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RESEARCH NOTES Analysis of the Variability of Parameters in the Brunauer-Emmett-Teller Solution Model for the Nitric Acid-Water System William O. Rains* and Robert M. Counce Department of Chemical Engineering, University of Tennessee, 419 Dougherty Hall, Knoxville, Tennessee 37996-2200
Barry B. Spencer Oak Ridge National Laboratory, P.O. Box 2008, Building 7601, Room 033, MS 6306, Oak Ridge, Tennessee 37831-6306
The Brunauer-Emmett-Teller (BET) model was applied to the nitric acid-water system in order to determine the variability of the model constants. The available data were interpreted to obtain values for the entropic and energetic parameters from 9 to 225 m. The values of those parameters and the manner in which they vary are entirely consistent with the known chemistry of the system. The main conclusion is that the BET model should be modified in a fashion which maintains its simplicity and chemical significance. The possibility of such a simple approach to model highly nonideal systems forms the basis for future study. Introduction The Pitzer, Bromley, Chen, and similar methods and extensions to these methods are perhaps the most widely used methods for predicting electrolyte activities.1 In practice, most of these models require a large number of adjustable parameters which are cumbersome to evaluate if they are not available. At >6 m or so, most of these models lose their predictive capability, and the addition of more parameters reduces the physiochemical significance of those parameters. Many of the effects present in concentrated electrolytes such as suppression of dissociation, complex formation, and hydration interactions may be described through the known chemistry of the system. In predicting activities in concentrated electrolyte solutions, it would be advantageous to have a model that required fewer parameters, some of which might be obtained from the known chemistry of the system. The Brunauer-Emmett-Teller (BET) isotherm was applied by Stokes and Robinson to model the adsorption of water to a solute.2 This was based on the postulate that a concentrated electrolyte solution could be viewed as an ionic lattice with irregularities introduced as a result of hydration. The result is a two-parameter model for the water activity,
maw 55.51(1 - aw)
)
1 c-1 + a cr cr w
(1)
where m is the molality of the solute, aw is the water * To whom correspondence should be addressed. Telephone: 423-974-2421. E-mail:
[email protected].
activity, r is the number of molecules of water in the monomolecular hydration layer when complete, and c is an energetic parameter related to the heat of adsorption with a temperature dependence of c ) exp(e/RT) where e ) E - EL, in which EL is the heat of liquefaction of pure water and E is the heat of adsorption. By plotting eq 1, Stokes and Robinson obtained values for the parameters for a variety of salts, NaOH, HCl, and HClO4, over a range of about 10-30 m, depending on the electrolyte. These parameters lead to a good fit of the data over the ranges considered and, surprisingly, fit some of the data to concentrations as low as 2 m despite the fact that the lattice model is not expected to hold in dilute conditions. More recently, Ally and Braunstein have applied the BET model to many of the systems approached by Stokes and Robinson and demonstrated that this model is thermodynamically consistent and that it may be extended to multicomponent systems for simple salts.3,4 Background Since the model was first proposed by Stokes and Robinson, numerous studies have indicated the need for improvement. One of Anderson’s modifications to the model is to add a third parameter which allows for the heat of adsorption to be less in each consecutive layer of hydration.2,5 This was originally coupled with the idea that the parameter r must be an integer to truly be a coordination number. Stokes and Robinson revisited their original model and suggested an improvement based on characterizing the energy of adsorption in each consecutive layer.6 Nesbitt proposed a modification of the model which is based on an extension to DebyeHu¨ckel theory like the most common models used
10.1021/ie990211g CCC: $19.00 © 2000 American Chemical Society Published on Web 12/02/1999
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today.7 In these modifications, both Stokes and Robinson and Nesbitt recognized that the r parameter was not identical with a tightly bound hydration number. This is based on the simple argument that for some electrolytes at fairly low concentrations (∼15 m) nearly all of the water must be bound in the original model; this conflicts with the data which show that the water activity is nonzero at much higher concentrations. This then leads to the postulation that water molecules are shared by the soluble species at high concentrations, perhaps lending validity to the lattice argument. From these researchers’ work and from simple chemical arguments, it is feasible for both parameters to be considered concentration dependent. This work investigates how these parameters change in a highly nonideal system and demonstrates how the variability of these parameters can be rationalized from the chemistry of the system. The aqueous nitric acid system is chosen as the basis of study for several reasons. Nitric acid does not have a solubility or miscibility limit in water. This attribute allows the investigation of the utility of the BET model into regions where there is very little water available for hydration. This region of concentration, on the molal scale, is an order of magnitude higher than the usual limit of the BET model. There are several distinct molecular species formed in concentrated nitric acid as its degree of dissociation declines. Like water, nitric acid is capable of forming strong hydrogen bonds and forms an azeotrope with water at about 68.5 wt %. These considerations make this system highly nonideal and well suited for the modeling of activities. In addition, a recent study concluded that the BET model may prove useful in this complex system with further parametrization.8 In the analysis of the results, several assumptions about the solution are made for simplification. The species in solution change with concentration, and the present analysis restricts this to the following species: the nitrate ion, the hydrogen ion (or hydronium), the nitric acid trihydrate, and the nitric acid monohydrate. The existence of these species is well established.9-11 There are likely to be other hydrates present in the solution, but for present purposes the analysis is restricted to those hydrates which have been isolated by precipitation. This restriction is based on the assumption that species which precipitate are thermodynamically more stable and the other hydrates may be transitional species or weakly bound combinations of the monohydrate and trihydrate. For the present purposes, nitrous acid and dissolved nitrogen oxides are considered negligible (i.e., the acid is not “fuming”). Addison and Miles point out some characteristics of this system which are relevant to the interpretation of the results.9,11 An important point to note is that 100% nitric acid undergoes a self-dehydration which produces 97% nitric acid, NO2+, NO3-, and water. This concentration of acid is equivalent to around 466 m. This establishes the upper limit of concentration considered in this study. As the concentration increases, the degree of dissociation decreases and the concentration of hydrates increases. Miles provides some information concerning the distribution of species in solution.11 At a concentration of 2.16 m (12%), about 10% of the total nitric acid is bound as a hydrate. At a concentration of 12.9 m (45%), the fractions of the solution which exist as ions and hydrates are equivalent. At around 36.5 m
Figure 1. Fit of the data for use with eq 2.
(70%) the hydrate concentration is maximized, accounting for around 75% of the nitric acid in solution, with the remaining 25% being equivalent amounts of ions and molecular nitric acid. Results and Discussion The data of Davis and DeBruin were used for values of the water activity and the degree of dissociation of the acid.12 The molar concentrations of nitric acid were converted to a molal basis using a density correlation by Spencer.13 Using these data, the left-hand side (lhs) of eq 1 was plotted against the water activity from 3.3 to 226 m as shown in Figure 1. The data are clearly not linear as required by eq 1 for the determination of the values of the two parameters r and c. A previous study showed that a linear fit over this region would give the parameters as r ) 2.42 and c ) 7.76.8 Although using these values may be somewhat useful in approximating water activities, they create significant error in the corresponding mean molal ionic activity coefficients and present a difficulty in assigning the required BET reference state.8 For this study the parameters r and c were allowed to vary with concentration. Denoting the lhs of eq 1 as f(aw) and the first derivative of the lhs with respect to aw as f ′(aw) and solving for the parameters gives
c)
f ′(aw) [f(aw) - awf ′(aw)]
+1
(2a)
and
r)
c-1 cf ′(aw)
(2b)
Equation 2 may be used to determine the parameters from the data by obtaining f ′(aw) by an appropriate method. As shown in Figure 1, a cubic fit represents the data nearly perfectly over this range. The inflection point of the curve occurs at about 15.9 m. The values of r and c are obtained from the curve fit utilizing eq 2. The results, normalized by their highest value, are shown
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Figure 2. Normalized results.
in Figure 2. The ionic molality, mi, is defined as
mi ) mνR where m is the stoichiometric molality, R is the fractional degree of dissociation, and ν is the number of moles of ions produced upon complete dissociation of the compound. There are several interesting features of this analysis. Because the parameter r represents a hydration number, it must always be positive, and this requires a positive value for eq 2a. Although the fit in Figure 1 begins at a molality of 3.3, the curve is too steep at this point for eq 1 to remain valid. Beginning at a concentration of 9 m and increasing, the intercept of eq 1 and consequently the value of eq 2a are greater than zero. The maximum value of r is around 2.9, and the maximum value of c is about 30. The maximum ionic molality occurs at about 15.5mi corresponding to a stoichiometric concentration of around 20 m. Figure 2 shows that the extrema in r and c correlate well with the maximum in the ionic molality. These extremes are close to the inflection point found at 15.9 m of Figure 1 as expected. It should be noted that the data only extend to a molarity of 22 with a water activity of 0.008; this corresponds to a molality of 225 or 93 wt %. Conclusions This brief analysis supports several important conclusions. First, the values of the parameters obtained are in agreement with values that would be expected from the chemistry. At low concentrations there is a mixture of ions and the trihydrate which average out at the maximum ionic concentrations to give values of r slightly less than 3. At the highest concentration of hydrates (70%), the value of r is close to 2 and decays off to around 1.2 at the highest concentration. The value of r does not drop to zero. This may be explained by the fact that solute molecules may be “shared” by solvent molecules in the BET model. Second, at these high concentrations there is less justification for several of the previously proposed extensions to the model. This is because these exten-
sions attempt to ascribe a varying heat of adsorption to subsequent water layers; in these high concentrations subsequent layers are unlikely to exist. Modifications to the model should probably not attempt to give too much weight to parametrization of adsorption energies beyond the monolayer. Third, averaging the hydration effects by using a linear fit in eq 1 hinders the potential to recover information about the solution. In addition, it does not indicate the region of validity of the model as does the analysis presented here. The main conclusion is that the variation of the parameters is consistent with the known chemistry of the system. This lends a new validity to the BET model and to the physical significance of its parameters. While the current analysis focuses on the aqueous nitric acid system, it is likely the BET model may be extensible to other concentrated, hydrating, electrolyte solutions. Current models require up to 10 or 20 parameters to describe these highly nonideal solutions over a broad concentration range. An extended BET model, with two parameters whose concentration dependence is chemically identifiable, would be very useful. As noted in the Introduction, the original model may accommodate mixtures of simple salts. In the development of an extended model, careful attention should be made to its ability for including multicomponent mixtures. The validity of the approach would be tested not only against the data but also against the known chemistry of the multicomponent system. The above conclusions form the foundation and the need to develop the BET model further while retaining its simplicity and straightforward approach. Acknowledgment The authors gratefully acknowledge the support of the DuPont Committee on Educational Aid. Nomenclature aw ) solvent activity, water activity c ) BET energetic parameter E ) heat of adsorption EL ) heat of liquefaction of pure water m ) molality mi ) ionic molality r ) BET entropic parameter Greek Symbols R ) fractional dissociation of electrolyte ) equivalent to E - EL ν ) number of ions formed upon complete dissociation of the solute
Literature Cited (1) Zemaitis, J. F., Jr.; et al. Handbook of Aqueous Electrolyte Thermodynamics; American Institute of Chemical Engineers: New York, 1986. (2) Stokes, R. H.; Robinson, R. A. Ionic Hydration and Activity in Electrolyte Solutions. J. Am. Chem. Soc. 1948, 70, 1870-1878. (3) Ally, M. R.; Braunstein, J. BET Model for Calculating Activities of Salts and Water, Molal Enthalpies, Molar Volumes, and Liquid-Solid Phase Behavior in Concentrated Electrolyte Solutions. Fluid Phase Equilib. 1993, 87, 213-236. (4) Ally, M. R.; Braunstein, J. Activity Coefficients in Concentrated Electrolytes: A Comparison of the Brunauer-EmmettTeller (BET) Model with Experimental Values. Fluid Phase Equilib. 1996, 120, 131-141.
Ind. Eng. Chem. Res., Vol. 39, No. 1, 2000 239 (5) Anderson, R. B. Modifications of the Brunauer, Emmett and Teller Equation. J. Am. Chem. Soc. 1946, 68, 686-691. (6) Stokes, R. H.; Robinson, R. A. Solvation Equilibria in Very Concentrated Electrolyte Solutions. J. Solution Chem. 1973, 2 (2/ 3), 173-191. (7) Nesbitt, W. H. The Stokes and Robinson Theory: A Modification with Application to Concentrated Electrolyte Solutions. J. Solution Chem. 1982, 11 (No. 6). (8) Rains, W. O. et al. Application of the Brunauer-EmmettTeller Isotherm to the Water-Nitric Acid System for the Determination of Mean Ionic Activity Coefficients. Chem. Eng. Commun. 1999, 171, 169-180. (9) Addison, C. C. Dinitrogen Tetroxide, Nitric Acid, and Their Mixtures as Media for Inorganic Reactions. Chem. Rev. 1980, 80, 21-39. (10) Hogfeldt, E. The Complex Formation between Water and Strong Acids. Acta Chem. Scand. 1963, 17, 785-796.
(11) Miles, F. D. Nitric Acid, Manufacture and Uses; Oxford University Press: London, 1961. (12) Davis, W., Jr.; DeBruin, H. J. New Activity Coefficients of 0-100 Percent Aqueous Nitric Acid. J. Inorg. Nucl. Chem. 1964, 26, 1069-1083. (13) Spencer, B. B. Simultaneous Determination of Nitric Acid and Uranium Concentrations in Aqueous Solution from Measurements of Electrical Conductivity, Density and Temperature. American Nuclear Society, Fourth International Conference on Facility Operations-Safeguards Interface, Sept 29-Oct 4, 1991.
Received for review March 22, 1999 Revised manuscript received September 22, 1999 Accepted September 27, 1999 IE990211G
