ p p dT dT T T T Tr Tr Z P Z P 1 1 V ðQÞdP V ðASÞdP þ P

Z T Z T mðQÞ C 0p ðQ; T Þ C 0p ðAS; T Þ mðASÞ Tr Tr "Z # Z T 0 T C 0 ðQÞ C p ðASÞ p dT dT þT T T Tr Tr

Z ¼ RT ln

ðP 1 P Þ ½V ðQÞ V ðASÞ ¼ a þ b ðT T r Þ;

ðA5Þ ðA6Þ

where ðA4Þ

P

where T r = 298.15 K and P = 0.1 MPa, C 0p and V stand for the known (Robie and Hemingway, 1995) heat capacities and volumes of solid R P R P phases. As the contribution of the term P 1 V ðQÞdP P 1 V ðASÞdP is very small, the temperature and pressure variations of the molar volumes of solid phases were ignored. For a linear regression analysis it is convenient to define the auxiliary function Z as follows:

a ¼ Df G0 ðT r ; QÞ Df G0 ðT r ; ASÞ

ðA7Þ

and b ¼ S 0 ðT r ; QÞ þ S 0 ðT r ; ASÞ

ðA8Þ

The values of the function Z were calculated using 105 experimental data points for the solubility of amorphous silica in water at saturated water pressure over the temperature range 273–573 K (Hitchen, 1935; Alexander et al., 1954; Krauskopf, 1956; Greenberg and Price, 1957; Kitahara,

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1960b; Siever, 1962; Morey et al., 1964; Martynova et al., 1975; Fournier and Rowe, 1977; Marshall, 1980; Marshall and Warakomski, 1980; Chen and Marshall, 1982; Dandurand and Schott, 1987; Gallup, 1998; Gunnarsson and Arno´rsson, 2000; Hamrouni and Dhahbi, 2001). In a few cases experimental data were presented only graphically (Kitahara, 1960b; Fournier and Rowe, 1977; Hamrouni and Dhahbi, 2001), and were read from enlarged plots. In addition, Fournier and Rowe (1977) measured solubility at a number of pressures between 200 and 1030 bar, and the solubility values at the saturated water pressure were obtained by extrapolation from higher pressures. All solubility data points employed are shown in Fig. A1. The unweighted linear regression resulted in Df G0 ðT r ; QÞ Df G0 ðT r ; ASÞ=-6.04 ± 0.06 kJ mol1, S 0 ðT r ;QÞþ S 0 ðT r ; ASÞ = 6.75 ± 0.42 J K1 mol1, and therefore, Df H 0 ðT r ; QÞ Df H 0 ðT r ; ASÞ = 8.05 ± 0.14 kJ mol1, uncertainties are given as 2r. The resulting values of the thermodynamic functions of amorphous silica are presented in Table A1, the residuals of the data points used to calculate the function Z are presented in Fig. A2. The obtained entropy, 48.21 ± 0.47 J K1 mol1, is in excellent agreement with the value of 48.5 ± 1.0 J K1 mol1 selected by Robie and Hemingway (1995). Aqueous solubility data give for the reaction Q = AS the value of Dr H 0 ðT r Þ = 8.05 ± 0.14 kJ mol1, which is about 1 kJ mol1 lower than the mean of 10 literature calorimetric data on the enthalpy of this reaction, Dr H 0 ðT r Þ = 9.03 ± 0.42 kJ mol1, see Table A2. As discussed by Richet et al. (1982) it is likely that the systematic errors in the experimental values are greater than the random errors. Therefore, calorimetric and water solubility values for the enthalpy change for the reaction Q = AS appear consistent, supporting the conclusion of Gunnarsson and Arno´rsson (2000) that is it the enthalpy of formation of amorphous silica that needs the adjustment. REFERENCES Alexander G. B., Heston W. M. and Iler R. K. (1954) The solubility of amorphous silica in water. J. Phys. Chem. 58, 453– 455. Allendorf M. D., Melius C. F., Ho P. and Zachariah M. R. (1995) Theoretical study of the thermochemistry of molecules in the Si–O–H system. J. Phys. Chem. 99, 15285–15293. Chase M. W. Jr. (1998) NIST-JANAF Thermochemical Tables. Fourth ed. Published by the American Chemical Society, the American Institute of Physics, and the National Institute of Standards and Technology. J. Phys. Chem. Ref. Data, Monograph No. 9, p. 1961. Cox J. D., Wagman D. D. and Medvedev V. A. (1989) CODATA Key Values for Thermodynamics. Hemisphere Publishing Corporation, New York. Chen C.-T. and Marshall W. L. (1982) Amorphous silica solubilities. IV. Behavior in pure water and aqueous sodium chloride, sodium sulfate, magnesium chloride, and magnesium sulfate solutions up to 350 °C. Geochim. Cosmochim. Acta 46, 279–287. Dandurand J.-L. and Schott J. (1987) New data on the solubility of amorphous silica in organic compound–water solutions and a new model for the thermodynamic behavior of aqueous silica in aqueous complex solutions. J. Solut. Chem. 16, 237–256.

Fournier R. O. (1960) Solubility of quartz in water in the temperature interval from 25 °C to 300 °. Geol. Soc. Am. Bull. 71, 1867–1868. Fournier R. O. and Rowe J. J. (1977) The solubility of amorphous silica in water at high temperatures and pressures. Am. Mineral. 62, 1052–1056. Gallup D. L. (1998) Aluminum silicate scale formation and inhibition (2): scale solubilities and laboratory and field inhibition tests. Geothermics 27, 481–501. Gerya T. V., MareschW V., Burchard M., Zakhartchouk V., Doltsinis N. L. and Fockenberg T. (2005) Thermodynamic modeling of solubility and speciation of silica in H2O–SiO2 fluid up 1300 °C and 20 kbar based on the chain reaction formalism. Eur. J. Mineral. 17, 269–283. Greenberg S. A. and Price E. W. (1957) The solubility of silica in solutions of electrolytes. J. Phys. Chem. 61, 1539–1541. Gunnarsson I. and Arno´rsson S. (2000) Amorphous silica solubility and the thermodynamic properties of H4SiO4o in the range of 0 to 350 °C at Psat. Geochim. Cosmochim. Acta 64, 2295– 2307. Hamrouni B. and Dhahbi M. (2001) Analytical aspects of silica in saline waters – application to desalination of brackish waters. Desalination 136, 225–232. Harvey A. H. and Bellows J. C. (1997) Evaluation and correlation of steam solubility data for salts and minerals of interest in the power industry. NIST Technical Note 1387, p. 88. Harvey A. H. and Lemmon E. W. (2004) Correlation for the second virial coefficient of water. J. Phys. Chem. Ref. Data 33, 369–376. Hashimoto A. (1992) The effect of H2O gas on volatilities of planetforming major elements: I. Experimental determination of thermodynamic properties of Ca-, Al-, and Si-hydroxide gas molecules and its application to the solar nebula. Geochim. Cosmochim. Acta 56, 511–532. Heitmann H. G. (1964) Solubility of silicic acid in water and steam and its effect on silica deposits in turbines (in German). Chemiker-Zeitung 88, 891–893. Heitmann H. G. (1965) Solubility of silica in water and water vapor (in German). Glastechnische Berichte 38, 41–54. Hildebrand D. L. and Lau K. H. (1994) Thermochemistry of gaseous SiO(OH), SiO(OH)2, and SiO2. J. Chem. Phys. 101, 6076–6079. Hitchen C. S. (1935) A method for the experimental investigation of hydrothermal solutions, with notes on its application to the solubility of silica. Trans. Inst. Mining Metal. 44, 255– 280. Hnedkovsky L., Majer V. and Wood R. H. (1995) Volumes and heat capacities of H3BO3(aq) at temperatures from 298.15 K to 705 K and at pressures to 35 MPa. J. Chem. Thermodyn. 27, 801–814. Holland T. J. B. and Powell R. (1998) An internally consistent thermodynamic data set for phases of petrological interest. J. Metamorphic Geol. 16, 309–343. Jacobson N. S., Opila E. J., Myers D. L. and Copland E. H. (2005) Thermodynamics of gas species in the Si–O–H system. J. Chem. Thermodyn. 37, 1130–1137. Kennedy G. C. (1950) A portion of the system silica–water. Econ. Geol. 45, 629–653. Kitahara S. (1960a) The solubility of quartz in water at high temperatures and high pressures. Rev. Phys. Chem. Japan 30, 109–114. Kitahara S. (1960b) The polymerization of silicic acid obtained by the hydrothermal treatment of quartz and the solubility of amorphous silica. Rev. Phys. Chem. Japan 30, 131–137. Krauskopf K. B. (1956) Dissolution and precipitation of silica at low temperatures. Geochim. Cosmochim. Acta 10, 1–26.

Thermodynamics of H4SiO4 in the ideal gas state Lakshtanov L. Z., Shmonov V. M., Dadze T. P., van der Erden A. and Jansen B. (1994) Free surface energy in the system quartzwater at high temperatures and pressures (in Russian). Geokhimiya 1479, 1488. Mackenzie F. T. and Gees R. (1971) Quartz: Synthesis at earth surface conditions. Science 173, 533–535. Manning C. E. (1994) The solubility of quartz in H2O in the lower crust and upper mantle. Geochim. Cosmochim. Acta 58, 4831– 4839. Marshall W. L. (1980) Amorphous silica solubilities. I. Behavior in aqueous sodium nitrate solutions; 25–300 °C, 0–6 molal. Geochim. Cosmochim. Acta 44, 907–913. Marshall W. L. and Warakomski J. M. (1980) Amorphous silica solubilities. II. Effect of aqueous salt solutions at 25 °C. Geochim. Cosmochim. Acta 44, 915–924. Martynova O. I., Fursenko V. F. and Popov A. S. (1972) The study of the dissolved silica distribution between water and steam (in Russian). Teploenergetika 12, 51–53. Martynova O. I., Popov A. S. and Fursenko V. F. (1975) Boundary lines of phase equilibrium diagrams of the silicon dioxide–water system (in Russian). Teploenergetika 5, 66–68. Mills I., Cvitasˇ T., Homann K., Kallay N. and Kuchitsu K. (1993) Quantities, Units and Symbols in Physical Chemistry. Blackwell Science, Oxford. Morey G. W., Fournier R. O. and Rowe J. J. (1962) The solubility of quartz in water in the temperature interval from 25 to 300 °C. Geochim. Cosmochim. Acta 26, 1029–1043. Morey G. W., Fournier R. O. and Rowe J. J. (1964) The solubility of amorphous silica at 25 °C. J. Geophys. Res. 69, 1995–2002. Morey G. W. and Hesselgesser J. M. (1951) The solubility of quartz and some other substances in superheated steam at high pressures. Trans. ASME 73, 865–875. Newton R. C. and Manning C. E. (2002) Solubility of enstatite + forsterite in H2O at deep crust/upper mantle conditions: 4 to 15 kbar and 700 to 900 °C. Geochim. Cosmochim. Acta 66, 4165–4176. Newton R. C. and Manning C. E. (2003) Activity coefficient and polymerization of aqueous silica at 800 °C, 12 kbar, from solubility measurements on SiO2-buffering mineral assemblages. Contrib. Mineral. Petrol. 146, 135–143. Newton R. C. and Manning C. E. (2008) Thermodynamics of SiO2–H2O fluid near the upper critical end point from quartz solubility measurements at 10 kbar. Earth Planet. Sci. Lett. 274, 241–249. Opila E. J., Fox D. S. and Jacobson N. S. (1997) Mass spectrometric identification of Si–O–H(g) species from the reaction of silica with water vapor at atmospheric pressure. J. Am. Ceram. Soc. 80, 1009–1012. Prausnitz J. M., Lichtenthaler R. N. and de Avezedo E. G. (1999) Molecular thermodynamics of fluid-phase equilibria, third ed. Prentice-Hall, New York. Richet P., Bottinga Y., Denielou L., Petitet J. P. and Tequi C. (1982) Thermodynamic properties of quartz, cristobalite and

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amorphous SiO2: drop calorimetry measurements between 1000 and 1800 K and a review from 0 to 2000 K. Geochim. Cosmochim. Acta 46, 2639–2658. Rimstidt J. D. (1997) Quartz solubility at low temperatures. Geochim.Cosmochim. Acta 61, 2553–2558. Robie R. A. and Hemingway B. S. (1995) Thermodynamic Properties of Minerals and Related Substances at 298.15 K and 1 Bar (105 Pascals) Pressure and at Higher Temperatures. U.S. Geological Survey Bulletin 2131. U.S. Gov. Printing Office, Washington. Rutz L. K. and Bockhorn H. (2005) Theoretical studies on the mechanism of formation of silicon dioxide (SiO2). Fourth Joint Meeting of the U.S. Sections of the Combustion Institute: Western States, Central States, Eastern States, Philadelphia, PA, United States, Mar. 20–23, 2005, F38/1-F38/6. Publisher: Combustion Institute, Pittsburgh, PA. Siever R. (1962) Silica solubility 0–200 °C, and the diagenesis of siliceous sediments. J. Geol. 70, 127–150. Singh P. and Vora S. D. (2005) Vapor phase silica transport during SOFC operation at 1000 °C. Ceramic Eng. Science Proc. 26, 99– 110. Straub F. G. (1946) Steam turbine blade deposits. Bulletin University of Illinois, Engineering Experiment Station, No. 364, p. 92. Straub F. G. and Grabowski H. A. (1945) Silica deposition in steam turbines. Trans. ASME 67, 309–316. Styrikovich M. A. and Martynova O. I. (1963) Contamination of the steam in boiling-water reactors by impurities from the water (in Russian). Atomnaya Energiya 15, 214–218. Van Lier J. A., de Bruyn P. L. and Overbeek J. T. G. (1960) The solubility of quartz. J. Phys. Chem. 64, 1675–1682. Wagner W. and Pruß A. (2002) The IAPWS formulation for the thermodynamic properties of ordinary water substances for general and scientific use. J. Phys. Chem. Ref. Data 31, 387–535. Walther J. V. and Helgeson H. C. (1977) Calculation of the thermodynamic properties of aqueous silica and the solubility of quartz and its polymorhs at high pressures and teperatures. Am. J. Sci. 277, 1315–1351. Wendlandt H.G. (1963) Reactionen zwischen Oxiden und Wasser bei ho¨heren Temperaturen und verschiedenen Dichten. Dissertation: Universita¨t Go¨ttingen (quoted from Harvey and Bellows (1997)). Wendlandt H. G. and Glemser O. (1963) Reaction of oxides with water at high pressures and temperatures (in German). Angew. Chem 75, 949–957. Zotov N. and Keppler H. (2000) In-situ Raman spectra of dissolved silica species in aqueous fluids to 900 C and 14 kbar. Amer. Miner. 85, 600–604. Zotov N. and Keppler H. (2002) Silica speciation in aqueous fluids at high pressures and temperatures. Chem. Geol. 184, 71–82. Associate editor: Dimitri A. Sverjensky