A Systematic Investigation of the Thermodynamic Properties of

Article pubs.acs.org/jced

A Systematic Investigation of the Thermodynamic Properties of Aqueous Barium Sulfate up to High Temperatures and High Pressures Essmaiil Djamali,* Walter G. Chapman, and Kenneth R. Cox Department of Chemical and Biomolecular Engineering, Rice University, 6100 S. Main Street, Houston, Texas 77005, United States S Supporting Information *

ABSTRACT: Standard state thermodynamic properties for completely dissociated aqueous barium sulfate, BaSO4(aq), were fixed by an ionic additivity relationship using the data from other completely dissociated electrolytes, barium chloride, sodium sulfate, and sodium chloride, from 298.15 to 598.15 K and at steam saturated pressure, psat. The solubility of barium sulfate in pure water was then predicted a priori from 273.15 to 598.15 K and pressures up to 140 MPa, and the results were compared with the literature data. These predicted solubility values for barium sulfate in pure water, when corrected for ion 2− pair association of Ba2+(aq) ion with SO2− 4 (aq) ion and hydrolysis of SO4 (aq) ion, are in very good agreement with the literature data to well within the uncertainties of the experimental data at all temperatures and pressures considered. Using the experimental data from this study and auxiliary data from the literature, the logarithm of the molar dissociation constants of BaSO°4 associated ion pair were also calculated up to 598.15 K and up to 150 MPa.

1. INTRODUCTION Thermodynamic data for chemical reactions in high temperature and high pressure aqueous solutions are important in several scientific and industrial processes.1−3 There is considerable economic and scientific interest in the mineralogy and geochemistry of sulfate minerals in high temperature aqueous solutions.4 For instance, the knowledge of the high temperature and pressure data for thermodynamics properties of solution of the solid barium sulfate (barite), one of the most common scale minerals in oil and gas production, decreases the likelihood of risks to human safety, the environment, and the economics of production from deep-water development. In nature, barite is deposited under diverse geochemical conditions.5 Therefore, the high temperature and pressure thermodynamic properties for aqueous solutions of barium sulfate are needed to understand the causes of precipitation and/or dissolution of barite in diverse natural processes. The main purpose of the manuscript is to present the standard state thermodynamic properties for completely dissociated aqueous barium sulfate, BaSO4(aq). For the ionic species the standard state adopted for the thermal properties of solutes (enthalpy, heat capacity, and volume) is infinite dilution and for the free energies and entropies, the hypothetically ideal one molal (mole of solute/kg of solvent) solution exhibiting infinitely diluted properties.6 The standard-state thermodynamic properties of aqueous ions over wide ranges of temperature and pressure are required for modeling most of the physicochemical processes in aqueous solutions related to a number of important applications. In general, determination of standard state requires extrapolation of experimental data to infinite dilution. Since in aqueous solutions of barium sulfate at © 2016 American Chemical Society

high temperatures, even in dilute solutions, the associated ion pairs, BaSO°4 (aq), are the dominant barium species in solution, the standard state thermodynamic properties for BaSO4(aq) cannot be obtained directly by the usual extrapolation to infinite dilution.7 However, more reliable standard state partial molar thermodynamic functions can be derived for completely dissociated BaSO4(aq) by ionic additivity relationships from other completely dissociated electrolytes. These properties were generated from 298 to 598 K by combining the data on BaCl2(aq),8 Na2SO4(aq),7 and NaCl(aq)9 according to ionic additivity principle: X °(BaSO4 , aq) = X °(BaCl 2, aq) + X °(Na 2SO4 , aq) − 2X °(NaCl, aq)

(1)

where X° represents the standard state partial molar thermodynamic properties of interest, H°2 , G°2 , C°p,2 and V°2 . The accurate values for the standard state partial molar Gibbs free energy, G2°(T, p), of aqueous barium sulfate from this study when combined with the available values for the molar Gibbs free energy, G°(T, p), of solid barium sulfate (barite) from the literature10 will provide directly the values for the very important solubility product, KSP: Δsol G°(T , p) = G2◦(T , p) − G°(T , p) = −RT ln KSP (2) Received: June 18, 2016 Accepted: August 29, 2016 Published: September 7, 2016 3585

DOI: 10.1021/acs.jced.6b00506 J. Chem. Eng. Data 2016, 61, 3585−3594

Journal of Chemical & Engineering Data

Article

Table 1. Standard State Thermodynamic Properties for Completely Ionized BaSO4(aq) at psata T K 298.15 323.15 348.15 373.15 398.15 423.15 448.15 473.15 498.15 523.15 548.15 573.15 598.15 a

H2°(T) − H2°(Tr) −1

G2°(T) − G2°(Tr) −1

kJ mol

kJ mol

0.000 −7.166 −13.435 −19.934 −28.032 −39.364 −55.852 −79.576 −114.054 −152.530 −224.577 −339.268 −568.128

0.000 −0.354 −0.192 0.418 1.503 3.175 5.642 9.191 14.205 21.164 30.676 43.555 61.129

S2°(T) −1

J mol

K

Cp,2 ° (T) −1

−1

J mol

V2°(T) −1

K

−319.4 −262.3 −247.0 −282.1 −377.7 −545.4 −799.8 −1165.5 −1690.6 −2482.4 −3906.7 −7248.1 −18219.7

29.7b 6.0 −12.9 −30.9 −51.9 −79.4 −117.1 −168.5 −239.4 −315.3 −450.3 −652.9 −1027.7

(∂V2°/∂T)pc −1

3

3

cm mol

cm mol

−1

K

(∂V2°/∂p)Tc

−1

cm mol−1 MPa−1 3

−0.0681 −0.1518 −0.2142 −0.2988 −0.4233 −0.6148 −0.9211 −1.4378 −2.3704 −4.2270 −8.4610 −20.3380 −57.1000

6.6 3.2 −1.8 −8.6 −17.7 −31.1 −50.1 −79.3 −125.1 −202.1 −344.0 −641.5 −1425.0

0.1106 0.1209 0.1373 0.1627 0.2012 0.2583 0.3620 0.5581 0.9685 1.9394 4.6522 14.4640 70.7500

Tr = 298.15 K. bNBS tables.11 cDjamali and Cobble.16

where R is the gas-constant. The KSP values from this study, calculated from high dilution (10−4 mol/kg) calorimetric measurements, are compared below with the corresponding values reported in the literature. At high temperatures, most of the reported equilibrium constants for solubility of minerals, KSP, are derived from solubility measurements, especially for sparingly soluble salts. These calculated equilibrium constants from experimental solubility sensitively depend on the model chosen for speciation and the activity coefficients of the ionic species involved. But the experimental activity coefficients of most species in aqueous solutions of simple and multicomponent electrolytes at very high temperatures and pressures are themselves not well characterized. Furthermore, in order to quantitatively describe the distribution of equilibrium concentration of species in aqueous barium sulfate solution, the equilibrium constant for each assumed species will be required, however, these equilibrium constants at high temperatures and pressures are usually not available. We first present the standard state thermodynamic properties for completely dissociated aqueous barium sulfate, BaSO4(aq), up to 598.15 K and at steam-saturation pressure, psat. Next, a model for an a priori prediction of the solubility of Barite as a function of temperature and pressure is presented and the results are compared with the available literature at temperatures up to 573.15 K and pressures up to 150 MPa. In the last section, the effect of association between Ba2+(aq) ion and 2− SO2− 4 (aq) ion and hydrolysis of SO4 (aq) ion on the solubility of barite at higher temperatures is investigated.

Table 2. Standard State Enthalpies of Solution of Barium Sulfate at psata T K 298.15 323.15 348.15 373.15 398.15 423.15 448.15 473.15 498.15 523.15 548.15 573.15 598.15 a

−1

kJ mol

H°(T, cr) − H°(Tr, cr)b kJ mol

0.000 −7.166 −13.435 −19.934 −28.032 −39.364 −55.852 −79.576 −114.054 −152.530 −224.577 −339.268 −568.128

−1

0.000 0.764 1.490 2.218 2.980 3.804 4.705 5.691 6.763 7.910 9.115 10.351 11.582

ΔsolH°(T) kJ mol−1 26.290c 18.360 11.365 4.138 −4.722 −16.878 −34.267 −58.977 −94.527 −134.150 −207.402 −323.329 −553.420

Tr = 298.15 K. bBarin et al.10 cNBS tables.11

Table 3. Standard State Entropies of Solution of Barium Sulfate at psata T K 298.15 323.15 348.15 373.15 398.15 423.15 448.15 473.15 498.15 523.15 548.15 573.15 598.15

2. CALCULATIONS AND RESULTS 2.1. Standard State Thermodynamic Properties of Aqueous Barium Sulfate. The thermodynamic properties for completely ionized BaSO4(aq) at steam-saturated pressure, psat, calculated from ionic additivity (eq 1) are summarized in Table 1. The calculated enthalpy, ΔsolH°, entropy, ΔsolS°, and Gibbs free energies, ΔsolG°, of a solution of barium sulfate from 298.15 to 598.15 K and at psat are also given in Tables 2−4, respectively. The thermodynamic properties of solution of barium sulfate refers to the following reaction BaSO4 (cr) = BaSO4 (aq)

H2°(T) − H2°(Tr)

a

(3)

S°2 (T, aq)

S°(T, cr)c

ΔsolS°(T)

−1

−1

(J mol−1 K−1)

(J mol

−1

K )

29.7 6.0 −12.9 −30.9 −51.9 −79.4 −117.1 −168.5 −239.4 −315.3 −450.3 −652.9 −1027.7

b

(J mol

−1

K )

62.4 64.7 66.9 69.0 71.0 73.0 75.1 77.2 79.4 81.7 83.9 86.1 88.2

−32.7 −58.7 −79.8 −99.9 −122.9 −152.4 −192.2 −245.7 −318.8 −397.0 −534.2 −739.0 −1115.9

Tr = 298.15 K. bNBS tables.11 cBarin et al.10

namic properties for solution of barium sulfate up to 598.15 K and at psat from this study are compared in Figure 1 with the corresponding values reported in the literature. In Figure 2, the

The thermodynamic functions of the solid salt are from the standard reference tables.10,11 The above calculated thermody3586



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thermodynamic properties for BaSO4(aq) obtained from SUPCRT9214 are based on the revised model of Helgeson, Kirkham, and Flowers (HKF),15 whose parameters are estimated from low temperature data. The experimental values of ΔsolH°(T, psat) for barium sulfate, obtained from ionic additivity of the original high dilution calorimetric measurements (Table 2), are used with eq 4 to calculate the values for Gibbs free energy of solution, ΔsolG°, at 298.15 K and 0.1 MPa from each reported values of ΔsolG°(T, psat) at T > 298.15 K from the literature.

Table 4. Standard State Gibbs Free Energies of Solution of Barium Sulfate at psata T

a

G2°(T, aq) − G2°(Tr, aq) −1

G°(T, cr) − G°(Tr, cr)b

ΔsolG°(T)

(K)

(kJ mol )

(kJ mol−1)

(kJ mol−1)

298.15 323.15 348.15 373.15 398.15 423.15 448.15 473.15 498.15 523.15 548.15 573.15 598.15

0.000 −0.354 −0.192 0.418 1.503 3.175 5.642 9.191 14.205 21.164 30.676 43.555 61.129

0.000 −3.410 −7.031 −10.854 −14.865 −19.058 −23.422 −27.947 −32.620 −37.432 −42.370 −47.421 −52.572

56.900c 59.956 63.739 68.172 73.268 79.133 85.964 94.038 103.725 115.496 129.946 147.876 170.601

Δsol G°(298.15K , 0.1MPa) 298.15 = Δsol G°(T , psat ) T T Δsol H °(T ′, psat ) dT ′ + 298.15 298.15 T ′2 T Δsol V °(T ′, psat ) ⎛ ∂p ⎞ ⎜ ⎟ dT ′ − 298.15 ⎝ ∂T ′ ⎠sat 298.15 T′

∫

∫

Tr = 298.15 K. bBarin et al.10 cNBS tables.11

standard state partial molar heat capacities, Cp,2 ° , for completely dissociated barium sulfate, BaSO4(aq), at psat from this study are compared with the related values reported in the literature. At temperatures above 423.15 K, large differences are observed between the thermodynamic properties of BaSO4(aq) from this study and the corresponding values reported in the literature. The Raju and Atkinson12 calculated thermodynamic properties of BaSO4(aq) are based on Criss and Cobble13 corresponding principle whose parameters are estimated from low temperature data. Blount5 estimated the thermodynamic properties of solution of barium sulfate from solubility measurements. The

The value for

∂p ∂T ′ sat

( )

(4)

in eq 4 is calculated from Hill’s

equation of state for water.17 The required molar volume change for the process of solution of the sample is calculated from eq 5 Δsol V °(T , psat ) = V 2◦ − V °(cr)

(5)

where V2° is the standard state partial molar volume of the electrolyte (see Table 1) and V°(cr) is the molar volume of solid solute calculated from its density.18 The calculated values

Figure 1. Comparison of the standard state partial molar entropy (a), enthalpy (b), and Gibbs free energy (c) of solution of BaSO4(cr) with the corresponding values from the literature at psat: ..., Blount;5 ---, Raju and Atkinson;12 bold , SUPCRT92;14 , this study. 3587



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Figure 2. Comparison of the standard state partial molar heat capacity of BaSO4(aq) from this study at psat with the corresponding values reported in the literature: ..., Raju and Atkinson;12 ---, SUPCRT92;14 , this study. Up to 473 K (a) and up to 598 K (b).

Figure 3. Analysis of the standard state Gibbs free energies of aqueous solution of barium sulfate at 298.15 K and 0.1 MPa from data at higher temperatures: △, Malinin;19 □, Templeton;20 ○, Blount;5 ◇, Jiang;21 +, Melcher;22 ●, Rosseinsky;23 ···, SUPCRT92;14 , this study. Up to 448 K (a) and up to 598 K (b).

for ΔsolG°(298.15 K, 0.1 MPa) from each reported value of ΔsolG°(T, psat) at temperatures above 298.15 K from the literature are compared with the best available corresponding value at 298.15 K and 0.1 MPa from NBS tables11 in Figure 3. At all temperatures greater than 423 K, large differences are again observed between the values of ΔsolG°(298.15 K, 0.1 MPa) calculated in this study from the reported Gibbs free energies of solution (eq 4) in the literature and the best available data from NBS tables. The uncertainty associated with the calculation of ΔsolG°(298.15 K, 0.1 MPa) from eq 4 is mainly due to the uncertainty in estimation of the standard state partial molar volume. Fortunately, even a 10% uncertainty in V2° at the higher temperatures will result in less than 0.1% error in calculated ΔsolG°(298.15 K, 0.1 MPa) from this study at the highest temperature because the contribution of V°2 to the second and third terms in eq 4 has opposite sign and comparable magnitude. At T > 423 K, the reported thermodynamic properties of barium sulfate from the literature are considerably less negative than the corresponding values obtained from this study, undoubtedly because above that temperature barium sulfate is no longer a strong electrolyte even at low concentrations. We show below that at temperatures above 423 K, the experimental results on thermodynamic properties of BaSO4(aq) are not consistent with the assumption of complete dissociation, that is, BaSO4(aq) is no longer acting as a strong electrolyte.

Therefore, the larger discrepancies at T > 423 K may be due to the lack of ion association (or complete dissociation) in the model chosen in the literature to represent BaSO4(aq). 2.2. Solubility of Barium Sulfate. For a given temperature T and pressure p, at the equilibrium of a solid with its saturated solution, the solubility (msat, mol of solute/kg of solvent) is related to the standard state Gibbs free energy of the solution of the electrolyte, ΔsolG°(T, p), according to the following thermodynamic cycle

from which Δsol G°(T , p) = G2◦(T , p , aq) − G°(T , p , cr) = −νRT ln asat(T , p) = −νRT ln[msat (T , p)γsat(T , p ; msat )/m°] (6)

where asat(T, p) is the activity of the saturated solution, G°2 (T, p, aq) is the standard state partial molar Gibbs free energy (the chemical potential) of the solute electrolyte, G°(T, p, cr), is the molar Gibbs free energy of the solid, ν is the stoichiometric 3588



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number of ions in the solute, R is the ideal gas constant, γsat and msat are the stoichiometric mean molar activity coefficient and the solubility (mol of solute/kg of solvent), respectively, of the solid-saturated solution at temperature T and pressure p, m° is equal to 1 mol/kg. From rearrangement of eq 6 ⎛ ⎞ ⎡ Δ G°(T , p) ⎤ 1 ⎟⎟ exp⎢ − sol msat (T , p)/m° = ⎜⎜ ⎥ ⎦ ⎣ υRT ⎝ γsat(T , p) ⎠

to select one set of data over the other. However, a more detailed analysis indicates that the solubility data reported by Blount5 is in better agreement with the independently measured enthalpy of solution of barium sulfate as follows. From a thermodynamic analysis it can be shown that the change in the solubility with temperature at constant pressure is directly proportional to the enthalpy of solution and in turn may be estimated from the following relationship:

(7)

⎛ ∂lnmsat ⎞ Δsol H ⎜ ⎟ = 2 ⎝ ∂T ⎠ p νRT (1 + m(∂ lnγ /∂m)T , p )

For prediction of solubility from eq 7, the value for the corresponding stoichiometric activity coefficient of the solid saturated solution, γsat, is required. Unfortunately, the values of γsat for aqueous solution BaSO4(aq) at low concentrations and at high temperatures are not available. However, these values for the activity coefficients of BaSO4(aq) are estimated from the Lindsay equation:24−26

For sparingly soluble minerals, when m ≪1, the above identity can be estimated as ⎛ ∂ ln msat ⎞ Δ H° ⎜ ⎟ ≅ sol 2 ⎝ ∂T ⎠ p νRT

2

γ|z|(T , p , I ) = γ±z,NaCl(T , p , I )

(9)

(8)

(10)

The availability of the values for standard state enthalpy of solution, ΔsolH°, (Table 2) allows a reasonable estimation of the slope of the solubility with temperature from eq 10. The values for the slope of the solubility with temperature calculated from eq 10 and plotted in right-hand side y-axis in Figure 5 are

where z is the ion charge, I = 1/2∑imiz2i is the ionic strength, and γ±, NaCl is the mean molar stoichiometric activity coefficient of NaCl(aq). These latter values were calculated from Archer.27 Considering the low solubility of Barite (msat ≤ 10−5 mol kg−1), the estimation of the values of γsat for aqueous solution BaSO4(aq) from eq 8 is reasonable. The predicted values for the solubility of barium sulfate from this study at psat, calculated from eqs 7 and 8 and the values for ΔsolG°(T, psat) from Table 4, are compared with the corresponding experimental values from the literature in Figure 4. The comparison indicates an excellent agreement between

Figure 5. Comparison of the predicted solubility values for barium sulfate from this study with the literature at psat: □, Templeton;20 ○, Blount (near 10 MPa);5 △, Strubel;28 , this study (not corrected for ion pair association). Important to note that the slope of the solubility with temperature, d ln msat/dT, calculated from calorimetric enthalpy of solution measurements (....) changes sign near 373 K, in agreement with Blount data. Figure 4. Comparison of the predicted solubility of barium sulfate in water from this study with the corresponding experimental values from the literature at psat: △, Malinin;19 ○, Templeton;20 □, Strubel;28 ×, Jiang;21 +, Melcher;22 ●, Rosseinsky;23 , this study.

in a significantly better agreement with the slope of the solubility with temperature extracted from the experimental data reported by Blount.5 From Figure 5, the slope of the solubility data of Blount with temperature changes sign near 373 K, in agreement with the related value calculated from eq 10 in this study. While the corresponding values for the slope calculated from Strubel28 solubility data indicate the sign change at much higher temperature (near 423 K), this is in disagreement with the direct calorimetric results for enthalpy of solution measurements. In Figure 5, the predicted values for the solubility of barium sulfate in pure water from this study are also compared with the available related data in the literature up to 573.15 K. These predicted values from this study are not corrected for contribution due to ion pair formation between Ba2+(aq) ion and SO2− 4 (aq) ion and for other side reactions (e.g., hydrolysis of SO2− 4 (aq)). More on this will follow (see section 2.4).

the predicted solubility values for barium sulfate from this study and those of the related values from the literature up to 373.15 K to well within the combined uncertainties of the experimental solubility data and uncertainties in the predicted solubility due to estimation of the activity coefficient in this study from eq 8. At temperatures above 373.15 K, there are two sets of data near steam-saturated pressure. Blount5 reported experimental solubility data for barium sulfate in pure water near 10 MPa at temperatures up 523.15 K. Strubel28 also reported experimental solubility data for barium sulfate in pure water at psat up to 623.15 K. Ignoring the small corrections for the pressure differences from psat to 10 MPa, these two sets of data disagree to more than 100% from each other. There is no a priori reason 3589



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Figure 6. Comparison of the predicted solubility values for barium sulfate from this study with the literature at 10 MPa (a), 50 MPa (b), 100 MPa (c), and 140 MPa (d): ○, Blount;5 ---, Blount (smooth fit); □, Strubel;28 , this study (no correction for ion pairing).

2.3. Properties of Barium Sulfate at High Pressures. The standard state thermodynamic properties for barium sulfate above psat were calculated from the unified theory of electrolytes, UTE,16,29 up to 140 MPa. The UTE16 requires two temperature and pressure independent constants, CH and CS, for each electrolyte, and once these two constants are fixed from low temperature experimental data, the model can then be used to predict the standard state thermodynamic properties to extreme temperatures and pressures. From the unified theory, the standard state Gibbs free energies of solution at temperature T and pressure p relative to reference temperature Tr, 298.15 K, is given by

and F1(D) is a function of dielectric constant, D(T, p), of the solvent30

F1(D) =

= Δsubl [G°(T , p) − F1(D)G°(Tr , p)]

(11)

⎡ ⎣⎢1 −

⎤ 1 ⎥ D(Tr , p) ⎦

(13)

Δsubl G°(T , p) = Δf G°(M+X−(g), T , p)

and the adjusted Gibbs free energy of solution at the reference temperature 298.15 K, ΔsolG⊕(Tr, p), is Δsol G⊕(Tr , p) = Δsol G°(Tr , p) − C H + CSTr ⎛ RT d m° ⎞ + υRTr ln⎜ r r ⎟ ⎝ 1000p° ⎠

1 ⎤ ⎥ D(T , p) ⎦

The above constants CH and CS represent the enthalpy and entropy loss of the solvent molecules in the primary solvation shell (inside the effective electrostatic radius); they are related to short-range ion solvent interactions and further compensate for the errors in the electrostatic representation of the nearest solvent molecules. The CH and CS constants are solute specific and have been shown to be independent of T and p.16,29,7,9,31−34 The values for the constants, CS and CH, for barium sulfates are also obtained from ionic additivity (eq 1).16,29 The required value for the Gibbs free energy of sublimation ,ΔsublG°(T,p), is calculated from the Gibbs free energy of formation from the available literature:10,35

Δsol [G°(T , p) − G°(Tr , p)] + Δsol G⊕(Tr , p)[F1(D) − 1] − CSΔT ⎛ RTdm° ⎞ ⎛ RT d m° ⎞ + υRT ln⎜ ⎟ − υRTr ln⎜ r r ⎟ ⎝ 1000p° ⎠ ⎝ 1000p° ⎠

⎡ ⎣⎢1 −

− Δf G°(MX(cr), T , p)

(14)

It is important to note that in eq 11, the thermodynamic properties of barium sulfate are required at only 298.15 K. Similarly as above, the predicted values for the solubility of barium sulfate (eqs 7 and 8) up to 140 MPa are compared with the related data from the literature in Figure 6. The differences above 373 K between the predicted solubility for barium sulfate from this study and the corresponding experimental values

(12)

where d is the density of the pure solvent in g/cm3,17 dr is the density of the solvent at the reference temperature 298.15 K, 3590



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Figure 7. Equilibrium constant for dissociation of barium sulfate ion pair at psat: ○, calculated from Jiang data;21 □, Felmy et al.;38..., Monnin39 (also calculated from Jiang data); , this study. Up to 598 K (a) and up to 373 K (b).

from Blount5 are mainly due to the lack of ion association in the results from this study (see section 2.4). It is also important to note that the relative differences between the two sets of the solubility data mentioned above decrease as the pressure increases since ion association is expected to decrease with increase in pressure. 2.4. Effect of Ion Association. Here we examine the effect of ion pair association, which becomes more significant at higher temperatures on the solubility data. To correct the observed solubility for contributions from side reactions, the following reactions were considered in the solution of barite, BaSO4(cr), in water. 2+

SO24 −(aq)

estimated from eq 8 and the solution of the nonlinear chemical system was repeated until the desirable convergence was achieved. Experimental values for the equilibrium constant for dissociation of barium sulfate ion pairs (reaction 19) above 298.15 K are not available. These equilibrium constants, Kd, BaSO4° were estimated in this study from eqs 2 and 20 as follows. The change in Gibbs free energy for dissociation of barium sulfate (reaction 19) relative to reference temperature 298.15 K is calculated from eq 20: Δd [G°(T , p) − G°(Tr , p)]

(15)

= [G2◦(T , p) − G2◦(Tr , p)]Ba 2+SO24−(aq)

H 2O = H+(aq) + OH−(aq)

(16)

− [G°(T , p) − G°(Tr , p)]BaSO◦4 (aq)

SO24 −(aq) + H 2O(l) = HSO−4 (aq) + OH−(aq)

(17)

Ba 2 +(aq) + H 2O(l) = BaOH+(aq) + H+(aq)

(18)

BaSO◦4 (aq) = Ba 2 +(aq) + SO24 −(aq)

(19)

BaSO4 (cr) = Ba (aq) +

(20)

where the values for [G°2 (T, p) − G°2 (Tr, p)]Ba2+SO2− are from 4 (aq) Table 1 and the corresponding values for the ion pair, BaSO°4 (aq), are calculated from eq 21: [G°(T , p) − G°(Tr , p)]BaSO◦4 (aq) ◦ ◦ = −SBaSO (Tr , p)ΔT + 4 (aq)

The equilibrium constant for solution of barium sulfate (reaction 15) is from this study, and the corresponding value for dissociation of water (reaction 16) is from Bandura and Lvov.36 The equilibrium constants for hydrolysis of SO2− 4 (aq) ion (reaction 17) were obtained from addition of the equilibrium constant for dissociation of water and the corresponding values for the second dissociation constants of sulfuric acid, Kd, HSO−4 . The values for Kd, HSO−4 at different temperatures and pressures are calculated previously from UTE.16,29 There are no experimental values for the first step hydrolysis of Ba2+(aq) ion (reaction 18) above 298.15 K. But, these values can be estimated from either UTE16 and/or the recently available electrical conductivities of solutions of aqueous strontium hydroxide37 at high temperatures. However, in the present study since the concentrations of Ba2+(aq) at higher temperatures are low, contribution from reaction 18 is negligible. Reactions 15−19 together with an equation imposing electroneutrality and mass balance defines a system of nonlinear equations that can be solved to obtain the distribution of all the species, provided that the equilibrium constant for the dissociation of barium sulfate (reaction 19), Kd, BaSO4°, is known (see below). The required values for activity coefficients of the electrolytes at the final ionic strength were

∫T

T

◦ ◦ Cp,BaSO dT ′ 4 (aq)

r

−T

∫T

T

r

◦ ◦ Cp,BaSO d ln T ′ 4 (aq)

(21)

The value for the entropy of the associated barium sulfate ion pair at 298.15 K, S°BaSO4(aq), equal to 136.7 J/(mol K), is estimated from the experimental data of Jiang21 (see Supporting Information), and the corresponding required values for Cp,BaSO ° were from SUPCRT9214 (Table S1 in 4(aq) Supporting Information). The equilibrium constants for dissociation of barium sulfate ion pairs, log Kd, BaSO4°, at psat calculated from eqs 2, 20, and 21 are plotted in Figure 7 and compared with the corresponding values estimated from the experimental solubility of barium sulfate in aqueous solutions of sodium sulfate reported by Jiang21 up to 353.15 K (see Supporting Information) and Felmy et al.38 at 298.15 K. In Figure 7, the calculated values for log Kd,BaSO4° reported by Monnin39 up to 353.15 K are also compared with the related values from this study. Monnin39 reported values for log Kd,BaSO4° were similarly calculated from Jiang21 data but with a different model. The values for 3591



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log Kd,BaSO4° from this study as a function of temperature at psat were empirically fitted by a least-squares method to eq 22, and the resulting parameters are summarized in Table 5. log Kd,BaSO◦4 = a1 + a 2T + a3T 2 + a4 /(a5 − T )2

(22)

Table 5. Empirical Parameters for Calculating the Values for log Kd,BaSO°4 at psat from eq 22 a1 a2 a3 a4 a5

−4.94067 1.84764 −4.42071 2.80367 9.80054

× × × × ×

1000 10−02 10−05 1006 1002

Figure 9. Distribution of species in aqueous solution of barium sulfate at different temperatures and at psat.

The equilibrium constants for dissociation of barium sulfate ion pairs, log Kd,BaSO4°, at psat calculated above are extended to higher pressures using UTE16 and summarized in Figure 8a. The equilibrium constant for dissociation of bisulfate ion, log Kd,HSO−4 , as a function of temperature and pressure are also calculated from UTE16,29 and plotted in Figure 8b. Comparison of the predicted values of log Kd,HSO−4 from UTE16,29 with the corresponding experimental values reported by Hnedkovsky et al.40 indicates excellent agreement (Figure 8b). The equilibrium concentrations of the principal species in aqueous solutions of barium sulfate from this study were then calculated and are summarized in Figure 9. The importance of temperature on dissociation noted above can be seen in the distribution of species in aqueous solution of barium sulfate up to 573.15 K and at steam saturated pressure (Figure 9). The main feature of the distribution plot is that the concentration of the associated species BaSO4°(aq) increases dramatically with increase in temperature as expected, and it accounts for the major species at higher temperatures (T > 473 K). Finally we compare the predicted values for solubility of barium sulfate in water from this study with those in the literature in Figure 10. This figure again reiterates the importance of including ion pair association in modeling solubility data.

from ionic additivities from other strong electrolytes up to 598.15 K. The unified theory of electrolytes16,29 is used to extend these standard state thermodynamic properties for BaSO4(aq) to higher pressures up to 200 MPa. Above 423.15 K, a comparison of the thermodynamic properties of barium sulfate from this study with the corresponding values from the literature indicates large disagreement. At high temperatures, most of the reported standard state thermodynamic properties for BaSO4(aq) are either derived from solubility measurements with no correction for side reactions (e.g., ion pair formation BaSO°4 (aq) and hydrolysis of SO2− 4 (aq) ion) and/or extrapolated from lower temperature data. The solubility of barium sulfate in water was predicted from the present study and compared with the available literature data up to a temperature of 573.15 K and a pressure up to 140 MPa. Furthermore, equilibrium constants for the associated species BaSO°4 (aq) and second dissociation of sulfuric acid were also calculated for the same temperature and pressure range. The effect of temperature and pressure on the distribution of species in an aqueous solution of barium sulfate was investigated up to 573.15 K. The result shows that the concentration of the associated species BaSO4°(aq) increases considerably with increase in temperature, and it accounts for the major species at high temperature. This study also demonstrates the necessity of incorporating the associated species in modeling aqueous solutions at high temperatures and that the speciation and concentrations in solution must be known before mineral precipitation can be understood,

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CONCLUSION Standard state thermodynamic properties for completely dissociated aqueous barium sulfate at psat were calculated

Figure 8. Equilibrium constant for dissociation of barium sulfate ion pair (a) and bisulfate ion (b) as a function of pressure at different temperatures: ○, □, Hnedkovsky et al.;40 (---, 473.15 K; ..., 523.15 K; , 573.15 K), this study. 3592



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Figure 10. Comparison of the predicted solubility values for barium sulfate from this study with the literature at 10 MPa (a), 50 MPa (b), 100 MPa (c), and 140 MPa (d): ○, Blount;5 □, Strubel;28 ----, Blount (smooth fit);5 ..., this study (no correction for ion pairing); , this study (corrected for ion pair association).

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particularly under high temperature and high pressure conditions. The standard state thermodynamic properties for aqueous barium sulfate from this study are recommended, since they are based on experimental results down to very dilute solutions (10−4 mol/kg), are over a much wider temperature range, and do not actually involve direct measurements on BaSO4(aq).

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(1) Brock, T. D. Life at high temperatures. Science 1985, 230, 132− 138. (2) Chen, X.; Izatt, R. M.; Oscarson, J. L. Thermodynamic data for ligand interaction with protons and metal ions in aqueous solutions at high temperatures. Chem. Rev. (Washington, DC, U. S.) 1994, 94, 467− 517. (3) Zemaitis, J.; Newman, S.; Barner, H.; Klein, M. Thermodynamics of Aqueous Systems with Industrial Applications. ACS Symp. Ser. 1980, 133, 227−246. (4) Rye, R. O. A review of the stable-isotope geochemistry of sulfate minerals in selected igneous environments and related hydrothermal systems. Chem. Geol. 2005, 215, 5−36. (5) Blount, C. Barite solubilities and thermodynamic quantities up to 300 degrees C and 1400 bar. Am. Mineral. 1977, 62, 942−957. (6) Majer, V.; Sedlbauer, J.; Wood, R. H. Calculation of standard thermodynamic properties of aqueous electrolytes and nonelectrolytes. Aqueous Systems at Elevated Temperatures and Pressures: Physical Chemistry in Water, Steam and Hydrothermal Solutions. Palmer, D. A., Fernandez-Prini, R., Harvey, A. H., Eds.; Elsevier: 2004; pp 99−147. (7) Djamali, E.; Chen, K.; Cobble, J. W. , Standard State Thermodynamic Properties of Completely Ionized Aqueous Sodium Sulfate Using High Dilution Calorimetry up to 598.15 K. J. Phys. Chem. B 2009, 113, 11688−11695. (8) Djamali, E.; Chen, K.; Murray, R. C.; Turner, P. J.; Cobble, J. W. Standard State Thermodynamic Properties of Ba2+(aq), Co2+(aq), and Cu2+(aq) up to 598.15 K, and Temperature Effect on Ligand Field. J. Phys. Chem. B 2009, 113, 2404−2408. (9) Djamali, E.; Cobble, J. W. Standard State Thermodynamic Properties of Aqueous Sodium Chloride Using High Dilution Calorimetry at Extreme Temperatures and Pressures. J. Phys. Chem. B 2009, 113, 5200−5207.

ASSOCIATED CONTENT

S Supporting Information *

. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.6b00506. Standard partial molar heat capacity of BaSO°4 (aq) at psat; values of log Kd for the dissociation of barium sulfate ion pair from Jiang data up to 353.15 K; calculated value for the standard state entropy of barium sulfate ion pair at 298.15 K and 0.1 MPa (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Funding

The authors acknowledge RPSEA/DOE 10121-4204-01 for their financial support. Notes

The authors declare no competing financial interest. 3593



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A Systematic Investigation of the Thermodynamic Properties of

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