Concentrated Aqueous Electrolyte Solutions

Journal of Solution Chemistry, Vol. 29, No. 7, 2000

Concentrated Aqueous Electrolyte Solutions: Analytical Equations for Humidity— Concentration Dependence Serge I. Prokopiev1,* and Yuri I. Aristov2 Received June 23, 1999; Revised February 18, 2000 The Henry’s law, Freundlich, and Dubinin–Astakhov, Brunauer–Emmett–Teller equations, and a new modification of the BET equation are used for the thermodynamic analysis of the activity of water over wide ranges of temperature (260–600 K) and water content (mole fraction 0.3–0.8) in concentrated aqueous solutions of electrolytes. The solutes considered are H2SO4, CaCl2, LiBr, LiCl, LiI, MgCl2, and NaOH in bulk solutions, and CaCl2 and LiBr dispersed in porous matrixes. The generalized Henry’s law equation is found to afford a correct extrapolation of bulk solution properties measured at low and moderate salt concentrations to the region of low water content. The suggested modification of the BET equation as well as the Dubinin–Astakhov equation give an accurate (relative error less than 1%) description of literature data on the humidity–concentration dependence and can be used for the reliable prediction of this dependence in the region of intermediate conditions. KEY WORDS: Electrolyte solutions; thermodynamics; water activity; humidity.

1. INTRODUCTION Concentrated aqueous electrolytes are used for the reversible heat sorption in a number of energy transformation devices.(1) The higher the solute concentration, the greater is the sorptivity of the solution. Recently, it has been found(2–5) that aqueous CaCl2 and LiBr solutions confined to microand mesopores of silica gels remain liquid at higher salt concentration than

1

Institute for Water and Environmental Problems, 630090 Novosibirsk, Russia. email: [email protected]. 2 Boreskov Institute of Catalysis, 630090 Novosibirsk, Russia. email: [email protected]. 633 0095-9782/00/0700-0633$18.00/0 q 2000 Plenum Publishing Corporation

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Prokopiev and Aristov

they do in the normal bulk state. For instance, in silica micropores at 1008C, the system CaCl2 1 H2O demonstrates the water sorption equilibrium of a divariant type (typical for liquid solutions), even at high salt concentrations up to 84 wt. %, whereas in the bulk system of the same composition, the salt forms solid hydrates and no liquid phase is observed.(2,3) As composites comprising concentrated electrolyte solution confined to a porous matrix, known as selective water sorbents (SWS), are promising materials for water sorption(2–5) and cooling/heating applications,(6) data on their thermodynamic properties are of great theoretical and practical interest, and analytical equations describing the water vapor pressure over concentrated solutions of CaCl2, LiBr, LiCl, and other hygroscopic salts are needed. Several models have been developed to describe the activity a of water in concentrated salt solutions. This property is of great importance, because at moderate temperatures (less than 2008C) the water activity is equal to the humidity h 5 p/ps, i.e., the ratio of vapor pressure p over a solution to the pressure ps over pure water. The Debye–Hu¨ckel–Pitzer (DHP) Eq.(7) was developed on the basis of the classical theory of dilute electrolytes.(8) For a solution of a single electrolyte, the osmotic coefficient w is expressed as(7) w 5 1—Am1/2 /(1 1 bm1/2) 1 Bm 1 Cm2 where A and b are the original Debye terms,(8) B 5 b(0) 1 b(1) exp(2am1/2), a 5 2.0 for 1:1 electrolytes and 6.0 for 1:2 electrolytes. The water activity can be calculated from w as a 5 exp (2n w m/m0), where n is the number of ions per dissociated salt molecule, m is the salt molality, and m0 5 55.5 mol-kg21. The DHP equation contains three empirical parameters [b(0), b(1), and C ], and adequately describes the thermodynamic properties of the aqueous electrolytes up to about 6 mol-kg21.(7) Stokes and Robinson(9) suggested adoption of the multilayer adsorption theory of Brunauer, Emmett, and Teller (BET)(10) to describe of electrolyte solutions. A suitably rewritten BET equation provides a relation between the water activity and the mole ratio N 5 m0/m of water to solute a/(1 2 a)N 5 [1 1 a(c 2 1)]/cr

(1)

Here r is the number of molecules of water in a complete monomolecular hydration layer; c is a constant related to the heat DU of adsorption of the adsorbate molecule in the first monolayer by c 5 exp(DU/RT ); R is the universal gas constant; and T is the absolute temperature. DU is the excess of the energy of absorption over the “latent heat” of vaporization of water,(11) which is equal to the difference in energy between water bound to an ion and water in bulk water, and is close to the differential enthalpy of dilution DHd. Equation (1) has two empirical parameters, c and r, and gives a good

Concentrated Aqueous Solutions

635

fit of a values for solutions of nitrates and chlorates(12) over a wide range of water content (at least N 5 0.4–8.0). However, for halide solutions, the approximation with Eq. (1) is satisfactory only in a more restricted region of concentration, e.g., 6–11 mol-kg21 (N 5 5.0–9.1) for CaCl2,(13) 9–24 molkg21 (N 5 2.2–6.4) for LiBr.(14) Moreover, using this model leads to significant overestimation of the water activity in LiI solution for N , 2.0 (i.e., at m . 27 mol-kg21).(14) According to the classical approach to the thermodynamics of aqueous electrolyte, based on the Debye–Hu¨ckel theory,(8) an infinitely dilute solution of an electrolyte in water is commonly considered as the standard state. On the contrary, a concentrated salt solution is a mixture where the mole fraction of the solute may exceed 0.5. Hence, it is reasonable to consider such a mixture as a solution of water in a molten electrolyte and to choose as a new standard state an infinitely dilute solution of water in the major component. Such an approach has been discussed, e.g., in Ref. 15. For solubility of gases (or vapors) in liquids, the well-known Henry’s law states that the mole fraction x of gas in a dilute solution is proportional to the gas partial pressure p, i.e., x 5 Kp, where K is the Henry’s law constant characterizing the affinity of the gas for the solvent.(16) A generalized form of the law is given in Ref. 17 for solubility of gases and vapors that dissociate in solution x 5 Kpa

(2)

where the exponent a is a kinetic order of the dissolution reaction. Equation (2) is similar to the Freundlich adsorption isotherm:(18) N 5 kpa

(3)

since, for a concentrated solution (as x → 0), N is proportional to x. The equations given above are applied in this paper to extrapolate the water activity measured on electrolyte solutions in the bulk at low and moderate solute concentrations to higher solute content (for concreteness, let us say that solute concentration is low when the water mole fraction x . 0.75 and high when x , 0.50, although these limits are not to be regarded as strong conditions). Our second aim in this work concerns electrolyte solution confined to a porous host matrix. The primary experimental data for confined solutions are presented in the literature(2,4,5) as isobaric charts N(T ) rather than sorption isotherms. Thus, isotherms like Eqs. (1–3) cannot be applied directly to the data. When water sorption in a dispersed solution increases, volume filling of pores by the solution can take place.(2,4,5) Thus, it might be reasonable to use the theory of volume filling of micropores (TVFM)(19) for an analytical description of water sorption in electrolyte confined in nanopores. TVFM postulates the temperature invariance of the relative adsorption plotted as a

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function of the adsorption potential DF (differential change of free energy during a reversible isothermal transfer of the adsorbate from a bulk liquid to an adsorbent) that is a direct consequence of the Polanyi potential theory of adsorption.(20) This approach suggests just one independent parameter DF instead of two (P and T ). The potential DF 5 2RT ln h is defined by the relative humidity h and presumably corresponds to the Helmholtz free energy of sorption from a bulk liquid.(19) An analytical equation for the sorbed volume V was suggested by Dubinin and Astakhov (DA)(21) for sorption of gases and vapors in micropores V 5 V0 exp(2(DF/E )g), where V0 is the volume of micropores, E is the characteristic energy of sorption, proportional to the mean potential energy of interaction between sorbate molecule and pore surface (thus, it depends on the affinity between adsorbate and adsorbent), and g is an empirical heterogeneity factor with integer values from 2 to 6. The DA equation has been successfully used for sorption of gases and vapors on many simple (noncomposite) microporous sorbents, such as active carbons,(22) synthetic zeolites,(23) and dehydrated inorganic gels (see Ref. 24 and literature cited there). The first attempt to apply a modified form for “composite,” calcium chloride solution confined to silica aerogel pores, was discussed in Ref. 25. Here we present a detailed study on other confined solutions using an equivalent form of the Dubinin– Astakhov equation rewritten in term of N N 5 N0 exp[2(DF/E )g],

(4)

where N0 is the limiting N for a solution that completely fills pores of a host matrix. The BET model can be applied to treat nonisothermal data a(N,T ) if one replaces the c parameter in Eq.(1) with its temperature dependence via parameter DU. N5

exp(DU/RT ) ar 1 2 a 1 1 a [exp(DU/RT ) 2 1]

(5)

In this paper we have tested the capability of the DHP, Henry’s law, Freundlich, DA, and BET equations, and their modifications to describe the behavior of concentrated solutions of single electrolytes, both in bulk and in confined states. Special attention is paid to the following questions. Can these equations be applied for a reliable prediction of (i) the water activity in concentrated solutions from experimental data for solutions at lower concentrations (extrapolation test) and (ii) intermediate water sorption isobars for systems of electrolyte confined to pores when starting from just two isobars


637

measured at extreme pressures (interpolation test)? Since the subject of our investigation is the equilibrium between water vapor and water in liquid solution, throughout this paper the term “activity” will be applied to the property of water, but not the salts. 2. EXTRAPOLATION FOR BULK SOLUTIONS Water activity data for bulk electrolyte solutions were taken from the references listed in Table I. As has been already mentioned, the DHP and BET equations are widely used to describe the water activity in aqueous electrolytes in bulk. Here we test the applicability of these equations as well as the generalized Henry equation for extrapolating the water activity to high solute concentration. Indeed, as most experimental thermodynamic data in the literature have been obtained for low and moderately concentrated solutions, it is of great interest to establish whether the equations under test can be used for reliable extrapolation of the activity to the region of high concentration. To answer this question we have considered the water activity data for aqueous solution of sulfuric acid because of its unlimited solubility in water. To deal with this solute at very high molality, it is convenient to use the water mole fraction x 5 m0/(nm 1 m0) instead of m. Taking from Ref. 26 the isothermal dependence a(x) measured for moderately concentrated H2SO4 solutions at 0.6 , x , 0.7, we have determined the empirical parameters of the DHP and BET equations, providing the best fit of the calculated a values to the experimental ones (Fig. 1). Then, we substitute the parameter values so obtained into the respective equations to predict the water activity a at higher electrolyte concentration (x , 0.6). The a values found show a deviation by some orders of magnitude Table I. Parameters of Eq. (7) Providing the Best Fit to Experimental Data and the Relevant Uncertainties in Approximation for Bulk Electrolyte Solutions

Electrolyte NaOH H2SO4 LiCl LiBr LiI MgCl2 CaCl2

ln k0

n0

A (K)

B (K)

Error d (%)

x range

Refs.

20.327 20.651 24.356 21.543 22.361 20.720 10.118

20.771 20.667 27.056 23.589 28.167 22.558 21.555

1944 2115 3605 3141 5501 3609 2014

411 481 1474 772 1198 627 124

3.9 3.5 0.9 1.4 3.0 1.2 1.0

0.2–0.7 0.2–0.7 0.4–0.8 0.4–0.8 0.3–0.8 0.5–0.8 0.4–0.8

26, 32 26 33 33 33 34–36 37, 38

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Fig. 1. Water activity a as a function of water mole fraction x in aqueous solution of H2SO4 at 258C (V) (data from Ref. 26). Approximations are: (1) DHP equation (M); (2) BET, Eq. (1) (n); (3) Henry’s law Eq. (6) (solid line).

from the relevant experimental data(26) (Fig. 1). Thus, it is evident that neither model is appropriate for extrapolation of the a(x) dependence to higher solute concentrations. The Henry’s law Eq. (2) was specifically derived for high concentrated salt solutions(17) and can be rewritten in term of water activity a: a 5 kxn,

(6)

where n 5 1/a and k 5 1/( ps/K1/a). Our analysis for a concentrated CaCl2 solution in bulk has shown that over a wide range of x the function a(x) obeys Eq. (6) with k and n dependent on temperature (Fig. 2). Unfortunately, the equation gives a poor description at x . 0.85; however, for a solution this dilute, the deviation from linear behavior looks reasonable, since at x → 1 the water activity should also tend to 1. We have found that the isothermal dependence of ln a vs. ln x is linear also for bulk aqueous solutions of MgCl2, LiBr, LiI, and LiCl. To introduce the temperature dependence of the parameters k and n in Eq. (6), we use the following expressions n(T ) 5 n0 1 B/T;

ln k(T ) 5 ln k0 1 A/T


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Fig. 2. Experimental isotherms of water activity in aqueous solution CaCl2 in log(a) vs. log(x). The dashed line corresponds to an ideal solution (data from Refs. 37, 38).

thus obtaining Henry’s law in a modified form: ln a 5 (ln k0 1 A/T ) 1 (n0 1 B/T ) ln x

(7)

with four empirical parameters, k0, A, n0, and B. To carefully analyze the applicability of Eq. (7) to concentrated electrolyte solutions, we first use this equation to fit experimental data from the literature for selected bulk electrolytes (Table I) over a wide concentration range (0.2 , x , 0.8). The relative deviation of the calculated activities aiC from the experimental ones aiE has been estimated using the value of the standard root mean-square deviation d according to the formula(27) d 5 [(oi (1/aiC)2(aiC 2 aiE)2)/(M 2 3)M ]1/2 3 100% where M is the number of data points (typically M ' 100), i 5 1, 2, . . . , M. In other words, d is the average error in the predicted activity of water expressed as a percentage deviation from the calculated value. For practically interesting solutions of CaCl2, d is as low as 1.0%, and, moreover, the major part (about 95%) of the data obtained with Eq. (7) lies within the band where the relative (not average) error is less than 5% (Fig. 3). This is strong evidence for the reliability of the equation. Similarly, small errors are also found for the other salts under consideration (see Table I). The Table shows that the

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Fig. 3. Water activity in bulk CaCl2 solution: calculated from Eq. (7) vs. experimental. The data are at temperatures from 2208 to 11758C. The relative error is 0% along the solid line and 5% along the dashed lines.

accuracy of the activity predicted by Eq. (7) is close to that obtained by the DHP and BET equations, commonly used for bulk electrolytes. Indeed, the error d of the DHP approximation is 3.0% for H2SO4 and 0.5% for CaCl2. The BET isotherm, Eq. (1), can be used for the electrolytes studied here exclusively at x . 0.5 (N . 3.0) and give d 5 3.0% for H2SO4 and 1.5% for CaCl2. After a successful approximation of the experimental data has been achieved, it is of interest to estimate whether the Henry’s law isotherm Eq. (6) can be used to predict the a(x) dependence for very concentrated solutions on the basis of experimental data obtained at low and moderate solute concentrations. For the analysis we followed the methodology that has been described above for the DHP equation and, first, applied Eq. (6) to describe the behavior of solution of sulfuric acid, because it is the only solute that can reach very low activities (1024 and less). We considered the literature data for H2SO4 at 0.5 , x , 0.6 to find the parameters k and n and then used their values to calculate a(x) at x , 0.5. Figure 1 shows that Eq. (6) provides much better agreement between calculated and experimental water activities than the DHP and BET equations and offers a satisfactory extrapolation to the range 0.0001 , a , 0.5, i.e., over almost four orders of magnitude. The relative error d of about 9% has been estimated for water activity in the range x 5 0.2–0.5 chosen for extrapolation. For CaCl2 and LiBr solutions, we took the water activity at 0.71 , x , 0.84 as initial data for the extrapolation and then followed the same procedure. It has been found that the error is less than 3% at xs , x ,0.71,


641

where xs is the mole fraction of water in the saturated salt solution. Thus, for bulk electrolyte solutions, Eq. (6) allows extrapolation with satisfactory accuracy of the water activity data obtained for low and moderate salt concentrations to higher salt content. 3. APPROXIMATIONS FOR DISPERSED SOLUTIONS The water activity data for CaCl2 solution confined to porous silica gel (sample S) and LiBr confined to porous carbon and alumina (samples C and A, respectively) were taken from the Refs. 2,4,5 as isobaric curves N(T ). As we have mentioned above, the salt concentration in a dispersed solution can exceed that in the bulk saturated solution. This phenomenon is probably a size effect and appears in small pores where crystallization cannot occur. First, we check the accuracy of the equations, discussed in the Introduction, to describe the whole set of literature data in the available concentration range. The average relative error in the predicted N value is calculated as the standard root mean-square deviation of the predicted N iC from experimental N iE data: d 5 [(oi(1/N iC)2 (N iC 2 N iE)2)/(M 2 3) M ]1/2 3 100% The results are collected in Table II. The parameters Nmin and Nmax in Table II give the range of experimental sorption data taken into consideration. The Nmin value for bulk solutions is about 2–4, because at lower N solid crystalline hydrates form, whereas for solutions in pores, N values less than 1 for humidity as low as 0.003. After the approximation gives good agreement between calculated and experimental data, the extrapolation of the experimental data to higher solute concentration is done. One more problem we tried to resolve is the correct calculation of a set of intermediate sorption isobars, starting from just two isobars measured at the extreme vapor pressures. 3.1. Approximation by the Freundlich Equation Here, for the sake of convenience, Eq. (3) is written in the form N 5 f(T ) ln N 5 (ln h 2 (ln k0 1 A/T ))/(n0 1 B/T )

(8)

and applied to describe the isobaric curves N(T ) measured for confined CaCl2 and LiBr solutions.(2,4,5) This approach is found to give an accurate approximation (d , 2%) of the literature data in the range of h 5 0.006–0.3, but, for humidity values beyond this interval, the calculation results in overestimated N values. For this reason, we did not use this equation for extrapolation to higher salt concentrations.

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Prokopiev and Aristov Table II. Best-Fit Parameters and Standard Deviations of Experimental Data from Calculations with Freundlich,a Second-Order Polynomial,b DA,c BET Isotherm,d Temperature-Dependent BET,e and Modified BETf,g Equations CaCl2 solution

Equation Parameter Nmin Nmax

a

Bulk 2.0 23.3

0.9 8.2

Bulk 1.3 8.0

in Carbon (C) in Alumina (A) 0.2 5.1

0.2 4.3

(8)

d% ln k0 n0 A B

(9)

d%

0.4

0.5

2.7

0.4

0.6

(4)

d% N0 g Eh

0.9 117 0.54 72

1.5 9.5 0.63 182

2.4 88 0.42 62

0.6 5.1 1.38 525

0.9 8.0 0.68 288

(1)

d% r c

2.9 6.48 10.2

1.1 3.99 22.7

1.6 3.63 34.8

1.2 2.64 43.7

0.8 2.40 51.0

(5)

d% r DU h

0.9 4.76 554

1.3 3.55 562

0.6 3.61 499

0.7 2.39 662

1.1 2.05 779

(10)

d% r DU h q

0.4 8.05 483 1.73

0.8 3.08 360 0.66

0.6 3.78 539 1.11

0.7 2.37 606 0.93

0.5 1.94 515 0.69

(11)

d%

0.4

1.2

0.6

0.7

1.0

1.4 0.5 1.4 10.04 20.29 20.77 10.31 13.07 10.18 21069 11592 21210 234 2163 592

Equation (8). Equation (9). c Equation (4). d Equation (1). e Equation (5). f Equation (10). g Equation (11). h E and DU in J-g21 of water. b

in Silica (S)

LiBr solution

0.6 20.02 21.94 21507 1492

0.6 21.12 10.01 21190 960


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3.2. The TVFM Approximation Figure 4 presents solution–vapor equilibrium data as N vs DF for the confined salt solutions. It demonstrates that these data can be described under the assumptions of the TVFM and that temperature-independent characteristic curves can be obtained.(28) Therefore, the free energy DF may be used as a single independent parameter for describing the equilibrium. The simplest way to obtain an analytical formula for the N(DF ) curves is to use a polynomial approximation ln N 5 a 1 b DF 1 c DF 2

(9)

with empirical coefficients a, b, c. For instance, for the aqueous LiBr solution confined to carbon nanopores, the approximating equation is N 5 exp(1.592 2 1.545 DF 2 0.992 DF 2) with DF in kJ-g21. The polynomial approximation gives a satisfactory description of isobaric charts for all solutions studied. The corresponding deviations d are presented in Table II, however, the empirical coefficients a, b, c giving the best fit are omitted as they have no physical meaning. To assign the description a physical meaning, we tried to approximate the characteristic curves N(DF ) by the DA equation, allowing parameter g to be fractional. Best-fit parameters for the bulk and confined solutions studied are displayed in Table II, together with the deviations d. The fitting procedure reveals the following. The accuracy of DA fitting is always worse than that of the polynomial one. Moreover, the greater the Nmax, the higher the optimal N0 value, whereas E and g become lower. The parameter N0 is 25–50% lower than that measured by a conventional technique. The heterogeneity factor g for the systems studied is commonly lower than 1.0, which is quite different from most data in the literature.(21–24) As the value of E can change for different matrixes,

Fig. 4. Characteristic curves for the solutions at T 5 23–1498C in the N vs. DF presentation and fitting by Eq. (4). (M with dashed line, for bulk solution; V with solid line, for solutions confined to nanopores.)

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it indicates an apparent character of this parameter or a strong effect of the host matrix. Thus, we conclude that at intermediate humidity (h 5 0.006–0.3) the characteristic curves N(DF ) obey the Dubinin–Astakhov equation with relative error d less than 1.5%, although the best-fit parameters are empirical rather than having a real physical meaning. 3.3. Approximation by the BET Equations In Refs. 2, 4, 5, experimental data on water content for CaCl2 and LiBr solutions confined in porous solids are presented as a function of relative humidity h. This makes it possible to locate experimental points measured at various pressures and temperatures along a single curve called a universal sorption isotherm, although the data are actually nonisothermal. Thus, the number of independent parameters is reduced from two (pressure P and temperature T ) to one (humidity h). The sorption vs. humidity plot gives another type of temperature-independent characteristic curve.(28) Currently, no analytical equation has been suggested for the N(h) curve, although such an equation would be the most convenient for further calculations.(5) With this target in mind we first used the two-parameter BET equation (1). In Fig. 5, the experimental data for both bulk and dispersed solutions over a wide temperature range are presented as plots of h/(1 2 h)N vs. h, which result in straight lines at moderate values of h. Best-fitting values of r and c are displayed for the solutions in Table II, together with the deviations d. It should be noted that the values found here are rather different from those reported in Refs. 9 and 13 because their authors took isothermal data at 258C, while we used all the data from 25 to 1458C. Nevertheless, using the approximation of a universal sorption isotherm with a two-parameter BET, Eq. (1) is somewhat worse than fitting the N(DF ) curve with a three-

Fig. 5. Characteristic curves for solutions at T 5 23–1498C in the h/(1 2 h)N vs. h presentation and linear fitting. (M with dash line, for bulk solution; V with solid line, for solutions confined to nanopores.)


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parameter DA Eq. (4) or applying the temperature-dependent Freundlich expression with four parameters to N(h,T ) data. The BET equation with temperature dependence, Eq. (5), is found to give at least no worse an approximation of the N(h) dependence than the DA Eq. (4) and for humidity h 5 0.006–0.3 it provides an analytical description with an error d of only 0.6–1.3% (see Table II where the best-fit values of the parameters r and DU are collected). The DU value is almost constant for CaCl2, but changes slightly for LiBr; although in both cases, it is close to the true heat of dilution of the solutions. The BET approximation can be somewhat improved if we add a third parameter q, analogous to the kinetic order n in the Henry or Freundlich isotherms, and replace h in Eq. (5) by hq N 5 crhq/{(1 2 hq)[1 1 (c 2 1)hq]}.

(10)

This parameter takes into account the possibility of multicenter interaction of an adsorbate (water) molecule, which could occur in concentrated salt solutions. The form of this equation bears a strong resemblance to the Abraham equation proposed in Ref. 29 for the activity of salt rather than water, because both contain the power hq. Application of the Eq. (10) to the systems studied here results in errors that are even less than for the polynomial Eq. (9) (see Table II and Fig. 6a). It is interesting to note that the r values obtained are close to the integer values of 4 and 8 stated in Ref. 26 as the most probable for bulk solutions of LiBr and CaCl2, respectively. Comparison of the results for the bulk and confined solutions reveals a reduction of the coordination number r with a decrease in the average pore diameter of the host matrix. This seems reasonable because in small pores it may be sterically impossible for salt ions located near a pore wall to have normal water environment. A good justification for Eq. (10) is that DU values obtained for LiBr solutions are close to each other. It seems reasonable to inquire whether are there cases when the suggested relation is better than any other three-parameter equation, e.g., that deduced by Anderson (30) from the BET model, by taking into account the variation of heat of adsorption in successive layers: N 5 cra/{(1 2 aZ ) [1/Z 1 (c 2 1)a]},

(11)

where Z is the additional parameter. As is shown in Table II, for bulk solutions, both equations give equal errors of approximation, but the accuracy of Eq. (10) with respect to a dispersed solution is much better than Anderson’s equation. The best approximation is for solutions confined in the hydrophilic host matrixes of oxides (silica and alumina) and, in these cases, the exponent q ,, 1. Thus, the introduction of the exponential parameter q appears to

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Fig. 6. Isobaric curves of vapor-solution equilibrium for LiBr aqueous solution confined in nanoporous alumina. experimental (open symbols) and values calculated with Eq. (4) (lines with filled symbols). The data taken into account for the fitting are experimental isobars at all available pressures (a) and just two extreme pressures p 5 6.2 and 52.6 mbar (b).

be very good idea, especially when prediction of the water activity in very concentrated (N , 2) and supersaturated solutions is necessary. 4. INTERPOLATION FOR DISPERSED SOLUTIONS As has been mentioned in the Introduction, it is of interest to calculate the intermediate isobaric curves of water sorption for electrolytes confined to pores, starting from just two isobars measured at extreme pressures. The procedure for the N(T) data interpolation was organized as follows. Two experimental isobars for a confined solution measured at extreme pressures (commonly about 6 and 50 mbar) are approximated by an equation chosen to provide the best fit. The equation parameters, which ensure this fit, are then reintroduced in the equation chosen to calculate the sorption isobars at several intermediate vapor pressures. Finally, the curves obtained are compared with experimental ones measured at the same intermediate pressures. It turns out that for all the equations under study, the error d of such interpolation does not exceed 3% over the whole range of N. For restricted ranges of N (0.55–3.8 for LiBr and 1.1–6.6 for CaCl2), the accuracy of the prediction is much better (d 5 0.5–1.5%). If we compare Eqs.


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(4, 9, 10), we can see that for LiBr solutions the average deviation for Eq. (4) is about 1.8–2.5 % over the entire range of N, which is worse than for the typical approximation (described in Section 3) but still close to the uncertainty of experimental measurements; at intermediate N, the error is less than 0.4–0.7 %. For Eqs. (9, 10) the accuracy over the whole range of N is much better: d 5 0.8% and 1.0%, respectively, and smaller by a factor of 2 for 0.55 , N , 3.8. Figure 6b demonstrates good agreement between the experimental isobaric curves at intermediate pressures and those calculated with the recommended Eq. (10). Apart from the possibilities discussed, some of the equations considered can, in principle, allow the computation of the differential heat (enthalpy) of dilution, as described in the Appendix. 5. CONCLUSIONS 1. A quantitative description of water content versus relative vapor pressure of water over bulk concentrated solutions of H2SO4, CaCl2, LiBr, LiCl, LiI, MgCl2, and NaOH by the Henry’s law equation adapted for aqueous electrolytes allows both a satisfactory approximation of experimental data from the literature for moderate salt concentrations and a correct extrapolation of these data to the region of high solute concentration. 2. When applying the Freundlich, TVFM, and BET adsorption models to describe of water relative pressure over CaCl2 and LiBr solutions dispersed in micropores of inert solid matrices, the uncertainty in the approximation is found to be less than 1.2% in the humidity range 0.006–0.3. The Dubinin– Astakhov equation is found to be the most reliable for precise calculation and prediction of isobars for water sorption by dispersed solutions. 3. A new equation with three empirical parameters is developed from the BET and Freundlich equations. Like the DA equation, the new equation is valid for the dispersed solutions and allows prediction of the water sorption at moderate humidity with high accuracy.

ACKNOWLEDGMENT This study was supported in part by the Russian Foundation for Basic Research (Grants 99-03-32312a and 99-03-32356a) and NATO (Grants HTECH.LG 970330 and CNS 972771). APPENDIX Some of the equations considered above, in principle, allow the computation of the differential heat (enthalpy) of dilution. For example, rearranging

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the BET Eq. (1) and differentiating ln a with respect to 1/T at constant N, one can derive the following expression(31) DHd 5 2R [dlna/d(1/T )]N 5 cDU (1 2 r/N 2 a)/{(1 2 r/N ) c 2 2 [1 1 (c 2 1) a]} 5 DU/{1 1 [a 1 (2/c)(1 2 a)]/[r/N 2 (1 2 a)]} from the TVFM we have DHd 5 2[d(DF/T )/d(1/T )]N 5 DF while Eq. (7) gives DHd 5 2R (A 1 B ln x) and the suggested three-parameter Eq. (10) results in DHd 5 (DU/q)/{1 1 [aq 1 (2/c)(1 2 aq)]aq21 /[r/N 2 (1 2 aq)aq21]} The dilution enthalpy DHd is connected with the isosteric heat of water desorption from the solution DHis 5 DHE 1 DHd, where DHE is the heat of evaporation of water. For bulk solutions these equations give an approximation with satisfactory accuracy (6 0.3 RT at N . 2.7). However, for solutions confined in nanopores, the immediate comparison of experimental and calculated DHis is hardly reasonable because the experimental evaluation of DHd using a numerical differentiation of ln a, as done in Refs. 2–5, is rather unreliable. The main reason is that small errors in the determination of a function often affect dramatically the accuracy of the derivative. For example, although the DHP model can be used for precise activity calculations at x . 0.5, it can be reliably applied for evaluation of DHd only at x . 0.6, while in the x range 0.5–0.6, the predicted and experimental DHd can differ tenfold. Using simple statistical methods,(27) the relative error in DHis calculation from the experimental data a(N,T ), within the temperature range from T1 to T2, can be evaluated via the error da in the activity as follows: dH . 2!3 da R/DHE(1/T1 2 1/T2) Thus, at the typical values T2 2 T1 5 50K, da 5 10%, and DHE 5 4 3 104 J/mol, the expected error is dH . 16%. REFERENCES 1. K. E. Herold, R. Radermacher, and S. A. Klein, Absorption Chillers and Heat Pumps (CRC Press, Boca Raton, FL, 1996). 2. Yu. I. Aristov, M. M. Tokarev, G. Restuccia, and G. Cacciola, React. Kinet. Catal. Lett. 59, 335 (1996).


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