Ind. Eng. Chem. Res. 2008, 47, 1283-1287
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Boiling Point Rise Calculations in Sodium Salt Solutions Marta Bialik,* Peter Sedin, and Hans Theliander Forest Products and Chemical Engineering, Chalmers UniVersity of Technology, SE-412 96 Go¨teborg, Sweden
The boiling point rise (elevation) of aqueous industrial solutions is often regarded as an important property with respect to chemical process design. This work shows an application of the Pitzer method for calculating the activity coefficients to the estimation of the boiling point rise of single-component and multicomponent electrolyte solutions. Good agreement between experimental and predicted values of the boiling point elevations of solutions of several salts (NaOH, Na2CO3, Na2SO4, NaCl, Na2S2O3, Na2S, mixed NaOH-Na2CO3, and mixed Na2CO3-Na2SO4) was obtained. A method for using the boiling point rise data to obtain ionic interaction parameters for the Pitzer method is also shown. Introduction
Theory
Boiling point rise, also called boiling point elevation, has long been recognized as an important property of numerous industrial solutions. Boiling point rise is defined as the difference in boiling temperatures of a given solution and its pure solvent when measured at the same pressure. The estimation of a true boiling temperature is an important step in the design of boiling, evaporation, and heat-exchange processes such as black liquor evaporation for pulp and paper production, industrial crystallization, liquid waste handling, etc. For this reason, many attempts have been made to develop a reliable method for estimating the boiling point from measurable solution properties.1 In the case of multicomponent industrial solutions, especially those containing various organic compounds (e.g., black liquor from the pulp and paper industry), empirical or semi-empirical methods relating density or solid content to boiling point elevation have frequently been used.2 Du¨hring’s rule, which involves plotting the boiling temperature of a solution versus the temperature of the pure solvent at different pressures, has also been applied.3 Estimating the boiling point elevation for multicomponent industrial solutions is usually a challenging task. Both rigorous thermodynamic methods and empirical methods based on easily measurable solution properties have been used for this purpose. For the rigorous thermodynamic methods, the calculation procedure is often very complicated because of the large number of chemical compounds present in the solution. Moreover, the exact composition of many industrial solutions (e.g., black liquor) is often difficult to determine with sufficient accuracy, and the appropriate thermodynamic parameters for less typical compounds are frequently not available. The empirical methods, on the other hand, usually have no support in solution theory and thus have very limited applications, modeling only one type of solution with acceptable precision. The purpose of this work is to present a comprehensive set of boiling point rise data from single-component and multicomponent salt solutions and to suggest a computational method for predicting boiling point rise values for industrial salt systems of great interest. For this application, the Pitzer method for calculating the activity coefficients4,5 with interaction coefficients from Pitzer5 and Weber6 was chosen, and the boiling point elevation was calculated using Pitzer’s osmotic coefficient.
Many properties of solutions, including boiling point elevation, freezing point depression, and osmotic pressure, can be related to the vapor pressure of the solvent and hence to its activity. These properties are often called “colligative” properties of a solution. In the rational system, the chemical potential, µl, of a liquid solvent in a solution is expressed in terms of the chemical potential of the pure liquid, µ0l , and its activity, al, as7
* To whom correspondence should be addressed. E-mail: bialik@ chalmers.se. Fax: +46-31-772-2995.
µl(T,p) ) gl(T,p) + RT ln ai ) µ0i (T,p) + RT ln ai
(1)
where al might be a function of temperature, pressure, and composition. If the liquid solvent is in equilibrium with its vapor, the equilibrium conditions can be written as an equality in chemical potential7
µ0i (T,p) + RT ln ai ) µvap(T,p)
(2)
Because the difference between the chemical potential of vapor, µvap, and the chemical potential of pure liquid, µ0l , at temperature T corresponds to the free energy of vaporization, ∆G0vap, of the pure liquid at this temperature, eq 2 can be rearranged to7
-∆G0vap ln a ) RT
(3)
Differentiating eq 3 with respect to activity and applying the Gibbs-Helmholtz equation gives7 0
( )
∆Hvap ∂T 1 )al RT2 ∂al
(4)
p
where ∆H0vap is the heat of vaporization. This expression can then be integrated with the lower integration limit corresponding to pure liquid, boiling at temperature T0, and the upper limit to a solution with solvent activity a, boiling at temperature T. For simplicity, the heat of vaporization is often assumed to be independent of temperature within the small temperature range between T0 and T. Thus, the final equation takes the form7
( )
[
]
∆H0vap ∆H0vap 1 1 1 1 ) ln a ) R T T0 R (T0 + θ) T0
10.1021/ie070564c CCC: $40.75 © 2008 American Chemical Society Published on Web 01/23/2008
(5)
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where “0” refers to the properties of the pure solvent (water) and θ is the boiling point elevation. If the activity of the solvent is known, it is theoretically possible to use the eq 5 to predict the boiling point rise of the solution. There are numerous methods for calculating water activity in aqueous solutions; however, only a few of them can be applied to high-temperature, concentrated, multicomponent systems. In this work, the semi-empirical Pitzer activity coefficient method4,5 was chosen. This method has been widely used for estimating activity coefficients in applications ranging from the simple characterization of salt solutions to multifunctional commercial simulation packages. The method, combining the virial equation of state with the equation for potential energy between ions, utilizes empirical interaction parameters of the species present in the system. Consequently, the method requires a set of well-established interaction parameters valid for the desired temperature, pressure, and concentration conditions. The basic interaction parameters from Pitzer5 and Weber,6 presented in the Supporting Information, were used in this work (see below). For electrolyte solutions, the water activity calculation is based on the definition of the practical osmotic coefficient, φ8
φ≡
-1000 ln a Mw
(6)
νimi ∑ i)1
where Mw is the molecular mass of the solvent (water), mi is the molality of the ith solute, and νi is the number of ions into which the solute dissociates. Pitzer5 proposed the following equation for the simplified calculation of the osmotic coefficient
(φ - 1) ) (2/
∑i mi)
[
-AφI3/2
1 + 1.2I1/2
+
∑c ∑a Intca +
∑ ∑ Intcc′(a) + ∑ ∑ Intaa′(c) c< c′ a< a′
]
(7)
where AΦ is the Debye-Hu¨ckel constant describing solvent properties and I is the total ionic strength. The first summation term inside the square brackets represents the ionic interactions in terms of Debye-Hu¨ckel theory. The second term summarizes double-interaction terms, involving ions of different signs (c, cation; a, anion). In analogy, the third and fourth terms inside the square brackets summarize interactions of ions of the same sign and also involve triple interaction terms (c-c′-a or a-a′-c). The equations for the subsequent interaction terms are given in the Supporting Information, and a detailed description of the calculation procedure can be found elsewhere.4,5 Ionic activities in a typical electrolyte system are highly temperature-dependent; the activity coefficients in an ambienttemperature solution can be as much as an order of magnitude higher than those in a concentrated salt system boiling at atmospheric pressure or higher. It is thus crucial that the activity coefficient calculation method account for this temperature dependence correctly. The Pitzer method addresses this problem in two ways:5 by introducing the temperature dependence into the solvent properties used in the calculation of the DebyeHu¨ckel constant and by also adjusting the empirical parameters responsible for the interaction of two oppositely charged ions. (The parameters are usually defined as temperature-dependent polynomials.) The Debye-Hu¨ckel constant is defined as
Aφ )
( )x
1 e 3 xDkT
3
2πd0NA 1000
(8)
where D is the dielectric constant of the solvent, d0 is the solvent density, T is the temperature, k is the Boltzmann constant, NA is Avogadro’s number, and e is the unit charge. In addition to its direct temperature dependence, AΦ includes two other temperature-dependent quantities: D and d0. It is thus fairly easy to calculate AΦ for a desired temperature. The temperaturedependent polynomials for the interaction coefficients in eq 7 are often tabulated for common salt systems. This work uses the temperature-dependence coefficients for the Pitzer parameters tabulated by Weber;6 for the systems whose coefficients were not available, only the Debye-Hu¨ckel constant modification was applied. For the sake of simplicity, Chen’s formula,4 which approximates the temperature dependence of DebyeHu¨ckel constants excellently, was used in this work
273.15 (T -273.15 )+ T - 273.15 + 2.864468[exp( 273.15 )]
AΦ ) -61.44534 exp
2
T - 0.6820223(T - 273.15) + 273.15 273.15 0.0007875695(T2 - 273.152) + 58.95788 (9) T
183.5379 ln
Because the entire calculation procedure is thoroughly explained by Pitzer,5 it is relatively simple to program it for easy use with a desktop computer or as part of more complex simulation software. This work used a simple program written for MATLAB (Mathworks, Inc., Natick, MA) software. Estimation of Pitzer Parameters. The boiling point rise calculation procedure presented above is very interesting. Because eq 5 shows a mutual relationship between the boiling point elevation and the corresponding solvent activity, it is possible to use this procedure to calculate water activity in a given aqueous solution if the value of a corresponding boiling point rise is known. Once the water activity is known, it is possible to calculate the osmotic coefficient in the Pitzer equation according to eq 6. Consequently, reversing the Pitzer equation for the osmotic coefficient (eq 7) makes it possible to convert a set of well-established boiling point rise data into a set of Pitzer empirical interaction parameters for a given salt solution. This procedure could be applied for estimating the Pitzer interaction parameters of complex solutions, especially those involving atypical salts for which the original Pitzer parameters are unknown (e.g., black liquor) and multicomponent mixtures for which the higher-order interaction parameters have not been identified. Literature Data In this work, two literature sources of experimental data for boiling point elevations were used: the work of Makarov and Krasnikov,9 containing a set of Na2CO3-Na2SO4 solubility data for systems boiling at atmospheric pressure, and that of Theliander,10 which includes the experimental boiling point rise results from research on the physicochemical data of model solutions for white liquor. The data include boiling point elevation values for NaOH, Na2CO3, Na2SO4, NaCl, Na2S2O3, Na2S, mixed NaOH-Na2CO3, and mixed Na2CO3-Na2SO4 solutions. The values of the experimental and modeled boiling point rise values are summarized in the Supporting Information.
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Figure 1. Boiling point rise as a function of molality: NaOH solution.
Figure 2. Boiling point rise as a function of molality: Na2CO3 solution.
Figure 3. Boiling point rise as a function of molality: Na2SO4 solution.
Figure 4. Boiling point rise as a function of molality: NaCl solution.
Figure 5. Boiling point rise as a function of molality: Na2S2O3 solution.
Results and Discussion Single-Salt Solutions. Figures 1-5 show comparisons between the modeled and calculated boiling point rise values and
the experimental data obtained for single-salt systems. The modeling was done in two steps: First, the temperature extension was applied only for the Debye-Hu¨ckel constant (with temperature-independent interaction parameters from Pitzer5), and then, the temperature-dependent interaction coefficients from Weber6 were introduced. (For the systems not tabulated by Weber,6 only the first step was performed). The results from both steps are shown for comparison. As can be seen, for both approaches, the overall fit is remarkably good, especially for moderate concentration levels (2-3 molal): the difference between the experimental and predicted values for moderately concentrated NaOH, Na2CO3 and Na2SO4 solutions is approximately equal to 0.2 °C, which can be compared to the experimental error estimated to be 0.1 °C. At higher concentrations, the predictions deteriorate somewhat, especially in the case of the NaOH solution modeled without including the temperature dependency in Pitzer parameters, for which a significant discrepancy between the experimental and predicted values can be observed at m ) 9. This is, however, not surprising given that it is very difficult to make a proper thermodynamic description of concentrated salt solutions at elevated temperatures. Moreover, the application of the temperature-independent Pitzer interaction parameters for the NaOH-H2O system is actually limited to 6 molal solutions.5 The model version that uses temperature-dependent interaction parameters performs significantly better at higher NaOH concentrations, which can be expected based on the fact that the temperature behavior of highly concentrated NaOH solutions has long been studied and evaluated. The accuracy of the prediction of boiling point elevation increased significantly with the introduction of the temperature dependency into the interaction parameters for only two of the four examined single-salt systems for which the temperature extensions were available: NaOH and Na2SO4. For the NaCl solution, no major improvement was observed within the examined composition range, whereas for the Na2CO3 solution, the prediction was somewhat worse. Several possible reasons for such behavior can be mentioned: First, both NaOH and Na2SO4 are well-known and thoroughly examined salt systems with well-tested parameters, whereas studies of Na2CO3 solutions are less common. Second, the parameters provided by Weber6 seem to pertain mostly to saturated solutions, so the advantage of using them might not appear clearly at lower concentration levels, as in the case of the examined NaCl solution. As can be seen in Figures 1-5, the accuracy of the predictions of boiling point rises differs between the univalent and divalent salt systems examined: The predictions for NaOH and NaCl tend to be slightly more exact for the same molality level than are those for Na2CO3, Na2SO4, and Na2S2O3. This trend can be explained by the fact that the two univalent salts are among the most comprehensively examined systems within the scope of solution chemistry: their interaction parameters are thoroughly verified. Moreover, the solutions of salts with divalent anions have ionic strengths that are 3 times higher than those of univalent-anion systems. Thus, relating the magnitude of error in their boiling point rise prediction to molality alone might not be a truly fair measure of the efficiency of the prediction. Multicomponent Solutions. The model was also used to predict the boiling point rises in multicomponent electrolyte solutions, resulting in a remarkably good fit, especially for moderate concentrations. Figures 6 and 7 show the experimental and predicted values for boiling point elevation plotted versus ionic strength, adopted as a total concentration measure. The
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Figure 6. Boiling point rise as a function of ionic strength: NaOH-Na2CO3 solution.
(pK1 ) 7.05)
Thus, the single-salt solution is actually a multicomponent system under practical conditions and needs to be modeled as such, which involves a need for higher-order interaction parameters. The total salt concentration also changes slightly by the amount of water molecules involved in the hydrolysis reaction, so the first step in modeling a Na2S-H2O solution is to recalculate the ionic concentrations. Three different approaches to modeling of the Na2S-H2O system were used in this work. The first approach involved modeling the solution as if it did not contain any hydrosulfide ions (i.e., only Na+ and OH- from water dissociation were assumed to be present). The purpose of this approach was to assess the magnitude of the influence of the hydrosulfide ions on the system’s behavior. As can be seen in Figure 8, ignoring the presence of SH- ions in the solution leads to an underestimation of the boiling point rise by roughly one-half of its value; this suggests that these ions play a significant role in the behavior of the system. The second approach involved modeling the multicomponent system in the usual manner, but ignoring the higher-order terms for interactions between the two like-charged ions and for tripleion interactions (empirical parameters for these interactions were not known). Because no temperature-dependence coefficients for Na+-HS- Pitzer interaction parameters were available, the behavior of OH- anions was, for consistency, also modeled using only a simple temperature dependence in the form of a modification of the Debye-Hu¨ckel constant. As can be seen in Figure 8, this modeling approach offers a remarkably good fit, with the absolute difference between the experimental and calculated values never exceeding the experimental error. Such good agreement suggests that higher-order interactions do not play any significant role in the system behavior at these temperature and concentration levels. The third, and final, approach consisted of modeling the behavior of all of the system anions as though they were hydroxide ions. In other words, the presence of SH- ions was ignored, and the amount of OH- ions was doubled (cf. the stoichiometry of the hydrolysis reaction). Surprisingly good agreement was obtained; again, the differences between the experimental and calculated values were almost negligible and never exceeded the level of the experimental error. Such an excellent fit can probably be explained by similarities in the behavior of OH- and SH- ions.
(pK2 ) 19)
Conclusions
Figure 7. Boiling point rise as a function of ionic strength: Na2CO3Na2SO4 solution.
NaOH-Na2CO3 system shows an excellent fit, especially when the temperature-dependent Pitzer parameters were used. Up to an ionic strength of 3, the differences between the experimental values and predictions from both versions of the model are still smaller than the estimated experimental error. However, such a good fit might possibly be due to the canceling of errors in the individual parameters of the two single salts. For the Na2CO3-Na2SO4 solution data from Makarov and Krasnikov,9 the total concentration of salt was always kept at the saturation level; only the proportion between the two salts was varied. Given the fact that the solubility of sodium carbonate in boiling water is slightly higher than that of sodium sulfate, the increasing ionic strength in Figure 7 roughly reflects the increasing proportions of carbonate ions. Also, the model predictions for this system compare favorably with the experimental data. The greatest discrepancies can be observed at both ends of the plot, which correspond to pure-salt solutions at saturation. It is worth noting that, whereas use of the temperature-independent Pitzer parameters tends to underestimate the boiling point rises for solutions with higher Na2SO4to-Na2CO3 ratios, calculations including the temperaturemodified parameters overestimates them. For solutions rich in sodium carbonate, the model predictions behave in exactly the opposite way. Na2S Solution. Aqueous solutions of sodium sulfide merit special attention. When dissolved in water at 25 °C, Na2S dissociates according to the following reaction scheme11
Na2S S Na+ + NaSNaS- S Na+ + S2-
Figure 8. Boiling point rise as a function of molality: Na2S solution.
Because of the high pK value of the second dissociation step, the free S2- ion is virtually never present in a solution within the operationally practical pH range. A hydrolysis reaction takes place instead, which can be summarized as
Na2S + H2O S 2Na+ + HS- + OH-
A technique for the calculation of boiling point elevation in single-salt and multicomponent aqueous solutions was developed in this work. This routine is based on the Pitzer ionic activity coefficient method for the calculation of water activity and on an osmotic coefficient calculation. Predictions obtained using the method were compared with the experimental boiling point elevation values for NaOH,
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Na2CO3, Na2SO4, Na2S, mixed NaOH-Na2CO3, and mixed Na2CO3-Na2SO4 solutions. Excellent predictions were observed for moderate concentrations in both single-component and multicomponent salt solutions. At significantly higher concentrations (solutions close to the saturation point), the predictions deteriorate somewhat; nevertheless, the results can still be recommended as a good first approximation in complex boiling point rise calculations. It was also noticed that the predictions for mixed systems were relatively better than those for pure salts; this fact was especially true for the saturated Na2CO3Na2SO4 solution. The modeling results for the complex solution of hydrolyzed Na2S salt suggest a very weak influence of higher-order ionic interactions on the solvent activity and the boiling point rise at the concentration levels analyzed, as well as a strong similarity between hydroxide and hydrosulfide ions. The technique presented for estimating boiling point rises also offers a method for calculating the empirical interaction parameters for the Pitzer method for systems for which the original parameters are unknown. Nomenclature a ) water activity AΦ ) Debye-Hu¨ckel constant d0 ) solvent density, kg/m3 D ) dielectric constant of water e ) unit charge; e ) 1.6022 × 10-19 C g ) free energy term in the definition of chemical potential ∆Gvap ) free energy change of vaporization, J/mol ∆Hvap ) enthalpy change of vaporization, J/mol I ) ionic strength Int ) ionic interaction term for the Pitzer method k ) Boltzmann constant; k ) 1.3807 × 10-23 J/K m ) salt molality, mol/(kg of solvent) Mw ) molecular mass of water; Mw ) 18.02 kg/kmol NA ) Avogadro’s number; NA ) 6.022 × 1023 p ) pressure R ) gas constant; R ) 8.314 J/(mol K) T ) temperature, K Greek Letters θ ) boiling point elevation, K (°C) µ ) chemical potential φ ) osmotic coefficient ν ) stoichiometric number of ions from dissociation
a ) anion i ) ith solute (ion) l ) liquid (solvent) vap ) vaporization Superscript 0 ) properties of the pure solvent Supporting Information Available: Tables showing the Pitzer parameters for ionic interactions used in this work; comparison of experimental and calculated boiling point rise values for single-salt systems; comparison of the experimental and calculated boiling point rise values for multicomponent salt systems; and summary of the Pitzer method for calculation of the osmotic coefficient (PDFs). This material is available free of charge via the Internet at http://pubs.acs.org. Literature Cited (1) Meranda, D.; Furter, W. F. Elevation of the Boiling Point of Water by Salts at Saturation: Data and Correlation. J. Chem. Eng. Data 1997, 22, 315. (2) Frederick, W. J.; Sachs, D. G.; Grady, H. J; Grace, T. M. Boiling Point Elevation and Solubility Limit for Black Liquors. Tappi J. 1980, 63 (4), 151. (3) Bujanovic, B.; Cameron, J. H. Effect of Sodium Metaborate on the Boiling Point Rise of Slash Pine Black Liquor. Ind. Eng. Chem. Res. 2001, 40, 2518. (4) Zemaitis, J. F., Jr.; Clark, D. M.; Rafal, M.; Scrivner, N. C. Handbook of Aqueous Electrolyte Thermodynamics; American Institute of Chemical Engineers (AIChE): New York, 1986. (5) Pitzer, K. S. ActiVity Coefficients in Electrolyte Solutions, 2nd ed.; CRC Press: Boca Raton, FL, 1991. (6) Weber, C. F. Phase Equilibrium Studies of Savannah River Tanks and Feed Streams for the Salt Waste Processing Facility; Report ORNL/ TM-2001/109, Oak Ridge National Laboratory, Oak Ridge, TN, 2001. (7) Castellan, G. W. Physical Chemistry; Addison-Wesley: Reading, MA, 1971. (8) Tester, J. W; Modell, M. Thermodynamics and Its Applications, 3rd ed.; Prentice Hall PTR: Upper Saddle River, NJ, 1997. (9) Makarov, S. Z.; Krasnikov, S. N. The three-component equilibrium conditions at the boiling point in the four-component system Na2SO4Na2CO3-NaCl-H2O. IzV. Sekt. Fiz.-Khim. Anal., Inst. Obshch. Neorg. Khim., Akad. Nauk SSSR 1956, 27, 367. (10) Theliander, H. Omro¨rning i fast-fas Va¨tska, speciellt sla¨ckningsoch kausticeringsoperationerna; Chalmers University of Technology: Go¨teborg, Sweden, 1988 (in Swed.). (11) Handbook of Chemistry and Physics 2006-2007, 87th ed.; CRC Press: Boca Raton, FL, 2006 (electronic version).
Subscripts
ReceiVed for reView April 23, 2007 ReVised manuscript receiVed October 23, 2007 Accepted November 27, 2007
c ) cation
IE070564C
