Ind. Eng. Chem. Res. 2000, 39, 2627-2630
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CORRELATIONS Reactive Gas Solubility in Water: An Empirical Relation Mohammed A. Islam,*,† Mohammed A. Kalam,‡ and Maksudur R. Khan† The Department of Chemical Technology and Polymer Science and The Department of Chemistry, Shahjalal University of Science and Technology, 3114 Sylhet, Bangladesh
An empirical equation is proposed to describe the solubility vs pressure relation for some reactive gases. The systems under consideration are ammonia/water, sulfur dioxide/water, chlorine/water, and hydrogen chloride/water. It is assumed that the reactive gases are dissolved in water by two mechanisms: one described by Henry’s law and the other by a Langmuir type equation. Out of the four systems only hydrogen chloride showed no Henry’s type of sorption, and a threshold concentration term was added to validate the model. The proposed equation fits well to the solubility/pressure data from the literature. A possible variation in the values of the empirical coefficients with the change in the water chemistry (pH, ionic strength) is also discussed. Henry’s law states that the solubility of a gas in a liquid is proportional to its equilibrium partial pressure in the gaseous phase.1-3 The linear relationship is valid only for very dilute solutions of nonreacting systems. At higher concentrations, the solubility vs equilibrium partial pressure relationship deviates so much from linearity that practically it cannot be used in precise analysis of industrial mass transfer processes. With the growth of environmental consciousness of the people, attention has been paid to the removal of pollutants from waste gases, and absorption method appears to be an effective one. Solubility vs pressure data are a prerequisite in the design and analysis of the performance of an absorber. So far as our knowledge goes, no mathematical relation is developed yet describing the solubility of a gas in a liquid over a wide concentration range. Such a relation is necessary in modeling of absorption processes and in developing analytical methods for absorber and reactor designs. Once such a relation is established, it might easily be subjected to computer processing. In the present work, we have made an attempt to propose an empirical equation describing the solubility vs partial pressure relation over a wide range of concentration. For the development of the model, we have chosen four systems: ammonia/water, chlorine/ water, sulfur dioxide/water, and hydrogen chloride/ water. All of them react with water, partially ionize and change the pH of the medium. We drew the solubility vs partial vapor pressure curves based on the literature data.2,4 The curves look like the deformational curve of a plastic body showing some viscous-elastic properties.5 Having this similarity in mind, we present the solubility as a sum of two components. Such an approach is known † The Department of Chemical Technology and Polymer Science, Shahjalal University of Science and Technology, 3114 Sylhet, Bangladesh. ‡ The Department of Chemistry, Shahjalal University of Science and Technology, 3114 Sylhet, Bangladesh.
in the literature as a dual sorption model, which was developed to account for the negative deviation of Henry’s law behavior exhibited by some gases in glassy polymers.6-8 The dual sorption model postulates that two concurrent modes of sorption are operative in the dissolution of a gas in glassy polymers. Nonlinear sorption isotherms were decomposed into a linear part that accounts for normal dissolution and a nonlinear Langmuir-type curve that accounts for immobilization of penetrant molecules at fixed sites within the medium. It was assumed that glassy polymer network contains macrovoids capable of immobilizing a portion of the sorbed molecules by entrapment or by binding at high energy sites at the molecular periphery.9-11 In the present model also, the total solubility is resolved into two components. The one follows Henry’s law for the dissolution of the nonreactive gases. The other is described by a saturation type curve. This component is related to the immobilization of the dissolved gaseous molecules by chemical reactions with water. It is found that a well-defined quantitative relation exists between these two components. It is also shown that by variation of the water chemistry (pH, ionic strength) the reactive gas absorption might be intensified. For the systems ammonia/water, chlorine/water, and sulfur dioxide/water, it was found that three empirical constants a, b, and xe completely describe the solubility vs partial pressure curve over a wide range of concentration. The coefficient a accounts for the Henry-type of sorption, and b and xe account for the sorption in form of the reaction products (including ionization products). For the system hydrogen chloride gas/water, a threshold concentration term x0 appears at zero partial pressure, and the coefficient a characterizing the Henry-type of sorption completely disappears. Thus, three empirical constants x0, b, and xe describe the solubility vs partial pressure relation for a highly polar gas such as hydrogen chloride. It is concluded that the approach “resolution of the solubility into two components” might serve as a good base for a mathematical description of solubility
10.1021/ie990558j CCC: $19.00 © 2000 American Chemical Society Published on Web 06/14/2000
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x=
Figure 1. Typical solubility x vs partial pressure P curve (3) resolved into two components: a saturation curve (1) and a linearly increasing one (2).
vs partial pressure relation for many reactive gas/ solvent systems.
It is a usual practice that the partial pressure P is represented as a function of solubility x. However, for an easy mathematical treatment, we have presented the solubility x (mole fraction) as a function of partial pressure P (Pa) (Figure 1, curve 3). Obviously the solubility increases with an increase in partial pressure, however with a decreasing rate until it attains a constant slope. Conventionally at low concentrations (correspondingly low partial pressures), the curve is approximated by a straight line with slope k, and the reciprocal value of k is taken to be Henry’s constant. Such an approach does not take into proper account the mechanism of sorption. At higher partial pressure, the curve 3 is really a straight line with slope a, and intercepts xe on concentration axis. As will be discussed later, the reciprocal value of the slope a represents the true Henry constant, and xe corresponds to the maximum amount of dissolved gas as a result of chemical reaction. Development of an Empirical Model The solubility curve 3 in Figure 1 is resolved into two components. The solubility component xR (curve 1) represents a saturation curve showing a tendency of saturation at some high values of P. Such a curve might be described by a Langmuir type equation (eq 1). The solubility component xH (curve 2) is a straight line with slope a (eq 2). The sum of the solubility components xR and xH represents the total solubility x (eq 3). Thus
P xR ) x e b+P
(1)
xH ) aP
(2)
x ) xR + xe
(3)
where a, b, and xe are empirical constants. Combining eqs 1-3, we have
When P < < b, then
P + aP b+P
)
xe +a P b
(4)
(5a)
Equation 5a represents a straight line with a slope of k ) (a + xe/b) (Figure 1). It is a usual practice that at low concentrations (correspondingly low partial pressures) the solubility vs equilibrium partial pressure curve is approximated by a straight line, and the reciprocal value of the slope of the straight line is taken to be Henry’s constant for the system. Thus, the conventional method for the determination of Henry’s constant is reduced to the determination of the slope of the straight line described by eq 5a. At higher concentrations (correspondingly higher partial pressures), the concentration vs partial pressure curve is described by eq 4 and this is considered to be the deviation from Henry’s law. For P > > b, the concentration vs partial pressure relation is reduced to a straight line (eq 5b):
x = xe + aP
Illustration of a Gas Solubility Curve
x ) xe
(
(5b)
Although the approach in the present model is empirical, it draws a well-defined demarcation line between two solution mechanisms: (1) solution as a result of chemical and/or physicochemical interactions between the reactive gas molecules and water (represented by eq 1) and (2) solution of the gas in unreacted form (represented by eq 2). The reactive gases are dissolved in water by both the mechanisms simultaneously. Under equilibrium, replacing P from eqs 1 and 2 relates the solubility components xR and xH.
xR ) x e
xH ab + xH
(6)
The present model recognizes the validity of Henry’s law over a wide concentration range. The apparent deviation from Henry’s law is attributed to the solution by chemical or physicochemical interactions. Since Henry’s law is valid only for nonreacting systems, the true Henry constant H for reacting systems is proposed to be calculated from the slope of the curve representing the solution in unreacted form (eq 2). Thus, the value of H is given by
H ) 1/a
(7)
Determination of the Empirical Constants a, b, and xe The following procedure might be maintained in determining the parameters graphically: • Solubility x vs partial pressure P curve is drawn as curve 3 in Figure 1 • The linear section of the curve is extrapolated to P f 0. The slope of the straight line gives the value of a, and the intercept gives xe. • Equation 4 is reduced to eq 8 by simple algebraic manipulations.
b 1 1 1 ) + x - aP xe P xe
(8)
Now, from the slope and the intercepts of the straight line 1/(x - a P) vs 1/P, the parameters b and xe are
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creases. This result was expected as it is a well-known fact that Henry’s constant H (which is the reciprocal value of a) increases with an increase in temperature. However, with an increase in temperature, the fitted equilibrium value of xe of the solubility component xR decreases for sulfur dioxide, increases for chlorine and remains almost unchanged for ammonia. Equilibrium Processes in Reactive Gas/Water Systems
Figure 2. Solubility x (mole fraction) vs partial pressure P (Pa) curve for ammonia/water system (eq 4) at temperatures of 10 (1), 20 (2) and 40 °C (3). Data are taken from reference books.2, 4
On the basis of the empirical model, the solubility of a reactive gas in water might be represented by the following equilibrium processes:
Table 1. Fitted Values of the Parameters a, b, and xe, and the Calculated Values of Henry’s Constant H (Eq 7) system NH3/H2O temp (°C) a × 107 (Pa-1) b × 10-3 (Pa) xe × 102 x0 H × 10-5 (Pa)
10 10.0 18.3 38 0 10.0
20 7.2 29.2 37 0 13.9
40 2.65 77.6 40 0 37.7
SO2/ H2O 10 4.2 2.2 0.12 0 23.8
20 2.9 3.8 0.09 0 34.5
40 1.5 4.0 0.04 0 66.7
Cl2/ H2O 10 0.7 2.0 0.09 0 143
20 0.7 5.0 0.18 0 143
HCl/H2O 20 0 1.9 12 0.12
determined. Steps 2 and 3 of the procedure would give approximately the same values of xe. Model Validation The model is applied to describe the solubility x vs partial pressure P relationship for four systems: ammonia/water, chlorine/water, sulfur dioxide/water and hydrogen chloride/water. Figure 2 represents the solubility x (mole fraction) vs partial pressure P (Pa) curves (eq 4) for the ammonia/water system at different temperatures. The data are collected from reference books.2,4 Obviously the proposed empirical equation satisfactorily fits to the experimental data given in the literature. Figures representing the same relations for the other three systems are not included in the paper, as they are similar (with a bit exception for hydrogen chloride/water system) to that of ammonia/water system. The fitted values of the empirical constants a, b, and xe, as well as the calculated values of Henry’s constant H, are shown in Table 1. A threshold concentration x0 is introduced in eq 1 to describe the solubility xR vs partial pressure relation for hydrogen chloride/water system and consequently the eq 1 is modified to eq 9.
xR = x0 + xe
P b+P
(9)
Such a deviation of the reactive solubility component xR of hydrogen chloride/water from other three systems is attributed to the fact that hydrogen chloride is a highly polar gas and almost completely ionizes in water. At low concentration range its equilibrium partial pressure is so low that with a good approximation, it might be considered zero. Unlike the other three systems, hydrogen chloride/water does not show any Henry type sorption (a ) 0). Table 1 shows that temperature affects differently on different solubility components. With an increase in temperature, the value of a for all the systems de-
At the gas-liquid interface, there exists an equilibrium between the unreacted gas molecules in water and those in the gaseous phase (described by Henry’s law). A part of the dissolved gas reacts with water, and the reacted form of the gas from its own part undergoes ionic dissociation. Therefore, in the liquid phase, there exist two successive equilibrium processes. The sum of the concentrations of the undissociated (xR′) and dissociated (xR′′) forms of the reacted gas constitutes the value of xR in the proposed empirical relations. Influence of Water Chemistry on the Reactive Gas/Water System In developing the empirical model, any change in the water chemistry is neglected. The gases under discussion react with water and the reaction products undergo ionic dissociation. Thus, the ionic strength and the pH of the medium change. These factors do not affect the coefficient a (Henry type of sorption), but do influence the equilibrium of the ionic dissociation and hence the whole sorption equilibrium. The empirical constant xe will be highly affected with the change in the water chemistry. In removal of pollutants from waste gases, in most cases the water chemistry is changed by the addition of some reactants. Therefore, for the precise analysis of the performance of an absorber, the influence of the water chemistry on xe should be taken into account. For example, for the removal of SO2 from waste gas, an alkaline solution of water is used as the absorbent. The sorption process is represented by eqs 10a-c.
SO2(g) S SO2(l)
(10a)
SO2(l) + H2O S H2SO3
(10b)
H2SO3 + H2O S H3O+ + HSO3-
(10c)
Obviously, an increase in pH of the absorbent would shift the equilibrium processes to the right. For eq 10c, we can write
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f+f- [H3O+][HSO3-]
Ka )
[H2SO3]
(11)
where Ka is the constant of acidity, and f+ and f- are the activity coefficients of the corresponding ions. The activity coefficients might be calculated by the extended Debye-Hu¨ckel equation12 (eq 12a).
log fi ) -
Conclusion
0.5Zi2xI
(12a)
1 + xI
with
I ) 0.5
∑ CiZi2
(12b)
where I is the ionic strength of the solution, Ci (mol/L) and Zi are respectively the concentrations and charges of the ionic species. The total concentration of the reacted SO2 (dissociated and undissociated) is given by the following relation (eq 13).
CR ) [H2SO3] + [HSO3-]
(13)
Combining eqs 11 and 13, we have
(
CR ) 1 +
Ka f+f- [H3O+]
)
[H2SO3]
(14a)
In terms of mole fractions, eq 14a might be rewritten as follows:
(
xR ) 1 +
Ka f+f- [H3O+]
)
xR′
(14b)
When the reacted gas molecules saturate the water; eq 14b is rewritten as eq 14c with index e
(
xe ) 1 +
Ka f+f- [H3O+]
)
A similar algorithm might be applied for other systems also. Due to the unavailability of literature data, the described method for the correction of xe might be applied successfully with a good approximation. Intensification of an absorption process might be also predicted by eq 14c.
xR,e′
(14c)
The reactive gas solubility vs partial pressure relation is described by an empirical equation. Three empirical constants fully describe solubility vs partial equilibrium pressure for reactive gases such as ammonia, chlorine, and sulfur dioxide. For a highly polar gas such as hydrogen chloride, a threshold concentration at zero pressure was introduced to validate the model. The method “resolution of the solubility into two components” might serve as a good base for the mathematical description of solubility vs equilibrium partial pressure relation for many other reactive gas/solvent systems. The influence of the water chemistry on the gas solubility is also discussed. The empirical equations would enable modeling of absorption processes and development of satisfactorily precise analytical methods for absorber and reactor design. Literature Cited (1) Barrow, G. M. Physical Chemistry; McGraw-Hill: Singapore, 1988. (2) Perry, R. H.; Green, D. Perry’s Chemical Engineers’ Handbook; Sixth edition, International Edition. McGraw-Hill: Selangor, Malaysia, 1984. (3) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The properties of Gases and Liquids; McGraw-Hill: Singapore, 1988. (4) Geankoplis, C. J. Transport processes and unit operations; Prentice Hall of India: New Delhi, 1997. (5) Islam, M. A. A mechanical model for the deformational behavior of the Polymeric Membranes operating in Pressure-driven Processes. J. Appl. Polym. Sci. 1992, 46, 1215.
At a given temperature and partial gas pressure P, the equilibrium value of [H2SO3] does not change (eqs 10a and 10b), however the total concentration of the reacted SO2 might be regulated by varying the ionic strength and the pH of the absorbent (eqs 10c and 14). In analyzing the performance of an absorption column, the following procedure might be maintained to make a correction of the value of xe. • Considering that the gas molecules sorbed by Henry’s mechanism have no contributions to the change in the water chemistry, the pH of the solution for the concentration Ce, mol/L (corresponding to xe in the present model) is estimated by the following relation:
(10) . Meares, P. The diffusion of gases in polyvinyl acetate in relation to the second-order transition. Trans. Faraday Soc. 1957, 53, 101.
[H3O+] ) xKaCe
(11) . Meares, P. The solubility of gases in polyvinyl acetate. Trans. Faraday Soc. 1958, 54, 40.
(15)
- Applying eqs 12a, 12b, and 14c, the value of xR,e′ is calculated. This value of xR,e′ depends on the temperature only and is independent of the water chemistry. For weak electrolytes, to simplify the computations, the activity coefficient fi (eqs 12a and 12b) at this stage might be taken to be unity. • For the working pH and ionic strength, the corrected value of xe is calculated by eqs 12a, 12b, 14c.
(6) .Vieth, W. R.; Sladek, K. J. A Model for Diffusion in Glassy Polymer. J. Colloid Sci. 1965, 20, 1014. (7) . Wieth, W. R.; Tam, P. M.; Michaels, A. S. Dual Sorption Mechanisms in Glassy Polystyrene. J. Colloid Sci. 1966, 22, 360. (8) Wieth, W. R.; Howell, J. M.; Hsieh, J. H. Dual Sorption Theory. J. Membr. Sci. 1976, 1, 177. (9) . Meares, P. The diffusion of gases through polyvinyl acetate. J. Am. Chem. Soc. 1954, 76, 3415.
(12) Genov, I.; Popova, E. Chemical Reactions and Equilibrium in Solutions; Chemical Institute Publishing House: Bourgas, Bulgaria, 1990 (in Bulgarian).
Received for review July 28, 1999 Revised manuscript received February 9, 2000 Accepted April 3, 2000 IE990558J
