Henry s Law Constants of Methane and Acid Gases at Pressures above the Saturation Line of Water

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1 Henry s Law Constants of Methane and Acid Gases at Pressures above the Saturation Line of Water Josef Sedlbauer and Vladimir Majer 2* Department of Chemistry, Technical University of Liberec, 46 7 Liberec, Czech Republic 2 Laboratoire de Thermodynamique des Solutions et des Polymères, Université Blaise Pascal / CNRS, 6377 Aubière, France * vladimir.majer@univ-bpclermont.fr A correlation model proposed originally for the description of hydration properties of nonelectrolytes was used for the prediction of the Henry s law constants of main constituents of natural gas (CH 4, CO 2, H 2 S) up to the critical point of water and at pressures up to 0. Besides the solubility data, the standard derivative properties resulting from volumetric and calorimetric experiments were also included in the data fit. The new correlations for the three gases are consistent with the representation of the Henry s law constant along the saturation line of water accepted recently as an IAPWS guideline. Some differences were observed at high temperatures for H 2 S(aq) where no reliable solubility data are available. Possibility to describe the Henry s law constant as a function of pressure allows reliable determination of the Poynting correction, which is necessary for evaluating the solubility of gases at pressures above the saturation line of water. It is shown that simplifying approximations, currently used for this correction in the literature, can lead to substantial errors in calculating the Henry s law constant at high pressures.. Introduction Solubility and vapour-liquid distribution of CH 4, CO 2 and H 2 S in aqueous systems at temperatures at least to 523 K and pressures up to 0 are needed for modelling the geological sequestration of acid gases in deep aquifers after their separation from natural gas. These data can be conveniently represented by the Henry s law constant k H ; this property depending on both temperature and pressure is defined as ( f ) kh ( T, p) = lim 2 x2 () x2 0 where f 2 is the liquid-phase fugacity of the solute and x 2 its molar fraction. Since the Henry s law constant of gases is usually determined from the solubility data it is typically presented along the saturation line of the solvent (p = p ). Such representative correlation has been recently published for 4 gases in water (including CH 4, CO 2 and H 2 S) by Fernandez-Prini et al. [] based on careful evaluation of most available gassolubility data. This formulation is now adopted as the IAPWS Guideline for calculating the Henry s law constants of these gases up to the critical point of water. It is often necessary to convert k H (T, p ) values to pressures far above the saturation line of water where they are needed in applications. An exact relation for such conversion requires the integration of the partial molar volume of solute at infinite dilution V 2 k H p ( p) = kh ( p )exp V dp RT 2 (2) p where the exponential term is known as the Poynting correction. However, the values of V 2 as a function of temperature and pressure are not often available in an analytical form allowing easy calculations. In their data processing, Fernandez- Prini et al. applied estimates of V 2 from a hardsphere perturbation theory using the Percus-Yevick approximation. Since this approach is not easily implemented and still approximate, it would be useful to establish a more simple methodology, allowing to perform the pressure integration with an analytical formula, which is (if possible) based on experimental evidence. Several correlations for partial molar properties # of aqueous solutes have # The partial molar properties throughout this paper are at the standard state of infinite dilution

2 been proposed over the last years. In this study we have used the SOCW model [2], which proved to be successful in earlier studies. This relationship expresses V 2 as a function of temperature, water density and compressibility and complies with the correct ideal gas limit at low densities as well as with the divergence at the critical point of the solvent. Parameters of the model were determined in this study for acid gases (CO 2, H 2 S) and also for CH 4 as a major part of natural gas. The data used in correlations included besides the k H values also the derivative properties (the partial molar volumes and heat capacities, enthalpies of hydration), derived from experimental results. The Henry s law constants generated from the SOCW model are compared in the first step to the calculations of k H (T,p * ) from the equation of Fernandez-Prini et al. (denoted henceforth as Guideline). In the second step the Poynting corrections are listed at selected temperatures and pressures to demonstrate the impact of this term on k H and to visualize the errors, resulting from commonly applied approximations for the partial molar volume of a solute. 2. Procedure 2.. Theory The Henry s law constant is simply linked with the Gibbs energy of hydration, hyd G 2 corresponding to the transfer of a solute from an ideal gas state at the reference pressure (p ref = 0. ) to the infinitely dilute solution at the same temperature and pressure p kh ( T, p) ig RT ln = G2 ( T, p) G2 ( T, pref ) = hydg p 2 ref (3) where G 2 and G 2 ig are the solute standard chemical potential in the solution and molar Gibbs energy of an ideal gas, respectively. This equation presents a straightforward connection of the Henry s law constant to the rigorous system of thermodynamic state functions allowing also to relate k H to the standard derivative properties. The temperature derivatives of equation (3) lead to 2 ln kh RT = T hydh 2 p 2 ln kh RT = hydc T T p,2 p (4) (5) where hyd H 2 and hyd C p,2 are the enthalpy and heat capacity of hydration of a solute defined analogously to hyd G 2. The pressure derivative of equation (2) yields the direct relationship to the partial molar volume ln k H RT p T = V2 (6) This equation is just another expression for the Poynting correction consistent with equation (2). It follows from the above relations that the temperature and/or pressure evolution of k H can be calculated by integration of the standard derivative properties that are obtained from calorimetric and volumetric data extrapolated to infinite dilution. To make most of this additional experimental information, it is necessary to use a model that is flexible enough to describe simultaneously k H and its derivatives as a function of both temperature and pressure. An equation-of-state of this type is the SOCW model described recently in literature [2]. It was formulated in terms of the partial molar volume V 2 which can be analytically integrated with pressure to yield hyd G 2 and hence also its temperature derivatives: V 2 = RTκ + d( V RTκ) + RTκρ Q ( a + b ϑρ ] ) + c exp[ θ / T ] + δ (exp[ ] ) ) Q = (exp[ λρ (7) where V, ρ and M are molar volume, specific density and molar mass of water, respectively, and general coefficients valid for all solutes are υ = m 3 /kg, θ = 0 K, λ = -0.0 m 3 /kg. Adjustable parameters are a, b, c and d, and δ = 0.35a for neutral molecules. Equation (7) is in fact modeling a series of perturbation effects due to insertion of a point mass (ideal gas particle) into water (RTκ ), growing it to a water-like molecule with size adjusted to mimic the intrinsic volume of a solute (d (V -RTκ )), and then changing its potential field from solvent-solvent to solutesolvent interaction (the semi-empirical term denoted as Q). Although the SOCW model is theoretically-based, the resulting equation requires experimental data for a given solute to adjust the four parameters a, b, c and d. All available data can be used for this purpose through equations (3) to (6). The logarithm of the Poynting correction is

3 then obtained by appropriate integration between the saturation pressure of the solvent and a pressure of interest: p ρ V2 dp = ln RT p ρ + ( ρ ρ ) + b ϑ + d λ G ( G + d RT ( a + c exp[ θ / T ] b δ ) exp[ ϑρ] exp[ ϑρ ] ( exp[ λρ ] exp[ λρ ]) + ρ ln ρ + ) + (8) where G, ρ are the molar Gibbs energy and density of pure water at T, p and G, ρ are the same properties at T and water saturation pressure p Experimental Data Base The three aqueous solutes of our interest belong among experimentally quite well-studied systems with the data available for most properties characterizing hydration at elevated temperatures and extending in case of V 2 and hyd C p,2 to supercritical conditions. For the purpose of this study, and in accordance with the Guideline, we have limited the use of experimental data sets to sub-critical temperatures. The Henry s law constants at the water saturation pressure p evaluated from solubility data were obtained from Fernandez-Prini and constituted the same set of data that was used for establishing the Guideline []. Major sources of derivative properties are the two papers by Hnedkovsky et al. [3,4], reporting respectively the partial molar volumes at two isobars and the partial molar heat capacities at one isobar for all three gases at temperatures from ambient to 705 K. In addition, there are data at or close to the temperature of 298 K: for V 2 [5-8] and for C p,2 [8,9]. The enthalpies of hydration are available from combination of heats of solution and of vaporization at 298 K [- 2] and for CH 4 also at temperatures to 323 K originating from the measurements carried out by the groups of Wadsö and Gill [3-5]. The properties of the solutes in an ideal gas phase, needed for converting C p,2 to hyd C p,2, were taken from JANAF Thermochemical Tables [6]. Altogether we used for CH 4, CO 2 and H 2 S the following numbers of data points: (45, 80, 27) for k H, (22, 7, 9) for V 2, (6, 7, 9) for C p,2 and (5,, ) points for hyd H 2, respectively. 3. Results and Discussion Henry s law constants were correlated simultaneously with the derivative properties using the weighted least-squares procedure with weights reflecting expected experimental uncertainties. They were estimated for k H from 2 to 3 %, based on the root-mean-square deviations of the fit reported in reference []. For the derivative properties the expected error margins were to 3 % for V 2, 3 to 5 % for hyd C p,2 and to 3 % for hyd H 2. The objective minimized function S was defined as cal, j exp, j j 2 S = [( X i X i ) / σ X i ] (9) j i where X and σx stand for k H, V 2, hyd C p,2 or hyd H 2 and their errors, respectively. Explicit equations for all thermodynamic functions considered resulting from the SOCW model can be found in reference [7]. The correlation parameters of the model obtained for the three aqueous solutes are summarized in Table, presented in Appendix. All four thermodynamic properties considered were described by the SOCW model within the limits of estimated uncertainties as stated above. The only exception was the set of Henry s law constants for H 2 S, for which the average deviation was close to 4 %. Figures to 3 provide comparison of the new correlation for k H with the Guideline. It is obvious that for CH 4 and CO 2 there is very good agreement of both correlations, some deviations can be observed for H 2 S namely at higher temperatures. It should be, however, noted that experimental Henry s law constants used in [] as well as in this work extend only to about 4 K for this solute, because the high-temperature literature data were excluded from consideration due to their unreliability. Therefore the Guideline above 4 K is actually an extrapolation, while our correlation is constrained by experimental data on V 2 and C p,2 to 623 K. Discrepancy between the two predictions of k H at higher temperatures requires some consideration and can not be resolved before new accurate high-temperature solubility data for H 2 S become available. The main focus of this work was nevertheless the determination of the Poynting corrections. Their values, corresponding to the exponential in equation (2) calculated via equation (8), are provided in Table 2 of the Appendix at selected temperatures and three pressures above the saturation line of water. It might be surprising how

4 large this correction is; for the difference between p and 0 it changes the calculated Henry s law constant by a factor of four, and at pressure of it still can make up to 5 % of k H. Another interesting point is the relative insensitivity of the Poynting correction to temperature. While the partial molar volume is increasing with temperature for hydrophobic solutes and diverges positively at the critical point of water, this increase is to at least 0-5 K compensated by the /(RT) multiplication factor, see equation. (2). In addition to the true Poynting correction, Table 2 also provides the estimates of this value using two simplifying approximations. One of them is based on the fact that V 2 is not changing much at conditions not far removed from ambient and thus it can be considered constant; usually the value at 298 K is used for this purpose in literature. The other approximation applies temperature dependent V 2 calculated along the saturation line of water, so only the pressure dependence at a constant temperature is neglected. It can be seen from Table 2 that both approximations are fair to about, but they are both incorrect at high temperatures and pressures. The 298 K approximation underestimates the true Poynting correction, because it does not reflect the increase of V 2 with temperature, while the saturation approximation overestimates it due to neglecting the pressure effect, which is strongly decreasing V 2, especially at high temperatures. 4. Conclusions Figure 3. Henry s law constants along the saturation line of water; points were generated from the IAPWS Guideline, lines correspond to correlations with the SOCW model including the standard derivative properties. The conclusions can be summarized as follows: i) Correlation of the Henry s law constant for CH 4 and CO 2 in the Guideline is consistent with the values of the standard derivative properties. Some inconsistency was observed for H 2 S at temperatures above 4 K where no reliable solubility values are available in the literature. The data on V 2 and C p,2 resulting from calorimetric and volumetric experiments suggest for this solute at high temperatures somewhat different trend in k H compared to that from the Guideline. ii) Theoretically founded models allowing representation of hydration properties as a function of both temperature and pressure represent an alternative to equations correlating k H along the saturation line of water. Their advantages are the possibility of utilizing for correlation purposes also the experimental information on derivative thermodynamic properties such as V 2, C p,2 and hyd H 2. In principle, just one k H value at specified reference condition is sufficient to obtain prediction of this property up to supercritical temperatures as demonstrated in reference [2]. In addition, the

5 uncertainties in estimating the Poynting correction are avoided since k H can be calculated at any temperature and pressure of interest. Disadvantages are, however, higher complexity of the models and a need for calculating thermodynamic properties of pure water, i.e. user must incorporate into a code an equation-of-state for water. iii) When evaluating the gas solubility or airwater partitioning at pressures remote from the saturation line of water, the Poynting correction must be applied to the value of k H (T, p ). This correction is not negligible even at moderate pressures and reaches up to 400 % of k H (T, p ) at 0 for the three studied solutes. Its values only slightly depend on temperature except at T > 5 K where the divergence of the partial molar volume of solute at the critical point of solvent has an important impact. An expression for the Poynting correction is provided based on the SOCW equation-of-state that enables easy and accurate evaluation of this property up to the critical temperature of water and at pressures up to 0. Acknowledgements The authors thank Roberto Fernandez-Prini for sharing the database of Henry s law constants evaluated from solubility measurements. J.S. acknowledges support of the Czech Ministry of Education under project MSM References [] R. Fernandez-Prini, J.L. Alvarez and A.H. Harvey, J. Phys. Chem. Ref. Data, 32, 903 (2003). [2] J. Sedlbauer, J.P. O`Connell and R.H. Wood, Chem. Geology, 63, 43 (2000). [3] L. Hnedkovsky, R.H. Wood and V. Majer, J. Chem. Thermodyn., 28, 25 (996). [4] L. Hnedkovsky and R.H. Wood, J. Chem. Thermodyn., 29, 73 (997). [5] E.W. Tiepel and K.E. Gubbins, J. Phys. Chem., 76, 3044 (972). [6] J.C. Moore, R. Battino, T.R. Rettich, Y.P. Handa and E. Wilhelm, J. Chem. Eng..Data, 27, 22 (982). [7] R.E. Gibbs and H.C. van Ness, Ind. Eng.Chem. Fundam.,, 32 (97). [8] J.A. Barbero, K.G. McCurdy and P.R. Tremaine, Can. J. Chem., 60, 872 (982). [9] J.A. Barbero, L.G. Hepler, K.G. McCurdy and P.R. Tremaine, Can. J. Chem., 6, 29 (983). [] S. F. Dec and S.J. Gill, J. Soluion Chem., 3, 27 (984). [] RL. Berg and C.E. Vanderzee, J. Chem. Thermodyn.,, 3 (978). [2] J.D. Cox, D.D. Wagman and V.A. Medvedev, CODATA Key Values for Thermodynamics, Hemisphere Publishing Corp., 989. [3] G. Oloffson, A.A. Oshodj, E. Qvarnstrom and I. Wadsö, J. Chem. Thermodyn., 6, 4 (984). [4] S. F. Dec and S.J. Gill, J. Solution Chem., 4, 827 (984). [5] H. Naghibi, S. F. Dec and S.J. Gill, J. Phys. Chem., 90, 462 (986). [6] D.R. Stull and H. Prophet (Project Directors). JANAF Thermochemical Tables, NSRDS-NBS 37, 97. [7] J. Sedlbauer, G. Bergin and V. Majer, AIChE J., 48, 2936 (2002).

6 Appendix Table. Parameters of the SOCW hydration model for the three solutes. Solute a 3 b 4 c 6 e a (m 3.kg -.mol) (m 3.kg -.mol) (m 3.kg - d.mol) (J.K -2.mol - ) CH 4 (aq) CO 2 (aq) H 2 S (aq) a parameter needed for calculating the absolute value of Henry s law constant (see Ref. 7) Table 2. Poynting corrections calculated for the three solutes. CH 4 (aq) true correction 323 K 373 K 473 K 573 K V 2 at p constant V 2 (298 K) CO 2 (aq) true correction K 373 K 473 K 573 K V 2 at p constant V 2 (298 K) H 2 S(aq) true correction K 373 K 473 K 573 K V 2 at p constant V 2 (298 K)

The International Association for the Properties of Water and Steam

IAPWS G7-04 The International Association for the Properties of Water and Steam Kyoto, Japan September 2004 Guideline on the Henry s Constant and Vapor-Liquid Distribution Constant for Gases in H 2 O and