Heat capacities of aqueous solutions of sodium hydroxide and water ionization up to 300 °C at 10 MPa

Available online at www.sciencedirect.com

Geochimica et Cosmochimica Acta 72 (2008) 3124–3138 www.elsevier.com/locate/gca

Heat capacities of aqueous solutions of sodium hydroxide and water ionization up to 300 °C at 10 MPa Simon Schro¨dle, Erich Ko¨nigsberger, Peter M. May, Glenn Hefter * Chemistry Department, Murdoch University, Murdoch, WA 6150, Australia Received 17 September 2007; accepted in revised form 9 April 2008; available online 29 April 2008

Abstract A commercial (Setaram C80) calorimeter has been modified to measure the heat capacities of highly caustic solutions at temperatures up to 300 °C and pressures up to 20 MPa. The improvements have allowed more accurate determination of the isobaric volumetric heat capacities of chemically aggressive liquids at high temperatures. Test measurements with aqueous solutions of sodium chloride showed a reproducibility of about ±0.1%, with an accuracy of 0.3% or better, over the whole temperature range. Heat capacities of aqueous solutions of sodium hydroxide at concentrations from 0.5 to 8 mol/kg were measured at temperatures from 50 to 300 °C and a pressure of 10 MPa. Apparent molar isobaric heat capacities of NaOH(aq) were calculated using densities determined previously for the same solutions by vibrating-tube densimetry. Standard state (infinite dilution) partial molar isobaric heat capacities of NaOH(aq) were obtained by extrapolation using an extended Redlich–Meyer equation. Values of the standard heat capacity change for the ionization of water up to 300 °C were derived by combining the present results with the literature data for HCl(aq) and NaCl(aq). Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Thermodynamic properties; Heat capacity; Sodium hydroxide; Aqueous solutions; Water ionization; High temperature

1. INTRODUCTION Accurate thermodynamic properties for aqueous solutions of sodium hydroxide (and for the ionization of water that can be derived from them) at high temperatures and pressures are critical in many geochemical situations. Examples include modeling the processes occurring during the formation or dissolution of oxide and hydroxide minerals (Wood and Vlassopoulos, 1989; Ziemniak et al., 1993; Ziemniak and Goyette, 2004), hydrothermal ore formation from hyperalkaline fluids (Palandri and Reed, 2004), the behavior of various elements in crustal fluids (Tremaine and LeBlanc, 1980; Wesolowski, 1992; Castet et al., 1993; Diakonov et al., 1996, 1999) and aqueous Martian surface geochemistry (Marion et al., 2008).

*

Corresponding author. Fax: +61 8 9310 1711. E-mail address: [email protected] (G. Hefter).

0016-7037/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.gca.2008.04.016

Data for NaOH(aq), especially at high concentrations, are also important for process design and optimization in the plethora of industrial processes that utilize concentrated caustic solutions. The most important of these are the refining of alumina from bauxitic ores via the well-known Bayer process (Hudson, 1987), the storage of certain types of nuclear waste (Hummel, 2005) and the neutralization of acid mine tailings (Mahoney et al., 2007). Sodium hydroxide is also used as a starting material in the production of synthetic silicates such as zeolites (Auerbach et al., 2003). In recognition of the importance of the thermodynamic properties of NaOH(aq) for aqueous geochemistry, Pabalan and Pitzer (1987) made a thorough evaluation of the then available data. However, because reliable heat capacity data at elevated temperatures were lacking at the time, their Pitzer model, covering solutions up to m = 10 mol kg1, # = 350 °C and P = 400 bar, had to be parameterized primarily on water activities and densities. The heat capacity database for NaOH(aq) remains patchy. At atmospheric pressure, reliable values are available at

Heat capacity of NaOH(aq) to 300 °C

25 °C (Magalha˜es et al., 2002) and up to 55 °C from the Picker flow calorimetry measurements of Roux et al. (1984). At higher temperatures, the only reliable data available are those of Simonson et al. (1989) at m 6 4 mol/kg and of Conti et al. (1988) at m 6 1.5 mol/kg, both measured by flow calorimetry up to 250 °C, as well as some approximate measurements to 90 °C by Mal’tsev and Mashovets (1965). Heat capacities of liquids can be determined using flow or static methods. Flow methods are usually preferred for measurements at moderate concentrations but suffer from experimental problems at high concentrations and temperatures (Rogers and Duffy, 1989; Coxam et al., 1991; Magalha˜es et al., 2002). On the other hand, static measurements, nowadays almost invariably with differential (scanning) calorimeters, although much slower and generally less precise, are probably better for more concentrated solutions, especially at high temperatures. For both static and flow calorimetry the quantity normally measured is the isobaric heat capacity per unit volume. This means that it is necessary to know the density of the sample, so as to obtain the heat capacity per unit mass and hence the desired molar heat capacity. A general disadvantage of the static method is that the density of the sample must be known over the full temperature range of interest, whereas for flow calorimetry the density is usually required only at one temperature (typically 25 °C). Since the present work was focussed on highly concentrated solutions of NaOH(aq), it was decided to employ a static calorimetric method. The densities required for the calculation of the massic/molar heat capacities were determined over the full temperature range at the appropriate pressure by means of high temperature vibrating-tube densitometry and have been reported elsewhere (Hnedkovsky et al., 2007). 2. EXPERIMENTAL 2.1. Calorimeter The experimental arrangement (Fig. 1) used for the determination of volumetric heat capacities at constant pressure (r/J K1 cm3) was based on a commercial Tian–Calvet type differential microcalorimeter (Setaram, Lyon, France; Model C80, 5 lW sensitivity, 0.1 lW resolution). While the manufacturer offers a wide variety of cells for the C80, none is fully suitable for measurements of concentrated caustic solutions up to 300 °C. Hastelloy C276 (‘Ha’) vessels obtained from Setaram were used for a set of experiments below 225 °C but at higher temperatures Ha is unsuitable because it suffers from corrosion stress cracking in the presence of NaOH(aq). Therefore, special cells were machined from nickel 201, a low-carbon nickel grade certified for high pressure service. The cells (with internal volumes of 15 mL) were built to operate at temperatures up to at least 300 °C and are capable of withstanding pressures of at least 20 MPa. High-precision machining techniques (including spark erosion) were used for fabrication to guarantee good physical and thermal symmetry of the sample and reference cells, and wall-thick-

3125

Fig. 1. Schematic of the high pressure setup of the Setaram C80 calorimeter: calorimetric cell (A) located inside the calorimeter (not shown but represented by the thick vertical lines), filling capillary (B), pre-heater (C), filling port (D), expansion volume (E), draining outlet (F), buffer volume (G) and high-pressure nitrogen supply (H), pressure gauge (P).

ness uniformity. Two concentric nickel tubes were used for filling and emptying the cell (Fig. 1). The outer tube (1/800 OD) was welded to the calorimetric vessel; the inner tube (1/1600 OD) reached almost to the bottom of the cell to avoid air entrapment during filling. From the outer tube a stainless steel (ss) capillary (1/1600 OD) led to a ss expansion tube (1/400 OD, of approximate volume 6 mL) to allow for the thermal expansion of the liquid in the sample cell during the course of the experiment. The pressure within the sample cell and expansion tube was kept constant by use of a large buffer volume (two ss cylinders, each of 500 mL volume) connected in series and filled with high-purity nitrogen at the desired pressure. An electronic pressure sensor (PDCR 4000, GE Druck; calibration traceable to NMI standards, accuracy 0.1%) was used to monitor the system pressure during the experiment. Temperature fluctuations in the laboratory (which was thermostated to ±3 °C) produced pressure changes of the order of ±0.3% during a typical experiment. Such variations have an insignificant effect on the measured heat capacities. As pointed out by Coxam et al. (1991), the major causes of uncertainty in the measurement of liquid heat capacities using the present type of calorimeter setup are heat leakages from the cells through the filling tubes and irreproducible convection phenomena within the calorimeter. To minimize these effects, the design of Bessieres et al. (2000) was adopted. This design utilizes a pre-heater (C in Fig. 1) in contact with the filling tubes and insulation in the upper part of the calorimeter. The temperature of this element is kept slightly (1 K) below that of the calorimeter block by a differential PID regulator linked to platinum resistance temperature probes. In this way the heat flow along the tubes becomes negligible, resulting in a significant improvement of the reproducibility of the experiments.

3126

S. Schro¨dle et al. / Geochimica et Cosmochimica Acta 72 (2008) 3124–3138

As the volume of the liquid present in the active area of the calorimeter is a function of temperature, two standards were required to calculate the heat capacities of the samples. Nitrogen and water were used for this purpose as accurate heat capacities are readily calculated for them from internationally-accepted models (Span et al., 2000; Wagner and Pruss, 2002). 2.2. Calorimetric measurements Heat capacities of liquids can be measured in a Tian– Calvet type differential calorimeter in two distinct ways. The first uses a constant heating rate (the continuous method) and is commonly employed for rapid determinations requiring only modest accuracy. The second is based on step-by-step increases of temperature interspersed with isothermal equilibration periods. The latter is in general more time consuming but more accurate (Paramo et al., 2002) as it is less affected by long-term drifts of the baseline and by systematic errors due to heat-transfer-related variations. However, very occasionally the stability of the present calorimeter when used in this mode was found to be unsatisfactory at some temperatures. In such cases the results obtained from the immediately adjacent steps were discarded. The origin of these random effects is not known but may be related to minor instabilities in the main power supply. To further establish the reliability of the results, two different increments, of 5 and 20 K (10 K for the last two steps) measured at heating rates of 0.25 K/min and 1 K/ min, respectively, were used alternately throughout the investigated temperature range. Although the heat capacities of aqueous solutions change rapidly at high temperatures, Coxam et al. (1991) have shown that temperature increments of 10 K are adequate to obtain reliable r values for NaCl(aq). All calorimetric measurements were made isoplethically. A carbonate-free sample of NaOH(aq) at the desired concentration was introduced using a syringe into the N2-filled cell (A in Fig. 1) via the port D. The cell was then pressurized with high-purity N2 and stabilized at 50 °C (the lowest temperature studied) for several hours. The temperature scan was performed under automatic control using the stepwise increments and scan rates detailed above. After each increase of temperature, the cell was allowed to stabilize for 1.5–2 h, depending on the temperature increment. The typical time to complete a scan from 50 to 300 °C was therefore about 2 days. Upon completion of a scan the calorimeter was allowed to cool slowly (for several hours) to room temperature. The cell was then thoroughly flushed with high-purity water and dried with a stream of N2. Regular checks of the calorimeter performance were made by making scans at 50 6 #/°C 6 300 using air, water or 1 mol kg1 NaCl(aq). 2.3. Densities of sodium hydroxide solutions For many practical applications and for the calculation of molar quantities the massic heat capacities, cp (J K1 g1), are needed. These were obtained as cp = rq1, where q is the density of the sample at the appropriate tem-

perature, pressure and concentration. The densities required for conversion of the isobaric volumetric heat capacities of NaOH(aq) to their massic values were those reported recently by Hnedkovsky et al. (2007). These data were obtained by vibrating-tube densimetry at 10 MPa pressure on solutions prepared from the same stock solutions used for the present study and cover the same range of concentrations and temperatures. They are in excellent agreement with, but rather more precise than, other recent independent measurements on this system (Hnedkovsky et al., 2007). One significant difference between the present heat capacity measurements and the densities is that the latter were obtained isothermally whereas the former were measured, as noted above, isoplethically. For convenience of interpolation, the difference in density (q  qw) between the sodium hydroxide solutions and pure water (qw) at p = 100 bar was fitted to Eq. (1): ðq  qw Þ=q0 ¼ aðm=m0 Þ þ bðm=m0 Þ1:5 þ cðm=m0 Þ2

ð1Þ

where m is the molality of the solution, m0  1 mol (kg H2O)1, q0  1 g cm3 and a, b and c were treated as temperature dependent adjustable parameters, y ¼ y 1 þ y 2 ðT =T 0 Þ þ y 3 ðT =T 0 Þ2 þ y 4 =ð647  ðT =T 0 ÞÞ þ y 5 ðT =T 0 Þ lnðT =T 0 Þ

ð2Þ

where y = a, b or c and T is the thermodynamic temperature in Kelvin with T0  1 K. These parameters are summarized in Table 1. The standard error in the overall fit of (q  qw) via Eqs. (1) and (2) increases from 4  105 g cm3 at 50 °C to 1.4  104 g cm3 at 200 °C and from 3.0  104 g cm3 at 250 °C to 4.6  104 g cm3 at 300 °C. Note that Eqs. (1) and (2) were employed in the present study only because they are simpler than the more accurate expressions given by Hnedkovsky et al. (2007). The minor differences in the solution densities calculated from these two expressions have a negligible effect on the calculated heat capacities. 2.4. Data processing Evaluation of the isobaric volumetric heat capacity from _ the recorded temperature T(t) and heat-flow QðtÞ data as a function of time t, requires determination of the equilibrium temperature before and after each heating step j, Tj = T(tj) and T 0j ¼ T ðt0j Þ. With the present calorimeter system these temperatures could usually be determined to

Table 1 Parameters for Eqs. (1) and (2), valid at 10 MPa Parameter Index

a

b

c

1 2 3 4 5

4.428194  101 1.149916  102 2.154666  106 4.974165 1.888241  103

1.339236  101 3.975724  103 8.052291  107 2.522685 6.583212  104

1.275080  102 4.306343  104 9.669490  108 3.943206  101 7.213404  105

Heat capacity of NaOH(aq) to 300 °C

same as those used for the determination of the NaOH(aq) densities (Hnedkovsky et al., 2007), were filtered (0.45 lm) under high-purity nitrogen and analyzed to ±0.1% by titration against standard hydrochloric acid (NIST traceable). Carbonate impurities, as determined by titration and checked by Raman spectroscopy, were always found to be below 0.1% of the total alkalinity.

±0.5 mK, Furthermore, the integral heat of a given step, Ij, is calculated as: Z t0j Ij ¼ Q_ dt ð3Þ tj

Using the temperature increase during the step, DT j ¼ T 0j  T j , an instrument-specific volumetric heat capacity qj is obtained as qj ¼ I j =DT j

3. RESULTS AND DISCUSSION

ð4Þ 3.1. Calorimeter performance

Assuming a linear response for the calorimeter and small temperature increments, the average isobaric volumetric heat capacity of a sample (S), rSj , at an average temperature T j ¼ ðT 0j þ T j Þ=2 that corresponds to the midpoint of the temperature increment, can be calculated as rSj ¼ ðqSj  qW j Þ

3127

N rW j  rj þ rW j W N qj  qj

The performance of the calorimeter is mainly determined by the overall reproducibility of the instrument and the linearity of its heat-flow response. The repeatability was investigated by comparing the instrument-specific heat capacities qj obtained for water calibration measurements over several months and was found to be ±0.1% up to 200 °C and ±0.15% at higher temperatures (Fig. 2a). Variations between runs were random. The accuracy of the response of the instrument is more difficult to investigate, as a standard liquid (other than water) whose density and heat capacity are accurately known over the full range of interest is needed. No such liquid is currently available although NaCl(aq) has been reasonably well studied. The numerous literature data for these solutions have been critically reviewed by Archer (1992) who incorporated them into an extended Pitzer model. Accordingly, measurements on NaCl(aq) were carried out at two concentrations: 0.9524 and 5.585 mol/kg, at 10 MPa pressure over the temperature range 50 6 #/°C 6 300 (Fig. 2b). For the 0.95 mol/kg solution, the present experimental results agree within 0.2% with the

ð5Þ

where the superscripts W and N denote values obtained from calibration experiments performed at the same pressure with an identical heating profile, using liquid water (W) and gaseous nitrogen (N) standards. 2.5. Materials All solutions were prepared by weight from appropriate stock solutions without buoyancy corrections. To minimize contamination with carbonate, present even in NaOH of analytical quality, all stock solutions (8 mol/kg) were prepared using the procedure outlined by Sipos et al. (2000) with solid NaOH pellets (‘Univar’ analytical reagent, Ajax Finechem, Australia). The stock solutions, which were the 0.3

a

0.8

b

0.7 0.6

0.2

Δ

0.5 0.1

0.4

0.0

0.2

Δσ

0.3

0.1 -0.1

0.0 -0.1

-0.2

-0.2 -0.3

-0.3

-0.4 50

100

150

200

250

300

50

100

150

200

250

300

ϑ Fig. 2. Calorimeter performance over the temperature range 50 6 #/°C 6 300 at a pressure of 10 MPa using air at 0.10 MPa as the reference. (a) Typical reproducibility of the calorimetric measurements using water; deviations: Dqj = 100(qj  qj,av.)/qj,av.. (b) Comparison of the present isobaric volumetric heat capacities (r) of NaCl(aq) (circles: 0.9524 mol/kg; triangles: 5.585 mol/kg) with values (rA) calculated from the equation of Archer, 1992); deviations: Dr = 100(r  rA)/rA.

3128

S. Schro¨dle et al. / Geochimica et Cosmochimica Acta 72 (2008) 3124–3138

Table 2 Densities, q, isobaric volumetric (r in J K1 cm3), massic (cp in J K1 g1), and apparent molar (Cp,U in J K1 mol1) heat capacities of NaOH(aq) at 10.0 MPa as a function of temperature and concentration; Ni cells (m0  1 mol kg1) r

cp

Cp,U

#/°C

q/g cm3

r

cp

Cp,U

m/m = 0.5000 51.91 1.01198 64.27 1.00561 76.63 0.99852 88.99 0.99077 101.36 0.98239 113.72 0.97342 126.09 0.96387 138.45 0.95376 163.18 0.93180 175.54 0.91992

4.1158 4.1040 4.0809 4.0606 4.0311 4.0133 3.9859 3.9614 3.9186 3.8937

4.0671 4.0811 4.0870 4.0985 4.1034 4.1229 4.1353 4.1535 4.2054 4.2326

23.0 6.2 9.1 4.8 19.1 9.5 21.8 30.4 45.0 67.1

187.90 200.27 212.64 225.00 237.37 262.11 274.49 281.91 291.81

0.90740 0.89421 0.88027 0.86554 0.84990 0.81545 0.79626 0.78397 0.76648

3.8767 3.8611 3.8474 3.8333 3.8249 3.8214 3.8362 3.8488 3.8746

4.2723 4.3179 4.3707 4.4288 4.5004 4.6862 4.8178 4.9093 5.0551

77.5 92.4 111.9 144.7 180.2 295.3 375.3 448.3 582.9

m/m0 = 1.000 51.91 64.27 76.63 88.99 113.72 126.09 138.45 150.81 163.18 175.54

1.03162 1.02510 1.01794 1.01019 0.99302 0.98365 0.97377 0.96336 0.95240 0.94089

4.1176 4.1062 4.0870 4.0682 4.0179 3.9960 3.9684 3.9394 3.9227 3.9007

3.9914 4.0056 4.0150 4.0272 4.0461 4.0624 4.0753 4.0893 4.1187 4.1457

8.9 0.1 2.3 5.4 2.2 4.1 13.4 26.4 28.6 39.3

187.90 200.27 212.64 225.00 237.37 249.74 262.11 274.49 281.91 291.81

0.92879 0.91606 0.90265 0.88852 0.87357 0.85774 0.84090 0.82291 0.81149 0.79540

3.8782 3.8619 3.8449 3.8288 3.8105 3.8022 3.8103 3.8270 3.8172 3.8409

4.1756 4.2158 4.2596 4.3092 4.3620 4.4328 4.5312 4.6505 4.7039 4.8288

53.9 66.0 84.1 108.1 144.0 180.7 215.1 265.2 339.6 425.7

m/m0 = 2.000 51.97 64.32 76.69 89.05 101.41 113.77 126.13 138.50 150.86 163.23 175.59

1.06886 1.06208 1.05478 1.04699 1.03876 1.03008 1.02097 1.01142 1.00142 0.99094 0.97998

4.1448 4.1384 4.1238 4.1030 4.0808 4.0535 4.0300 4.0068 3.9889 3.9585 3.9336

3.8778 3.8965 3.9097 3.9188 3.9286 3.9351 3.9472 3.9616 3.9833 3.9946 4.0139

14.0 21.2 24.6 24.7 23.9 19.8 17.0 13.3 11.3 1.0 8.0

187.96 200.32 212.71 225.10 237.47 249.84 262.21 274.58 282.01 291.89

0.96849 0.95646 0.94381 0.93052 0.91658 0.90190 0.88644 0.87012 0.85990 0.84577

3.9160 3.8971 3.8765 3.8560 3.8381 3.8214 3.8155 3.8135 3.8118 3.8188

4.0434 4.0745 4.1073 4.1439 4.1874 4.2370 4.3043 4.3828 4.4329 4.5152

15.0 25.1 39.3 57.5 79.4 107.9 140.1 185.0 223.1 286.7

m/m0 = 3.000 51.93 64.28 76.65 89.01 101.37 113.73 126.09 138.46 150.82 163.18 175.55

1.10384 1.09683 1.08940 1.08157 1.07338 1.06482 1.05589 1.04657 1.03686 1.02674 1.01617

4.2001 4.1888 4.1739 4.1542 4.1330 4.1080 4.0786 4.0617 4.0461 4.0119 3.9863

3.8050 3.8190 3.8314 3.8409 3.8505 3.8580 3.8628 3.8809 3.9022 3.9074 3.9229

33.9 37.2 39.3 39.6 39.2 36.9 32.4 31.6 30.4 21.4 14.3

187.92 200.28 212.65 225.01 237.38 249.75 262.12 274.49 281.91 291.81

1.00513 0.99360 0.98153 0.96889 0.95565 0.94178 0.92726 0.91208 0.90266 0.88978

3.9642 3.9481 3.9175 3.9047 3.8742 3.8469 3.8429 3.8355 3.8383 3.8340

3.9440 3.9735 3.9913 4.0300 4.0540 4.0847 4.1444 4.2052 4.2522 4.3089

6.9 0.0 14.6 25.4 46.7 72.0 95.3 130.7 156.4 207.3

m/m0 = 4.000 51.98 64.34 76.70 89.07 101.43 113.79 126.16 138.52 150.89

1.13666 1.12945 1.12189 1.11401 1.10583 1.09734 1.08853 1.07938 1.06989

4.2655 4.2602 4.2403 4.2183 4.1913 4.1774 4.1576 4.1219 4.1071

3.7527 3.7719 3.7796 3.7866 3.7902 3.8069 3.8194 3.8187 3.8388

48.3 52.4 52.8 52.4 50.4 51.4 50.4 44.4 43.4

200.25 212.65 225.04 237.43 249.82 262.19 274.56 281.98 291.89

1.02795 1.01626 1.00405 0.99130 0.97798 0.96411 0.94973 0.94088 0.92888

3.9891 3.9625 3.9445 3.9234 3.8962 3.8922 3.8813 3.8761 3.8544

3.8806 3.8991 3.9286 3.9579 3.9839 4.0371 4.0868 4.1197 4.1495

12.8 2.2 8.2 22.4 42.6 61.4 90.6 113.6 159.1

#/°C

q/g cm3

0

Heat capacity of NaOH(aq) to 300 °C

3129

Table 2 (continued) #/°C

q/g cm3

r

cp

Cp,U

#/°C

q/g cm3

163.26

1.06001

4.0709

3.8404

35.6

r

cp

Cp,U

m/m0 = 6.000 51.95 64.31 76.67 89.04 113.73 126.09 138.46 150.82 163.19 175.55

1.19652 1.18897 1.18120 1.17322 1.15662 1.14796 1.13904 1.12983 1.12029 1.11041

4.4116 4.4068 4.3937 4.3709 4.3219 4.3013 4.2659 4.2393 4.2061 4.1782

3.6870 3.7064 3.7196 3.7255 3.7366 3.7469 3.7452 3.7522 3.7545 3.7627

68.7 71.7 73.2 72.8 70.5 69.5 65.4 62.2 57.2 52.5

187.91 200.28 212.66 225.05 237.40 249.76 262.13 274.50 281.92 291.81

1.10015 1.08948 1.07836 1.06678 1.05476 1.04227 1.02933 1.01606 1.00799 0.99724

4.1464 4.1224 4.0831 4.0564 4.0307 3.9966 3.9763 3.9648 3.9367 3.9230

3.7689 3.7838 3.7864 3.8025 3.8214 3.8346 3.8630 3.9022 3.9055 3.9338

46.1 40.2 30.2 20.8 9.6 6.0 22.9 43.9 64.8 95.0

m/m0 = 8.000 51.97 64.34 76.70 89.06 101.43 113.79 126.16 138.52 150.89 163.26 175.62

1.24898 1.24119 1.23325 1.22516 1.21692 1.20852 1.19992 1.19111 1.18206 1.17273 1.16308

4.5647 4.5600 4.5456 4.5154 4.4892 4.4678 4.4405 4.3985 4.3818 4.3273 4.2954

3.6548 3.6739 3.6859 3.6855 3.6890 3.6969 3.7006 3.6928 3.7069 3.6899 3.6931

83.0 85.5 86.5 85.2 84.3 83.7 82.0 77.8 76.7 69.8 65.5

187.99 200.36 212.73 225.10 237.48 249.85 262.22 274.59 282.01 291.91

1.15310 1.14276 1.13202 1.12089 1.10937 1.09748 1.08529 1.07296 1.06558 1.05595

4.2610 4.2335 4.1902 4.1657 4.1251 4.1043 4.0676 4.0470 4.0263 4.0001

3.6953 3.7046 3.7015 3.7164 3.7185 3.7397 3.7480 3.7718 3.7785 3.7882

60.1 54.9 46.4 39.4 28.4 18.1 2.3 15.6 30.7 56.2

values calculated from Archer’s equation up to 300 °C. However, somewhat larger discrepancies (up to 0.5%) were found for the 5.585 mol/kg solution above 200 °C. These deviations can almost certainly be attributed to the lack of reliable experimental data that were available at these temperatures and pressures for incorporation into Archer’s model (Archer, 1992). Interestingly, Coxam et al. (1991), using a similar calorimetric setup to the present one, also found discrepancies (of up to 1%) between their results and the rather few published data. No noticeable differences were found between the heat capacities obtained from measurements using the Ha and Ni cells, despite a significant difference in their wall thicknesses, which might be counteracted by the substantially higher thermal conductivity of nickel: 79.3 W m1 K1 (Specialmetals, 2006) vs. 9.8 W m1 K1 for Ha (Specialmetals, 2004). Thus it appears that the physical and thermal properties of the cells do not cause significant systematic errors in the measured liquid heat capacities. Equally importantly, no systematic differences were observed between the results obtained from the two different heating rates, 0.25 and 1.0 K/min. 3.2. Volumetric and massic heat capacities of NaOH(aq) Using the procedures outlined above, two sets of heat capacity data at a pressure of 10 MPa were acquired for NaOH(aq) using the Ni and Ha cells (Tables 2 and 3). The isobaric volumetric heat capacities, determined at the average temperatures T j (see Section 2.4) given in Tables 2 and 3, were converted to the massic values using the densities of Hnedkovsky et al. (2007), interpolated to the pres-

ent experimental conditions via Eqs. (1) and (2). For convenience these quantities are also listed in Tables 2 and 3. Values of the apparent molar heat capacity C p;U ¼ Mcp  1000ðcW p  cp Þ=m

ð6Þ

were derived from the present cp data combined with the massic heat capacity of water, cW p , as given in the IAPWS-95 formulation (Wagner and Pruss, 2002), where M is the molar mass (in g mol1) of the solute and m (in mol kg1) its concentration (molality) in the solution. Based on the calorimeter performance during the test measurements with NaCl(aq), and including the small additional error arising from the uncertainty of the density measurements (
ð7Þ

was derived from the experimental data, measured over intervals of 5 K, 10 K and 20 K. It was found that the temperature dependence of A could be well-described by the function Að#Þ ¼

a þ c# þ e#2 1 þ b# þ d#2

ð8Þ

3130

S. Schro¨dle et al. / Geochimica et Cosmochimica Acta 72 (2008) 3124–3138

Table 3 Densities, q, isobaric volumetric (r in J K1 cm3), massic (cp in J K1 g1), and apparent molar (Cp,U in J K1 mol1) heat capacities of NaOH(aq) at 10.0 MPa as a function of temperature and concentration; Ha cells (m0  1 mol kg1) r

cp

Cp,U

#/°C

q/g cm3

r

cp

Cp,U

m/m = 0.5023 51.88 1.01209 64.24 1.00573 76.60 0.99864 88.95 0.99089 101.31 0.98251 113.67 0.97355 126.04 0.96401

4.1159 4.0997 4.0809 4.0582 4.0353 4.0090 3.9839

4.0668 4.0764 4.0865 4.0955 4.1071 4.1180 4.1327

22.8 14.9 9.3 10.0 10.5 18.7 26.2

138.40 150.77 163.14 175.50 187.87 200.23 212.60

0.95389 0.94321 0.93193 0.92006 0.90754 0.89435 0.88042

3.9620 3.9404 3.9153 3.8964 3.8756 3.8600 3.8427

4.1535 4.1776 4.2012 4.2349 4.2704 4.3160 4.3646

29.4 35.1 52.3 61.1 80.0 94.6 122.6

m/m0 = 1.001 51.88 64.23 76.59 88.96 101.32 113.68 126.04

1.03167 1.02517 1.01800 1.01025 1.00194 0.99310 0.98373

4.1234 4.1093 4.0901 4.0679 4.0479 4.0225 4.0010

3.9968 4.0084 4.0177 4.0266 4.0400 4.0505 4.0672

3.1 3.2 5.3 5.0 6.9 2.5 1.2

138.40 150.76 163.12 175.49 187.86 200.22 212.59

0.97385 0.96344 0.95250 0.94098 0.92888 0.91615 0.90275

3.9740 3.9527 3.9274 3.9065 3.8865 3.8738 3.8509

4.0807 4.1027 4.1232 4.1515 4.1841 4.2283 4.2657

7.6 12.2 23.5 32.8 44.7 52.5 77.2

m/m0 = 2.002 51.87 64.23 76.59 88.94 101.30 113.66 126.02

1.06899 1.06221 1.05491 1.04713 1.03890 1.03023 1.02113

4.1471 4.1370 4.1191 4.0988 4.0792 4.0527 4.0291

3.8795 3.8948 3.9047 3.9143 3.9264 3.9338 3.9457

15.1 20.4 22.0 22.4 22.9 19.3 16.4

138.39 150.75 163.12 175.49 187.86 200.22 212.59

1.01158 1.00158 0.99111 0.98014 0.96866 0.95663 0.94401

4.0052 3.9833 3.9571 3.9331 3.9122 3.8938 3.8722

3.9593 3.9770 3.9926 4.0127 4.0388 4.0704 4.1019

12.4 8.2 0.2 8.3 17.0 26.9 41.7

m/m0 = 2.995 51.87 64.23 76.59 88.95 101.32 113.69 126.05

1.10370 1.09669 1.08926 1.08144 1.07324 1.06468 1.05575

4.2108 4.1938 4.1789 4.1555 4.1362 4.1115 4.0840

3.8152 3.8240 3.8364 3.8426 3.8540 3.8617 3.8683

37.5 38.9 41.0 40.1 40.3 38.1 34.4

138.42 150.78 163.14 175.51 187.88 212.65

1.04643 1.03672 1.02660 1.01603 1.00499 0.98134

4.0614 4.0412 4.0133 3.9897 3.9668 3.9215

3.8812 3.8980 3.9094 3.9268 3.9471 3.9961

31.5 28.6 21.9 15.5 7.8 13.1

m/m0 = 3.992 51.87 64.23 76.60 88.96 101.33 113.69 126.05

1.13646 1.12926 1.12170 1.11383 1.10565 1.09716 1.08835

4.2658 4.2539 4.2385 4.2162 4.1970 4.1691 4.1451

3.7536 3.7670 3.7786 3.7854 3.7959 3.7999 3.8086

48.4 50.8 52.3 51.9 51.9 49.2 47.1

138.41 150.78 163.14 175.51 187.87 200.24 212.61

1.07921 1.06972 1.05984 1.04957 1.03887 1.02770 1.01603

4.1181 4.0978 4.0701 4.0564 4.0201 3.9993 3.9656

3.8159 3.8308 3.8403 3.8648 3.8697 3.8915 3.9031

43.5 40.9 35.5 32.9 22.9 15.7 3.1

m/m0 = 4.983 51.89 64.25 76.61 88.97 101.34 113.70 126.06

1.16707 1.15969 1.15202 1.14410 1.13591 1.12747 1.11875

4.3335 4.3273 4.3079 4.2882 4.2659 4.2419 4.2184

3.7131 3.7315 3.7394 3.7481 3.7555 3.7623 3.7706

58.9 62.1 62.5 62.7 62.0 60.6 58.9

138.42 150.79 163.15 175.51 187.87 200.24 212.61

1.10973 1.10040 1.09072 1.08067 1.07021 1.05932 1.04796

4.1895 4.1638 4.1379 4.1154 4.0799 4.0586 4.0277

3.7752 3.7839 3.7937 3.8082 3.8123 3.8313 3.8434

55.4 52.0 47.8 43.5 35.3 29.0 19.2

m/m0 = 5.978 51.91 64.27 76.63 88.99 101.35

1.19592 1.18839 1.18062 1.17264 1.16445

4.4123 4.4020 4.3882 4.3647 4.3437

3.6895 3.7042 3.7168 3.7221 3.7303

68.9 71.0 72.3 71.8 71.5

138.43 150.79 163.15 175.52 187.88

1.13844 1.12923 1.11970 1.10982 1.09956

4.2610 4.2416 4.2036 4.1751 4.1388

3.7429 3.7562 3.7543 3.7620 3.7640

64.6 62.8 56.9 52.0 44.8

#/°C

q/g cm3

0

Heat capacity of NaOH(aq) to 300 °C

3131

Table 3 (continued) #/°C

q/g cm3

r

cp

Cp,U

#/°C

q/g cm3

r

cp

Cp,U

113.71 126.07

1.15603 1.14737

4.3146 4.2865

3.7323 3.7360

69.4 67.0

200.24 212.61

1.08889 1.07777

4.1052 4.0800

3.7701 3.7856

37.0 29.6

m/m0 = 6.969 51.90 64.26 76.62 88.97 113.69 126.06 138.44

1.22287 1.21520 1.20734 1.19931 1.18269 1.17407 1.16521

4.4915 4.4782 4.4672 4.4440 4.3894 4.3542 4.3284

3.6729 3.6851 3.7000 3.7055 3.7114 3.7087 3.7147

77.0 78.4 80.1 79.7 76.9 73.7 71.5

150.80 163.17 175.53 187.89 200.26 212.63

1.15608 1.14666 1.13691 1.12680 1.11630 1.10539

4.3012 4.2661 4.2417 4.2021 4.1742 4.1377

3.7205 3.7204 3.7310 3.7292 3.7393 3.7432

68.6 63.9 60.3 53.4 47.5 39.1

m/m0 = 7.996 51.90 64.25 76.62 88.98 101.34 113.70 126.06

1.24893 1.24114 1.23320 1.22512 1.21689 1.20849 1.19989

4.5641 4.5536 4.5435 4.5168 4.4945 4.4599 4.4289

3.6544 3.6689 3.6843 3.6868 3.6934 3.6905 3.6911

83.0 84.6 86.2 85.4 85.0 82.6 80.4

138.43 150.79 163.16 175.53 187.89 200.25 212.62

1.19109 1.18203 1.17270 1.16306 1.15309 1.14275 1.13202

4.3954 4.3630 4.3299 4.3053 4.2587 4.2252 4.1878

3.6902 3.6911 3.6923 3.7017 3.6933 3.6974 3.6994

77.4 74.1 70.2 66.9 59.8 53.7 46.1

Table 4 Parameters for the calculation of the parameter A (103 J K1 mol1) in Eq. (8) at concentrations m (mol kg1)a and 10 MPa valid for the temperature range 4 6 #/°C 6 300 unless otherwise stated m/m0 0.5 1 2 3 4 5b 6 7b 8 a b

a 0.346417798 0.287175133 0.216208490 0.173539837 0.146822153 0.126654344 0.113249547 0.101254119 0.094071830

b 0.038220892 0.029377073 0.023834654 0.018931087 0.019734067 0.017699742 0.018982136 0.015622098 0.022171007

c

d 3

2.514282  10 1.535827  103 1.232764  103 7.26984  104 7.77946  104 5.77595  104 5.31414  104 2.49352  104 5.26794  104

r2

e 4

1.2133  10 9.2649  105 7.5994  105 6.064  105 6.2538  105 5.8090  105 5.9019  105 4.6244  105 6.6388  105

6

4.07269  10 3.54311  106 1.35806  106 1.51680  106 1.02407  106 8.41005  107 1.30768  106 2.03411  106 1.47868  106

0.99936 0.99779 0.99977 0.99946 0.99925 0.99787 0.99948 0.99707 0.99933

m0  1 mol kg1. Valid for 4 6 #/°C 6 225.

for each concentration studied. Also included in the fit were data published by Roux et al. (1984) at 4 6 #/°C 6 55. Although they determined their heat capacities at p = 0.1 MPa, the pressure dependence of A was found to be only small (0.3% MPa1) at temperatures up to 55 °C. Thus these data can be used to extend the temperature range of the present data. The empirical parameters a to e in Eq. (8) were determined using the Levenberg–Marquardt algorithm and are listed in Table 4, together with their correlation (r2) coefficient. The cp values calculated from the fit lay well within the estimated accuracy of the data (Figs. 3 and 4a). Table 5 summarizes the present isobaric massic heat capacities interpolated to rounded concentrations and temperatures. 3.3. Apparent molar heat capacities of NaOH(aq) The quantity A (Eq. (7)) can also be conveniently used to calculate the apparent molar heat capacities

C p;U ¼ McW p  A  ð1000 þ MmÞ

ð9Þ

The present heat capacities were obtained, as already noted, from temperature scans of samples at fixed concentrations. However, it is desirable to derive an expression for the heat capacity isotherms, both for interpolation and to determine the apparent molar heat capacities of NaOH(aq) at infinite dilution, C 1 p;U , which is equal to the standard partial molar isobaric heat capacity of the solute in the solvent. For convenience, an extended Redlich/Meyer-type equation (Millero, 1979) 1=2 C p;U ¼ C 1 þ BC m þ C C m3=2 þ DC m2    p;U þ AC m

ð10Þ

was used for fitting the apparent molar heat capacities as a function of concentration. Such equations have been used frequently for this purpose (see for example, Roux et al., 1984; Magalha˜es et al., 2002) but other formulations, such as that of Pitzer (1991), are also suitable although they

3132

S. Schro¨dle et al. / Geochimica et Cosmochimica Acta 72 (2008) 3124–3138 5.6

a

5.4

100

5.2

0

5.0

-100

4.8

-200

4.6 -300 -400

4.2 4.0

-500

3.8

-600

b

3.6

Φ

4.4

-700

3.4 0

50

100

150

200

250

300

0

50

100

150

200

250

300

ϑ Fig. 3. Isobaric massic heat capacities (a) and isobaric apparent molar heat capacities (b) of NaOH(aq) obtained in the present study at 50 6 #/°C 6 300 and p = 10 MPa, along with the data reported by Roux et al. (1984) at 4 6 #/°C 6 55 and p = 0.1 MPa. Concentrations are m/mol kg1 = 0.5, 1, 2, 3, 4, 6 and 8 (a, top to bottom; b, bottom to top). The dotted line in (a) represents the massic heat capacity of water according to the IAPWS-95 formulation (Wagner and Pruss, 2002); the dashed line in (b) gives the standard partial molar volume C 1 p;U of NaOH(aq).

It should be noted that in Eq. (10), whereas BC, CC and DC were treated as empirical fit parameters, AC is derived from the Debye–Hu¨ckel theory (Fernandez et al., 1997). The value of AC has little influence on the interpolation of Cp,U values but it can have a significant effect on the magnitude of C 1 p;U , which is determined by extrapolation of

usually contain more adjustable parameters. It was found that only four parameters, AC to DC, were required for an accurate description of the present data using Eq. (10). Table 6 lists the Cp,U values interpolated from the present experimental data via Eqs. (7)–(10) to rounded temperatures and concentrations.

0.8

0.8

a

b

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6 -0.8

-0.8 50

100

150

200

250

300

50

100

150

200

250

300

Fig. 4. Fit residuals (a) and deviations from the model of Simonson et al. (1989) (b) for the present experimental isobaric massic heat capacities of NaOH(aq) at m/mol kg1 = 1 (squares), 2 (circles), 4 (triangles up), 6 (triangles down) and 8 (diamonds), all at p = 10 MPa; measurements using Ha cells (empty symbols) and Ni cells (filled symbols).

Heat capacity of NaOH(aq) to 300 °C Table 5 Massic heat capacities of NaOH(aq) at rounded concentrations and temperatures at 10 MPa pressure (m0  1 mol kg1) #/°C

50

m/m0

cp/J K1 g1

100

0.5 1 2 3 4 6 8

4.067 3.993 3.878 3.802 3.748 3.683 3.648

4.105 4.035 3.924 3.851 3.793 3.727 3.687

150

200

250

275

300

4.176 4.098 3.976 3.893 3.828 3.748 3.694

4.316 4.217 4.070 3.967 3.887 3.779 3.705

4.587 4.444 4.242 4.098 3.993 3.837 3.730

4.822 4.638 4.382 4.206 4.077 3.886 3.759

5.206 4.957 4.600 4.383 4.218 3.987 3.842

Table 6 Apparent molar heat capacities of NaOH(aq) at rounded concentrations and temperatures at 10 MPa pressure (m0  1 mol kg1) #/°C

50

100

150

200

250

275

3.4. Comparison with previous studies

22.3 12.7 36.6 94.4 230.2 383.4 740.2 6.9 3.3 15.9 63.6 171.8 286.2 525.1 14.6 22.4 8.4 26.8 106.1 188.7 356.3 33.0 39.7 27.8 2.2 67.8 133.1 257.1 47.0 51.7 40.9 15.1 40.5 95.1 197.0 67.9 71.3 61.6 39.6 5.9 48.5 122.8 82.0 84.2 74.9 55.1 16.3 18.6 76.1

Cp,U data to infinite dilution. This is problematic because the value of AC depends upon the dielectric properties of the solvent and these are not known to a sufficiently high level of accuracy at the temperatures and pressures of the

100

present study. An extensive survey of these properties by Bradley and Pitzer (1979) has been widely used. However, more recently, Archer and Wang (1990), Pitzer (1991) and Fernandez et al. (1997) have re-evaluated the Debye–Hu¨ckel coefficients and reported somewhat different results both from those of Bradley and Pitzer (1979) and from each other. As the magnitude of AC remains controversial at the temperatures and pressures of interest here, two fits are given (Table 7): one using the AC values recommended by Fernandez et al. (1997) and the other using those of Pitzer (1991). The values of C 1 p;U so obtained are in excellent agreement at low temperatures (#/°C 6 100) but diverge at higher temperatures where the magnitude of AC becomes larger (and hence more critical) and more uncertain because the dielectric properties of water are less well known. Nevertheless, even at 300 °C, where the absolute variation in C 1 p;U is quite large (39 J K1 mol1), the relative difference is still less than 3%.

300

m/m0 Cp,U/J K1 mol1 0.5 1 2 3 4 6 8

3133

a

At low temperatures (Figs. 3 and 5) there is a seamless fit between the present values (at p = 10 MPa) and the data of Roux et al. (1984) measured by Picker flow calorimetry at 4 6 #/°C 6 55 and p = 0.1 MPa in spite of the pressure difference. At high temperatures only two previous studies of the heat capacities of NaOH(aq) have been published to our knowledge, both using Picker-type flow calorimeters. Conti et al. (1988) presented values for cp up to 250 °C and 1.5 mol/kg. Unfortunately, they did not report the pressure, which makes quantitative comparison with the present data impossible.

100

b

0

50 -100 0 -1

-200 -300

-1

-50

-400 -100 p,

-500 -600

-150

-700 -200 -800 -250 0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

1.5

2.0

2.5

-900 3.0

Fig. 5. Apparent molar isobaric heat capacities of NaOH(aq). (a) At temperatures below 100 °C: #/°C = 4 (1), 10 (2), 25 (3), 40 (4), 55 (5), Roux et al. (1984) at p = 0.1 MPa; and #/°C = 50 (open triangles down), 100 (bullets), this study at p = 10 MPa. (b) At temperatures P100 °C: #/°C = 100 (bullets), 150 (circles), 200 (solid squares), 250 (open squares), 275 (solid triangles up), 300 (open triangles up), this study at p = 10 MPa.

3134

S. Schro¨dle et al. / Geochimica et Cosmochimica Acta 72 (2008) 3124–3138

0

a

b

c

d

-200

-400

-600

-800

-1

-1000

-1

-1200

p,Φ

-1400

-1600

200

0

-200

-400

-600

-800 50

100

150

200

250

300 50

100

150

200

250

300

Fig. 6. Apparent molar isobaric heat capacities at p = 10 MPa and m/mol kg1 = 0 (a), 0.1 (b), 1 (c) and 6 (d) of aqueous solutions of NaOH (this study, solid line), NaCl (Archer, 1992; dashed lines) and HCl (Sharygin and Wood, 1997 corrected to 10 MPa; dotted lines) at 50 6 #/°C 6 300.

The work of Simonson et al. (1989) was more comprehensive, covering a concentration range from 0.1 to 4 mol/kg, mainly at 7 MPa pressure, but again limited to temperatures below 250 °C. A comparison of the present results with the Pitzer model given by Simonson et al. (1989), in the form of a residual plot (Fig. 4b), shows excellent agreement, with deviations below 0.3% for m 6 6 mol/kg. However, as often found with Pitzer models, even at temperatures only slightly above their parameterization limit (250 °C in this case) their model becomes unsatisfactory. 3.5. Comparisons with other electrolytes So few reliable data exist for the heat capacities of the aqueous solutions of the other alkali metal hydroxides at

elevated temperatures that no useful comparisons can be made with the present results for NaOH(aq). For example, Patterson et al. (2001) have recently carried out measurements on KOH(aq) and reviewed the existing experimental data. However, their measurements did not extend beyond 120 °C; nor could they identify any reliable data at higher temperatures in the literature. On the other hand, the heat capacities of aqueous solutions of sodium chloride and hydrochloric acid have been reasonably well characterized at high temperatures and pressures. Fig. 6 plots the apparent molar heat capacities of NaCl(aq) at 10 MPa, calculated from the equation of Archer (1992). Also shown in Fig. 6 are the corresponding HCl values along with the present values for NaOH(aq), calculated using Eq. (10) and the AC values of Fernandez

Heat capacity of NaOH(aq) to 300 °C

3135

Table 7 1 Standard partial molar heat capacities, C 1 mol1), Debye–Hu¨ckel constants, AC (J K1 mol1.5 kg0.5) and fitting parameters for the p;U (J K calculation of apparent molar heat capacities (in J K1 mol1) using Eq. (10) C1 p;U

AC

BC

CC

DC

r2

AC from Fernandez et al. (1997) 50 54.63 100 51.33 150 93.70 200 192.4 250 440.2 275 745.7 300 1532

39.596 59.730 93.206 166.51 381.00 681.76 1543.8

9.8625 9.6146 21.0214 45.9791 139.524 279.876 685.580

1.6409 4.7668 6.0684 9.5044 30.584 62.476 147.335

0.2710 1.0051 1.0047 1.1106 3.0800 5.8738 11.865

0.99986 0.99977 0.99987 0.99995 0.99998 0.99998 0.99925

AC from Pitzer (1991) 50 54.71 100 51.95 150 95.31 200 193.5 250 436.5 275 735.6 300 1493

39.813 61.451 97.661 169.76 370.67 653.68 1435.3

9.6579 11.2385 25.2248 49.0444 129.685 253.127 582.181

1.5613 5.3986 7.7040 10.6971 26.719 51.971 106.724

0.2820 1.0920 1.2296 1.2747 2.5445 4.4181 6.2376

0.99986 0.99977 0.99987 0.99994 0.99999 0.99997 0.99909

#/°C

et al. (1997). The HCl(aq) values, measured at 28 MPa by Sharygin and Wood (1997), were converted to 10 MPa using their Pitzer model for the volumetric properties of HCl(aq) combined with Eq. (11).    2  @C p;U @ VU ¼ T ð11Þ @p T @T 2 p At infinite dilution (Fig. 6a), the standard partial molar heat capacities of all three electrolytes are negative, and show a broadly similar dependence on temperature, decreasing strongly above 250 °C. Even at high solute concentrations, Cp,U for NaOH(aq) and NaCl(aq) continue to show a broad similarity up to at least 300 °C (Fig. 6b–d). In contrast, Cp,U(HCl(aq)) differs increasingly from those of the other two electrolytes at higher temperatures and concentrations. Thus, for a 1 mol/kg solution at # J 200 °C, Cp,U(HCl(aq)) rapidly becomes positive (Fig. 6c) reaching a value of ca. +200 J K1 mol1 at 300 °C. This overall effect has been interpreted (Holmes et al., 1987; Sharygin and Wood, 1997) as being indicative of increasing association of HCl with rising temperature and concentration, coupled to a positive value of Cp,U(HCl0(aq)). Fig. 6a–d therefore reflects the transition of HCl(aq) from a strong-electrolyte to that of a largely associated molecular solute. 3.6. Heat capacity change for the ionization of water The thermodynamic parameters for the ionization of water (Scheme I below) are of great importance in the investigation of almost all ionic equilibria in aqueous solutions and, as noted in Section 1, especially at the high temperatures and pressures relevant to many geochemical calculations. Such data can be obtained more or less directly, e.g., by the measurement of the equilibrium constant for Eq. (10) as a function of temperature and pressure, or they can be derived from neutralization reactions such as that shown in Scheme II.

H2 OðlÞ ¼ OH ðaqÞ þ Hþ ðaqÞ NaOHðaqÞ þ HClðaqÞ ¼ H2 O þ NaClðaqÞ

ðIÞ ðIIÞ

While many of the thermodynamic parameters for Equilibrium (I) are well known over wide ranges of temperature and pressure, the standard (infinite dilution) molar heat capacity change for the ionization of water, DC1 p;m , is not particularly well characterized at temperatures above 100 °C. Above 200 °C the disagreement becomes considerable between the values of Sweeton et al. (1974), obtained from potentiometric measurements of the ionization constant of water, and those of Chen et al. (1994), using flow calorimetric measurements of heats of mixing of NaOH(aq) and HCl(aq). The present results shed some light on this important quantity. If the standard partial molar heat capacities of NaCl(aq), NaOH(aq) and HCl(aq) are known for the same conditions, then DC1 p;m can be derived by assuming electrolyte additivity. This gives the relation 1 1 1 DC1 p;m ¼ Cp;U ðNaOHÞ þ Cp;U ðHClÞ  Cp;U ðNaClÞ

 Cp ðH2 OÞ

ð13Þ

where C p (H2O) is the isobaric molar heat capacity of pure water (Wagner and Pruss, 2002). The C 1 p;U values for NaCl(aq) were calculated from the equation of Archer (1992), those for HCl(aq) were obtained from the data of Sharygin and Wood (1997) as described above, and the NaOH(aq) values were taken from Table 7 (using the AC values of Fernandez et al., 1997). All values refer to a pressure of 10 MPa. The results of these calculations are compared with the literature values of DC1 p;m at the saturation vapor pressure of water, psat, or at 0.1 MPa (# 6 100 °C) in Table 8 and Fig. 7. At lower temperatures (6150 °C) the agreement with previous studies is excellent. At higher temperatures, despite the differences in pressure, the present results are in good agreement with the values obtained by Sweeton et al. (1974) and, where available, those of Palmer and

3136

S. Schro¨dle et al. / Geochimica et Cosmochimica Acta 72 (2008) 3124–3138 -150

-300

-450

-600

-750

-900

-1050 0

50

100

150

200

250

300

Fig. 7. Standard molar isobaric heat capacity change, DC 1 p;m , for the ionization of water (Table 8) at #/°C < 100 (p = 0.1 MPa, literature data) and at #/°C P 50 (p = 10 MPa, present study; or psat at #/°C P 100, literature data). This study (bullets), Sweeton et al. (1974, diamonds); Palmer and Drummond (1988, open triangles down), Chen et al. (1994, circles); Sharygin and Wood (1997, open triangles up) and Patterson et al. (2001, crosses). Dotted line is included only as a visual guide; error bars are estimates (corresponding to an overall uncertainty in the present results of 6%).

Table 8 Standard molar heat capacity change for the ionization of water (Equilibrium (I)) at saturation pressure (psat) or (# 6 100 °C) at p = 0.1 MPa unless otherwise indicated 50 0.012

#/°C psat/MPaa b

This study Sweeton et al. (1974) Palmer and Drummond (1988) Chen et al. (1994) Sharygin and Wood (1997) Patterson et al. (2001)d a b c d

100 0.101

1 DC 1 mol1 p;m /J K 186 ± 5 182 ± 5 186 ± 5 174 ± 5 183 ± 7 167 ± 5 182 182 185 ± 4 183 ± 5 185.9 187.3

150 0.476

200 1.55

250 3.98

275 5.95

300 8.59

224 ± 10 233 ± 13 227 ± 7 230 236 ± 14

314 ± 26 341 ± 30 341 ± 21 352 344 ± 25

476 ± 34 511 ± 45 545 ± 39 690 547 ± 46

629 ± 41 661 ± 55

984 ± 59 964 ± 93

1121c

2164c

Saturation vapor pressure of pure water (Wagner and Pruss, 2002). At p = 10 MPa. Not shown in Fig. 7. At p = 0.35 MPa.

Drummond (1988) and Sharygin and Wood (1997). Collectively, these results suggest that the much more negative values reported by Chen et al. (1994) are incorrect. In particular it should be noted that the DC1 p;m values of Sweeton et al. (1974) and Palmer and Drummond (1988), being based on the second temperature derivative of the ionization constant of water determined from potentiometric measurements, are experimentally independent of the present electrolyte heat capacities. The extrapolations given by Olofsson and Hepler (1975) and the study of Chen et al. (1994) both suffer from the large uncertainties in the thermodynamic data then available for NaOH(aq) at high temperatures. Exact agreement with the values given by Sharygin and Wood (1997) at higher temperatures would

not be expected because of differences in reference pressures and the uncertainties that inevitably arise from the different procedures used in the extrapolation to infinite dilution. On the other hand, it appears that pressure, at least up to 10 MPa, has little effect on DC1 p;m at temperatures up to ca. 150 °C. At higher temperatures, overall uncertainties in the present DC1 p;m values are estimated to ca. ±6%. The conclusions presented above would still apply even if these uncertainties were considerably larger. 4. CONCLUSIONS The modified calorimetric apparatus used in the present study has been shown from test measurements with Na-

Heat capacity of NaOH(aq) to 300 °C

Cl(aq) to yield heat capacities of concentrated electrolyte solutions with an accuracy of better than 0.3% at temperatures up to 300 °C at a pressure of 10 MPa. Measurements on NaOH(aq) in purpose-built Ni cells have extended the range of reliable heat capacity data for this industrially important electrolyte to concentrations up to 8 mol/kg and temperatures up to 300 °C at a pressure of 10 MPa. Combination of these results with published data for NaCl(aq) and HCl(aq) have resolved a significant disagreement in the literature concerning the heat capacity change for the ionization of water at high temperatures. ACKNOWLEDGMENTS This work was funded through the Australian Mineral Industries Research Association by the Australian alumina industry (Alcoa World Alumina, Alcan, Comalco Aluminium, Queensland Alumina, and Worsley Alumina), and the Australian Government through the Australian Research Council (Linkage Grant No. LP0349107). The initial design and construction of the high pressure system of the calorimeter was by Dr. Sally-Anne Rowlands. The authors thank Dr. Vladimir Majer and Dr. Jean-Yves Coxam (Universite´ Blaise Pascal, Clermont-Ferrand, France) for design advice and hospitality, and Dr. Ju¨rgen Seidel (TU Bergakademie Freiberg, Germany) and Dr. Lan-Chi Ko¨nigsberger (Murdoch University) for experimental assistance. We are grateful to Associate Editor Dr. David Wesolowski and three anonymous reviewers for their constructive comments.

REFERENCES Archer D. G. (1992) Thermodynamic properties of the NaCl + H2O system. II. Thermodynamic properties of NaCl(aq), NaCl2H2O(cr), and phase equilibria. J. Phys. Chem. Ref. Data 21, 793–829. Archer D. G. and Wang P. (1990) The dielectric constant of water and Debye–Hu¨ckel limiting law slopes. J. Phys. Chem. Ref. Data 19, 371–411. Auerbach S. M., Carrado K. A. and Dutta P. K. (2003) Handbook of Zeolite Science and Technology. Marcel Dekker Inc., New York, Basel, 928 pp. Bessieres D., Saint-Guirons H., Daridon J.-L. and Coxam J.-Y. (2000) Apparatus for simultaneous determination of the densities and heat capacities of liquids and of liquids with dissolved gas under an extended range of pressure (0.1–100 MPa). Meas. Sci. Technol. 11, N69–N72. Bradley D. J. and Pitzer K. S. (1979) Thermodynamics of electrolytes. 12. Dielectric properties of water and Debye– Hu¨ckel parameters to 350 °C and 1 kbar. J. Phys. Chem. 83, 1599–1603. Castet S., Dandurand J. L., Schott J. and Gout R. (1993) Boehmite solubility and aqueous aluminum speciation in hydrothermal solutions (90–350 °C): experimental study and modelling. Geochim. Cosmochim. Acta 57, 4869–4884. Chen X., Oscarson J. L., Gillespie S. E., Cao H. and Izatt R. M. (1994) Determination of enthalpy of ionization of water from 250 to 350 °C. J. Solution Chem. 23, 747–768. Conti G., Gianni P., Papini A. and Matteoli E. (1988) Apparent molar heat capacity and relative enthalpy of aqueous NaOH between 323 and 523 K. J. Solution Chem. 17, 481–497. Coxam J.-Y., Quint J. R. and Grolier J.-P. E. (1991) Modification of a C-80 Setaram calorimeter for measuring heat capacities of liquids at temperatures up to 548 K and pressures up to 20 MPa. J. Chem. Thermodyn. 23, 1075–1083.

3137

Diakonov I., Pokrovski G., Schott J., Castet S. and Gout R. (1996) An experimental and computational study of sodium – aluminum complexing in crustal fluids. Geochim. Cosmochim. Acta 60, 197–211. Diakonov I. I., Schott J., Martin F., Harrichourry J.-C. and Escalier J. (1999) Iron(III) solubility and speciation in aqueous solutions. Experimental study and modeling: part 1. Hematite solubility from 60 to 300°C in NaOH–NaCl solutions and thermodynamic properties of FeðOHÞ 4 ðaqÞ. Geochim. Cosmochim. Acta 63, 2247–2261. Fernandez D. P., Goodwin A. R. H., Lemmon E. W., Levelt Sengers J. M. H. and Williams R. C. (1997) A formulation for the static permittivity of water and steam at temperatures from 238 K to 873 K at pressures up to 1200 MPa, including derivatives and Debye–Hu¨ckel coefficients. J. Phys. Chem. Ref. Data 26, 1125–1166. Hnedkovsky L., Ko¨nigsberger E., Cibulka I., Ko¨nigsberger L.-C., Schro¨dle S., May P. M. and Hefter G. T. (2007) Densities of NaOH(aq) at temperatures from (323 to 573) K and 10 MPa pressure. J. Chem. Eng. Data 52, 2237–2244. Holmes H. F., Busey R. H., Simonson J. M., Mesmer R. E., Archer D. G. and Wood R. H. (1987) The enthalpy of dilution of aqueous hydrogen chloride to 648 K and 40 MPa. Thermodynamic properties. J. Chem. Thermodyn. 19, 863–890. Hudson L. K. (1987) Alumina production. In Production of Aluminium and Alumina, vol. 20 (ed. A. R. Burkin), pp. 11– 46. Critical Reports on Applied Chemistry. Wiley, New York. Hummel W. (2005) Solubility equilibria and geochemical modeling in the field of radioactive waste disposal. Pure Appl. Chem. 77, 631–641. Magalha˜es M. C. F., Ko¨nigsberger E., May P. M. and Hefter G. (2002) Heat capacities of concentrated aqueous solutions of sodium sulfate, sodium carbonate, and sodium hydroxide at 25 °C. J. Chem. Eng. Data 47, 590–598. Mahoney J., Slaughter M., Langmuir D. and Rowson J. (2007) Control of As and Ni releases from a uranium mill tailings neutralization circuit: solution chemistry, mineralogy and geochemical modeling of laboratory study results. Appl. Geochem. 22, 2758–2776. Mal’tsev G. Z. and Mashovets V. P. (1965) Specific heat of sodium aluminate solutions in the 25–90° temperature range. J. Appl. Chem. (USSR) 38, 85–90. Marion G. M., Kargel J. S. and Catling D. C. (2008) Modeling ferrous–ferric iron chemistry with application to Martian surface geochemistry. Geochim. Cosmochim. Acta 72, 242–266. Millero F. J. (1979) Effects of pressure and temperature on activity coefficients. In Activity Coefficients in Electrolyte Solutions, vol. 2 (ed. R. M. Pytkowicz). CRC Press, Boca Raton, FL, USA, pp. 63–151. Olofsson G. and Hepler L. G. (1975) Thermodynamics of ionization of water over wide ranges of temperature and pressure. J. Solution Chem. 4, 127–143. Pabalan R. T. and Pitzer K. S. (1987) Thermodynamics of NaOH(aq) in hydrothermal solutions. Geochim. Cosmochim. Acta 51, 829–837. Palandri J. L. and Reed M. H. (2004) Geochemical models of metasomatism in ultramafic systems: serpentinization, rodingitization, and sea floor carbonate chimney precipitation. Geochim. Cosmochim. Acta 68, 1115–1133. Palmer D. A. and Drummond S. E. (1988) The molal dissociation quotients of water in sodium trifluoromethanesulfonate solutions to high temperatures. J. Solution Chem. 17, 153–164. Paramo R., Zouine M. and Casanova C. (2002) New batch cells adapted to measure saturated heat capacities of liquids. J. Chem. Eng. Data 47, 441–448.

3138

S. Schro¨dle et al. / Geochimica et Cosmochimica Acta 72 (2008) 3124–3138

Patterson B. A., Call T. G., Jardine J. J., Origilia-Luster M. L. and Woolley E. M. (2001) Thermodynamics for ionization of water at temperatures from 278.15 K to 393.15 K and at the pressure 0.35 MPa: apparent molar volumes of aqueous KCl, KOH, and NaOH and apparent molar heat capacities of aqueous HCl, KCl, KOH, and NaOH. J. Chem. Thermodyn. 33, 1237–1262. Pitzer K. S. (1991) Ion interaction approach: theory and data correlation. In Activity Coefficients in Electrolyte Solutions (ed. K. S. Pitzer), second ed. CRC Press, Boca Raton, FL, USA, pp. 75–153. Rogers P. S. Z. and Duffy C. J. (1989) Comparison of calibration methods for flow heat-capacity calorimeters and heat capacities of concentrated NaCl(aq) to 598 K. J. Chem. Thermodyn. 21, 595–614. Roux A. H., Perron G. and Desnoyers J. E. (1984) Capacite´s calorifiques, volumes, expansibilite´s et compressibilite´s des solutions aqueuses concentre´es de LiOH, NaOH et KOH. Can. J. Chem. 62, 878–885. Sharygin A. V. and Wood R. H. (1997) Volumes and heat capacities of aqueous solutions of hydrochloric acid at temperatures from 298.15 K to 623 K and pressures up to 28 MPa. J. Chem. Thermodyn. 29, 125–148. Simonson J. M., Mesmer R. E. and Rogers P. S. Z. (1989) The enthalpy of dilution and apparent molar heat capacity of NaOH(aq) to 523 K and 40 MPa. J. Chem. Thermodyn. 21, 561–584. Sipos P., May P. M. and Hefter G. T. (2000) Carbonate removal from concentrated hydroxide solutions. Analyst 125, 955–958. Span R., Lemmon E. W., Jacobsen R. T., Wagner W. and Yokozeki A. (2000) A reference equation of state for the thermodynamic properties of nitrogen for temperatures from

63.151 to 1000 K and pressures to 2200 MPa. J. Phys. Chem. Ref. Data 29, 1361–1433. Specialmetals. (2004) Inconel alloy C-276, Report SMC-019, Special Metals Corporation, Huntington, WV. Specialmetals. (2006) Nickel 200&201, Report SMC-061, Special Metals Corporation, Huntington, WV. Sweeton F. H., Mesmer R. E. and Baes C. F. (1974) Acidity measurements at elevated temperatures. VII. Dissociation of water. J. Solution Chem. 3, 191–214. Tremaine P. R. and LeBlanc J. C. (1980) The solubility of magnetite and the hydrolysis and oxidation of iron(+2) ion in water to 300 °C. J. Solution Chem. 9, 415–442. Wagner W. and Pruss A. (2002) The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. J. Phys. Chem. Ref. Data 31, 387–535. Wesolowski D. J. (1992) Aluminum speciation and equilibria in aqueous solution: 1. The solubility of gibbsite in the system NaK-Cl-OH-Al(OH)4 from 0 to 100 °C. Geochim. Cosmochim. Acta 56, 1065–1091. Wood S. A. and Vlassopoulos D. (1989) Experimental determination of the hydrothermal solubility and speciation of tungsten at 500 °C and 1 kbar. Geochim. Cosmochim. Acta 53, 303–312. Ziemniak S. E., Jones M. E. and Combs K. E. S. (1993) Solubility behavior of titanium(IV) oxide in alkaline media at elevated temperatures. J. Solution Chem. 22, 601–623. Ziemniak S. E. and Goyette M. A. (2004) Nickel(II) oxide solubility and phase stability in high temperature aqueous solutions. J. Solution Chem. 33, 1135–1159. Associate editor: David J. Wesolowski