Temperature Dependent Thermodynamic Model of the System H+

J. Phys. Chem. A 2010, 114, 11595–11631

11595

Temperature Dependent Thermodynamic Model of the System H+-NH4+-Na+-SO42--NO3--Cl--H2O Elmar Friese* and Adolf Ebel Rhenish Institute of EnVironmental Research, UniVersity of Cologne, Cologne, Germany ReceiVed: February 3, 2010; ReVised Manuscript ReceiVed: August 10, 2010

A thermodynamic model of the system H+-NH4+-Na+-SO42--NO3--Cl--H2O is parametrized and used to represent activity coefficients, equilibrium partial pressures of H2O, HNO3, HCl, H2SO4, and NH3, and saturation with respect to 26 solid phases (NaCl(s), NaCl · 2H2O(s), Na2SO4(s), Na2SO4 · 10H2O(s), NaNO3 · Na2SO4 · H2O(s), Na3H(SO4)2(s), NaHSO4(s), NaHSO4 · H2O(s), NaNH4SO4 · 2H2O(s), NaNO3(s), NH4Cl(s), NH4NO3(s), (NH4)2SO4(s), (NH4)3H(SO4)2(s), NH4HSO4(s), (NH4)2SO4 · 2NH4NO3(s), (NH4)2SO4 · 3NH4NO3(s), H2SO4 · H2O(s), H2SO4 · 2H2O(s), H2SO4 · 3H2O(s), H2SO4 · 4H2O(s), H2SO4 · 6.5H2O(s), HNO3 · H2O(s), HNO3 · 2H2O(s), HNO3 · 3H2O(s), and HCl · 3H2O(s)). The enthalpy of formation of the complex salts NaNH4SO4 · 2H2O(s) and Na2SO4 · NaNO3 · H2O(s) is calculated. The model is valid for temperatures j263.15 up to 330 K and concentrations from infinite dilution to saturation with respect to the solid phases. For H2SO4-H2O solutions the degree of dissociation of the HSO4- ion is represented near the experimental uncertainty over wide temperature and concentration ranges. The parametrization of the model for the subsystems H+-NH4+-NO3--SO42--H2O and H+-NO3--SO42--Cl--H2O relies on previous studies (Clegg, S. L. et al. J. Phys. Chem. A 1998, 102, 2137-2154; Carslaw, K. S. et al. J. Phys. Chem. 1995, 99, 11557-11574), which are only partly adjusted to new data. For these systems the model is applicable to temperatures below 200 K, dependent upon liquid-phase composition, and for the former system also to supersaturated solutions. Values for the model parameters are determined from literature data for the vapor pressure, osmotic coefficient, emf, degree of dissociation of HSO4-, and the dissociation constant of NH3 as well as measurements of calorimetric properties of aqueous solutions like enthalpy of dilution, enthalpy of solution, enthalpy of mixing, and heat capacity. The high accuracy of the model is demonstrated by comparisons with experimentally determined mean activity coefficients of HCl in HCl-Na2SO4-H2O solutions, solubility measurements for the quaternary systems H+-Na+-Cl--SO42--H2O, Na+-NH4+-Cl--SO42--H2O, and Na+-NH4+-NO3--SO42--H2O as well as vapor pressure measurements of HNO3, HCl, H2SO4, and NH3. I. Introduction Atmospheric aerosol particles play an important role for several aspects of air pollution. They interfere directly with climate due to absorption and scattering of solar radiation1 or indirectly acting as cloud condensation nuclei.2 Heterogeneous reactions at the surface of polar stratospheric clouds lead to an enhancement of ozone depletion over the Arctic and Antarctic.3,4 Effects of aerosol particles on human health5,6 caused the enacting of a EU directive, which is aimed at environmental compliance.7 The water-soluble inorganic component of atmospheric aerosol particles is mainly composed of sulfate, nitrate,65 and ammonium amended by see salt within the marine planetary boundary layer.8 In polluted air over coastal cities sulfuric and nitric acid replace the sea salt chloride, which is released as HCl to the gas phase. Thus, in such air masses, sea salt particles can contain a considerable amount of sulfate and nitrate already after few hours.9,10 Aerosol particles with a sea salt component are not restricted to coastal areas because they are also produced within the size range, which is subject to long-range transport.11 The impact of an aerosol depends on the phase of the particles, their liquid water content, and the partitioning of volatile components between the gas phase and aerosol phase. * To whom correspondence should be addressed: E-mail: ef@ eurad.uni-koeln.de.

For determination of these properties models are needed, which ideally cover temperatures occurring from the lower troposphere up to the polar stratosphere. Such models require the knowledge of equilibrium constants of solid and liquid components as well as of activity coefficients of any occurring solute ion and of the solvent water. Bromley12 and Kusik and Meissner13 have established empirical relations for the calculation of activity coefficients in aqueous solutions, which are widely used in models of the inorganic atmospheric aerosol.14–22 The methods of Bromley and Kusik and Meissner have a number of disadvantages that are likely to limit their general application.23 First, these models require values of the activity coefficients of each cation-anion combination in pure aqueous solution at the same ionic strength as the mixture. This may require extended extrapolations where the mixture is concentrated but contains components that are relatively insoluble in pure solution. Second, there is no straightforward method of treating association equilibria, sulfuric acid being the most important example for atmospheric applications. Third, the mixing rules do not directly account for the specific interactions that arise between solution components in mixture. Last, the methods of Bromley and Kusik and Meissner have to be paired with a further empirical method of estimating water activity: the ZSR relation,24,25 the methods according to Bromley12 or Kusik and Meissner,13 or the RWR relation.26 Because of its simplicity the ZSR relation is widely

10.1021/jp101041j  2010 American Chemical Society Published on Web 10/13/2010

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used in models of the inorganic atmospheric aerosol.16–22 While this relation can be successfully applied for a multitude of aqueous solutions, it is inaccurate if association equilibria occur, as showed by Park et al.27 for aqueous NH4HSO4 solutions. Application of an extended ZSR-relation28 cannot completely solve this problem. The above-described difficulties are avoided by application of semiempirical, thermodynamical self-consistent models for aqueous solutions.29–31 Pitzer29 assumes that the excess Gibbs energy can be expressed as the sum of terms for the shortrange and long-range forces between ions and molecules in solution. For short-range and long-range forces an extended Debye-Hu¨ckel term and a third-order expansion of the excess Gibbs energy is used. Pitzer’s molality based model has been parametrized for a multitude of solutions29 including natural waters.32–37 The model can be applied to dilute to moderately concentrated solutions up to a molality of about 6 mol kg-1. Archer38,39 enlarged the range of validity of Pitzer’s model up to about 13 mol kg-1 due to introduction of a additional model parameter for the third virial coefficient. However, concentrations of atmospheric aerosol particles can be much higher.40,41 Pitzer and Simonson42 present a multicomponent, molefraction-based, thermodynamic model, which was generalized by Clegg and Pitzer31 (in the following called PSC model or PSC equations). The PSC model uses a single generalized expression for the excess Gibbs energy. This expression is a sum of an extended Debye-Hu¨ckel term42 for long-range contributions and a fourth-order Margules expansion for shortrange contributions to the excess Gibbs energy. The model is applicable to the entire concentration range; e.g., it has been used to describe the thermodynamic properties of hydrochloric acid for 0-100% HCl.43 The local composition model of Chen et al.,30 which is available for aqueous mixtures44 as well as for miscible solutions,45 also uses an extended Debye-Hu¨ckel term42 for long-range contributions to the excess Gibbs energy. The treatment of short-range contributions is based on the concept of “local compositionc”.46 While the model of Chen and Evans44 is suited for dilute to moderately concentrated solutions, it was found to represent the properties of highly concentrated solutions less well.47 The PSC approach has been used in previous studies to build thermodynamic models of the H+-SO42--NO3--Cl--Br-H2O23 and H+-NH4+-SO42--NO3--H2O48 systems, which are valid for temperatures from below 200 to about 330 K. A parametrization of the PSC equations for the H+-NH4+-Na+SO42--NO3--Cl--H2O system,49 valid at 298.15 K only, is part of several recent models of the inorganic atmospheric aerosol.50–54 Using the PSC equations, we build on previous studies23,55,56 and on the available literature data a model of the system H+-NH4+-Na+-SO42--NO3--Cl--H2O, which is valid for temperatures from about 263-330 K and concentrations from infinite dilution to saturation. II. Model Description Mole fractions are calculated on the basis of complete dissociation of all salts. Thus the mole fraction xi of species i is given by xi ) mi/Σjmj, where mj is the molality of species j. The symbol fi is used for the mole fraction activity coefficient of species i on the basis of a reference state of infinite dilution with respect to the solvent.42 The mole fraction activity coefficient fi is related to the corresponding molal value γi by fi ) γi(1 + M1jmj), where M1 (kg mol-1) is the molecular weight

Friese and Ebel of the solvent (water), and the sum is over all solute species. Clegg et al.47,57,58 present equations for the activity coefficients of the solvent, cations, and anions. They are derived due to differentiation of an expression for the excess Gibbs energy, which is assumed to consist of a sum of short-range and longrange contributions. The expression for the short-range forces between components (dominant in concentrated solutions) is based on a fourth-order Margules expansion. Long-range forcesmost important in dilute solutions-are accounted by an extended Debye-Hu¨ckel term42 and a higher order electrostatic term arising from the unsymmetrical mixing of ions of the same sign but different charge magnitude.59 A complete description of the model equations in given in Appendix I. III. Phase Equilibria If the heat capacity change ∆rCp (J mol-1 K-1) of a chemical reaction is expressed as ∆rCp ) ∆a + ∆bT + ∆cT2 + ∆dT3, the general relation for the equilibrium constant K is given by60

(

)

∆rH° 1 1 + R Tr T Tr ∆a T T ∆b - 1 + ln + Tr - 1 + T - Tr + R Tr Tr 2R T Tr ∆c 2Tr2 - 1 + T2 - Tr2 + 6R T Tr ∆d 3Tr3 - 1 + T3 - Tr3 (1) 12R T

ln K(T) ) ln K(Tr) +

(

[ (

)

)

[(

[ (

)

)

]

]

]

where T is the temperature, Tr is a reference temperature, ∆rH° (J mol-1) is the enthalpy change for the reaction at Tr, and R (8.3144 J mol-1 K-1) is the gas constant. Equilibrium constants x K on the mole fraction scale are related to those on the molal scale (mK) by p-νr) K ) mKM(ν 1

x

(2)

where νr is the sum of the stoichiometric numbers of the reacting liquid-phase solutes and νp is the equivalent sum for any products which are also solutes.48 (1) Solid-Liquid Equilibrium. The activity product on the mole fraction scale for a saturated solution in equilibrium with the solid phase AνABνBCνC · zH2O(s) is given by (xAfA)νA(xBfB)νB(xCfC)νCa1z and is equivalent to the equilibrium constant of the reaction

AνABνBCνC · zH2O(s) h νAA(aq) + νBB(aq) + νCC(aq) + zH2O (3) which is solved numerically using the Newton-Raphson or the Van Wijngaarden-Dekker-Brent method.61 (2) Vapor-Liquid Equilibrium. The solubility of a volatile acid HY such as HNO3 or HCl in aqueous solution is expressed as the equilibrium

HY(g) h H+(aq) + Y-(aq)

(4)

with the associated Henry constant xKH (atm-1) on the mole fraction scale

Thermodynamic Model of H+-NH4+-Na+-SO42--NO3--Cl--H2O

J. Phys. Chem. A, Vol. 114, No. 43, 2010 11597

TABLE 1: Henry’s Law Constants of Several Vapor-Liquid Equilibria K(Tr)a HNO3 HCl NH3 H2SO4 a

∆rH°

2.628563 × 10 2.04 × 106 1.066696 × 1011 1.463243 × 1017

∆a

-72300 -74852.0 -86251.4 -180383

6

-577.8992 -543.315 34.347 -1877.57

∆b 1.508666 1.3

ref

∆c 3.10414 × 10

-5

4.8685

158 23 48 51

The reference temperature Tr is 298.15 K. Henry’s law constants are calculated with eq 1.

KH )

x

xHfHxYfY pHY

(5)

where pHY (atm) is the equilibrium partial pressure of gas HY. For strong electrolytes the explicit treatment of the undissociated molecule HY(aq) in the liquid phase can be neglected. Its influence is integrated into the activity coefficients of the ions. For the weak electrolyte NH3(aq), the consideration of the equilibrium NH3(g) h NH3(aq) is appropriate. The dissolution of ammonia is described by transition to the liquid phase and subsequent dissociation:

where a1 ) f1x1 is the water activity of the solution and p°H2O (atm) is the vapor pressure of pure water. An expression for the calculation of p°H2O is given in Appendix III. (3) HSO4- Dissociation Equilibrium. Spectral studies have shown that the first dissociation of sulfuric acid

H2SO4(aq) h H+(aq) + HSO4-(aq)

is mainly complete for concentrations lower than 40 mol kg-1 at 298.15 K.62 We do assume that the first dissociation is also essentially complete at other temperatures within this concentration range. The second dissociation

(6)

NH3(g) h NH3(aq)

HSO4-(aq) h H+(aq) + SO42-(aq)

NH4+(aq) h H+(aq) + NH3(aq)

(7)

For weak acidic aerosol particles an equilibrium may be reached between gas-phase ammonia and dissolved NH3 in the liquid phase. However, in atmospheric aerosol particles the concentration of aqueous ammonia is very low compared to the amount of water present, so that it is not necessary to treat the solutions as being based on a mixed solvent. Furthermore, within the context of the PSC equations, dissolved molecular NH3 will not influence the activities of other components. Therefore we describe the solubility of ammonia in terms of the equilibrium +

NH3(g) + H (aq) h NH4 (aq)

KHSO4- )

x

xH+fH+xSO42-fSO42xHSO4-fHSO4-

KNH3 )

γNH4+m(NH4+) γH+m(H+)pNH3

(9)

is derived from the quotient of the equilibrium constants of reactions 6 and 7. The equilibrium partial pressure of H2SO4 over aqueous solutions is calculated using the equilibrium constant of the reaction H2SO4(g) h 2H+(aq) + SO42-(aq) according to Wexler and Clegg.51 All parameters required for calculation of equilibrium constants KH2SO4, KNH3, KHCl, and KHNO3 are summarized in Table 1. The equilibrium water vapor pressure pH2O (atm) over an aqueous solution is calculated using the relation

pH2O ) a1p°H2O

(10)

(13)

on the mole fraction scale is explicitly considered.55 apply the expression

log xKHSO4- ) 560.9505 - 102.5154 ln T -

(8)

whose equilibrium constant

(12)

with associated dissociation constant

1.117033 × 10-4T2 + 0.2477538T +

(11)

13273.75 T

(14)

for the second dissociation constant of sulfuric acid. This expression is based on the value 0.0105 mol kg-1 for the molar dissociation constant KHSO4- at 298.15 K63 and on the temperature dependence of KHSO4- proposed by Dickson et al.64 Knopf et al.65 have noted that the second dissociation of sulfuric acid is endothermic at temperatures lower than about 233 K if expression 12 is used, even though the reaction is exothermic at room temperature and above. They propose on the basis of their own measurements of the degree of dissociation of HSO4- an expression for the dissociation constant that ensure an exothermic reaction over the temperature range from about 100 to 500 K. However, our calculations show that osmotic coefficients and in particular the apparent relative molar enthalpy of H2SO4(aq) cannot adequately be represented with our model, if the dissociation constant proposed by Knopf et al.65 is used. Thus, we use eq 14 for the second dissociation of sulfuric acid. IV. Regression Model Model parameters Bca, B1ca, U1ca, Ucc′a, Uaa′c, W1ca, Wcc′a, Waa′c, V1ca, Q1cc′a, and Q1aa′c appearing in eq A17 (in the following also called activity parameters) are determined by fitting this equations to available thermodynamic data of binary and ternary

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aqueous solutions such as osmotic coefficients, emf measurements, vapor pressure measurements, or solubilities of solid phases. If the thermodynamic data for an aqueous solution are mainly available for a certain temperature To, it is appropriate to express the temperature dependence of a model parameter P as

P(T) ) Po + To2

(

)(

)

PoJ PoJ 1 1 + (T2 - To2) To - PoL 3 T To 6 (15)

where Po ) P(To). PoL ) PL(To) and PoJ ) PJ(To) are the enthalpy and heat capacity parameters at the reference temperature To, respectively. These parameters are given by

PL )

∂P ∂T

(16)

TABLE 2: Estimation of Observation Minus Model Differences ro for Several Thermodynamic Properties of Aqueous Solutions propertya

ro

propertya

ro

Φ, a1 R L1 ∆solH Φ Cp

0.002 0.05 5 150 1

γ E Φ L ∆dilH ∆mixH

0.005 0.00013 20 20 2

a Φ, osmotic coefficient; a1, water activity; γ, activity coefficient; E, emf (V); R, degree of dissociation of HSO4-; ΦL, apparent relative molar enthalpy (J mol-1); L1, partial molar enthalpy of H2O (J mol-1); ∆dilH, enthalpy of dilution (J mol-1); ∆solH, enthalpy of solution (J mol-1); ∆mixH, enthalpy of mixing (J mol-1); ΦCp, apparent molar heat capacity (J mol-1 K-1).

weight so that each weighted equation has a similar variance during the minimization. If L data sets each with Nl measurements are used, the weight wl of data set l is defined as

and Nl

2

1 ∂ 2 L ∂P 2 ∂P (T P ) ) 2 + T ∂T T2 ∂T ∂T

PJ )

(17)

a2 a3 p (T - T◦) + p3(T - T◦)2 + 2 2 4 a4 a 5 1 1 + p (T - T◦)3 + p5(T - T◦)4 + a6p6 6 4 8 T T◦ a7 1 1 1 1 1 - 2 + p8 + p 2 7 T2 2T T T T Tr T r r ◦

P(T) ≡ p1 +

(

°

(

)

(

)

)

a9 1 1 + p 2 9 (T - T )2 (T Tr)2 r ◦ a10 1 1 p 6 10 (T - T )3 (T Tr)3 r ◦

(

for l ) 1, ..., L

(19)

PL° and PJ° are obtained fitting model equations for the apparent relative molar enthalpy ΦL and the apparent molar heat capacity ΦCp to calorimetric data such as the enthalpy of dilution, or enthalpy of solution and the heat capacity of aqueous solutions. Model equations for ΦL and ΦCp are given in Appendix III. If thermodynamic data for an aqueous solution is available over a wider temperature range, the expression below is used to parametrize the temperature dependence of an activity parameter:

(

∑

1 1 ≡ w (y - y(xkl,P1, ...,PM))2 wl Nl k)1 kl kl

)

)

where (xkl, ykl) is the kth measurement from data set l, y(xkl,P1, ...,PM) is its model equivalent depending on the model parameters P1, ..., PM, and wkl is the weight of measurement value ykl. 1/wl1/2 can be considered as standard deviation σl of data set l. Hence, the problem is related to χ2 fitting, where the sum L

χ ) 2

Nl

∑∑

l)1 k)1

(

ykl - y(xkl, P1, ...,PM) σl

)

2

(20)

is minimized. Since the weights wl are not known a priori, the fit is conducted in two steps. First, an estimated value ro (see Table 2) is assigned to each difference between measurement and model equivalent so that the weight for data set l is given by

ro2 Nl 1 w ) Nl k)1 kl wlo

∑

(21)

(18)

where To and Tr are reference temperatures, a2 ) 0.1, a3 ) 0.01, a4 ) a5 ) 0.001, a6 ) a9 ) 100, a7 ) 1000000, and a10 ) 10000. In this case enthalpy and heat capacity parameters are derived applying the derivatives in eqs 16 and 17 to eq 18. The computer program NL2SOL66,67 is used to fit the PSC equations to available thermodynamic data for an aqueous solution. NL2SOL uses a nonlinear regression algorithm based on an adaptive “full Newton-type” least-squares method. Bunch et al.67 upgraded the program to account for simple constraints such as K(T) > 0 for an equilibrium constant K. Thermodynamic properties of aqueous solutions can vary in size over several orders of magnitude, e.g., from O(10-4) for emf measurements to O(104) for enthalpy data. Each equation representing a measurement value should have an adequate

The fit conducted with this weights lead to an intermediate parameter set P′1, ..., P′M. Second, given the intermediate model parameters, the final weights are calculated using eq 19 and the fitting procedure is repeated to determine the final set of model parameters P1, ..., PM. V. Parameterization of the Model Model parameters for binary and ternary systems and equilibrium constants for the formation of solid phases are required as functions of temperature. In this section new parametrizations for single solute aqueous solutions and ternary mixtures are described that are part of the system H+-NH4+-Na+SO42--NO3--Cl--H2O. Partly revised or fully adopted parametrizations for binary and ternary solutions, which have been treated in previously published studies on the systems H+-NH4+-SO42--NO3--H2O48 and H+-SO42--NO3-Cl--Br--H2O,23 are discussed in Appendix II. Table 3 lists



TABLE 3: Fitted Model Parameters for Ternary Aqueous Solutions p1a WNO3,SO4,H UNO3,SO4,H Q1,NO3,SO4,H WHSO4,NO3,H Q1,HSO4,NO3,H WNO3,SO4,NH4 Q1,NO3,SO4,NH4 BNa,HSO4 W1,Na,HSO4 U1,Na,HSO4 V1,Na,HSO4 WH,Na,SO4 UH,Na,SO4 Q1,H,Na,SO4 WH,Na,HSO4 UH,Na,HSO4 Q1,H,Na,HSO4 WSO4,HSO4,Na WH,NH4,Cl Q1,H,NH4,Cl WH,Na,Cl Q1,H,Na,Cl WCl,SO4,NH4 Q1,Cl,SO4,NH4 WCl,NO3,NH4 UCl,NO3,NH4 Q1,Cl,NO3,NH4 WCl,SO4,Hb

p2 0.473332

6.76969 -0.371651 -6.50598 2.60541 0.549035 2.58908 30.1442 -2.91079 -5.69007 6.23307 20.3014 18.9425 -12.0233 -7.18857 -3.93068 2.33868 -11.2799 -12.1476 5.87371 3.65224 -1.07133 1.3162 -0.957827 1.06856 1.52434 -1.40903 3.0

0.200294 -0.166925 -0.405404 -0.0317003 -0.0431800

-0.0765717 0.0396802 0.109513 0.0900980 0.516955 -0.282849 -0.021465 0.026288 0.0081773 -0.0314794

p1

p2

WH,NH4,NO3 UH,NH4,NO3 Q1,H,NH4,NO3

-3.49315 -0.282789 0.482245

0.0999678 0.0949601 -0.000690698

WH,Na,NO3 UH,Na,NO3 BNH4,HSO4 W1,NH4,HSO4 U1,NH4,HSO4 WH,NH4,SO4 WH,NH4,HSO4 Q1,H,NH4,HSO4 WSO4,HSO4,NH4 USO4,HSO4,NH4

0.676372 -0.147105 32.6487 -1.50189 0.431267 -3.23219 -16.3181 6.36458 -8.35898 12.1628

0.0114893 -0.0555459 -0.660106 0.00659269

-2.10909 1.22901 0.0516512 -0.347441 -3.53964 1.70900 -7.57075 -1.89951 4.21135 -6.63165 3.64398

WNa,NH4,SO4 Q1,Na,NH4,SO4 WNa,NH4,NO3 Q1,Na,NH4,NO3 WNa,NH4,Cl Q1,Na,NH4,Cl WNO3,SO4,Na UNO3,SO4,Na Q1,NO3,SO4,Na WCl,NO3,Na Q1,Cl,NO3,Na

p3 0.000766304

0.00228375

0.119687 -0.0444511 0.111909 -0.236403

-0.0124888 0.0171845 -0.0277552 0.239345 -0.144758 -0.258111 0.0864602 0.181504 0.144157 -0.100356

-0.00101421 0.000918004 0.000730282 -0.000441390 -0.00882252 0.00517359

-0.000646480 0.000526748

Parameters p1-p3 are coefficients of eq 18. The reference temperature To for the system HNO3-H2SO4-H2O is equal to 273.15 K. For all other systems To is equal to 298.15 K. b Fixed. a

TABLE 4: Equilibrium Constants for Solid Phasesa NaCl(s) NaCl · 2H2O(s) Na2SO4(s) Na2SO4 · 10H2O(s) Na3H (SO4)2(s) NaHSO4(s) NaHSO4 · H2O(s) NaNH4SO4 · 2H2O(s) NaNO3(s) NaNO3 · Na2SO4 · H2O(s) NH4Cl(s) NH4NO3 (III)e NH4NO3 (IV)g NH4NO3 (V)h (NH4)2SO4(s) (NH4)3H(SO4)2(s) NH4HSO4(s) (NH4)2SO4 · 2NH4NO3(s) (NH4)2SO4 · 3NH4NO3(s)

To (K)

Ko

∆rHo (J mol-1)

∆a (J mol-1 K-1)

298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 305.38 298.15 256.2 298.15 298.15 298.15 298.15 298.15

38.4372 37.1127 0.538394 0.0595821 0.185993 4.1089 0.753343 0.146791 12.1261 2.57114 17.4970 15.5719 12.2260 1.93357 1.093739 0.15815 1.96777 84.33 825.348

4016.94 18301.0 -2430.00 79450.0 -21385.8 -19137.3 -5757.47 23750.3 20350.4 30036.8 16821.6 23260.0 25690.0 30410.0 6084.0 -3119.48 -14961.6 58850.0 84860.0

-342.831 -2021.39 -1301.11 -14.8929 -351.370 7.99323 2471.61 -973.103 -780.026 319.213

-6.393 -508.688 -222.996

∆b (J mol-1 K-2) 0.704029 6.69664 3.19915

-9.5116 2.72068 -1.18749

ref this work b c d this work this work this work this work this work this work this work f f f i this work this work f f

a Calculation of equilibrium constants with eq 1. Activities are on the molal scale. b Ko is calculated from the standard state molar Gibbs energy of solution G°sol,r given by Archer and Carter76 using the relation Ko ) exp(-(G°sol,r)/(RTo)). The enthalpy change of the dissolution reaction is calculated from the enthalpy of formation of Na+ and Cl- and of NaCl · 2H2O(s) tabulated in Wagman et al.98 and given by Archer,39 respectively. The heat capacity change of the reaction is fitted by the model. c ∆rH° is calculated from the enthalpy of formation of the ions and of the solid tabulated in Wagman et al.98 Ko and the heat capacity change of the dissolution reaction are fitted by the model. d The value of Ko of Clegg et al.49 is used. ∆rH° is calculated from the enthalpy of formation of the ions and of the solid tabulated in Wagman et al.98 The heat capacity change of the dissolution reaction is calculated from the standard state heat capacity of Na+ and SO42- at 298.15 K given in Wagman et al.98 and Clegg and Brimblecombe,55 respectively, and from the heat capacity of the solid given in Pabalan and Pitzer.182 e Rhombohedric form between 305.38 and 357.25 K. f Clegg et al.48 g Rhombic form between 256.2 and 305.38 K. h Orthorhombic form below 256.2 K. i Clegg et al.56

the obtained model parameters for ternary systems. Equilibrium constants of salts and acid hydrates are given in Tables 4 and 5, respectively.

(1) H2SO4-H2O. Clegg and Brimblecombe55 have developed a model of this system from 0 to 40 mol kg-1 sulfuric acid and temperatures from below 200-328.15 K. Later, two

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TABLE 5: Equilibrium Constants for Acid Hydratesa H2SO4 · H2O(s) H2SO4 · 2H2O(s) H2SO4 · 3H2O(s) H2SO4 · 4H2O(s) H2SO4 · 6,5H2O(s) HNO3 · H2O(s) HNO3 · 2H2O(s) HNO3 · 3H2O(s) HCl · 3H2O(s)

A

B

C

D

ref

996.167 15.0848 109.202 93.8083 27.3474 37.2355 4.557 18.3084 27.5098

-45592.2 -870.435 -12805.6 -11956.8 -5719.47 -4061.51 3320.2 -3408.14 -2478.44

1.64244

-76.4537

this work this work this work this work this work ref 23 ref 158 ref 23 ref 23

-0.200569 -0.163049 -0.045788 -0.033934

a Calculation of equilibrium constants with the expression ln K(T) ) A + B/T + CT + DT1/2, where T is temperature. Activities are on the molal scale.

studies have been published that present measurements of the degree of dissociation of the HSO4- ion over wide ranges of concentration and temperature.65,68 Especially for high molalities and low temperatures the model of Clegg and Brimblecombe55 yields values of the degree of dissociation that are low compared to the results of these two studies. We build our model on almost all of the thermodynamic data used by Clegg and Brimblecombe55 to parametrize their model. Furthermore, we use the measurements of the degree of dissociation of HSO4- by Knopf et al.65 and Lund Myhre et al.,68 some newer isopiestic water activity measurements of Rard69 at 298.15 K, measurements of the enthalpy of dilution at 283.15, 293.15, and 313.15 K of Ru¨tten et al.,70 and heat capacity data of Socolik71 at 313.15 K. Since the degree of dissociation of HSO4- is a priori unknown, the model equations are changed to completely dissociated HSO4-.55,72 The derivatives of excess Gibbs energy with respect to temperature needed to calculate the apparent relative molar enthalpy and the apparent molar heat capacity of aqueous H2SO4 on the basis of complete dissociation of HSO4- ion (see eqs 19-25 of Clegg and Brimblecombe55) are numerically computed using simple centered differences with a step size of 0.01 K. Figure 1 exhibits a comparison of measurements of the degree of dissociation of HSO4- by Knopf et al.65 with our own model results and results of the model of Clegg and Brimblecombe.55 Especially at low temperatures and high H2SO4 concentrations the degree of dissociation is better reflected by the newly parametrized model. The underprediction of results of Knopf et al. by both models around 290 K may be justified because their measurements at 298.15 K are too high compared to results of other studies.62,68,73–75 Apart from representation of HSO4- dissoziation, the well established model of Clegg and Brimblecombe55 is confirmed by the recent work. Our manifold results of the fit of the PSC equations to the data are comparable to those of Clegg and Brimblecombe. We therefore refer to their Figures 2-7 for comparison between model results and measurements. Smaller differences near the lower temperature limit of both models may be ascribed mainly to the use of newly available low temperature heat capacity measurements for pure water76,77 in the present work (see Appendix III.2). Six studies,78–83 whose results are not fitted, provide water vapor pressure measurements over aqueous H2SO4. With the exception of the works of Daudt78 and Collins79 the main focus of these studies is on low temperatures. The data of Daudt78 and of Zhang et al.80 are partially systematically inaccurate.82 Figure 2 shows percent deviations between predicted and measured water vapor pressure for the remaining four studies. The model agrees with the data of Collins79

Figure 1. Degree of dissociation of HSO4- in aqueous H2SO4 solutions with concentrations between 0.54 and 15.23 mol kg-1 plotted as a function of temperature. Symbols: data of Knopf et al.65 Dashed lines: predicted by the model of Clegg and Brimblecombe.55 Solid lines: fitted model.

within (2% up to about 53 wt % H2SO4. For higher concentrations the model predictions are systematically up to 4% lower than the measurements. The agreement of (15% with measurements of Becker et al.81 for a solution with 35 wt % H2SO4 is satisfactory in view of the low temperature range from 200 to 230 K. Large deviations between model predictions and water vapor pressure measurements of Becker et al. over a solution with 16 wt % H2SO4 may be explained due to ice precipitation since the solution is supersaturated with respect to ice within the considered temperature range. Except for their results for 24.61 mol kg-1, the water vapor pressure measurements of Massucci et al.82 for nine different concentrations between 5.05 and 26.18 mol kg-1 agree with model predictions above and below about 235 K within the range (8% and (15%, respectively. For 24.61 mol kg-1 the measurements below 230 K are up to 80% higher than the model predictions. Massucci et al.82 have obtained similar deviations comparing their measurements for this concentration with calculations based on tabulated thermodynamic properties84 and attributed these deviations to potential measurement errors. Except for small systematic deviations



Figure 2. Percent deviation between model predictions and measurements of the equilibrium water vapor pressure over aqueous H2SO4 solutions plotted as function of temperature. (a) For the data of Collins79 for 29.9 (plus signs), 38.23 (crosses), 43.36 (stars), 48.12 (open squares), 52.68 (solid squares), 58.03 (open circles), and 65.47 wt % (solid circles) H2SO4. (b) For the data of Becker et al.81 for 35 wt % H2SO4. (c) and (d) For the data of Massucci et al.82 for 5.05 (plus signs), 5.94 (crosses), 6.1 (stars), 7.01 (open squares), 7.07 (solid squares), 8.05 (open circles), 10.02 (solid circles), 24.61 (triangles), and 26.18 mol kg-1 (diamonds) H2SO4. (e) For the data of Yao et al.83 for 10.19 mol kg-1 H2SO4.

the model agree well with water vapor pressure measurements over a solution with 10.19 mol kg-1 H2SO4 for the temperature range from about 241 to 298 K of Yao et al.83 The authors have compared their measurements with predictions

of the model of Clegg et al.49 They have obtained similar deviations of about 3% between measurements and model predictions and attributed them to a possible systematic measurement error.

TABLE 6: Fitted Model Parameters for Aqueous H2SO4 Solutionsa BH,SO4 U1,H,SO4 V1,H,SO4 W1,H,SO4 BH,HSO4 U1,H,HSO4 V1,H,HSO4 W1,H,HSO4 WSO4,HSO4,H USO4,HSO4,H Q1,SO4,HSO4,H

BH,SO4 U1,H,SO4 V1,H,SO4 W1,H,SO4 BH,HSO4 U1,H,HSO4 V1,H,HSO4 W1,H,HSO4

p1

p2

p3

p4

-35.2100285 5.82101432 -14.9993592 -5.95331738 24.5633629 -1.69437917 -2.67302875 -11.3761295

-15.3372551 1.18407067 -0.417415775 0.953958781 -24.5324780 -0.0385965615 0.407262554 0.681282473 -1.99734782 -1.31249629 1.31526118

-5.88243070 -0.225147849 0.211756355 -0.105562709 4.20128719 -0.235672433 0.211541809 -0.0759843531 -0.223293849 -0.485887548 0.132966185

0.936327610 0.0491750602 -0.0659871525 -0.000173813385 -0.212244677 0.0768174054 -0.0797974016 0.0190269679 0.0737494333 -0.0131706692 -0.0530602070

p5

p8

p9

-0.00519186637 -0.000442944316 0.000357416157 -0.000172540921 0.00110572570 -0.000283785580 0.000570553533 0.000103511989

-1445.58657 -15.9961788 3.39203503 -26.3879139 -3369.81076 66.2256049 -122.199570 -17.1312051

143.324567 28.1963836 -19.7827379 15.6651820 -403.566094 24.9304164 -13.7251402 9.60280213

The parameters RH,SO4 and RH,HSO4 are equal to 9.5 and 17. Parameters p1-p9 are coefficients of eq 18. The reference temperatures To and Tr are 298.15 and 160 K. a

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TABLE 7: Sources of Thermodynamic Data for Aqueous NH4Cl Solutionsa Φ(iso) Φ(iso) Φ(iso) Φ(iso) Φ(gp) Φ(edb) Φ(edb) Φ(hm) ∆solH Φ L Φ Cp Φ Cp Φ Cp Cp Cp sol sol

Nmeasb

mminc

mmaxc

d

weight

T (K)

ref

14 (0) 10 (0) 11 (4) 8 (0) 33 (2) 64 (0) 54 (0) 14 (0) 50 (45) 51 (0) 36 (0) 8 (0) 10 (0) 24 (21) 10 (0) 17(0) 9 (0)

0.203 5.043 1.904 2.332 0.0037 6.365 6.258 0.200 0.203 0.0001 0.0006 0.037 0.054 0.0058 1.014 4.586 6.897

4.647 7.390 7.418 6.209 0.102 22.76 21.62 6.000 22.76 7.000 7.401 0.216 0.404 11.44 5.469 14.45 10.09

KCl NaCl NaCl NaCl

1.0 1.0 1.0 1.0 1.0 0.1 0.1 0.1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 258.15-373.15 293.15-333.15

183 183 184 185 92 89 90 91 96 93 93 95 186 96 94 107 97

e f f f e, f f

a Φ, osmotic coefficient from isopiestic (iso), hygrometric (hm), freezing point depression (fp), or electrodynamic balance (edb) measurement; ∆solH, integral enthalpy of solution of NH4Cl(s); ΦL, apparent relative molar enthalpy; ΦCp, apparent molar heat capacity; Cp, heat capacity; sol, solubility of NH4Cl(s) in NH4Cl(aq). b Number of measurements. The value in parentheses is the number of rejected measurements. c Molality of NH4Cl. d Isopiestic reference standard. e Five and three values of the integral heat of solution of NH4Cl(s) and heat capacity of NH4Cl(aq) for supersaturated solutions at 298.15 K are derived fitting the data of Mishchenko and Ponomareva96 to an expression of the form ∆solH ) a + bT + cm + dT2 + em2 + fTm, where T is temperature in Kelvin and m is NH4Cl molality. f The value -56.6 J mol-1 K-1 is used for the apparent molar heat capacity at infinite dilution.187

The activity product of a saturated solution in equilibrium with the hydrate H2SO4 · nH2O(s) is given by aH+aSO42-a1n. Activity products of the five different hydrates occurring below 40 mol kg-1 H2SO4 are calculated using the parametrized model and fitted to the expression

ln(aH+aSO42-an1) ) a +

1 b + cT + dT /2 T

(22)

for each of the freezing points measured by Gable et al.85 and Kunzler and Giauque.86 The melting points measured by Beyer et al.87,88 are not used for determination of activity products because they exhibit large scatter. Parameters of expression 22 for the five hydrates are given in Table 5. Model parameters for the system H2SO4-H2O are listet in Table 6. (2) NH4Cl-H2O. Activity parameters for 298.15 K are determined by fitting the PSC equations to available isopiestic, hygrometric, and edb measurements of water activity (see Table 7). A low weight is assigned to the edb data of Liang and Chan89 and Ha et al.90 as well as to the hygrometric measurements of El Guendouzi et al.91 since they show relatively large scatter. Osmotic coefficients at low concentrations are derived by applying eq A23 to freezing point data of Garnsey and Prue.92 Values of the relative partial molar enthalpy and partial molar heat capacity of water needed for this purpose are determined fitting eqs A31 and A35 to available calorimetric data.93–95 Measured and fitted osmotic coefficients are compared in Figure 3. In a second step, enthalpy and heat capacity parameters at 298.15 K are determined by fitting the PSC equations to calorimetric data and salt solubilities. Mishchenko and Ponomareva96 have measured the integral heat of solution of NH4Cl(s) and the heat capacity of NH4Cl(aq) at 298.15, 323.15, and 348.15 K for concentrations up to saturation. The increasing solubility of NH4Cl(s) with increasing temperature (see Figure 4) allow an extrapolation of data of Mishchenko and Ponomareva to determine the apparent relative molar enthalpy and apparent molar heat capacity of supersaturated solutions at

Figure 3. Molal osmotic coefficient of NH4Cl(aq) at 298.15 K plotted as a function of square root of NH4Cl molality. Symbols: data of Wishaw and Stokes183 (crosses), Shul’ts et al.184 (diamonds), Kirgintsev and Luk’yanov185 (open squares), Garnsey and Prue92 (solid squares), Liang and Chan89 (circles), Ha et al.90 (triangles), and El Guendouzi et al.91 (dots). Line: fitted model.

Figure 4. Solubility of NH4Cl(s) and ice in NH4Cl(aq) as a function of temperature plotted as function of NH4Cl molality. Symbols: data of Silcock107 (plus signs) and Ji et al.97 (squares). Lines: fitted model.

298.15 K. Model parameters for the system NH4Cl-H2O are given in Table 8. Ji et al.97 have evaluated available thermodynamic data to estimate the equilibrium constant KNH4Cl of the dissolution reaction

NH4Cl(s) h NH4+(aq) + Cl-(aq)

(23)

between 273.15 and 400 K. They have obtained the values 17.272 mol2 kg-2 for the equilibrium constant, 16.2 kJmol-1



TABLE 8: Fitted Model Parameters for Aqueous NH4Cl Solutions at 298.15 Ka parameter BNH4,Cl W1,NH4,Cl U1,NH4,Cl V1,NH4,Cl

value

parameter

8.63482 -0.398856 2.67936 -1.82137

L BNH 4,Cl L W1,NH 4,Cl L U1,NH 4,Cl L V1,NH 4,Cl

value

parameter

value

0.0847867 -0.0454883 -0.0983014 0.0584381

J BNH 4,Cl J W1,NH 4,Cl J U1,NH 4,Cl J V1,NH 4,Cl

0.000188580 0.000444588 0.000801184 -0.000571597

a Coefficient RNH4,Cl is equal to 15. The temperature dependence of activity parameters is obtained from the parameters given in the table using eq 15.

Figure 6. Differences between fitted and measured emf in Na2SO4(aq) plotted as a function of square root of molality for the data of Shibata and Murata105 (plus signs), Rondinini et al.104 (open squares), and Harned and Hecker106 (solid squares).

Figure 7. Solubility of Na2SO4(s), Na2SO4 · 10H2O(s), and ice in Na2SO4(aq) as a function of temperature plotted as a function of square root of Na2SO4 molality. Symbols: data of Indelli100 (crosses), Holmberg239 (squares), and Silcock107 (plus signs). Lines: fitted model.

Figure 5. Osmotic coefficient of Na2SO4(aq) (a, b) at temperatures indicated and at the freezing temperature (ft) and differences between fitted and measured osmotic coefficients (c) plotted as a function of square root of Na2SO4 molality. Lines in (a, b): fitted model. Symbols: data of Wu et al.206 (solid diamonds), Platford230 (pentagons), Downes and Pitzer231 (open squares), Rard and Miller232 (solid squares), Filippov et al.233 (dots) at 298.15 K, and for the data of Randall and Scott234 (plus signs, 272.7-273.15 and 298.15 K), Gibson and Adams235 (crosses, 298.15 and 300.65 K), Indelli100 (stars, 272.27-273.12 and 298.15 K), Humphries et al.236 (open triangles, 333.13 K), Childs and Platford101 (inverted open triangles, 288.15 K), Moore et al.237 (open diamonds, 353.37 K), Platford238 (solid triangles, 273.15 K), Holmberg239 (inverted solid triangles, 383.14 K), and Rard et al.99 (open circles, 298.15 and 323.15 K).

for the enthalpy change ∆rH°, and -136.83 J mol-1 K-1 for the heat capacity change ∆rCp of reaction 21 at 298.15 K. In the present work, the equilibrium constant is simultaneously fitted with the enthalpy and heat capacity parameters. The obtained values 17.497 mol2 kg-2 for KNH4Cl and 16.822 kJmol-1 for ∆rH° are in good agreement with the results of Ji et al.97 In

contrast, the obtained value of -34.8 J mol-1 K-1 for ∆rCp deviates from the result of Ji et al. From data compiled by Wagman et al.98 the values ∆rH° ) 14.761 kJ mol-1, and ∆rCp ) -140.6 J mol-1 K-1 can be derived. (3) Na2SO4-H2O. The data used to parametrize the PSC equations for this system are mainly selected according to the critical evaluation of Rard et al.99 (see their Tables IV-VII). Figure 5 shows fitted and measured osmotic coefficients and the differences between them for several temperatures. Except for minor deviations for osmotic coefficients derived from freezing point data of Indelli100 and for isopiestic measurements at 288.15 K of Childs and Platford101 an excellent agreement between fitted and measured osmotic coefficients is achieved. Test calculations show that the inclusion of edb data for supersaturated solutions102,103 lead to large systematic deviations between the model and all measured osmotic coefficients over the entire concentration range. Therefore we limit our model for aqueous Na2SO4 to concentrations up to saturation. Rard et al.99 have mentioned that the emf measurements of Rondinini et al.104 are internally consistent even though they exhibit relatively large scatter. Since their emf data lead to activity coefficients at low concentrations over a wide range of temperature we use the data of Rondinini et al. in addition to

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TABLE 9: Fitted Model Parameters for Aqueous Na2SO4 Solutionsa BNa,SO4 UNa,SO4 VNa,SO4 WNa,SO4

∆solH°

p1b

p2

p3

p4

p6

p8

38.4723 -19.3736 8.50222 -10.6527

2650.31 10100.5 -6700.69 4310.16

-157.541 -598.546 396.592 -255.741

4.70760 17.7425 -11.7319 7.59891

123124 468105 -310736 199619

-19077.8 -63494.6 42478.4 -26729.2

r1c

r2

r3

-2217.12

-317.532

0.825249

d

s1 C°p

Φ

-193.7

s2

s3

s4

-14.9243

0.0828965

1494.68

Coefficient RNa,SO4 is equal to 13. p1-p8 are parameters of eq 18. The reference temperatures To and Tr are 298.15 and 215 K. c Parameters r1-r3 of the expression ∆solH° ) r1 + r2(T - To) + r3(T - To)2 for the integral heat of solution ∆solH° of Na2SO4(s) are simultaneously fitted with the model parameters. d Parameters s1-s4 of the expression ΦC°p ) s1 + s2(T - To) + s3(T - To)2 + s4(T - To)/(T - Tr2) for the apparent molar heat capacity at infinite dilution are simultaneously fitted with the model parameters. The temperature Tr2 is equal to 225 K. a

b

the emf measurements of Shibata and Murata105 and Harned and Hecker.106 Figure 6 show differences between fitted and measured emf data. The results of the fit of the PSC equations to calorimetric data listed in Table VII of Rard et al.99 are satisfactory but cannot be shown here. To determine the equilibrium constant of the reaction

Na2SO4(s) h 2Na+(aq) + SO42-(aq)

(24)

activity products for the salt solubilities compiled by Silcock107 are calculated using the model and are fitted to eq 1. The enthalpy of reaction at 298.15 K is fixed at -2430 J mol-1.98

For the heat capacity change of reaction 22 at 298.15 K the value -347.29 J mol-1 K-1 is obtained, whose absolute value is about 7% larger than that derived from the tabulation of Wagman et al.98 The model satisfactory represents solubilities of Na2SO4 · 10H2O(s) in aqueous Na2SO4, if the equilibrium constant at 298.15 K of Clegg et al.49 and the values for the enthalpy change and heat capacity change at 298.15 K derived from the tabulation of Wagman et al.98 are used. Figure 7 exhibits solubilities of ice, Na2SO4(s), and Na2SO4 · 10H2O(s) in aqueous Na2SO4. Model parameters are listed in Table 9. (4) NaNO3-H2O. Archer108 has used an extended Pitzer model38 for a description of the thermodynamic properties of aqueous NaNO3 for temperatures between 236 and 425 K and concentrations from infinite dilution to saturation. For the

Figure 8. Osmotic coefficients of NaNO3(aq) (left) and differences between fitted and measured values (right) plotted as a function of the square root of molality. (a), (b) At 298.15 K. Symbols: data of Scatchard and Jones240 (crosses), Robinson241 (stars), Pearce and Hopson242 (open squares), Kangro and Groeneveld243 (solid squares), Kirgintsev and Luk’yanov185 (open circles), Kirgintsev and Luk’yanov131 (dots), Shpigel’ and Mishchenko244 (plus signs), Bezboruah et al.213 (open triangles), Tang and Munkelwitz102 (solid triangles), and El Guendouzi and Dinane109 (inverted triangles). Lines: (a) fitted model, (b) difference between the fitted model and the model of Archer.108 (c), (d) At other temperatures. Symbols: freezing point depression data of Scatchard and Jones240 (crosses), Rodebush245 (inverted solid triangles), Holmberg239 (open diamonds), Kangro and Groeneveld243 (solid squares) at 293.15 K, Shpigel’ and Mishchenko244 (plus signs) at 274.15, 323.15, and 348.15 K, and Voigt et al.246 (solid diamonds) at 373.45 K. Lines: (a) fitted model at freezing temperatures (ft) and temperatures indicated; (b) differences between the fitted model and the model of Archer108 for 263.15 (solid) and 323.15 K (dashed).



Peka´rek et al.111 have measured the enthalpy of solution of NaNO3(s) in NaNO3(aq). These measurements are an approximation of the differential enthalpy of solution ∆difH and are fitted as

∆difH ≈ ∆solHo +

Figure 9. Differences between the fitted and measured apparent molar heat capacity of NaNO3(aq) plotted as function of square root of molality. (a) At 298.15 K for the data of Epikhin and Stakhanova247 (plus signs), Puchkov et al.248 (solid squares), Petrov and Puchkov249 (crosses), Enea et al.113 (stars), and Roux et al.95 (open squares). Line: Differences between the fitted model and the model of Archer.108 (b) At other temperatures for the data of Puchkov et al.248 (solid squares, 323.15, 348.15, and 373.15 K) and Archer112 (circles, 234-285 K).

parametrization of our model we use mainly the thermodynamic data for temperatures below 373.15 K listed in Table 3 of Archer,108 and in addition hygrometric water activity measurements of El Guendouzi and Dinane,109 and edb measurements of the water activity of supersaturated solutions of Tang and Munkelwitz,102 both at 298.15 K. The fit of the PSC equations to the edb data prevent a strong increase of water activity beyond the saturation concentration (see Figure 8). Osmotic coefficients for 298.15 K tabulated in Wu and Hamer,110 and calorimetric properties calculated with the model of Archer108 are used to convert osmotic coefficients at the freezing temperature to their values at 298.15 K.

meΦL(me) - msΦL(ms) me - ms

(25)

where me is the final molality, if NaNO3(s) is solved in an aqueous NaNO3 solution of molality ms. The integral heat of solution at infinite dilution ∆solH° is expressed as ∆solH° ) r1 + r2(T - To). The value 20 235 J mol-1 obtained for 298.15 K is in very good agreement with the result of Archer108 (20213 J mol-1). Carter and Archer112 have used the heat capacity measurements of Archer and Carter76 of cold-stable and supercooled H2O to convert their heat capacity measurements of NaNO3(aq) for concentrations from 0.1 to 10 mol kg-1 and temperatures between 236 and 285 K to apparent molar heat capacities. The apparent molar heat capacity data of Carter and Archer112 can be fitted consistently since our calculation of the heat capacity of cold-stable and supercooled water is also based on the data of Archer and Carter76 (see Appendix III). For 298.15 K we obtain -31.2 J mol-1 K-1 for the apparent molar heat capacity at infinite dilution, which is slightly larger than the result of Archer108 (-35.4 J mol-1 K-1). However, our value agrees well with results of Roux et al.95 (-30.15 J mol-1 K-1) and Enea et al.113 (-28.9 J mol-1 K-1). Differences between fitted and measured apparent molar heat capacities are shown in Figure 9. Model parameters for the system NaNO3-H2O are given in Table 10. Parameters of the equilibrium constant of the reaction

NaNO3(s) h Na+(aq) + NO3-(aq)

(26)

are simultaneously fitted with the model parameters. For the equilibrium constant and the enthalpy change at 298.15 K we obtain the values12,13 mol2 kg-2 and 20350 J mol-1, which are in good agreement with the result of Archer108 (12.29 mol2 kg-2) and the value 20370 J mol-1 tabulated in Wagman et al.98 The value -162 J mol-1 K-1 obtained for the heat capacity change of reaction 24 at 298.15 K is 22% larger than the value derived from the tabulation of Wagman et al.98

TABLE 10: Fitted Model Parameters for Aqueous NaNO3 Solutionsa p1b BNa,NO3 1 BNa,NO 3 UNa,NO3 VNa,NO3 WNa,NO3

p2

22.3981 -14.4987 0.348567 -2.17531 0.410584

∆solH° C°p

p4

p5

p8

22.7141

-2.48905

0.141184

-0.000307712

8136.03

63.3881 -42.1588 25.5752

-9.02386 5.86559 -3.66023

0.666362 -0.422689 0.272325

-0.00214918 0.00131458 -0.000886132

18278.9 -12555.3 7394.62

r1c

r2

20234.5

-81.0962

s1d Φ

p3

-31.2358

s3

s4

s5

-11.4175

0.109233

-0.00620331

1108.12

-13505.9 -19521.4 13919.5 -7956.71

p10 3801.11 4871.50 -3531.65 1986.13

Coefficients RNa,NO3 and are equal to 15 and 13. p1-p10 are parameters of eq 18. The reference temperatures To and Tr are 298.15 and 210 K. c Parameters r1 and r2 of the expression ∆solH° ) r1 + r2(T - To) for the integral heat of solution ∆solH° of NaNO3(s) are simultaneously fitted with the model parameters. d Parameters s1-s5 of the expression ΦC°p ) s1 + s2(T - To) + s3(T - To)2 + s4(T - To)3 + s5(T - To)/(T - Tr2) for the apparent molar heat capacity at infinite dilution are simultaneously fitted with the model parameters. Temperature Tr2 is equal to 215 K. a

1 RNa,NO 3

s2

p9

b

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Figure 11. Solubilities of solid phases in NaCl(aq) as a function of temperature plotted as a function of NaCl molality. Symbols: data compiled by Silcock107 for ice (crosses), NaCl · 2H2O(s) (solid squares), and NaCl(s) (open squares). Lines: fitted model.

Figure 10. Mean activity coefficient of NaCl in NaCl(aq) plotted as function of temperature and the square root of NaCl molality. (a) Fitted model. (b) Deviations between the model of Archer and Carter76 and the fitted model.

(5) NaCl-H2O. Archer39 has used an extended version of Pitzer’s ion-interaction model to describe the thermodynamic properties of aqueous NaCl for concentrations from 0 to 6 mol kg-1 for the temperature range 250-600 K and pressures up to 100 MPa. Archer and Carter76 have revised this model including heat capacity measurements of NaCl(aq) in cold-stable and supercooled states. Because of the considerable quantity of available thermodynamic data for aqueous NaCl solutions we abandon a direct fit to these data. The PSC equations are fitted to 960 values each of the osmotic coefficient, mean activity coefficient, apparent relative molar enthalpy, and apparent molar heat capacity of NaCl(aq) calculated with the model of Archer and Carter76 for NaCl concentrations up to 6.1 mol kg-1 and temperatures between 245 and 373.15 K at standard pressure (101 325 Pa). Figure 10 shows exemplary mean activity coefficients of NaCl calculated with the fitted model. The differences between mean activity coefficients calculated with our model and with the model of Archer and Carter76 are sufficiently small. Comparable differences are obtained for the osmotic coefficient, the apparent relative molar enthalpy, and the apparent molar heat capacity.

Model parameters for aqueous NaCl solutions are listed in Table 11. Activity products for the solubilities compiled by Silcock107 are calculated with the model and fitted to eq 1 to determine the equilibrium constants of the dissolution reactions NaCl(s) h Na+(aq) + Cl-(aq) and NaCl · 2H2O(cr) h Na+(s) + Cl-(aq) + 2H2O(l). Figure 11 exhibits solubilities of solid phases in NaCl(aq). (6) H2SO4-Na2SO4-H2O. Pitzer’s ion-interaction model has been used to describe thermodynamic properties of this system for 298.15 K114 and for temperature ranges between 273.15 and 548.15 K.37,115 The model has only been parametrized for 298.15 K.49 For temperatures below 298.15 K and for the temperature range above 298.15 and below 383.46 K only salt solubilities are available for parametrization of a temperature dependent PSC model. However, the large number of measurements for 298.15 K along with the fact that the models for the subsystems H2SO4-H2O and Na2SO4-H2O cover wide temperature ranges should be adequate to build a temperature dependent model for H2SO4-Na2SO4 mixtures. The used thermodynamic data are listed in Table 12. Harned and Sturgis,116 Randall and Langford,117 and Covington et al.118 investigated the cell

(Pt)H2(g)|H2SO4(m1), Na2SO4(m2)|Hg2SO4(cr)|Hg(l) (27) whose emf is given by

E ) E° -

RT ln(γH+2γSO42-m(H+)2m(SO42-)) 2F

(28)

For the data of Harned and Sturgis116 the standard potential E° of the cell is determined by application of eq 28 to a pure 0.01

TABLE 11: Fitted Model Parameters for Aqueous NaCl Solutionsa BNa,Cl UNa,Cl VNa,Cl WNa,Cl

p1b

p2

p3

p4

p7

p8

p9

20.2120 -2.70657 -3.02331 -5.26167

3.81253 -4.46006 2.44701 -0.797114

0.523626 1.53381 -0.993737 0.585186

-0.0556459 -0.150788 0.0979532 -0.0609720

13.6714 -7.17438 2.71029 -1.58017

-1353.09 -487.196 385.487 -230.617

337.864 408.735 -294.200 165.241

s1 C°p

Φ

c

-85.3

s2

s3

s4

s5

-4.70449

0.0428740

-0.000267204

464.858

Coefficient RNa,Cl is equal to 5. p1-p9 are parameters of eq 18. The reference temperatures To and Tr are 298.15 and 215 K. c The apparent molar heat capacity at infinite dilution is expressed as ΦC°p ) s1 + s2(T - To) + s3(T - To)2 + s4(T - To)3 + s5(T - To)/(T - Tr2), where the reference temperature Tr2 is equal to 225 K. A value taken from Archer and Carter76 is used for ΦC°p at 298.15 K. a

b



TABLE 12: Sources of Thermodynamic Data for Aqueous H2SO4-Na2SO4 Solutionsa Φ(iso) Φ(iso) Φ(iso) Φ(iso) Φ(iso) Φ(iso) Φ(edb) Φ(iso) emfi emfi emf Rl Rl sol sol sol sol sol

Nmeasb

mminc

mmaxc

22 (1) 151 (0) 32 (0) 96 (3) 173 (0) 29 (0) 20 (0) 64 (0) 13 (0) 16 (0) 6 (0) 8 (1) 13 (0) 91 (44) 79 (16) 91 (41) 59 (35) 77 (55)

0.104 0.158 1.000 0.193 1.000 0.520 10.68 0.480 0.030 0.200 0.020 0.0004 0.0009 0.554 7.536 7.910 4.109 16.57

6.320 6.611 10.68 4.754 14.96 6.786 15.00 5.006 1.100 2.000 0.109 0.509 2.368 8.585 18.94 17.28 9.879 29.49

composition NaHSO4 e f g g NaHSO4 NaHSO4 h j k variable NaHSO4 NaHSO4 variable variable variable variable variable

d

weight

T (K)

ref

NaCl NaCl CaCl2 NaCl CaCl2 NaCl

1.0 1.0 1.0 1.0 1.0 1.0 0.1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

298.15 298.15 298.15 298.15 298.15 383.46 298.15 383.46 298.15 298.15 298.15 298.15 298.15 273.15-293.15 273.15-319.15 273.15-319.15 288.15-319.15 273.15-319.15

188 189 189 190 190 191 102 115 116 117 118 192 74 107 107 107 107 107

NaCl

1 2 3 4 5

a Φ, stoichiometric osmotic coefficient from isopiestic (iso) or electrodynamic balance (edb) measurement; emf, electromotoric force; R, degree of dissociation of HSO4-; sol, salt solubility. b Number of measurements. The value in parentheses is the number of rejected measurements. c Total molality m(H2SO4) + m(Na2SO4). d Reference standard for isopiestic measurements or composition of the solid phase: 1, Na2SO4 · 10H2O(s); 2, NaHSO4 · H2O(s); 3, Na3H(SO4)2(s); 4, Na2SO4(s); 5, NaHSO4(s). e (1 - y)H2SO4 + yNa2SO4 for y ∈ {0.8492, 0.70189, 0.55595, 0.50011}. f y ∈ {0.55595, 0.50011}. g y ∈ {0.37439, 0.24962, 0.12471}. h (1 - x)NaHSO4 + xNa2SO4 for x ∈ {0.35, 0.5, 0.65}. i Conversion of emf from international volt to volt. j m(H2SO4) ∈ {0.1, 0.01}. k m(H2SO4) + m(Na2SO4) ∈ {0.1, 0.5, 1,2}. l Concentrations in mol dm-3 are converted to molality using the density of NaHSO4(aq) tabulated in So¨hnel and Novotny´.193

Figure 12. Stoichiometric osmotic coefficient of {(1 - y)H2SO4 + yNa2SO4}(aq) at 298.15 K plotted as function of the square root of total molality. Symbols: data of Rard189 (crosses), Rard193 (open circles), Stokes188 (solid squares), and Tang and Munkelwitz102 (open squares). Lines: fitted model.

Figure 13. Stoichiometric mean activity coefficient of H2SO4 in aqueous H2SO4-Na2SO4 mixtures at 298.15 K plotted as a function of total molality (a) and H2SO4 mole fraction (b). Lines: fitted model. Symbols: data of Harned and Sturgis116 (a) for H2SO4 molalities and of Randall and Langford117 (b) for total molalities given in the legend.

mol kg-1 aqueous H2SO4 solution. The mean activity coefficient of H2SO4(aq) needed for this purpose is calculated using the present model for the system H2SO4-H2O. For the remaining studies the standard potential is simultaneously fitted with the model parameters, which lead to the values 0.6110 and 0.6122 V for the work of Covington et al.118 and Randall and Langford,117 respectively. The value 0.6110 V is 1.5 mV lower than the potential determined by Covington et al.,118 while the value 0.6122 V is in good agreement with the standard potential 0.61257 ( 0.00018 V determined by Rondinini et al.104 Results

Figure 14. Degree of dissociation of HSO4- in NaHSO4(aq) at 298.15 K plotted as a function of the square root of total molality. Symbols: data taken from Sherrill and Noyes192 (squares) and Lindstrom and Wirth74 (crosses). Lines: fitted model.

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Figure 15. Phase diagrams of H2SO4-Na2SO4-H2O: (a) 273.15 K; (b) 298.15 K; (c) 302.65 K; (d) 319.15 K. Compositions are given as molalities. Symbols: data compiled by Silcock107 for Na2SO4 · 10H2O(s) (plus signs), Na2SO4(s) (crosses), Na3H(SO4)2(s) (stars), NaHSO4 · H2O(s) (open squares), and NaHSO4(s) (solid squares). Solid lines: fitted model. Dashed lines: predicted equilibrium relative humidity in steps of 10% increasing from 10% up to 90% with decreasing H2SO4 molality.

TABLE 13: Sources of Thermodynamic Data for Aqueous HNO3-NaNO3 Solutionsa Φ γ( sold sole

Nb

mminc

mmaxc

T (K)

ref

3 3 28 12

17.14 17.14 6.652 6.652

25.05 25.05 28.81 8.667

298.15 298.15 273.75-338.15 226.75-255.05

119 119 194, 195 194

a

Φ, osmotic coefficient derived from water vapor pressure measurement; γ(, mean activity coefficient of HNO3 derived from vapor pressure measurement, sol:salt solubility. b Number of measurements. c Total molality m(HNO3) + m(NaNO3). d Saturation with respect to NaNO3(s). e Simultaneous saturation with respect to ice and NaNO3(s). The activity product of NaNO3 and the water activity both at the freezing temperature are fitted.

of the fit of the PSC equations to thermodynamic data at 298.15 K are shown in Figures 12-14. Measurements of the osmotic coefficient of several aqueous H2SO4-Na2SO4 mixtures at 383.46 K (see Table 12) are used to stabilize the model at temperatures above 298.15 K, although the range of validity of the model for the subsystem H2SO4-H2O is exceeded at this temperature. The following series of solids precipitate as aqueous H2SO4 concentration is increased: Na2SO4 · 10H2O(s), Na2SO4(s), Na3H (SO4)2(s), NaHSO4 · H2O(s), and NaHSO4(s).32 Parameters needed to determine the temperature dependence of equilibrium constants of Na3H(SO4)2(s), NaHSO4 · H2O(s), and NaHSO4(s) formation are simultaneously fitted with the model parameters. Figure 15 shows phase diagrams of aqueous H2SO4-Na2SO4 solutions at several temperatures. For H2SO4 concentrations about 7 mol kg-1 above the sodium sulfate concentration increases with increasing relative humidity. This behavior is

Figure 16. Top: solubility of NaNO3(s) in HNO3(aq) at several temperatures plotted as a function of HNO3 molality. Symbols: data of Kurnakov and Nikolajew194 (stars, open squares) and Saslawsky et al.195 (plus signs, crosses). Lines: fitted model. Bottom: simultaneous saturation of aqueous HNO3-NaNO3 mixtures with respect to ice and NaNO3(s) as a function of temperature and molalities of HNO3 and NaNO3. Symbols: data of Kurnakov and Nikolajew194 for HNO3 (solid circles) and NaNO3 (open circles) molality in mixture. Lines: fitted model. The arrow indicates the temperature and composition at which the solution is saturated with respect to HNO3 · 3H2O(s).

qualitatively similar to that of aqueous H2SO4-(NH4)2SO4 and is likely to be due to HSO4- formation.49

Thermodynamic Model of H+-NH4+-Na+-SO42--NO3--Cl--H2O TABLE 14: Sources of Thermodynamic Data for Aqueous HCl-NH4Cl Solutionsa d

emf emfe emfe emfe,f γ( solg solh

Nmeasb

mminc

mmaxc

T (K)

ref

55 (0) 95 (1) 5 (0) 113 (1) 4 (3) 113 (52) 5 (2)

0.025 0.100 1.000 0.086 10.39 3.910 3.557

3.000 3.000 1.000 1.343 16.73 25.61 8.645

298.15 293.15-308.15 298.15 298.15-313.15 298.15 273.15-298.15 198.15-254.55

196 197 125 198 107 107 107

a emf, electromotive force; γ(, mean activity coefficient of HCl derived from vapor pressure measurement; sol, solid-phase solubility. b Number of measurements. The value in parentheses is the number of rejected measurements. c Total molality m(HCl) + m(NH4Cl). d Standard potentials given in Robinson et al.196 are used for calculation of mean activity coefficients of HCl. e The tabulated mean activity coefficients of HCl are used. f Results for five molar fractions y ) m(NH4Cl)/(m(HCl) + m(NH4Cl)) of ammonium chloride (y ∈ {0.1, 0.3, 0.5, 0.7, 0.9}) at 298.15 and 313.15 K are reported. g Saturation with respect to NH4Cl(s). h Saturation with respect to ice. Converted to water activity at the freezing temperature.


of HCl activity, the dissociation constant of NH4+, solubilities of NH4Cl(s) and ice in HCl(aq), and a few determinations of the HCl vapor pressure (see Table 14). Figure 17 exhibits satisfactory agreement between the fitted model and mean activity coefficients of HCl derived from emf measurements. The modeled solubilities of NH4Cl(s) in terms of molality at 273.15 and 298.15 K are slightly higher than the data of Silcock107 (not shown). Maeda et al.120 have measured the dissociation constant Km of the reaction NH4+(aq) h H+(aq) + NH3(aq) for NH4Cl molalities to 7 mol kg-1 at 298.15 K. Km is given by

Km )

m(H+) m(NH3(aq))

(29)

and is related to the activity product of the dissociation reaction via

aH+aNH3 (7) HNO3-NaNO3-H2O. The available thermodynamic data for this mixture is listed in Table 13 and comprise very few HNO3 and H2O vapor pressure measurements for low NaNO3 molalities and solubility data over a wide range of temperature. Fitted and measured solubilities of NaNO3(s) in HNO3(aq) do agree well, while the predicted NaNO3 and HNO3 molalities at simultaneous saturation with respect to ice and NaNO3 are lower and higher than the observed values, respectively (see Figure 16). However, the model is able to predict composition and temperature at simultaneous saturation with respect to ice, NaNO3(s), and HNO3 · 3H2O(s). The results of the fit to the vapor pressure data of Flatt and Beng´uerel119 for 298.15 K are very similar to those of Clegg et al.49 (see their Figure 28). (8) HCl-NH4Cl-H2O. The available thermodynamic data for aqueous HCl-NH4Cl mixtures comprise emf measurements

m(NH4+)

aNH4+

) Km

γH+γNH3 γNH4+

(30)

The measurements of Km by Maeda et al. are not included in the fit but are compared with model predictions in terms of the ratio γNH4+/γH+, which can be derived from eq 30 using activity coefficients of γNH3 given in Maeda et al.120 Figure 18 exhibits an increasing difference with increasing NH4Cl molality between measured and predicted ratios γNH4+/γH+. Clegg et al.49 have obtained a similar deviation, though independent of NH4Cl molality. (9) HCl-NaCl-H2O. The emf over wide ranges of concentration and temperature, solubilities of NaCl(s) in HCl(aq), and the freezing point depression was determined for this system (see Table 15). The emf of the cell

Figure 17. Natural logarithm of the mean activity coefficient of HCl in aqueous HCl-NH4Cl mixtures plotted as a function of square root of NH4Cl molality for the temperatures 298.15 K (a) and (b), 293.15 K (c), 303.15 K (d), and 308.15 K (e). Symbols: data of Chan et al.,125 Downes,197 and Robinson et al.196 for the ionic strengths 0.025 (open circles), 0.05 (solid circles), 0.1 (plus signs), 0.25 (open triangles), 0.5 (crosses), 1 (stars, solid triangles), 2 (open squares), and 3 (solid squares). Lines: fitted model.

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Pt|H2(g,1 atm)|HCl(m1),NaCl(m2)|AgCl(cr)|Ag|Pt (cell A) is related to the mean activity coefficient γ( of HCl by

E ) E° Figure 18. Reciprocal of NH4+ and H+ activity coefficients in NH4Cl(aq) determined from measurements of the NH4+ dissociation constant at 298.15 K plotted as a function of NH4Cl molality. Symbols: data of Maeda et al.120 Lines: predicted by the model.

TABLE 15: Sources of Thermodynamic Data for Aqueous HCl-NaCl Solutionsa d

emf emfe emff emf emfg Φ(fp)h,i Φ(fp)i a

Nmeasb

mminc

mmaxc

T (K)

weight

ref

18 (1) 170 (0) 171 (0) 6 (0) 36 (6) 9 (5) 11 (4)

4.000 1.000 0.100 1.000 4.000 0.200 0.506

6.000 3.000 0.872 1.000 7.000 1.009 3.191

298.15 273.15-323.15 278.15-318.15 298.15 298.15 269.41-272.81 258.50-271.31

1.0 0.25 1.0 1.0 0.1 1.0 1.0

199 121 122 125 200 201 202

emf, electromotive force; Φ(fp), osmotic coefficient derived from freezing point depression. b Number of measurements. The value in parentheses is the number of rejected measurements. c Total molality m(HCl) + m(NaCl). d Measurements for ionic strengths I ∈ {4, 5, 6} mol/kg. Conversion from international volt to volt. e Ten values of the mean activity coefficient of HCl are calculated in steps of 10 K for temperatures between 273.15 and 323.15 K with the regression equations given in Harned121 for ionic strengths 1, 2, and 3 mol kg-1. The equation for 2 mol kg-1 at 323.15 K is not used. f Measurements for the ionic strengths 0.1, 0.3809, 0.6729, and 0.8720 mol kg-1 in steps of 5 K. g Measurements for the ionic strengths 4, 5, 6, and 7 mol kg-1. h Conversion from molarity to molality with the densities tabulated in So¨hnel and Novotny´.193 i Conversion to activity coefficients at the freezing temperature.

RT ln(m1(m1 + m2)γ(2) F

(31)

Harned121 have only reported regression equations for the mean activity coefficient. For the remaining studies γ( can be determined from the measured emf’s using eq 31. The value E° ) 0.22243 V, derived by Macaskill et al.122 from the emf of a pure 0.01 mol kg-1 HCl solution at 298.15 K, is 0.09 mV higher than the value of Bates and Bower.123 Following Macaskill et al.122 we assume that this difference is temperature independent and correct the standard potentials calculated with the expression of Bates and Bower for any temperature by 0.09 mV. Lietzke et al.124 have measured the emf of 29 for ionic strengths of 0.4 and 1 mol kg-1 at 298.15, 333.15 K, and higher temperatures. Their results are rejected since they are strongly deviant from other measurements121,125 at an ionic strength of 1 mol kg-1. Figure 19 shows results of the fit to emf measurements with good agreement between model and data. The model has been successfully tested on salt solubilities of NaCl(s) in HCl(aq) and on freezing temperatures of ice. (10) NH4Cl-(NH4)2SO4-H2O. Clegg et al.49 have built their model for this system at 298.15 K upon salt solubilities (see their Table 10). We additionally use hygrometric water activity measurements for three different molar fractions of NH4Cl (0.2, 0.5, and 0.8) at 298.15 K.126 Deviations between the fitted model and osmotic coefficients derived from hygrometric measurements are large for concentrations above about 5 mol kg-1. Figure 20 shows good agreement between measured and fitted simultaneous saturation with respect to NH4Cl(s) and

Figure 19. Natural logarithm of the mean activity coefficient of HCl in aqueous HCl-NaCl solutions plotted as a function of the square root of NaCl molality for the ionic strengths I given in the figure for 298.15 K (a and b), 283.15 K (c), and 313.15 K (d). Symbols: data of Hawkins199 (triangles), Harned121 (crosses), Macaskill et al.122 (solid squares), Chan et al.125 (open squares), and Jiang200 (circles). Lines: fitted model.



TABLE 16: Sources of Thermodynamic Data for Aqueous NH4Cl-NH4NO3 Solutionsa Φ(fp) Φ(hm)d sole solf

Nmessb

mminc

mmaxc

T (K)

weight

ref

6 (0) 30 (9) 222 (77) 65 (37)

3.555 0.200 6.032 9.186

9.186 10.00 52.87 86.02

253.15-263.15 298.15 253.15-338.15 253.15-348.15

1.0 0.25 1.0 1.0

107 203 107, 127, 128 107, 127, 128

a Φ, osmotic coefficient from freezing point determination (fp) or hygrometric measurement (hm); sol, salt solubility. b Number of measurements. The value in parentheses is the number of rejected measurements. c Total molality m(NH4Cl) + m(NH4NO3). d Hygrometric measurements for NH4Cl molar fractions 0.2, 0.5, and 0.8. e Saturation with respect to NH4Cl(s). f Saturation with respect to NH4NO3(s).

TABLE 17: Sources of Thermodynamic Data for Aqueous Na2SO4-(NH4)2SO4 Solutionsa Nmessb

Figure 20. Temperature at simultaneous saturation with respect to NH4Cl(s) and (NH4)2SO4(s) in NH4Cl-(NH4)2SO4-H2O plotted as function of total molality. The arrow indicates concentration and temperature where the system becomes saturated with respect to ice. Symbols: data of Gmelin,128 Hill and Loucks,250 and Silcock.107 Line: fitted model.

mminc

mmaxc

Φ(fp) 7 (0) 0.281 5.178 Φ(iso)d 43 (1) 1.685 5.707 Φ(hm)e 28 (5) 0.200 4.500 solf 152 (45) 4.987 10.18 solg 80 (33) 3.876 10.18 92 (24) 0.281 5.271 solh soli 192 (12) 4.130 10.18

T (K) 252.15-271.95 298.15 298.15 252.15-373.15 299.15-373.15 252.15-299.65 257.15-371.15

ref 107 130 129 107, 107, 107, 107,

130, 204 204 130, 204 130, 204

a Φ, osmotic coefficient from freezing point determination (fp), isopiestic (iso), or hygrometric (hm) measurement; sol, salt solubility. b Number of measurements. The value in parentheses is the number of rejected measurements. c Total molality m(Na2SO4) + m((NH4)2SO4). d Reference standard is NaCl, the isopiestic molalities are not reported. e Measurements for molar (NH4)2SO4 fractions of 0.2, 0.5, and 0.7. f Saturation with respect to (NH4)2SO4(s). g Saturation with respect to Na2SO4(s). h Saturation with respect to Na2SO4 · 10H2O(s). i Saturation with respect to NaNH4SO4 · 2H2O(s).

Figure 21. Osmotic coefficient of aqueous NH4Cl-NH4NO3 solutions at 298.15 K plotted as function of square root of total molality for molar fractions y of NH4Cl given in the figure. Symbols: Data of El Guendouzi and Errougui.203 Solid lines: fitted model. Dashed lines: osmotic coefficient of pure aqueous NH4Cl and NH4NO3 solutions.

(NH4)2SO4(s). The model is able to predict temperature and concentration at the eutectic point. (11) NH4Cl-NH4NO3-H2O. Clegg et al.49 have built a model for this system at 298.15 K upon salt solubilities tabulated in Silcock.107 We additionally use further salt solubility measurements at temperatures between 293.15 and 348.15 K in steps of 5 K127 and hygrometric water activity measurements at 298.15 K for three NH4Cl fractions. The thermodynamic data used to parametrize the PSC equations is listed in Table 16. Figure 21 exhibits results of the fit to the hygrometric measurements. Fitted salt solubilities do agree well with observations (not shown). Gmelin128 has reported NH4Cl and NH4NO3 concentrations of 3.53 and 5.49 mol kg-1 at 251.15 K for simultaneous saturation with respect to NH4Cl(s), NH4NO3(s), and ice. The model predicts concentrations of 3.44 and 5.67 mol kg-1 at 251.28 K in good agreement with the values given in Gmelin. (12) Na2SO4-(NH4)2SO4-H2O. The available thermodynamic data for this system are listed in Table 17 and comprise osmotic coefficients at 298.15 K, salt solubilities of (NH4)2SO4(s), Na2SO4(s), Na2SO4 · 10H2O(s), and of the double salt NaNH4SO4 · 2H2O(s) over wide temperature ranges, and a few freezing temperature determinations. Figure 22 shows

Figure 22. Deviations between fitted and measured osmotic coefficients for aqueous Na2SO4-(NH4)2SO4 solutions at 298.15 K plotted as function of total molality for data taken from Filippov et al.130 (squares) and Mounir et al.129 (crosses).

deviations between fitted and measured osmotic coefficients. The scatter of hygrometric measurements of Mounir et al.129 is comparable to those of isopiestic measurements of Filippov et al.130 The predicted concentrations of Na2SO4 and (NH4)2SO4 of 0.37 and 4.74 mol kg-1 at a temperature of 252.7 K for simultaneous saturation with respect to Na2SO4(s), (NH4)2SO4(s), and ice are in good agreement with the available data. The equilibrium constant of the reaction NaNH4SO4 · 2H2O(cr) h Na+(aq) + NH4+(aq) + SO42-(aq) + 2H2O(l) is simultaneously determined with the model parameters. (13) NaNO3-NH4 NO3-H2O. Clegg et al.49 have built a model of this system for 298.15 K upon isopiestic water activity measurements131 and salt solubilities107,132 (see their Table 13). Meanwhile available are edb measurements of the water activity of supersaturated solutions90 and hygrometric water activity measurements for NH4NO3 molar fractions of 0.2, 0.5, and 0.8,133 both at 298.15 K. The hygrometric measurements are

11612


Friese and Ebel

Figure 23. Temperature at simultaneous saturation with respect to NaNO3(s) and NH4NO3(s) (plus signs) and with of addition ice (solid boxes) in NaNO3-NH4NO3-H2O solutions plotted as a function of total molality. The solid and dashed arrows indicates the transition from orthorhombic to rhombic and from rhombic to rhombohedric NaNO3(s). Symbols: data of Gmelin,128 Bergman and Shulyak,132 and Silcock.107 Line: fitted model.

Figure 24. Equilibrium water vapor pressure above simultaneous saturated solutions with respect to NaNO3(s) and NH4NO3(s) in NaNO3-NH4NO3-H2O solutions as a function of temperature. Symbols: data taken from Prideux251 (solid boxes) and Dijkgraaf252 (plus signs). Line: predicted by the model.

rejected because they strongly deviate from the isopiestic measurements of Kirgintsev and Luk’yanov.131 Following Clegg et al.49 we assume that the double salt NH4NO3 · 2NaNO3(s), observed by Bergman and Shulyak132 and Timoshenko,134 is in fact NaNO3(s). Figure 23 exhibits the temperature at simultaneous saturation with respect to NaNO3(s) and NH4NO3(s). The model is successfully tested on the equilibrium water vapor pressure over such simultaneously saturated solutions (see Figure 24) and on the edb measurements by Ha et al.90 at high concentrations (not shown). (14) NaCl-NH4Cl-H2O. Table 18 lists the data used to parametrize the PSC equations for this system. Some data are additionally available to those used from Clegg et al.49 for the parametrization of their model for 298.15 K. Ji et al.135 have published emf measurements of the mean activity coefficient of NaCl for several NH4Cl fractions at 283.15, 209.15, and 313.15 K. Ji et al.97 have measured solubilities of NH4Cl(s) in

Figure 25. Osmotic coefficients in aqueous NaCl-NH4Cl solutions plotted as function of total molality. (a) At 298.15 K for the water activities given in the legend. Symbols: data of Kirgintsev and Luk’yanov.205 Lines: fitted model. (b) Deviations between the fitted model and measurements at 298.15 K for data of Kirgintsev and Luk’yanov205 (crosses), Maeda et al.137 (solid squares), and Dinane et al.136 (stars), and at freezing temperatures determined by Sawada202 (open squares).

NaCl-NH4Cl-H2O at temperatures between 293.15 and 333.15 K in steps of 5 K. Two studies are engaged in water activity at 298.15 K. First, Ha et al.90 have performed edb measurements of the water activity of supersaturated solutions of 1:1 and 3:1 molar Na: NH4 mixing ratio. Second, Dinane et al.136 have used the hygrometric method to determine water activities for three different molar NH4Cl fractions. Figure 25 shows osmotic coefficients of aqueous NaCl-NH4Cl solutions and deviations between fitted and measured values. Mean activity coefficients of NaCl are shown in Figure 26. Aside from systematic deviations from results of Maeda et al.,137 the agreement between model and observations is satisfactory. Fitted salt solubilities at 273.15 K are slightly higher than the observed values. Gmelin128 has reported NaCl and NH4Cl concentrations of 6.61 and 2.98 mol kg-1 at 248.05 K for simultaneous saturation with respect to NaCl(s), NH4Cl(s), and ice. The model predicts concentrations of 6.44 and 2.94 mol kg-1 at a slightly lower temperature of 246 K.

TABLE 18: Sources of Thermodynamic Data for Aqueous NaCl-NH4Cl Solutionsa Φ(fp) Φ(iso)e Φ(iso) Φ(hm)f emfg sol sol sol

Nmeasb

mminc

mmaxc

T (K)

12 (0) 88 (0) 4 (0) 21 (14) 170 (1) 79 (35) 81 (33) 77 (9)

1.554 2.264 2.167 0.300 0.080 6.040 6.049 7.083

2.160 6.829 5.937 6.000 2.800 12.10 12.10 11.04

265.85-267.86 298.15 298.15 296.15 283.15-313.15 248.05-333.15 248.05-333.15 293.15-333.15

d NaCl NaCl NaCl(s) NH4Cl(s) NH4Cl(s)

weight

ref

1.0 1.0 0.25 0.1 1.0 1.0 1.0 1.0

202 205 137 136 135 107, 128 107, 128 97

a Φ, osmotic coefficient derived from freezing point determination (fp), isopiestic (iso), or hygrometric (hm) measurement; emf, emf measurement of the mean activity coefficient of NaCl; sol, salt solubility. b Number of measurements. The value in parentheses is the number of rejected measurements. c Total molality m(NaCl) + m(NH4Cl). d Isopiestic reference standard or composition of the solid phase. e Measurements for eight different water activities between 0.787 and 0.9218. Isopiestic molalities are not reported. f Measurements for molar NH4Cl fractions of 0.33, 0.5, and 0.67. g Measurements for molar NH4Cl fractions of 0.2, 0.4, 0.6, and 0.8. Tabulated mean activity coefficients are fitted.



a slightly lower value of 2.23 mol5 kg-5. Chre´tien142 has reported NaNO3 and Na2SO4 concentrations of 7.14 and 0.014 mol kg-1 at 255.35 K for simultaneous saturation with respect to NaNO3(s), Na2SO4 · 10H2O(s), and ice. The corresponding values predicted by the model are 7.21 and 2.5 × 10-6 mol kg-1, respectively, at 255.85 K. (16) NaCl-Na2SO4-H2O. Rard et al.145 have used an extended Pitzer ion-interaction model38 for a temperature dependent description of this system and have critically evaluated the available thermodynamic data. We mainly follow the recommendations of Rard et al.145 and additionally use salt solubilities and the data of Marliacy et al.146 to parametrize the PSC equations (see Table 19). Marliacy et al.146 have measured the H2O equilibrium vapor pressure over aqueous NaCl-Na2SO4 mixtures with NaCl:Na2SO4 mixing ratios of 1:10 and 1:1 at temperatures between 298.31 and 362.26 K as well as the integral heat of solution of NaCl(s) and Na2SO4(s) for the mixing ratios 1:10, 1:1, and 10:1 at 297.15, 317.45, and 332.35 K. The thermal effect ∆solH associated wiht the n1 mole NaCl(s) and n2 mole Na2SO4(s) in nw mole water is given by Figure 26. Osmotic coefficients of aqueous NaCl-NH4Cl solutions plotted as a function of total molality. (a) For fractions y of NaCl indicated in the figure, at the temperatures given in the legend. Values are vertically shifted by offsets indicated in the plot. Symbols: data of Ji et al.135 Lines: fitted model. (b) Reciprocal of fitted and measured mean activity coefficient of NaCl for the data of Ji et al.135 for 283.15, 298.15, and 313.15 K.

(15) NaNO3-Na2SO4-H2O. The model for this system is based on solubilities of NaNO3(s), Na2SO4(s), Na2SO4 · 10H2O(s), and the double salt NaNO3 · Na2SO4 · H2O(s).107,138–144 The equilibrium constant of the reaction NaNO3 · Na2SO4 · H2O(cr) h 3Na+(aq) + NO3-(aq) + SO42-(aq) + H2O(l) is simultaneously fitted with the model parameters. We get the value 2.57 mol5 kg-5 at 298.15 K. Clegg et al.49 have obtained

∆solH ) z∆solH°1 + (1 - z)∆solH°2 + ΦL(m1,m2)

(32) where ∆solH°1 and ∆solH°2 are the integral heats of solution at infinite dilution per mole NaCl147 and Na2SO4,148 ΦL(m1,m2) is the apparent relative molar enthalpy of the ternary mixture, given by eq A33, and z ) m1/(m1 + m2) for molalities m1 ) n1/(nwM1) and m2 ) n2/(nwM1). Figures 27 and 28 show the results of the fit of the PSC equations to the data of Marliacy et al.146 The mean activity coefficient of NaCl in aqueous NaCl-Na2SO4 mixtures can be derived from emf measurements for four different cells145 using the relation

TABLE 19: Sources of Thermodynamic Data for Aqueous NaCl-Na2SO4 Solutionsa Φ Φ Φ Φ a1 emf emf emf emf emf ∆mixHe ∆mixHe ∆mixHe,h ∆mixHe ∆solH sol sol sol sol

Nmeasb

mminc

mmaxc

T (K)

15 (5) 4 (0) 33 (7) 119 (10) 210 (1) 17 (0) 21 (3) 220 (3) 45 (5) 32 (3) 69 (13) 29 (0) 9 (0) 12 (0) 89 (5) 173 (103) 187 (137) 213 (134) 28 (6)

0.861 0.883 1.187 0.120 0.273 0.387 0.250 0.199 0.225 0.080 0.200 0.050 3.000 1.030 0.394 6.019 3.504 0.672 5.072

4.647 1.639 4.520 3.801 2.099 0.622 3.900 5.000 3.600 2.410 5.000 1.000 3.000 1.030 5.636 6.862 6.862 6.248 6.182

298.15 298.15 298.15 298.15 298.31-362.26 298.15 298.15 298.15-318.15 278.15-318.15 288.15-303.15 293.15-298.15 298.15 298.15 303.15 297.15-332.35 273.05-373.15 291.65-373.15 251.45-288.15 251.45-273.25

d NaCl NaCl NaCl NaCl I II IV I III

NaCl(s) Na2SO4(s) Na2SO4 · 10H2O(s) NaCl · 2H2O(s)

weight 1.0 1.0 0.5 1.0 1.0 1.0 1.0 1.0 0.2 0.2 f, g f f f 1.0 1.0 1.0 1.0

ref 206 207 208 145 146 149 150 152 153 153 209 154 210, 211 212 146 107 107 107 107

a Φ, osmotic coefficient; a1, water activity derived from equilibrium water vapor pressure; emf, electromotive force; ∆mixH, molar enthalpy of mixing; ∆solH, integral heat of solution; sol, salt solubility. b Number of measurements. The value in parentheses is the number of rejected measurements. c Total molality m(NaCl) + m(Na2SO4). d Reference standard for isopiestic measurements, type of cell for emf measurements,145 or composition of the solid phase for salt solubilities. e Conversion of enthalpy of mixing per kilogram of solvent ∆mixH1 to molar enthalpy of mixing using ∆mixH ) ∆mixH1/(m1,f + m2,f), where m1,f and m2,f are the NaCl and Na2SO4 molalities after the mixing process.145 f Since the values of the enthalpy of mixing vary over 2 orders of magnitude, the weight w ) |∆mixH|-3/4 is used. g A zero weight is used for the enthalpy of mixing for ionic strength 0.2 mol kg-1 since the values differ from data of other studies. h Calculation of nine values of the enthalpy of mixing for ionic strength I ) 3 mol kg-1 using the mixing parameters ho and h1 given in Wood et al.210 with the relation ∆mixH1 ) y1y2I2(RTho + (y2 - y1)RTh1), where y1 and y2 are the ionic strength fractions of NaCl and Na2SO4.

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Figure 27. Differences between fitted and measured water activity in aqueous NaCl-Na2SO4 mixtures for the water vapor pressure measurements of Marliacy et al.146 (a) Plotted as a function of square root of the total molality. (b) Plotted as a function of temperature. Figure 29. Mean activity coefficient of NaCl in aqueous NaCl-Na2SO4 mixtures plotted as function of total molality. (a) For ionic strengths I ) 0.5, 1.0, 2.0, and 3.0 at the temperatures given in the legend. The graphs for 308.15 and 318.15 K are shifted vertically by +0.01 and +0.02. Symbols: data of Sarada and Ananthaswamy.152 Lines: fitted model. (b) Differences between fitted and measured natural logarithm of the mean activity coefficient for data taken from Gieskes149 (open squares), Storonkin et al.150 (crosses), Sarada and Ananthaswamy152 (solid squares), and Tishchenko et al.153 (open circles).

Figure 28. Enthalpy of solution of NaCl(s) and Na2SO4(s) in H2O for the molar NaCl: Na2SO4 mixing ratios r ) 0.1, r ) 1, and r ) 10 at temperatures given in the legend. Symbols: data of Marliacy et al.146 Lines: fitted model.

∆E )

(

(m1 + 2m2)γNa+m1γClRT ln F (m1,0γ(,0(NaCl))2

)

(33)

where ∆E is the difference in electrode potential, if the cell is filled with an aqueous mixture of NaCl and Na2SO4 with molalities m1 and m2, and a pure aqueous NaCl reference solution of molality m1,0. γ(,0(NaCl) is the mean activity coefficient of NaCl in the reference solution, which we calculate with the model of Archer and Carter.76 For the studies of Gieskes,149 Storonkin et al.,150 and Galleguillos-Castro et al.151 the mean activity coefficient of NaCl in the mixture can be derived from eq 33. Tabulated mean activity coefficients from the studies of Sarada and Ananthaswamy152 and Tishchenko et al.153 are directly fitted, since they did not report values of ∆E. Figure 29 shows the results of the fit of the PSC equations to the used emf measurements. Figure 30 exhibits the enthalpy of mixing for aqueous solutions of NaCl and Na2SO4 with the same ionic strength. For ionic strengths higher than 0.2 mol kg-1 the mixing process is exothermic, while below 0.2 mol kg-1 a transition from an endothermic process at low ionic strength fractions of NaCl to an exothermic

Figure 30. Thermal effect for the intermixture of an aqueous NaCl solution of ionic strength I with an aqueous Na2SO4 solution of the same ionic strength at 298.15 K for I ) 0.05 (plus signs), 0.1 (crosses), 0.2 (stars), 0.5 (open and solid squares), 1 (open circles), 2 (dots), 3 (open triangles), and 5 mol kg-1 (solid triangles) plotted as a function of ionic strength fraction of Na2SO4. Symbols: data of Cassel and Wood154 (plus signs, crosses, stars, open squares), Smith209 (solid squares, open circles, dots, and solid triangles), and Wood et al.210 and Srna and Wood211 (open triangles). Lines: fitted model.

process at high ionic strength fractions is observed.154 The fitted model is not able to reflect this transition. The modeled enthalpy of mixing, calculated using eq A32, is endothermic (independent of the NaCl fraction) for an ionic strength of 0.05 mol kg-1 and exothermic, if the ionic strength is greater than 0.1 mol kg-1. Fitted salt solubilities of NaCl · 2H2O(s), NaCl(s), Na2SO4 · 10H2O(s), and Na2SO4(s) agree very well with observations. Gmelin155 gives NaCl and Na2SO4 molalities of 5.15 and 0.014 mol kg-1 at 251.75 K for simultaneous saturation with respect to NaCl · 2H2O(s), Na2SO4 · 10H2O(s), and ice. The model yields 5.23 and 0.022 mol kg-1, respectively, at 251.74 K in good agreement with the observations.



TABLE 20: Fitted Model Parameters for Aqueous NaCl-Na2SO4 Solutionsa WCl,SO4,Na p1 p2 p3 p4 p5 p6 p8 p9

UCl,SO4,Na

-1.64343594 -43230.6635 2681.83402 -111.752607 0.295635141 -2009567.13 443828.682 -287755.733

-0.264024363

Q1,Cl,SO4,Na 1.12834897 30026.46718 -1861.65160 76.3162380 -0.191454528 1393236.32 -287567.860 176670.177

a Equation 18 is used to calculate the temperature dependence of model parameters. The reference temperatures To and Tr are equal to 298.15 and 215 K.

Model parameters for the system NaCl-Na2SO4-H2O are listed in Table 20. (17) NaCl-NaNO3-H2O. Sources of thermodynamic data for this system comprise osmotic coefficients, mean activity coefficients of NaCl, and salt solubilities (see Table 21). Figure 31 shows fitted and measured osmotic coefficients of aqueous NaCl-NaNO3 solutions and differences between them. At 298.15 K the calculated osmotic coefficients are too low while those at the freezing temperature show a large scatter but no systematic deviation from measurements. Fitted solubilities of NaCl(s) and NaNO3(s) do agree very well with observations. However, the model does not predict NaCl · 2H2O(s) solubilities very well. This is also the case for simultaneous saturation with respect to NaCl(s), NaCl · 2H2O(s), and ice for which Chre´tien142 has reported NaCl and NaNO3 molalities of 4.52 and 3.76 mol kg-1 at the temperature 248.85 K. The model predicts 4.24 and 3.47 mol kg-1, respectively, at a temperature of 247.04 K, which is 1.8 K lower than the observed value. VI. Results and Discussion (1) Mean Activity Coefficient of HCl in Aqueous HCl-Na2SO4 Solutions. Pierrot et al.156 have measured the emf of the cell Pt; H2(g)|HCl(m1)+Na2SO4(m2)|AgCl,Ag for concentrations to 6 mol kg-1 at temperatures between 278.15 and 323.15 K. The stoichiometric mean activity coefficient of HCl γ(,st can be obtained using the relation

E ) Eo -

RT ln(m(H+) m(Cl-)γ(,st) F

(34)

Since HSO4- ions can be present in the system HCl-Na2SO4-H2O, the molality of the hydrogene ion m(H+)

Figure 31. Osmotic coefficients of aqueous NaCl-NaNO3 solutions. (a) At 298.15 K for the water activities given in the legend plotted as function of total molality. Symbols: data of Kirgintsev and Luk’yanov.185 Lines: fitted model. (b) Differences between fitted and measured osmotic coefficients at for the data of Kirgintsev and Luk’yanov185 (plus signs) and Bezboruah et al.213 (crosses) plotted as function of total molality. (b) At the freezing temperatures for data compiled by Silcock107 plotted as a function of temperature.

has to be determined by solving the equilibrium HSO4-(aq) h H+ (aq) + SO42-(aq). The stoichiometric mean activity coefficient of HCl is then given by

γ(,st )

(

m(H+) γ γ m(HCl) H+ Cl-

)

1/2

(35)

Calculated and measured mean activity coefficients of HCl are shown in Figure 32. The overall agreement between predicted and measured activity coefficients is satisfactory but at low and high ionic strengths, respectively, the measurements are slightly underestimated and overestimated by the model. A small underestimation of the measured activity coefficients is obtained at temperatures below 298.15 K. However, the magnitude of deviations is within the range generally observed, if the model

TABLE 21: Sources of Thermodynamic Data for Aqueous NaCl-NaNO3 Solutionsa Φ(iso) Φ(iso) Φ(fp) emff,g emff emff,g sol sol sol

e

Nmessb

mminc

mmaxc

T (K)

54 (3) 17 (0) 49 (0) 24 (14) 60 (14) 61 (24) 210 (134) 156 (82) 61 (6)

2.470 0.981 1.549 1.000 0.500 1.000 6.552 7.138 5.422

5.980 6.164 7.376 5.000 3.000 6.000 21.54 21.54 9.203

298.15 298.15 248.65-268.55 298.15 298.15 298.15 267.30-373.15 248.65-373.15 248.65-273.05

d

ref

NaCl(s) NaNO3(s) NaCl · 2H2O(s)

185 213 107 214 215 216 107, 142, 143, 217 107, 142, 143, 217 107, 142

NaCl NaCl

a Φ, osmotic coefficient derived from isopiestic measurement (iso) or from freezing point determination (fp); emf, emf measurement; sol, salt solubility. b Number of measurements. The value in parentheses is the number of rejected measurements. c Total molality m(NaCl) + m(NaNO3). d Isopiestic reference standard or composition of the solid phase. e Measurements for 9 mixing ratios each at 6 constant water activities between 0.845 and 0.913. The isopiestic molalities are not reported. f The tabulated mean activity coefficients of NaCl are fitted. g The results of Haghtalab and Vera216 and Lanier214 for ionic strengths of 6 and 5 mol kg-1, respectively, are zero weighted because they are inconsistent with results of other studies.

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is applied to mean activity coefficients derived from emf measurements. (2) HCl and HNO3 Vapor Pressure Over Aqueous HCl-HNO3-H2SO4 Solutions at Low Temperatures. Elrod et al.157 have measured the effective Henry’s law constant H* for HCl in two aqueous HCl-HNO3-H2SO4 mixtures (47 wt % H2SO4, 3.5 wt % HNO3, 0.0039 mol dm-3 HCl, and 36.2 wt % H2SO4, 12.5 wt % HNO3, 0.0062 mol dm-3 HCl) at temperatures between 208 and 233 K. H* is related to the equilibrium vapor pressure pHCl of HCl and to the molarity of the chloride ion M(Cl-) by

H* )

M(Cl-) pHCl

(36)

For solutions containing H2SO4, H* is given by23

H* )

KHCl M(H2SO4) m γ 2 m(H2SO4)

(37)

H (

where KHCl is the molar Henry’s law constant for HCl, mH is the molality of hydrogen ion (not present as HSO4-), γ( is the mean molal activity coefficient of HCl, and M(H2SO4) and m(H2SO4) are the molarity and molality of H2SO4. The density of the solutions, which is required for obtaining molarity, is estimated as mass weighted mean of the densities of H2SO4 and HNO323 neglecting the HCl contribution. Figure 33 shows good agreement between model predictions and results of Elrod et al. A similar good agreement was obtained by Massucci et al.158 in a comparison of HCl vapor pressure derived from the H* measurements of Elrod et al.157 with predictions by the model of Carslaw et al.23 (see Figure 22d in Carslaw et al.23). Hanson159 has measured the HNO3 and HCl vapor pressure over four different aqueous HCl-HNO3-H2SO4 solutions at 205 K. Their results are compared with predictions of the model in Table 22. Modeled HNO3 vapor pressures are on average 9.1% lower than the results of Hanson, whereas the modeled and measured HCl vapor pressure disagree by a factor of about 2, with the exception for the mixture with the lowest H2SO4 fraction and highest HNO3 fraction, for which the experimental results are only 20% underestimated. Massucci et al.158 have obtained a minor underprediction of the HCl vapor pressure for solutions containing mainly H2SO4 and an overprediction for solutions containing mostly HNO3 in their comparison of measurements of Hanson159 with values predicted by the model of Carslaw et al.23 (see Figure 22a-c in Massucci et al.158). (3) NH3 Vapor Pressure at 298.15 K. The accuracy of equilibrium partial pressures of NH3 over aqueous solutions predicted by the model is restricted by the ability to determine the quotient aNH4+/aH+. Since the dissociation of H2O into H+ and OH- is not treated by the model, this ability is restricted to acid solutions with pH of about 5 and below, where the dissociation of water is suppressed.48 Maeda and Iwata160 have measured the dissociation constant pK*a of NH4+ and the activity coefficient of NH3(aq) in aqueous (NH4)2SO4 at 298.15 K. For this solution, where the dissociation of the HSO4- ion must be considered, the dissociation constant of NH4+ is expressed as a function of ionic strength by

) K*(I) a

mT(H+) m(NH3) m(NH4+)

(38)

Figure 32. Stoichiometric mean activity coefficient of HCl in aqueous HCl-Na2SO4 mixtures as a function of ionic strength I ) m(HCl) + 3m(Na2SO4) at several temperatures. (a) Lines: predicted by the model. Symbols: data of Pierrot et al.156 (b) Differences between modeled and observed activity coefficients.

Figure 33. Effective Henry’s law constant for HCl in aqueous HCl-HNO3-H2SO4 solutions plotted as a function of temperature. Lines: predicted by the model. Symbols: data of Elrod et al.157 for solutions with 47 wt % H2SO4, 3.5 wt % HNO3, 0.0039 mol dm-3 HCl (plus signs), and 36.2 wt % H2SO4, 12.5 wt % HNO3, and 0.0062 mol dm-3 HCl (crosses).

Figure 34. Negative logarithm of the dissociation constant of NH4+ in aqueous (NH4)2SO4 at 298.15 K as a function of ionic strength. Line: predicted by the model. Symbols: data of Maeda and Iwata.160



TABLE 22: HNO3 and HCl Vapor Pressure over Aqueous HCl-HNO3-H2SO4 Solutions T (K)

[HCl] (10-3 wt %)

p(HNO3)(meas)a (10-9 atm)

205.0 205.0 205.0

1.8 3.6 6.6

2.8 2.6 2.5

204.0 204.0 205.0 205.0 205.0

2.6 8.2 2.7 1.5 4.8

204.5 204.5 204.5 205.0 205.0 a

p(HNO3)(mod)b (10-9 atm)

p(HCl)(meas)a (10-10 atm)

p(HCl)(mod)b (10-10 atm)

44.6 wt % H2SO4, 4.4 wt % HNO3 2.2 2.2 2.2

2.3 3.6 7.3

1.2 2.4 4.3

2.0 2.0 3.5 3.8 3.2

39.6 wt % H2SO4, 7.9 wt % HNO3 2.1 2.1 2.5 2.5 2.5

1.4 4.0 1.7 0.76 2.8

0.68 2.2 0.82 0.46 1.4

1.9 4.8 5.5

2.3 2.5 3.7

30.0 wt % H2SO4, 16.8 wt % HNO3 2.8 2.8 2.8

0.5 1.4 1.2

0.28 0.72 0.83

3.8 8.7

2.7 3.5

20.3 wt % H2SO4, 25.6 wt % HNO3 2.8 2.8

0.45 1.0

0.36 0.83

Data of Hanson.159 b Predicted by the model.

where mT(H+) is total molality of the hydrogen ion (mT(H+) ) m(H+) + m(HSO4-)). Algebraic transformation of eq 38 leads to the relation

) pKa + log pK*(I) a

γH+γNH3 γNH4+

(

- log 1 +

)

I γH+γSO423 KHSO4-γHSO4(39)

between K*(I) and the activity coefficients of the ions. In eq 39 a Ka and KHSO4- are the equilibrium constants of the reactions

NH4+(aq) h H+(aq) + NH3(aq) and HSO4-(aq) h H+(aq) + SO42-(aq), for which according to Maeda and Iwata160 the values pKa ) 9.245161 and pKHSO4- ) 1.97972 at 298.15 K are used. Figure 34 exhibits a comparison between measured values of and predictions by the model. Despite a slight underespK*(I) a timation and overestimation of the experimental results by the model below and above an ionic strength of about 2.5 mol kg-1, respectively, the agreement is satisfactory. (4) Salt Solubilities in Quaternary Systems. We apply our model to salt solubilities in the systems H+-Na+-Cl-SO42--H2O (system 1), Na+-NH4+-Cl--SO42--H2O (sys-

Figure 35. Measured and predicted solution compositions at simultaneous saturation with respect to two or three solid phases for the system Na+-NH4+-Cl--SO42--H2O for the temperatures 273.15 (a), 298.15 (b), 313.15 (c), and 333.15 K (d). Lines: predicted by the model. Symbols: data from the compilation of Silcock.107 Solid phases: Na2SO4 · 10H2O(s) (open triangles), NaNH4SO4 · 2H2O(s) (open squares), Na2SO4(s) (dots), (NH4)2SO4(s) (solid triangles), NH4Cl(s) (open circles), and NaCl(s) (solid squares).

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Figure 36. Water activity of saturated solutions in the system Na+-NH4+-Cl--SO42--H2O for the temperatures 273.15 (a), 298.15 (b), 313.15 (c), and 333.15 K (d). Solid phases: Na2SO4 · 10H2O(s) (1), NaNH4SO4 · 2H2O(s) (2), Na2SO4(s) (3), (NH4)2SO4(s) (4), NH4Cl(s) (5), and NaCl(s) (6). In (a) a lower limit of 0.55 is applied to the water activity for reasons of illustration.

tem 2), and Na-NH4+-NO3--SO42--H2O (system 3). Solution compositions are predicted, for which the systems are simultaneously saturated with respect to two or three solid phases. The model predictions are compared with data compiled in ref 107. For System 1 the model results for 298.15 K are in very good agreement with the data except for the simultaneous saturation with respect to Na2SO4(s) and Na2SO4 · 10H2O(s). The data are potentially erroneous since the model is able to calculate salt solubilities in the systems H2SO4-Na2SO4-H2O and NaCl-Na2SO4-H2O with considerably lower deviations from measurements. Clegg et al.49 obtained similar results for comparison of their model results with the data of Silcock.107 Solubility data for System 1 at temperatures other than 298.15 K are not available to the best of our knowledge. Figure 35 shows compositions, for which system 2 is simultaneously saturated with respect to two or three solid phases at the temperatures 273.15, 298.15, 313.15, and 333.15 K. With the exception of simultaneous saturation with respect to NaCl(s) and Na2SO4(s) as well as NaCl(s) and NH4Cl(s) at 333.15 K, the agreement between model predictions and measurements is excellent. The deviations can possibly be attributed to difficulties of the model to predict the simultaneous saturation with respect to NaCl(s) and NH4Cl(s) at temperatures above 320 K. Wexler and Clegg51 have studied the role of the ammonium sulfate/ammonium nitrate double salts (NH4)2SO4 · 2NH4NO3(s) and (NH4)2SO4 · 3NH4NO3(s) at 270, 290, and 310 K as well as

the role of the sodium sulfate double salts NaNH4SO4 · 2H2O(s) and NaNO3 · Na2SO4 · H2O(s) at 298.15 K. According to their investigation the particle mass is overestimated by models of the inorganic aerosol, if the double salts are neglected. This behavior is qualitatively apparent from Figures 35 and 36 for system 2 and from Figures 37 and 38 for system 3. The sodium sulfate double salts are more important than the ammonium sulfate/ammonium nitrate double salts, since they cover a larger range of composition and they occur over larger intervals of relative humidity. At temperatures below and above about 298 K the sodium sulfate double salts become less important and they are replaced by Na2SO4 · 10H2O(s) and Na2SO4(s), respectively. (5) Gibbs’ Energy and Enthalpy of Formation. In Table 23 the Gibbs’ energy and enthalpy of formation of several salts at 298.15 K calculated with our model is compared with results of previous studies.39,48,49,98 The agreement is generally very good. Larger differences are existent between the compilation of Wagman et al.,98 the results of Clegg et al.,49 and computations with the recent model for NaHSO4(s) and NaHSO4 · H2O(s) as well as between the results of Clegg et al.,48 and the results of the recent model for the Gibbs’ energy of formation of (NH4)3H (SO4)2(s) and NH4 HSO4(s). To the best of our knowledge no further literature data are available for the enthalpy of formation of the double salts NaNH4SO4 · 2H2O(s) and NaNO3 · Na2SO4 · H2O(s).



Figure 37. Predicted solution compositions at simultaneous saturation with respect to two or three solid phases for the system Na+-NH4+-NO3--SO42--H2O at temperatures 273.15 (a), 298.15 (b), and 313.15 K (c).

Figure 38. Water activity of saturated solutions in the system Na+-NH4+-NO3--SO42--H2O for the temperatures 273.15 (a), 298.15 (b), and 313.15 K (c).

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TABLE 23: Gibbs’ Energy and Enthalpy of Formation for Several Salts at 298.15 Ka ∆fG° (kJ mol-1) NaCl(s) NaCl · 2H2O(s) Na2SO4(s) Na2SO4 · 10H2O(s) Na3H(SO4)2(s) NaHSO4(s) NaHSO4 · H2O(s) NaNH4SO4 · 2H2O(s) NaNO3(s) NaNO3 · Na2SO4 · H2O(s) NH4Cl(s) NH4NO3 (IV)e (NH4)2SO4(s) (NH4)3H(SO4)2(s) NH4HSO4(s) (NH4)2SO4 · 2NH4NO3(s) (NH4)2SO4 · 3NH4NO3(s)

∆fH° (kJ mol-1)

Wagman et al.

Clegg et al.

this work

Wagman et al.

-384.138

-384.097 -858.75c -1270.10b -3646.62b -2278.92b -1002.97b -1244.1b -1569.7b -367.00b -1876.6b -203.46b -184.35f -902.9f -1791.46f -852.12f -1273.28f -1458.18f

-384.078 -858.43 -1269.87 b -2278.94 -1002.93 -1244.3 -1564.8 -366.97 -1876.3 -203.44 f f -1731.56 -822.16 f f

-411.153

-1270.16 -3645.85 -992.8 -1231.6 -367.0 -202.87 -183.87 -901.67

b

Clegg et al. -997.24c

-1387.08 -4327.26 -1125.5 -1421.7 -467.85 -314.43 -365.56 -1180.85

-365.56f -1180.4f -2210f -1027f -1913f -2279f

this work -411.296 c d d -2517.5 -1130.3 -1429.5 -1877 -467.83 -2153 -316.49 f f -2213.0 -1026.8 f f

a Values of ∆fG° and ∆fH° are derived from the relations -RTlnKo ) Σjnj∆fG°j - Σini∆fG°i and ∆rH° ) Σjnj∆fH°j - Σini∆fH°i using the Gibbs’ energy and enthalpy of formation of the ions tabulated in Wagman et al.98 Ko and ∆rH° are the equilibrium constant and the enthalpy of reaction for the dissolution reaction of the corresponding salt at 298.15 K. The first and second sum on the right hand side of the above relations is over all reaction products and reactants, respectively. b Clegg et al.49 c Archer.39 d Wagman et al.98 e Rhombic form between 256.2 and 305.38 K. f Clegg et al.48

VII. Range of Validity The upper temperature limit of the model for the system H+-NH4+-Na+-SO42--NO3--Cl--H2O (system I) is definitely sufficient for atmospheric applications. For information about the maximum temperature of measurements used to parametrize the PSC equations for a certain system we refer to the various tables within this work and to the cited literature. Some limitations are to be discussed regarding the lower temperature limit. For the system H+-NO3--SO42-Cl--H2O (system II), the model is applicable to temperatures lower than 200 K. Depending upon liquid-phase composition this may be possible also for the system H+-NH4+NO3--SO42--H2O (system III). Caution may be advised, if the liquid phase is mainly aqueous NH4NO3-(NH4)2SO4. For this system only data down to 273.15 K are used for model parametrization. Clegg et al.49 have predicted the composition of the eutectic point at -21 °C with some uncertainty using their model for system III. We have obtained a similar result using our own model. Also for aqueous H2SO4-Na2SO4, NH4Cl-(NH4)2SO4, and NaNO3-Na2SO4 solutions data only down to 273.15 K are used for model parametrization. For the last two systems it is shown that the model is able to predict the eutectic point at -17.8 and -22.5 °C, respectively, with satisfying accuray (see sections 10 and 15). Uncertainties are largest for aqueous H2SO4-Na2SO4 solutions. The only available data below 273.15 K are few solubility measurements.107 Predicted values of 1.81 mol kg-1 H2SO4 and 0.35 mol kg-1 Na2SO4 for simultaneous saturation with respect to ice and Na2SO4 · 10H2O at 263.15 K are to some extent in agreement with the measurement, which is 1.91 mol kg-1 H2SO4 and 0.44 mol kg-1 Na2SO4. There are two data points at 243.15 K, which represent equilibrium with respect to Na3H (SO4)2: the first at 6.24 mol kg-1 H2SO4 and 0.27 mol kg-1 Na2SO4 and the second at 7.85 mol kg-1 H2SO4 and 0.20 mol kg-1 Na2SO4.107 The model predicts Na2SO4 molalities of 0.40 mol kg-1 and 0.38 mol kg-1 for the first and second data points, respectively, which are in moderate agreement with the data. We conclude that the

model for system I is valid to 263.15 K but may be extrapolated depending upon liquid-phase composition. Regarding solute concentration, the model for system I is at least applicable up to saturation. If the model is applied to systems II and III, the range of validity is expanded up to about 40 mol kg-1 total solute molality for solutions containing mainly H2SO4 and HNO3 for system II and up to supersaturated solutions for system III. VIII. Conclusions The thermodynamic model presented here enables the calculation of activity coefficients in H+-NH4+-Na+-NO3-SO42--Cl--H2O (system I) mixtures for the temperature range from j∼263.15 to 328 K and concentrations from infinite dilution to saturation. For the subsystems H+-NO3--SO42-Cl--H2O (system II) and H+-NH4+-NO3--SO42--H2O (system III) the model is applicable to temperatures below 200 K, dependent upon liquid-phase composition, and for system III to supersaturated solutions. The model is able to represent equilibrium partial pressures of HNO3, HCl, NH3, and H2O and saturation with respect to 26 solid phases, including the complex salts NaNH4SO4 · 2H2O(s) and NaNO3 · Na2SO4 · H2O(s). In comparison to the model of Clegg and Brimblecombe55 for aqueous H2SO4, the representation of the degree of dissociation of the bisulfate ion is significantly improved over wide ranges of temperature and concentration. This is not a necessary condition for the accurate calculation of solute and solvent activities, which determine gas/liquid/solid equilibrium.162 However, it is generally desirable that solution models for systems containing H+ and SO42- ions, represent well the observed HSO4- speciation. The low freezing point depression in aqueous Na2SO4 solutions combined with the lack of thermodynamic data at temperatures below freezing for this system inhibits the determination of activities and solubilities of acid hydrates of H2SO4, HNO3, and HCl with the model for system I. However, with the exception of H2SO4-Na2SO4-H2O mixtures, for all ternary

Thermodynamic Model of H+-NH4+-Na+-SO42--NO3--Cl--H2O subsystems the model allows with satisfying accuracy for the calculation of the temperature and concentration at the eutectic point. The model is parametrized using available measurements of vapor pressure, degree of dissociation, dissociation constant of NH3, and isopiestic and edb data as well as calorimetric properties such as enthalpy of solution, enthalpy of dilution, enthalpy of mixing, and heat capacity. The parametrization of the model for system II and system III relies largely on previous studies of Clegg et al.48 and Carslaw et al.23 Model parameters for (NH4)2SO4-H2O, HNO3-H2O, and HCl-H2O are adopted from Clegg et al.56 and Carslaw et al.23 For some of the single solute solutions and ternary mixtures that are part of systems II and III, the models presented in Clegg et al. and Carslaw et al. are partly adjusted to newly available thermodynamic data. However, the discussions of this systems in Appendix II confirm the models of Clegg et al. and Carslaw et al. Moreover, the models presented by Clegg et al. and Carslaw et al. represent solvent activities and equilibrium constants in a quality similar to that of the models for systems II and III. Differences are ascribed mainly to an improved description of HSO4- dissociation and to the use of newly available low temperature heat capacities of pure water76,77 in the present work. Acknowledgment. This work was funded by the BMBF projects AFS (Grant 07AF303C/2) and AFO2000 (Grant 07ATF02). We also acknowledge the support of the Landesamt fu¨r Natur, Umwelt und Verbraucherschutz NRW (LANUV). The plots within this work have been produced with the computer program gnuplot.


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