J. Phys. Chem. 1990, 94, 7675-7681
7675
Modellng of the Thermodynamics of Electrolyte Solutions to High Temperatures Including Ion Association. Application to Hydrochloric Acid J. M. Simonson,* H. F. Holmes, R.H. Busey, R. E.Mesmer, Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
D.C . Archer,**+and R. H. Wood Department of Chemistry, University of Delaware, Newark, Delaware 1971 6 (Received: March 24, 1989; In Final Form: April 24, 1990)
Two semiempirical models for the excess thermodynamic properties of electrolyte solutions are presented and applied to the available thermodynamic results for hydrochloric acid from 298 to 648 K, from near saturation pressure of water to 40 MPa, and from 2 mobkg-' to infinite dilution. Each treatment includes explicitly the formation of one or more associated solute species. The ion association-interaction model is based on the ion interaction treatment of Pitzer and includes an ion pairing equilibrium constant determined from electrical conductance measurements. The activity expansion-chemical equilibrium treatment is based on the activity expansion model of Wood,Lilley, and Thompson; formation constants for two associated solute species are determined by fitting to excess thermodynamic measurements. These two treatments are compared with each other and with the ion interaction treatment applied at high temperature by Holmes et al. Values of excess thermodynamic properties calculated from the three models are in reasonable agreement to 523 K. The two ion association treatments differ greatly in the low-temperaturevalues of Kk The wide range of assumed association behavior consistent with measured excess thermodynamic properties when K A is small is demonstrated by the difference in association thermodynamic quantities in the two association models. Differences in excess thermodynamicquantities calculated from the three treatments at temperatures above 523 K demonstrate the dependence of these quantities on the models used.
Introduction Recently, experimental methods have been developed which permit straightforward measurement of the thermodynamic properties of aqueous electrolyte solutions to high temperatures and pressures.'-s This experimental capability has increased the quantity and quality of electrolyte thermodynamic results under extreme conditions. The results obtained in these studies present a significant challenge in modeling. Semiempirical treatments6 which have been applied successfully to experimental results at temperatures below 523 K are less satisfactory at higher temperature~.~~' Above 523 K, the rapid decrease of the solvent bulk dielectric constant with increasing temperature favors formation of ion pairs. Models based on complete solute dissociation are not expected to represent quantitatively the observed thermodynamic properties of strongly associated electrolytes. In principle, treatments that include explicit consideration of solute speciation may be used to correlate electrolyte solution thermodynamic properties over wide ranges of temperature, pressure, and solute molality. However, when the amount of association is small, as it is for HCl(aq) at low temperatures, the application of association models to measured stoichiometric thermodynamic properties is ambiguous. If solute speciation at the conditions of interest is not known from independent experiments, relative species concentrations must be estimated from the overall treatment. Different assumed identities of associated species or different measurements give quite different values of KA. If a virial expansion or specific interaction model is used to calculate activity coefficients of solute species, the estimation of relative concentrations is complicated by the mathematical covariance of the association constant and the second virial coefficient. As discussed by Guggenheim,* association models may therefore not be unique. When the amount of association is large, the association constant becomes unambiguous. In this work two distinct treatments are used to represent the dilution enthalpy results for hydrochloric acid to 648 K and 2 mobkg-' reported recently by Holmes et aL9 The two models differ significantly in the solute species assumed and in the treatment of short-range solute interactions. The ion association-interaction (AI) model is based on the ion interaction 'Present address: Electrolyte Data Center, National Institute of Standards and Technology. Gaithersburg, MD 20899.
framework developed by Pitzer;Iosimilar treatments for associating electrolytes have been presented by Pitzer, Roy, and Silvester," Pitzer and Silvester,l* and Barta and Bradley.13 Weare and c o - ~ o r k e r s have ~ ~ * ~made ~ extensive use of this approach in modeling mixed electrolyte solutions a t 298.15 K. In this treatment the association constant is assumed to be known at any temperature and solvent density from extrapolation of the values determined from the conductance measurements at 673-1073 K of Frantz and Marshall.I6 Interaction parameters at the second and third virial coefficient levels for the ions and at the second virial coefficient level for the ion pairs are determined through least-squares fitting to the dilution enthalpy results. A chemical equilibrium (CE) model for use over wide temperature and pressure intervals has recently been developed and demonstrated by application to the available thermodynamic data for aqueous magnesium sulfate, an electrolyte whose behavior at 298.15 K is intermediate between that of a strong and a weak electr~lyte.'~This model originated from the activity expansion model developed by Wood, Lilley, and Thompson'*J9 and is ca(1) Busey, R. H.; Holmes, H. F.; Mesmer, R. E. J . Chem. Thermodyn. 1984, 16, 343.
(2) Smith-Magowan, D.; Wood, R. H. J. Chem. Thermodyn. 1981, 13,
.-. . .
1flA7
(3) Mayrath, J. E.; Wood, R. H. J . Chem. Thermodyn. 1982, 14, 15. (4) Holmes, H. F.; Baes, Jr., C. F.; Mesmer, R. E . J . Chem. Thermodyn. 1979, 11, 1035. (5) Rogers, P. S. Z.; Pitzer, K. S . J . Phys. Chem. 1981, 85, 2886. (6) Holmes, H. F.; Mesmer, R. E.J . Phys. Chem. 1983, 87, 1242. (7) Simonson, J. M.; Busey, R. H.; Mesmer, R. E. J . Phys. Chem. 1985, 89, 557. (8) Guggenheim, E. A. Trans. Faraday Soc. 1960, 56, 1159. (9) Holmes, H. F.;Busey, R. H.; Simonson, J. M.; Mesmer, R. E.; Archer, D. G.; Wood, R. H. J. Chem. Thermodyn. 1987, 19, 863. (10) Pitzer, K. S.J . Phys. Chem. 1973, 77, 268. (11) Pitzer, K. S.; Roy, R. N.; Silvester, L. F. J . Am. Chem. SOC.1977, 99, 4930. (12) Pitzer, K. S.;Silvester, L. F. J. Solution Chem. 1976, 5, 269. (13) Barta, L.; Bradley, D. J. J. Solution Chem. 1983, 12, 631. (14) Harvie, C. E.;M~iller,N.; Weare, J. H. Geochim. Cosmochim. Acta 1984, 48, 723. (15) Felmy, A. R.; Weare, J. H. Geochim. Cosmochim. Acta 1986, 50, -11,
LIII.
(16) Frantz, J. D.; Marshall, W. L. Am. J. Sci. 1984, 284, 651. (17) Archer, D. G.; Wood, R. H. J. Solution Chem. 1985, 14, 757. (18) Wood, R. H.; Lilley, T. H.; Thompson, P. T. J . Chem. Soc., Faroday Trans. I 1978, 74, 1301.
0022-3654/90/2094-7675%02.50/0 0 1990 American Chemical Society
7676
The Journal of Physical Chemistry, Vol. 94, No. 19, 1990
pable of representing the thermodynamic properties of aqueous carboxylic acids,20electrolytes which are decidedly weak in behavior. Preliminary application of this chemical equilibrium model to enthalpy of dilution results for HCl(aq) from 298 to 600 K2I showed that this model could indeed represent the observed transition from strong-teweak electrolyte behavior with increasing temperature, without including explicitly the electrical conductance results at high temperatures. Development of Semiempirical Models Ion Association-Interaction Model. For the uni-univalent electrolyte considered in this work, it is assumed that only the neutral ion pair is present due to ion association. The equilibrium constant K A for the association reaction is KA
= mNyN/m?gA2
(1)
where mi and are the equilibrium molalities in mol-kg-l of the charged and neutral solute; yNand g, are the neutral species activity coefficient and the real mean ionic activity coefficient, respectively. The charged and neutral species molalities are related to the stoichiometric molality m through the degree of dissociation a
mN = (1 - a ) m
(2a)
mi = a m
(2b)
The stoichiometric mean ionic activity coefficient yt and stoichiometric osmotic coefficient & are related to the corresponding real quantities as Y+ =
h = dCm,)/2m
(3a)
+
G E / R T = -(4A,I/b) In (1 + 2m;Gx + 2 m $ p ~ x+ "'ANN + 2 m i m ~ A ~( 4x) ~
where ionic short-range interactions are represented at the third virial coefficient (triple ion interaction) level and interactions involving ion pairs are included a t the second virial coefficient level. In this expression GErefers to a solution containing 1 kg of solvent, R is the gas constant and T the absolute temperature, A , is the limiting slope for the osmotic coefficient as defined by Bradley and Pitzer,22and I is the real ionic strength
(5)
The ion interaction coefficient pMx is a function of ionic strength given by BfMx = @cO)
+ 2/3("[1 - ( 1 + all/') e x p ( - ~ P / ~/u21 )]
bulk dielectric constant, and k is the Boltzmann constant. While Pitzer and Li23 have found it useful to include this dependence at very high temperatures (to 873 K), a constant value of b = 1.2 kg1/2.moll/2was used successfully in the comprehensive ion interaction of NaCl(aq) to 573 K by Pitzer, Peiper, and B ~ s e y . ~ ~ This constant value of b was also found to be adequate in this work to 647 K. Parametric expressions for the activity coefficients of charged and neutral solute species and the real osmotic coefficient may be obtained from appropriate differentiations of eq 4. The real mean ionic activity coefficient is given by In g, = - ( A , / b ) [ 2 In (1
(6)
where Po)and PI)are adjustable parameters; a is the a parameter of the ion interaction treatmentlo and is given a value of 2.0 kg'/2.mol-'/2, The ion size parameter b in eq 4 is predicted by Debye-Hiickel theory to be a function of temperature and dielectric constant having the form
b = q/DkT (7) where 9 is a temperature-independent parameter, D is the solvent (19) Wood,R. H.; Lilley, T. H.;Thompson, P. T. Fluids and Fluid Mixrures; I.P.C. Science and Technology Press: Guilford, U.K., 1979; pp
+
+ b11/2/(l +
+
m i a x + mi2ckx + ~ N A M X(8)N
where
ax= 2pc0) + 28(1)[1 - (1 +
- Y2a21)exp(-aP/2)] /a21 (9)
and
ckx
=3Gx (10) The expression for the neutral species activity coefficient is
In YN = 2"X"
+~
~ ~ A M x N
(1 1)
The real osmotic coefficient 4 is written as
4 - 1 = {-(2A&[13/2/(1
+ b11/2)] + 2 m ? a x + 2 m i 3 a x + mN2XNN
+
(12)
2mimNAMXNI/(Cmj)
where the summation is over all solute species and
ax= + /P)exp(-aIl/2)
(3b)
where the summation in eq 3b is over all salute species. In the ion interaction treatment developed by Pitzer, the excess Gibbs free energy of an associating 1:l electrolyte solution is written as a sum of long-range (electrostatic) and short-range interactions among solute species. Extending the ion interaction treatment to include ion pairs gives
I = '/zCmiz$
Simonson et al.
(13)
p(0)
CRX = 2 G x (14) The dependence of the relative soslute concentrations on temperature must be recognized in deriving an expression for the apparent relative enthalpy. The resulting expression is
+
L, = ( A H / b )In ( b I l l z / j ) - 2 m i R P [ B h X+ miCMx] (1 - a ) ( ~ H f / 2 b- ~ P [ 2 m , ( d ~ & ~ / a T4m,2CMx ), mN(2AhXN - XkN)]) - ( R T 2 / a ) ( a a / a T ) ( 2- a (1 - a ) [ ( A , / b ) ( 3 f - P ) - 4mj[p(')+ p(')(1a11/'/4) eXp(-dl/')] - 6 m ? a x + 2 m i ( 2 A ~ x- ~A N N ) ] ] (1 5 )
+
+
where
f= bP/(l
+ bN*)
(16)
In eq 15 all coefficients denoted with superscript L are derivatives with respect to temperature at constant pressure of the corresponding parameters of eq 4 . The quantity (a&x/aT), is a partial derivative in which the relative solute concentrations are held constant. The limiting slope A, follows the definition of Bradley and Pitzer.2z The temperature derivative of the degree of dissociation includes a dependence on the enthalpy of association of the solute at infinite dilution and may be obtained through differentiation of eq 1
aa
- = - a(l - a ) aT a-2
1
AAHo d In g, +2--RF
dT
d In
YN
dT
]
(17)
where AAHO is the standard-state enthalpy change for the ion association reaction. The expression for d a / d T in terms of the interaction coefficients may be obtained through differentiation of eqs 8 and 11. An expression for the excess heat capacity may in principle be obtained through temperature differentiation of eq 15. In
103-111.
(20) Harris, A. L.; Thompson, P. T.; Wood,R. H.J. Solurion Chem. 1980. 9, 305. (21) Archer, D. G. Ph.D. Dissertation, University of Delaware, 1984. (22) Bradley, D. J.; Pitzer, K. S. J . Phys. Chem. 1979, 83, 1599.
(23) Pitzer, K. S.; Li, Y.-g. Proc. Natl. Acad. Sei. U.S.A. 1983.80, 7689. (24) Pitzer, K. S.; Peiper, J. C.; Busey, R. H. J . Phys. Chem. ReJ Data 1984, 13, I .
The Journal of Physical Chemistry, Vol. 94, No. 19, 1990 7677
Thermodynamics of Electrolyte Solutions practice, this differentiation is quite cumbersome. To avoid using a very complicated equation for heat capacities may be calculated by numerical differentiation. Chemical Equilibrium Model. The equations used in the chemical equilibrium model are17
q,
ai = miTi = -yoi(m,- C n,K,aa/yon) n> I
(18)
The symbol n specifies a set n = nl, n2, ..., nu in which nl, n2, ..., n, are the numbers of component ions or molecules of the set n defining an n-mer species in solution. The product of the activities of the component ions or molecules of the set n is a' = nElap. A, is the Debye-Hiickel limiting law slope for the activity coefticient calculated form the equation of state for water of Haar, Gallagher, and Kellzs and the dielectric constant of Uematsu and Franck.26 The charges of species i and n respectively are zi and z,; K. is the equilibrium constant for formation of the cluster or n-mer of the set n. Eni is the sum of the numbers of molecules or ions (nl + n2 + ... + nu) of the set n. is an adjustable parameter approximating a volume exclusion term for the solute species.I7 The set of nonlinear equations (18-20) is solved iteratively with a Newton-Raphson algorithm. The osmotic coefficient of the solution is obtained from a Gibbs-Duhem integration of the "frozen" equilibrium mixture, with the proportions of all solute species held at their finite-molality equilibrium values. The result is
where 4j
=1
+ '/2@(Zmi) i
(A,z/2/81)[1
+ 2I1I2- 2 In (1 + 2I1I2)- (1 + 211/2)-1](22)
and
4n = 1 + !'KZmO(CnJ i i (A,z?/81)[1 + 2I1I2- 2 In (1
+ 2I1I2)- (1 + 211/2)-1](23)
The excess Gibbs energy per kilogram of solvent is
GE = umRT(1 - dSt+ In y*)
(24)
where u is the stoichiometric number of particles into which the solute dissociates ( u = 2 for 1-1 electrolytes). The excess enthalpy and heat capacity are obtained numerically from the relations
[W"TIp
=-L/P
(25)
q
(26)
and aL/m=
The relative apparent molar enthalpy, L,, and heat capacity, Cp,, - COpsQare L, = L / m
(27)
CPd - C O P , , = q / m
(28)
where COP, is the apparent molar heat capacity at infinite dilution and m is the stoichiometric molality. A Marquardt nonliner least-squares routine is used to fit the equations to measured values. The routine minimizes the sum of the squared weighted errors. (25) Haar, L.; Gallagher, J . S.; Kell, G . S.NBSINRC Sleam Tables; Hemisphere Publishing: Washington, DC, 1984. (26) Uematsu, M.; Franck, E. U.J . Phys. Chem. Ref. Dora 1980,9, 1291.
Application to Hydrochloric Acid Each of the semiempirical models described above combines explicit recognition of an assumed ion association equilibrium with parametric expressions for species activity coefficients in a consistent representation of excess thermodynamic properties. The significant differences in practical application of the models are illustrated by fitting each to the dilution enthalpy results for hydrochloric acid from 298 to 647 K, 7 to 40 MPa, and 0.008 to 2.0 molakg-I. The results fitted are the lower molality dilution enthalpies reported by Holmes et aL9 AI Model. To remove the correlation in values of the association constant K A and the enthalpy of association P A P at the temperature and pressure of interest with the composition-independent fitting parameters of eq 15, we adopt the expression used by Frantz and MarshallI6 to represent association constants for HCl(aq) In K A = C1 + C 2 / T C , In ( p w / p * ) (29)
+
Here pw is the density of water, p* = 1 g . ~ m - ~and , the empirical parameters have the values CI = -12.45, C2 = 8923.9, and C, = 13.93. The use of eq 29 near ambient temperature involves a long extrapolation in temperature of the results of the conductance measurements. As a test of the validity of this extrapolation, the value of the association constant at 298 K and 0.1 MPa calculated from eq 29 was compared with values calculated by Robinson2' and by Marsh and McElroyZ8from the partial pressures of HCl over the aqueous solution and the vapor pressure of pure HCI. Association constants calculated from eq 29 are about an order of magnitude lower at 298.15 K than those of refs 27 and 28. However, the effects of ion association are negligibly small to temperatures near 500 K the shorter extrapolation to temperatures at which ion association is important should give an adequate approximation for the association constant. Calculation of the degree of dissociation a using eq 1 requires values of the activity coefficients of all solute species a t the conditions of interest. If measurements of these activity coefficients are not available, the dilution enthalpy results must be integrated from a reference temperature a t which the values are known. Noting that the ion association-interaction model reduces to the ion interaction model when the solute is completely dissociated and K A at 298 K and 0.1 MPa calculated from eq 29 is very small, solute activity coefficients were calculated by adopting the ion interaction coefficients at this reference temperature and pressure from Holmes et aL9 Choosing suitable temperature-dependent forms for the interaction coefficients and fitting the dilution enthalpy results as a function of temperature gives values of the activity coefficients through implicit temperature integration. Thus, the fitting calculation is iterative, involving optimization of the model parameters by nonlinear least squares, calculation of activity coefficients through temperature integration, and determination of the degree of dissociation by iterative solution of eqs 1, 8, and 1 1 . In eq 15 the interaction parameters involving the neutral species appear as the linear combination 2A- ANN; the two parameters therefore cannot be determined uniquely through fits of dilution enthalpy results. The parameter AMXNand its temperature derivative were set to zero in this work. The four composition-independent parameters Po),PI),@, and ANN were determined as functions of temperature and pressure; following the example of Holmes et al.? the density of water has been used to express both temperature and pressure dependencies. The parameters are given by
Q= 41
+ 42(T - TR) + 43 In
(p/pR)
+ 44(p
- PR)
+ qSP + 46@ (30)
where Q is Po),PI),@, or ANN and the subscript R denotes values of the parameters at the reference temperature and pressure (27) Robinson, R. A. Trans. Faraday SOC.1936, 32, 743. (28) Marsh, A. R. W.; McElroy, W. J. Armos. Emiron. 1985, 19, 1075.
7678 The Journal of Physical Chemistry, Vol. 94, No. 19, 1990 TABLE 1: Parameters of the AI Model (E4 30)
Q 91 I 04q2 q3 44 io5q5 105q,
@'I'
@'O'
0.176 942 -4.436 43 -0.479 687
0.295 699
0.50 3.771 64
TABLE 11: Parameters of the CE Model (Eqs 32-34)
x
K@p
0.000 37 1
14.8I42 -1 5.0343
2.72 12.6443
c"
0.21 5 559 -1.15
-1.293 76 0.976 858
(298.15 K, 0.1 MPa), pR = 0.99707 g - ~ m -p~ is; the density of water at the experimental temperature and pressure and was calculated from the equation of state of Haar, Gallagher, and KelLz5 The exponential term contributes to an improved fit above 523 K, with x = (7'- 473.15)/20
Simonson et al.
(31)
The forms of eqs 30 and 31 were chosen empirically and are not unique. An effort was made to represent the experimental results at an acceptable level of accuracy with a minimum number of adjustable parameters. The parameters of eq 30 were determined by nonlinear leastsquares fitting to 185 measurements. Values of q1 and q5 for @(O), and c" were adjusted to reproduce the stoichiometric osmotic and activity coefficients at 298.15 K and 0.1 and 40 MPa tabulated by Holmes et aL9 within f0.002. Weights were assigned to the results by assuming an experimental uncertainty of 2% of the dilution enthalpy, or 20 Jemol-I in the case of small measured values. Values of the parameters, including the reference coefficients determined in the ion interaction study, are listed in Table I. The standard error of fit to the dilution enthalpy results was 130 Jmmol-', corresponding to an average deviation of 4.7%. This deviation is larger than the estimated experimental error but was accepted as a reasonable compromise between fit quality and number of parameters required. CE Model. The osmotic and activity coefficient values for NaCl(aq) from 0.001 to 6.0 mol-kg-I at 298.15 K given by Hamer and WuZ9were fitted to determine the set of K. and /3 that would best represent activity data for the general case of a strong electrolyte. The use of K , - , , K2-2 (Le., association constants for the association of one Na+ and one C1- and for the association of two Na+ and two C1-, respectively), and /3 clearly gave better representation of the values than did fits using K I - ]and 0,or K I - , , 8, and a /32(zmi)2(or /32Cni(xmi)2in y o n ) . Root-mean-square errors for the osmotic and activity coefficients were 6 X IO4 and 7 X IO4, respectively. The best fit to the activity coefficients for HCl(aq) given by Hamer and WuZ9at 298.15 K, 0.1 MPa, and 0.001-4.0 molskg-I was also obtained with this set of parameters and gave a root-mean-square error of 4 X IO4. Representation of the HCl(aq) results over the remainder of the temperature and pressure interval was accomplished by fitting equations for the temperature and solvent density dependencies of Kl-l and K2-2 and the temperature dependence of /3 to the experimental results. In addition to the Hamer and Wu activity coefficients and the enthalpy of dilution results through 2.0 mol-kg-I, the fitted results included a set of activity coefficients at 7.2 MPa calculated from the Hamer and Wu values and the volume data of miller^,^^ the activity coefficient values of Greeley et al.31 from 333.15 to 498.15 K, and the 298.15 K, 0.1 MPa dilution enthalpies of Falk and S ~ n n e r . ~ ~ The dependence of KI-Ion temperature and solvent density was given the general functional form In K1-I = In K T F + [ A In ( T / T R ) + B ( l / T - l/TR) + C ( I / P - I / T R 2 ) + ...I + I n ( p / p o ) [ F ( l / T - l / T R ) + G ( P - P o )+ H ( p - p o ) ( l / T - ]/TR) + ...I (32) (29) Hamer, W. J.; Wu, Y . C . J . Phys. Chem. Ref. Data 1972, 1 , 1047. (30) Millero, F. J. In Activity Coefficients in Electrolyte Solutions; Pytkowitz, R. M., Ed.;CRC Press: Boca Raton, FL, 1979; Vol. 11. (31) Greeley, R. S.;Smith, W. J.; Lietzke, M. H.; Stoughton, R. W. J . Phys. Chem. 1960,64, 1445. (32) Falk, B.;Sunner, S . J . Chem. Thermodyn. 1973, 5. 553.
A C F
G
0.710116 0.281 392 -1.80792 X IO4 5.03449 X IO' -40.535 65
H K%fR A2 A,
44
-2.51442 X IO4 0.094 9 16 1.957 170 0.206502 -5.228 18 X I O J
where po is the density of water at a reference temperature TR and reference pressure pR. T R and pR were arbitrarily chosen to be 298.15 K and 7.18 MPa, respectively, giving po = 1,00023 g . ~ m - ~KTrp ; is the value of Kl-l at TR and PR, and A, B, C, ... are the adjustable parameters. The temperature dependence of /3 was represented as a power series in temperature of the form
fl = A@+ Bo(T- TR)
+ c@(PTRz) + ...
(33)
where A,, Bo,... are the adjustable parameters. No solvent density dependence has been given to P because fits of the chemical equilibrium model to solution densities for NaCl(aq) to 6.0 mol-kg-l at 298.15 K and from 0.1 to 40 MPa showed that adequate representations of the data were obtained with (8/3/ap), = 0. Isothermal fits to the dilution enthalpy data to only 2.0 molekg-I and at the elevated temperatures could not produce values of dKz-2/dT that were significant with 95% confidence. In other words, the isothermal data were either not sufficiently precise or did not extend to high enough molalities to remove correlation of aK2-2/dTwith the other parameters. However, it was clear from the preliminary enthalpy fits2I and the activity coefficient fits that Kz-2 and were required to give a good representation of the experimental results. To minimize the number of adjustable parameters for its temperature and solvent-density dependence, K2-2 was represented as Thus, the change in K2-2with respect to T and p was assumed proportional to the change of Kl-l. An F-test showed the leastsquares estimated value of A2 to be significant with greater than 95% confidence. The least-squares routine minimized the quantity x((observed with the exception of the dilution value - calculated val~e),W,]~ enthalpies of Holmes et al.,9 which were included as C([1 {(calculatedvalue)/(observed value)),] K.).The W iwere chosen so that each set of data had the appropriate weight in determining the parameter estimates. The least-squares data fit required only 10 parameters, which are given in Table 11, to represent the 347 data points. Average absolute deviations of the dilution enthalpies of Holmes et al. from 298 to 600 K, at 622.0 K, and at 647.6 K were 3.1%, 7.7%, and 11.2%, respectively. Additional parameters could have been added to improve the fit to the dilution enthalpies at 622.0 and 647.6 K; however, at these temperatures, the error in L, arising from extrapolation of the dilution enthalpies from the lowest measured molality to infinite dilution is much larger than the error in the representation of the dilution enthalpies themselves. Thus, further refinement of the fit was unnecessary. The results of Greeley et al.jl were represented within their estimated experimental errors. Differences become larger at the highest temperatures where known experimental difficulties exi~ted.~ The quality of fit for the CE model is equivalent to that of the A I model, required one-third fewer adjustable parameters, and was achieved without prior knowledge of an equation for the equilibrium constant for ion pairing. Discussion At temperatures below 523 K, the measured excess thermodynamic properties of HCl(aq) are consistent with the assumption of complete dissociation of the solute. Extrapolation of the dilution enthalpies to infinite dilution, necessary to calculate activity coefficients by temperature integration, is straightforward where the composition dependence of the measured values smoothly approaches the behavior predicted by the limiting law. Thus, below 523 K excess thermodynamic properties calculated from the CE
The Journal of Physical Chemistry, Vol. 94, No. 19, 1990 7679
Thermodynamics of Electrolyte Solutions
TABLE IV; Stoichiometric Osmotic Coefficients of HCI(aa)
TABLE 111: Stoichiometric Activitv Coefficients of HCl(aa) 548.15 yat(CE)
K
598.15
Yat(AI)
'Yst(CE)
K
648.15
'Yst(A1)
K'
%l(CE)
7't(AI)
Sacturation PIressure
Saturation Pressure 0.01 0.05 0.10 0.20 0.30 0.50 0.75 1 .oo 1.50 2.00
0.782 0.608 0.523 0.441 0.396 0.344 0.308 0.285 0.258 0.242
0.791 0.618 0.529 0.441 0.394 0.340 0.304 0.281 0.252 0.232
0.628 0.400 0.314 0.241 0.205 0.166 0.140 0.125 0.106 0.095
0.647 0.413 0.321 0.242 0.202 0.161 0.134 0.118 0.099 0.088
0.282 0.140 0.102 0.073 0.060 0.046 0.038 0.032 0.026 0.023
0.269 0.133 0.097 0.069 0.057 0.045 0.037 0.032 0.026 0.023
0.01 0.05 0.10 0.20 0.30 0.50 0.75 1.oo 1.50 2.00
0.805 0.647 0.566 0.487 0.442 0.390 0.352 0.328 0.300 0.284
0.812 0.656 0.575 0.494 0.450 0.399 0.364 0.342 0.313 0.292
40 MPa 0.721 0.512 0.419 0.335 0.291 0.242 0.208 0.187 0.161 0.145
0.741 0.536 0.439 0.350 0.304 0.253 0.219 0.198 0.173 0.158
0.422 0.226 0.167 0.122 0.101 0.079 0.064 0.056 0.045 0.039
0.4 16 0.222 0.165 0.121 0.100 0.079 0.065 0.057 0.048 0.042
0.05 0.10 0.20 0.30 0.50 0.75 1.oo 1S O 2.00
0.920 0.850 0.816 0.784 0.767 0.750 0.742 0.739 0.744 0.756
0.924 0.853 0.814 0.775 0.755 0.738 0.733 0.732 0.733 0.727
0.842 0.733 0.690 0.654 0.636 0.617 0.606 0.601 0.600 0.605
0.851 0.734 0.683 0.636 0.612 0.587 0.574 0.570 0.573 0.579
0.652 0.578 0.556 0.538 0.528 0.5 16 0.508 0.502 0.496 0.495
0.645 0.574 0.555 0.540 0.533 0.526 0.521 0.518 0.515 0.514
0.01 0.05 0.10 0.20 0.30 0.50 0.75 1.oo 1S O 2.00
0.930 0.869 0.838 0.809 0.793 0.777 0.768 0.765 0.769 0.780
0.933 0.873 0.841 0.810 0.795 0.783 0.780 0.781 0.782 0.775
40 MPa 0.890 0.796 0.754 0.714 0.694 0.671 0.656 0.648 0.642 0.644
0.900 0.806 0.759 0.715 0.692 0.672 0.663 0.661 0.663 0.666
0.728 0.628 0.596 0.569 0.555 0.539 0.527 0.520 0.512 0.508
0.725 0.626 0.596 0.573 0.562 0.553 0.548 0.547 0.548 0.550
0.01
" p = 30 MPa.
" p = 30 MPa. 0
TABLE V Apparent Relative Enthalpies of HCl(aq) 548.15
K
598.15
L+(CE)
L+(AI)
L+(CE)
0.10 0.20 0.30 0.50 0.75 1.00 1.50 2.00
6.48 14.99 20.32 26.54 30.54 35.96 40.58 44.05 49.27 53.25
5.24 12.41 17.45 23.92 28.41 34.79 40.42 44.68 50.97 55.50
44.94 85.39 103.2 120.0 129.2 140.3 148.7 154.5 162.6 168.4
0.01 0.05 0.10 0.20 0.30 0.50 0.75 1.00 1.50 2.00
3.99 9.38 12.95 17.30 20.22 24.31 27.92 30.72 35.06 38.48
3.26 7.61 10.75 14.88 17.81 22.04 25.85 28.80 33.28 36.64
40 MPa 14.64 32.57 42.22 52.37 58.41 66.09 72.27 76.72 83.18 87.97
K
L+(AI)
648.15
K"
L+(CE)
L+(AI)
46.55 91.81 111.3 129.2 138.5 149.8 158.7 165.2 174.6 181.2
293.2 362.0 382.1 397.8 405.5 414.0 420.1 424.1 429.9 434.1
309.3 376.5 395.3 409.8 416.7 424.2 429.7 433.7 440.0 445.2
12.92 30.16 39.96 50.32 56.45 64.17 70.29 74.61 80.53 84.35
130.5 186.1 204.4 219.5 227.2 235.9 242.3 246.7 252.9 257.4
143.5 202.1 220.5 235.2 242.2 249.8 255.6 259.3 264.9 269.0
Saturation Pressure -1
0.01 0.05
," -2
-3
I 0.0
0.5
1.0
1.5
mfl/(mol.kg-')fl
Figure 1. Stoichiometric mean ionic activity coefficients (as In y) at saturation pressure: 0, A I model; 0 , CE model; A, ,9(*) model (ref 9).
and AI treatments are in reasonably good agreement with those tabulated by Holmes et al.9 Differences in calculated properties may be attributed to the intrinsic differences between the models, in the sets of measurements used to determine values of the adjustable parameters of each treatment, to the relative weights assigned in the least-squares procedures, and to the fit residuals. The maximum differences in $st and ystare as large as f0.02, although agreement is within fO.O1 over most of the range in T, p , and m. Differences in calculated L, are near the 4 5 % fit uncertainty. While the models presented here give results in the low-temperature region near those of Holmes et the values tabulated in that work are based on a fit of a more comprehensive set of results and are recommended as more reliable to 523 K. Above 523 K the observed thermodynamic properties are inconsistent with assumed complete solute dissociation. In this range Holmes et aL9 used the extended form of the ion interaction treatment developed by Pitzer and M a y ~ r g ato~describe ~ the thermodynamics of moderately associated 2-2 electrolytes near ambient temperature. This treatment includes an additional term (33) Pitzer,
K. s.; Mayorga, G. J . Solution Chem. 1974, 3, 539.
' p = 30 MPa.
in the composition-depenL,nt seconc tirial coefficient of the form @(2) is an adjustable parameter and f(az,I) is defined such that the function corresponds to that for the @ I ) term where a is replaced by u2. The f12) parameter takes relatively large, negative values, indicating strong attractive interionic interactions, and effectively describes the association behavior when the equilibrium constant is not too large. In the infinite dilution limit, @C2) is proportional to K A as discussed below. This treatment, model 111 of Holmes et al.? is therefore comparable to the models presented here in the high-temperature region. Stoichiometric activity and osmotic coefficients and apparent relative enthalpies calculated from the CE and AI models are given in Tables 111-V; calculated ystand L, are compared with those of Holmes et aL9 in Figures 1 and 2. The three models give L, and ystin reasonable agreement at 548.1 5 and 598.1 5 K. It should be noted that the differences in L, curves in Figure 2 are not due to failure of any of the models to fit the dilution enthalpy values. At 648.15 K the two association models give excess properties in good agreement, while values calculated from the @(2) treatment $z)f(nz,I), where
7680 The Journal of Physical Chemistry, Vol. 94, No. 19, 1990
400
f a E
S40
%m Ji
too
0
0.0
0.5
LO
1.5
mfl/(md.kg-9@
Figure 2. Apparent molar relative enthalpy at saturation pressure: 0, AI model; 0 , CE model; A, p2)model (ref 9).
differ significantly. The difference in ys,is a consequence of the large differences in L, calculated from the j3@)model as compared with the C E and AI models at the highest tempertures (623 and 648 K). On the scale of Figure 2, the L, calculated from the C E and AI models decrease slowly with decreasing molality at molalities greater than a few tenths and decrease strongly below 0.01 mol-kg-I. The values of L, are thus largely determined by the values of K A and AAH', with other parameters making relatively modest contributions at higher molalities. All three models give L, curves with molality that are concave up at very low molalities and ancave down over the rest of the range to 2 mo1.kg-I as shown in Figure 2. At molalities below 0.01 mol-kg-I, where no data are available, the /3(2) model gives a much less steep extrapolated L, dependence on molality compared with a treatment assuming an explicit ion association reaction. The values of /3c2) and its temperature derivative were determined solely from the dilution enthalpy data, which probably do not extend to sufficiently low molalities to allow a very accurate description of the thermodynamics of rather strongly associated solutes. Values of excess properties calculated from the CE and AI models are more reliable under conditions of relatively high ion association than those calculated from the f12)model, as both these models are consistent with the conductance results in the supercritical range. This point will be discussed below in more detail. Agreement of calculated stoichiometric thermodynamic properties at lower temperatures does not imply agreement of the ion association reaction thermodynamic properties or of the solute speciation calculated from the C E and AI treatments. K A and AAHo for the first association reaction (1-1 ion pair) are shown in Figure 3. The trend with increasing temperature is to closer agreement of the first association constants of the two equilibrium models, with very good agreement at the highest temperatures. Near ambient temperature the K A , while relatively small in each case, differ by several orders of magnitude; the reaction enthalpies also show large differences. Ruaya and S e ~ a r dhave ) ~ calculated reaction thermodynamic functions for ion pairing of HCl(aq) based on measurements of the solubility of AgCl in HCl(aq). The values obtained from these results depend to some extent on the choice of Ag complex species assumed present in the solution and assignment of activity coefficients for all species; also, the solubilities in HCl(aq) at temperatures below 473 K are very near those measured in NaCl(aq),35 indicating that there are small differences in the (34) Ruaya, J . R.; Seward, T. M . Geochim. Cosmochim. Acta 1987, 51, 121.
(35) Seward, T. M. Geochim. Cosmochim. Acta 1984. 38, 1664.
Simonson et al. association constants for NaCl(aq) and HCl(aq) in this lowtemperature range. Noting these sources of potentially large uncertainties in K A and A A H O calculated from the solubility results, the agreement of these values with those from the CE model at low temperatures is remarkably good. At high temperatures the large differences in the K A calculated by Ruaya and Seward from the conductance values may be due to the increased difficulty in identification of complex species.34 The degree of dissociation of HCl(aq) calculated from the two models at saturation pressure and temperatures to 648 K are shown in Figure 4. For a stoichiometric molality of 2.0 molvkg-I at 298.15 K and 0.1 MPa, the CE model gives a dissociation fraction of 0.67, compared with a dissociation fraction near unity for the AI model. The relatively high level of association calculated from the C E model and the disagreement of the association constant calculated from this treatment compared with that from vapor pressure measurement^^^*^* imply that the associated species of the CE model are not those which partition into the vapor phase. The second associated species of the C E model, H2CI2,is a significant fraction of the total associated solute at higher stoichiometric molalities. At 298.15 K and 0.1 MPa, the fraction of associated species represented by H2C12(aq)increases from 0.07 for mst = 1 .O to 0.18 for m,, = 2.0. The relative importance of the 2-2 species increases regularly with temperature; at 648.1 5 K and 30 MPa, the fractions of the 2-2 complex are 0.18 for m,, = 1.0 and 0.28 for m,,= 2.0. While including this species gives an improved fit of the thermodynamic measurements considered here, there is no independent experimental evidence for the existence of the 2-2 species in aqueous solution. Association constants at high temperatures may be calculated from the PC2) model with the limiting relation33 K A = -2P(2)
(35)
Values of K A calculated from eq 35 and the parameters of Model I11 of Holmes et aL9 are shown in Figure 3. At 650 K, K A and A A H O calculated from the Pc2) model are smaller than the corresponding quantities of the CE and AI models; the latter values are consistent with the conductance results. Because the association constants are large and the dilution enthalpy measurements extend only to about 0.01 molekg-', the /3(2) model probably underestimates the extent of ion association at the highest temperatures. The explicit association models are therefore more reliable in this range. It is of interest to note from the curves of Figure 4 that neither the C E nor AI model predicts redissociation of the solute with increasing stoichiometric molalities at any conditions considered here. D a ~ i e shas ) ~ described this effect, due to a very rapid drop in activity coefficient; Davies and Pitzer and Silvester)' have discussed redissociation in aqueous solutions of high-charge-type electrolytes. Similar behavior of 1-1 charge type electrolytes in polar solvents with lower dielectric constant has been described by Pitzer and S i m o n ~ o n . ) ~It should be noted that the present case differs from those previously considered in that the pure solute is a neutral molecule, rather than the (possibly supercooled) pure, ionized fused salt used as a pure solute reference state by Pitzer and Simonson. It is nevertheless worthwhile to compare the product of the dielectric constant and the temperature for the present case with that where apparent redissociation has been observed. For the picrate salt in 1-butanol studied by Pitzer and Simonson DT = 3.6 X lo3 K, while for the aqueous solution at 648.15 K and 30 MPa, DT = 7.8 X lo3 K. Even neglecting the qualitative differences in the forms of the pure solutes, the DT product is not quite small enough in the aqueous system to expect an apparent redissociation of HCl(aq) with increasing molality. The CE, AI, and /3(2) models are all based on the use of the Debye-Huckel limiting law in the excess Gibbs energy. This approximation must fail at the solvent critical conditions, where the limiting slope for the enthalpy is infinite but the dilution ( 3 6 ) Davies, C. W. Ion Association; Butterworths: Washington, 1962. (37) Pitzer, K. S . ; Silvester, L. F. J . Phys. Chem. 1978, 82, 1239. (38) Pitzer, K . S . ; Simonson, J . M . J . Am. Chem. SOC.1984, 106, 1973.
The Journal of Physical Chemistry, Vol. 94, NO. 19, 1990 7681
Thermodynamics of Electrolyte Solutions 5
I
,
I
400
300
0
lml
f
2
B
3 loo
-5
0
-lo
I
2so
554
550
4#)
L
650
T/k Figure 3. (a) Association constant for 1-1 ion pairs at saturation pressure: A, AI model; 0,CE model; 0, ref 34; V, calculated from ,9(*) of ref 9 and eq 35. (b) Enthalpy of association for 1-1 ion pairs at saturation pressure: A, AI model; 0,CE model; 0, ref 34. I
I
1
1
I
0.0
0.8
0.8
0.6
0.7
0.4
0.6
0.2
a
-A-0-
0
0.0
0.5
tO
t5
rn~/(mo~-kg-f)*
0.0
03
to
u
m*/(mo1-1q-9*
Figure 4. Degree of dissociation of HCl(aq) at saturation pressure: (a) 0, AI model, 498.15 K; other curves from CE model at 0,298.15 K; A, 398.15 K;V, 448.1 5 K; 0 ,498.15 K. (b) 0, AI model, 548.15 K; 0,A I model, 598.15 K; A, AI model, 648.15 K; 0, CE model, 548.15 K; W, CE model, 598.15 K; V, 648.15 K.
enthalpies are finite at nonzero solute molalities. The models presented here are thus unreliable in the very near vicinity of the solvent critical conditions and should be applied with caution at high temperatures at pressures below the range of the fitted experimental results.
Conclusions
Measured excess thermodynamic properties of HCl(aq), which are consistent with assumed complete solute dissociation near ambient temperature, must be analyzed with some recognition of ion association at high temperatures. The quantitative application of explicit ion association models to the available experimental results involves an ambiguous assignment of interparticle interactions between the association constant and the activity coefficients of the solute species. In this work two treatments which include ion association have been used to represent the dilution enthalpies from 298 to 650 K, although the reaction thermodynamic quantities and the identities of the as-
sociated species differ significantly. The CE and AI treatments, with pairwise association constants a t high temperatures either given by or in good agreement with those extrapolated from conductance measurements, give stoichiometric thermodynamic properties at the highest temperatures considered here in better agreement with each other than with the j3(*)treatment. The comparisons presented show the model dependence of high-temperature excess thermodynamic properties. Measured values in dilute solutions are critically important in developing models applicable at high temperatures. Acknowledgment. This research was sponsored by the Division of Chemical Sciences, Office of Basic Energy Sciences, US. Department of Energy, under Contract DE-AC05-840R21400 with Martin Marietta Energy Systems, Inc., by the National Science Foundation Grants CHE8009672 and CHE8712204, and by the Office of Standard Reference Data of the National Institute of Standards and Technology.
