4941
J . Phys. Chem. 1989, 93, 4941-4951
Calculation of the Thermodynamic Properties of Aqueous Electrolytes to 1000 OC and 5000 bar from a Semicontinuum Model for Ion Hydration John C. Tanger IVt and Kenneth S. Pitzer* Department of Chemistry and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720 (Received: November 16, 1988)
A semicontinuum approach for prediction of standard-state Gibbs free energies of ionic hydration, AhG, at high temperature steam conditions has been revised to permit predictions at temperatures from 0 to 1000 OC and any pressure up to 5 kbar.
The revised semicontinuum model provides for a more realistic identification of inner-shell and outer-shell contributions. The inner-shell term accounts for contributions to AhG from the first six H 2 0 molecules of hydration. Inner-shell contributions are calculated by using available experimental equilibrium constants for successive ionic hydration reactions and an empirical parameter that represents the effective volume increment per H20of hydration. The outer-shell term accounts for contributions to AhG from H20molecules outside the inner-shell region. Outer-shell contributions are obtained from the Born equation using the bulk solvent dielectric constant and an empirical parameter that represents the effective Born radius. The empirical parameters in the revised model are characterized in a manner that ensures reasonable consistency with experimental standard-state volumes and heat capacities for aqueous electrolytes. Calculations on alkali-metal halide and alkali-metal hydroxide electrolytes are considered, and the results for NaCl are used to illustrate the pressure and temperature behavior of standard-state properties. Evaluation of uncertainties in predicted values for AhG supports application of the revised model for rough quantitative estimation of standard-state properties at supercritical conditions. The validity of the revised model at supercritical conditions is also supported by a comparison of calculated and experimental apparent molal heat capacities for NaCI.
pressures and introduces two temperature-dependent empirical parameters. These model parameters are evaluated at selected Methods for calculating the standard-state thermodynamic temperatures from 0 to 700 'C by using independently predicted properties of aqueous ions and electrolytes at high pressures and values of AhG that are consistent with the available experimental at supercritical temperatures are greatly needed by geochemists data for standard-state properties. Regression analysis of caland chemical engineers. There have been several attempts to culated values for these two parameters provides empirical calculate Gibbs free energies for aqueous electrolytes at conditions equations that permit their close description and allow estimation other than 298 K and 1 bar. Several models for prediction of the of these parameters to 1000 OC. The pressure and temperature Gibbs free energy of hydration, AhG, for aqueous electrolyte^'-^ dependence of predicted values for AhG and standard-state volhave been applied at vapor-liquid saturation pressures of H 2 0 umes, entropies, and heat capacities is discussed using the propand from 0 to 350 "C. Equations for the standard-state partial erties of NaCl as examples. Evaluation of uncertainties in calmolal volumes and heat capacities of electrolytes proposed by culated values of AhG includes provision for the uncertainties in Tanger and Helgeson4 permit prediction of AhG at pressures to the dielectric constant and uncertainties in gas-phase ion hydration 5 kbar and temperatures to 450 'C. However, their equations are restricted to liquidlike H20densities greater than =0.6 g ~ m - ~ . data. We have not provided for solvent compressibility effects in our model, and these effects are important at conditions close or less) and temAt supercritical steam densities (~0.25g to the solvent critical point. Uncertainties in our calculated values peratures up to and perhaps exceeding 1000 OC, the treatment arising from the omission of compressibility effects are investigated proposed by Pitzer5 permits prediction of the self-ionization by using a comparison of experimental and calculated apparent constant of H 2 0 5as well as AhG for the ions Na' and C1-6 and molal heat capacities for NaCl in the critical region. AhG for the ion pair, NaClO.' The ion hydration models of Tremaine and Goldman' and Proposed Semicontinuum Model Pitzer6 as well as early efforts by Muirhead-Gould and Laidlers and Goldman and Batesgare called semicontinuum models because The proposed model for the standard state Gibbs free energies the solvent in the region nearest the ion is treated as discrete of hydration of ions, AhG, at P and T is given by molecules, and outside this region the solvent is treated as a dielectric continuum. The successful application of these semicontinuum models at widely different conditions of temperature where Ah@, represents the inner-shell contribution, which acand H 2 0density suggested to us that the semicontinuum approach counts for contributions to AhG from the first six H 2 0 molecules could provide a truly comprehensive treatment. of hydration, AhGo.s.stands for the outer-shell contribution, which The purpose of this study is to explore development of a semicontinuum model that permits reliable prediction of AhG for aqueous electrolytes at any temperature and H 2 0 density of (1) Tremaine, P. R.; Goldman, S.J. Phys. Chem. 1978, 82, 2317. practical interest. We have restricted estimation of AhG in the (2) Sen, U. 1.Chem. SOC.,Faraday Trans. 1 1981, 77, 2883. present study to aqueous alkali-metal halide and alkali-metal (3) Abraham, M. H.; Matted, E.; Liszi, J. J . Chem. Soc.,Faraday Trans. hydroxide electrolytes from 0 to 1000 OC and pressures of 5 kbar 1 1983, 79, 2781. (4) (a) Tanger, J. C., IV; Helgeson, H. C. Am. J . Sci. 1988, 288, 19. (b) or less. These restrictions are imposed by the available gas-phase Tanger, J. C., IV Ph.D. Thesis 1986, Department of Geology and Geophysics, ion hydration data and by the available equations for estimation UC Berkeley. of the dielectric constant of H 2 0 . Nevertheless, calculations in (5) Pitzer, K. S.J. Phys. Chem. 1982, 86, 4704. the present study cover solvent conditions ranging from gaseous (6) Pitzer, K. S.J. Phys. Chem. 1983, 87, 1120. (7) Pitzer, K. S.; Pabalan, R. T. Geochim. Cosmochim. Acta 1986, 50, to condensed states and include the solvent critical region. 1445. Revision of the inner-shell and outer-shell contributions in (8) (a) Muirhead-Gould, J. S.;Laidler, K. J. Trans. Faraday SOC.1967, Pitzer's semicontinuum model6 extends this treatment to high 63, 944. (b) Muirhead-Gould, J. S.; Laidler, K. J. In Chemical Physics of
Introduction
~~~
'Chemical Thermodynamics Division, Center for Chemical Physics, National Institute of Standards and Technology. Gaithersburg, MD 20899.
0022-3654/89/2093-4941$01.50/0
~
Ionic Solutions; Conway, B. E., Barradas, R. G., Eds.; Wiley: New York, 1966. (9) Goldman, S.;Bates, R. G. J . Am. Chem. SOC.1972, 94, 1476.
0 1989 American Chemical Society
4942
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989
accounts for contributions to AhG from the H 2 0 molecules outside the inner shell region, and AG, accounts for the change in standard states in the hydration process which occurs in the absence of ion-solvent interactions (hypothetical 1 bar ideal gas to hypothetical 1 m ideal solution). The volume of 1 kg of water is lOOO/p with p the density in g while the volume of an ideal gas at 1 bar is R T with R in bar cm3 K-' mol-] and T i n K. It follows that the standard-state conversion term1.* is given by
AGss = R T In ( R T p / 1 0 0 0 )
(2)
The subdivision of AhG into inner-shell and outer-shell contributions is consistent with the widely used conceptual models proposed by Frank and Wenlo and Gurney.I1 X-ray and neutron diffraction studies as well as Monte Carlo and molecular dynamics simulations of dilute aqueous solutions at 1 bar and 298 K indicate that coordination numbers for the inner hydration shell are in the range 5-7 for alkali-metal and halide Determination of the coordination number for infinite dilution in these studies is not unambiguous. Nevertheless, our provision for up to six H 2 0 molecules to describe ion-solvent interactions in the inner-shell region should be reasonably realistic. The inner-shell contribution, AhG'.S.,is written as n=6
A ~ / R =T -In (x,/xg) = -In [ I
+ C 1x , / x , ] n=
(3)
where x denotes the mole fraction, n represents the number of inner-shell H 2 0 molecules hydrating the ion, and the subscripts g and t stand for the pure anhydrous gaseous solute species and the total solute, respectively. Equation 3 is consistent with previous theoretical expressions used to calculate AhG at low pressure steam
condition^.^*^ The equilibrium constant, K,, for the ion (A) hydration reaction A,
+ nHzO = A(H,O),
is related to the fugacities,f, of the reaction species as given by K , = f"/f$"H o. Consequently, by appropriate definition of the fugacity coefficient for the ith solute species as Qli =.f;.x;'PI, where C#Ji 1 as P 0, we can write
- -
xn/xg = PH,O (dg/dn)Kn It is convenient for calculation of (x,/x,)
(4)
to express Kn as
j=n
K, = n K j j= 1
where KJ represents the equilibrium constant for the successive hydration reaction A(H20),-1
+ H 2 0 = A(H2O),
It follows from eq 3-5 that AhGIs is given by
AhGIs / R T = -In (1 +
n=6 n= I
Lf"Hp(dg/d,)iiiKJI} J=1
Tanger and Pitzer TABLE I: Experimental Enthalpy and Entropy Changes for Successive Hydration of Alkali-Metal Cations and Halide and Hydroxide Anions' ion
1
2
3
4
5
6
13.9 12.3 10.7 10.5 (9.9) 13.2 (10.7) (10.5) (8.7) 14.1
12.1 10.7 10.0 (9.8) (9.4) (12.5) (10.4) (10.2) (8.5) (13.4)
31.4 28.1 25.2 25.7 (25.4) 30.7 (25.8) (26.8) (21.3) 33.2
32.0 26.0 25.7 (25.7) (25.4) (30.7) (25.8) (26.8) (21.3) (33.2)
-AHjo, kcal molw1
Lit Na+
K' Rb' Cs+
F C1-
BrIOH'
(34.0)b 24.0 17.9 15.9 13.7 23.3 13.1 12.6 10.2 25.0
25.8 19.8 16.1 13.6 12.5 16.6 12.7 12.3 9.8 17.9
(23.0)b 21.5 21.6 21.2 19.4 17.4 16.5 18.4 16.3 20.8
21.1 22.2 24.2 22.2 22.2 18.7 20.8 22.9 19.0 21.2
20.7 15.8 13.2 12.2 11.2 13.7 11.7 11.5 9.4 15.1
16.4 13.8 11.8 11.2 10.6 13.5 11.1 10.9
(9.0) 14.2
-ASj', cal mol-' K-' Lit Na'
K' Rb' Cs'
F CIBrIOH-
24.9 21.9 23.0 24.0 23.7 20.4 23.2 24.8 21.3 24.8
29.9 25.0 24.7 24.8 25.4 36.9 25.8 26.8 (21.3) 29.5
References 16-19. bParentheses indicate estimates.
However, the uncertainties in these calculated K j values generally exceed those of measured values. Consequently, we have restricted our present calculations to ions for which experimental Kj data are available in order to more rigorously test the validity of the semicontinuum approach. Experimental values of Kj for alkali-metal, halide, and hydroxide ions are available from gas-phase mass spectrometric studies.I6l9 The temperature dependence of these experimental values for Kj is consistent with In Kj = A S j o / R - A H j o / R T
(7)
where ASj. and AH,. represent the entropy and enthalpy of the jth hydration reaction. Values of K j required for the present study are calculated from eq 7 using the reported or estimated values of ASj" and AHj. given in Table I. Alternate equations to eq 7 were used in earlier steam-phase s t ~ d i e sbecause ~ , ~ theoretical considerationsS suggested that the heat capacity changes of successive hydration reactions, ACpoj,are in excess of 2R. Although empirically determined values of 3R and 3.5R for ACpojwere adopted in these earlier studies, similar analysis of ACpojin the present study did not support the need for a nonzero ACpoj;hence, we used the simple eq 7 . Prediction of AhG1.s.from eq 6 and 7 requires evaluation of (C#J,/&). It follows from the above definition of di and the thermodynamic relation, Vi= R T (8 lnf;./8P)T,x,, that
(6)
In earlier versions of eq 6,6v7maximum values of n were not limited to 6 and the term ~LJ.#J~was taken as unity. However, these earlier studies were restricted to steam conditions where ( 1 ) the fugacities, J , can be closely approximated by the partial pressures, P,, and ( 2 ) contributions to hhG from hydration steps with n > 6 are generally small or negligible. Frequently used theoretical approaches for computation of KJ values are the quantum mechanical method, both ab initio and ~emiempirical,'~and the liquid drop theory treatment.14,1s ( I O ) Frank, H. S.; Wen, W. Y. Faraday Soc. Discuss. 1957, 24, 133. ( I I ) Gurney, R. W. Ionic Processes in Solution; Dover: New York, 1953. (12) Chandrasekhar, J.; Spellmeyer, D. C.; Jorgensen, W. L. J . Am. Chem. SOC.1984, 106, 903. (13) Marcus, Y . Ion Soluotion; Wiley: Chichester, UK, 1985. (14) Heicklen, J. J . Phys. Chem. 1978, 82, 2136.
where P stands for the partial molal volume. We evaluate the quantity V,, - V, by first defining an effective molal volume increment per H 2 0 of hydration as given by VH20*
(Vn- V , ) / n
(9)
where VH20* is independent of n. We then assume that VH20* is independent of pressure because we expect inner-shell H 2 0 (15) Castleman, A. W., Jr.; Holland, P. M.; Keesee, R. G. J . Chem. Phys. 1978, 68, 1760. (16) Dzidic, I.; Kebarle, P. J . Phys. Chem. 1970, 74, 1466. (17) Arshadi, M.; Yamdagni, R.; Kebarle, P. J . Phys. Chem. 1970, 74, 1475. (18) Payzant, J. D.; Yamdagni, R.; Kebarle, P. Can. J . Chem. 1971, 49, 3308. (19) Arshadi, M.; Kebarle, P. J . Phys. Chem. 1970, 7 4 , 1483.
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989 4943
Thermodynamic Properties of Aqueous Electrolytes molecules to be nearly incompressible. Consequently, it follows from eq 8 and 9 that we can write In
($g/$n)
= -~PVH',,O*/RT
(10)
vHH1o*
Empirical evaluation of is discussed below. It is now generally accepted that ionsolvent interactions outside the inner-shell region are principally due to long-range electrostatic forces. In common with several other ion hydration models, we use the Born equationZoto represent this outer-shell contribution, AhCo.s.,for the ith ion as
where 7 stands for 83 549 A K, E denotes the dielectric constant or relative permittivity of H 2 0 , Zistands for the ionic charge, and Ri* represents the effective electrostatic radius of the spherical cavity in the dielectric continuum which contains the inner-shell region. For the kth electrolyte, AhCy. is given by
The effective electrostatic radius for the kth aqueous electrolyte is defined by Rk* 2(CionsviZ~/Ri*)-l, where vi denotes the stoichiometric coefficient of each ion. Calculation of the total electrostatic contribution to the free energy of hydration using the Born equation and the bulk solvent dielectric constant is frequently criticized because this computation neglects the effects of dielectric saturation and solvent compressibility on the local dielectric constant. Although these higher order, continuum effects are quite large near the ion, electrostatic calculation^^^-^^ suggest that these effects are small in the outer-shell region at P-T conditions where pure H 2 0 has liquidlike densities. Nevertheless, our restricted application of the Born equation to the outer-shell region would permit some implicit incorporation of these higher order effects because we treat the Born radius term as an adjustable parameter. Consideration of other ion hydration model^'-^-^ that have applied the Born equation at high temperatures and liquidlike densities suggested that Rk* could be represented as a pressureindependent function of temperature. However, in the critical region, large solvent compressibility effects are p r e d i ~ t e dand ~~.~~ implicit incorporation of these effects in the Rk* parameter should require it to become also pressure dependent. The implications of these solvent compressibility effects for our model's uncertainties in the critical region are discussed in following sections. We also considered noncontinuum, structural effects in the outer-shell region in choosing our model for AhG0.'.. Monte Carlo simulations at 1 bar and 298 K12vZ6support the structural changes in the solvent proposed by Frank and Wenlo and indicate that the degree of hydrogen bonding in water decreases significantly as the innermost part of the outer-shell region is approached. Structural contributions to AhC and standard-state volumes and heat capacities are expected to decrease with increasing temperature and pressure in response to the decrease in the hydrogen-bonded structure of bulk ~ a t e r . ~ ,Semiempirical ~' analysis of these structural contributions indicates that they are relatively large contributions only for the P-T derivatives of AhG at liquid densities and temperatures below -50 "C4Comparison of uncertainties in Ah@. values from our model with estimates for (20) Born, M. Z . Phys. 1920, I , 45. (21) Bucher, M.; Porter, T.L. J . Phys. Chem. 1986, 90, 3406. (22) Ehrenson, S.J . Phys. Chem. 1987, 91, 1868. (23) Hatano, Y . ;Saito, M.; Kakitani, T.; Mataga, N. J . Phys. Chem. 1988, 92, 1008. (24) Wood, R. H.; Quint, J. R.; Grolier, J.-P. E. J . Phys. Chem. 1981,85, 3944. (25) Quint, J. R.; Wood, R. H. J . Phys. Chem. 1985, 89, 380. (26) Mezei, M.; Beveridge, D. L. J . Chem. Phys. 1981, 74, 622. (27) Cobble, J. W.; Murray, R. C . , Jr.; Sen, U. Nature 1981, 291, 566.
TABLE 11: Conventional Values of AhC/R for Alkali-Metal Cations and Halide and Hydroxide Anions at P. ( 1 bar) and T,(298 K)' AhGpr.Tr/R
ion
Li'
69.266 81.88 1 90.634 9 3.28 5
Na+
K*
R b+
ion A~JP,.T,IR CS+ 96.027
I-
F
OH-
C1-
Br-
-184.828 -169.358 -166.245
ion
AhGP,.T,IR
-162.060 -184.695
"Calculated by using data in ref 31 and 32 and R = 1.987 19 cal
mol-' K-'.
structural contributions to AhG obtained from ref 4 suggested that separate provision for these contributions in our Ah@'' model was unwarranted. For our calculations with eq 11 we obtained values of E from the equation given by Uematsu and FranckZ8for the conditions (1) 360 OC 1 T I 0 "C and 5 kbar I P > vapor-liquid saturation of H 2 0 and (2) 550 "C 1 T 1 360 "C and 5 kbar 1 P 1 P,,,, where P,,, in bars is given by P,,, = -9798 15.777. At all other pressures and temperatures considered in this study, we used the Kirkwood equation with the parameters and the model for the Kirkwood correlation factor given by P i t ~ e r . Differences ~~ between E values obtained from these two equations are generally less than 0.1 along the common boundary curve given by P,,,. With respect to vapor-liquid coexistence conditions, the Uematsu and Franck equation is used for coexisting liquids at 0-360 "C and the Kirkwood equation is used above 360 "C and for all coexisting vapors. It follows from the ion additivity rule and eq 1, 2, 6 , and 10 for the ith ion together with eq 11 for the kth electrolyte that AhGk at P and T is given by
+
AhCk/RT = -E lolls
VI
In (1 +
n=6
c IfB@eXp(-nPvH20*/RT)
x
n=l
where vk is the total number of ions in the electrolyte; vk ~ i , , m v i . Equation 12 represents the adopted general statement of our model. Densities and fugacities of H 2 0 required for the present study are calculated from the equation of state given by Haar et The model parameters VH@* and Rk* are evaluated by using eq 12 and independently calculated values of AhGk which are consistent with the available experimental data for standard-state properties. Evaluation of Model Parameters
Independently calculated values of AhG for the kth 1:l aqueous electrolyte at P and T were obtained by using the relation Ahck
= C AhGP,,T, + (Gk" - Gk"Pr,T,) lolls
,c
ions
(Gg"
-
Gg'T,)
(13) where C," represents the Gibbs free energy of the pure gaseous ion at T and PI and TI stand for the reference pressure (1 bar) were caland temperature (298 K). Values of (Gko- Gk",,) culated by using the TH88 equations (TH88 = ref 4). Values of (G," - GgoT,)were computed by using the relation
G - GTr = T [ ( G - HTo)/q - Tr[(GTt - HTo)/T1l
(I4)
where Todenotes 0 K. Equations and coefficients for (G - HTJ/T were taken from Chase et aL3I Consequently, all gaseous ions considered in this study, except the hydroxide ion, are treated as ideal monatomic gases and their values of (C - HT,)/T are taken (28) Uematsu, M.; Franck, E. U. J. Phys. Chem. R e j Data 1980, 9, 1291. (29) Pitzer, K. S.Proc. Natl. Acad. Sci. U.S.A. 1983, 80, 4515. (30) Haar, L.; Gallagher, J.; Kell, G. NBSINRC Steam Tables; Hemisphere: Washington, DC, 1984. (31) Chase, M. W., Jr.; Davies, C. A,; Downey, J. R., Jr.; Frurip, D. J.; McDonald, R. A,; Syverud, A. N. J . Phys. Chem. Ref Data 1985, 14, Suppl. 1, I .
4944
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989
Tanger and Pitzer
5K
a
1 0 ” B e s t F i t ” Value
4
‘w, 1 critical point Hpy
0
0
250
500
750
Temperoture,
1000
O C
Figure 1. Pressure-temperature regions denoted by I, 11, 111, and IV considered in the present study. In regions I, 11, and 111 independently calculated values of AhG from eq 13 are used to determine empirical parameters for the semicontinuum model.
0
I
2
4
3
5
6
9
8
7
IO I /
;, Ion’ i3
TABLE 111: Coefficients for Eu 15 and 17a,b
n 1
2 3
4 5 6
4
b,”
17.84 7.583 X -3.718 X lo7 1.064 X lo7 1.343
1.0291 2.0019 X lo4 -7751 7.2155 X lo-’ 1.1995 X -8.9049 X lo-’
For NaC1. to be purely translational. Finally, our accepted conventional values for AhGpr,Tt of ions31v32 are given in Table 11. The TH88 treatment has been applied to the estimation of (Gk’ - Gkopr,T,) in the pressure-temperature regions denoted by I and I1 in Figure 1. In common with our model, the TH88 treatment uses the Born equation and an effective electrostatic radius, re, to represent electrostatic ionsolvent interactions. The re parameter is a salt-dependent constant in region I, and in this region standard-state partial molal volumes and heat capacities predicted from the TH88 model are in good agreement with the available measurements. However, in region 11, the re parameter must be treated as a function of pressure and temperature in order to provide accurate predictions. Extrapolation of the TH88 model to 1000 ‘C, as proposed in their study, keeps r, constant at its region I value for pressures greater than 2 kbar. However, application of the TH88 equations to description of the dissociation constant for the ion pair, NaC1°,33 suggests that re is no longer constant at temperatures greater than those corresponding to the dashed curve in Figure 1. Consequently, we assume that re is constant in the more restricted region 111. Using eq 13 and the TH88 model, we found that the calculated AhGk-P-T surfaces for 1:1 electrolytes are quasi-planar in regions I and 111. This simple, quasi-planar shape suggests that our extrapolation of the TH88 treatment into region 111 is acceptable for estimation purposes. We evaluated the vHZo* and Rk* parameters in our model (eq 12) using a trial and adjustment procedure and independently calculated values of AhG obtained from eq 13. Our first step was and to calculate pressure-independent, “best fit”va1ues of Rk* at selected temperatures from 0 to 700 OC (regions I, 11, and 111) for the electrolytes listed in Table 111. These calculations indicated that both vH29* and Rk*increased systematically with temperature and that VH20*also increased systematically with ion size. This behavior seems reasonable because both increasing temperature and ion size should favor a more open configuration of inner shell H 2 0 molecules. “Best fit” values for both parameters
vH20*
(32) CODATA J . Chem. Thermodyn. 1978, 10, 903. (33) Sverjensky, D. A. In Reviews in Mineralogy; Ribbe, P. H . , Ed.; BookCrafters: Chelsea, MI, 1987; Vol. 17.
*I>
zN
l6r
0
,
,
,
,
,
,
,
200 400 600 Temperoture, ‘ C
1 800
Figure 2. Graphic representation of eq 15 for the effective molal volume increment per H20of hydration (rHZo*)as a function of the crystallographic radius cubed (r:) at selected temperatures (a) and as a function of temperature for the K+ aqueous ion (b). Symbols represent “best fit” values obtained by using eq 12 and 13. TABLE IV: Calculated Values of the Effective Electrostatic Radius, Ra*, in A for Alkali-Metal Halide and Hydroxide Electrolytes at T , (298 9) solute ( k ) Rk*7. solute ( k ) Rk*T. solute ( k ) Re*T. CsCl 2.745 KBr 2.805 LiCl 2.649 KF 2.486 KI 2.899 NaCl 2.731
KC1 RbCl
2.754 2.734
KOH KCI
2.580 2.754
are essentially linear functions of temperature at high temperatures (i.e., from 300 to 700 ‘c for Rk*and from 400 to 700 ‘c for When we use “best fit” values for and Rk*, the differences between AhGkfrom our model (eq 12) and calculated values of AhGk from eq 13 are generally much less than the estimated uncertainties for our model (see below). Consequently, these calculations supported taking Rk* and as pressure independent and we accepted these “best fit” values for evaluation of the temperature dependence of ~ H ~ oand * Rk*. Regression analysis of our “best fit“ VH2,* values using various empirical functions to represent VHfl* as a function of temperature and ionic radius indicates that the “best fit” values are closely consistent with
vHzo*).
vH20*
vH20*
VH2,,* = d l + d2t + d 3 / ( d 4+ t 3 ) + d5r:
(15)
where dnstands for an empirical constant, r, denotes the octahedral crystallographic (Pauling’s) radius of the ion as reported by A h r e n ~ and , ~ ~t represents OC. Regression values of dl-d5 are given in Table 111. Representative comparisons of “best fit” values for VHz0* with calculated values from eq 15 are shown in Figure 2. Note that the “best fit” and calculated values for the K+ ion (Figure 2b) vary almost linearly with temperature above -400 OC. Equations 12 and 15, ion additivity relations, and AhG values for ions at 1 bar and 298 K (Table 11) permit direct calculation (34) Ahrens, L. H . Geochim. Cosmochim. Acta 1952, 2 , 1 5 5
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989 4945
Thermodynamic Properties of Aqueous Electrolytes
I .25
I .20
1.15
T
1.10
I .05
0.950
I .oo
100 200 300 400 500 Temperature, O C
Figure 3. Graphic representation of eq 17a,b (curve) for the model parameter, T, for NaCl (Table 111) as a function of temperature. Symbols denote "best fit" values for alkali-metal chloride electrolytes obtained by using eq 12, 13, and 16 and Tables I and IV.
0.95 0
100 200 300 400 500 Temperature, C'
Figure 4. Comparison of eq 17a,b (curve) with "best fit" values (symbols)
for potassium halide and potassium hydroxide electrolytes (see Figure of the values for Rk* at Tr (298 K) given in Table IV. It follows that the value of Rk*T, for any electrolyte composed of the ions given in Table 11 can be calculated from the Rk*T, values given in Table IV by using additivity relations such as l/RNaBr*T,
=
/RNaCI*T,
-
/RKCI*T,
+
/RKBr*Tr
Any empirical model for description of Rk* as a function of temperature must be consistent with Rk* additivity relations. The simple assumption that Rk* at a given temperature is given by
Rk* = R k * TrT
(16)
where T is an electrolyte-independent, temperature function and T = 1 at Tr satisfies the Rk* additivity constraint. It follows from the definition of Rk*,the ion additivity principle, and eq 16 that Ri* = Ri*Tr? for each ion. Consequently, determination of R!*T, values consistent with our Rk*Tr values would permit estimation of absolute hydration properties for individual ions using our model. However, we will not attempt the calculation of Rj*T, values in the present study. Representative "best fit" T values calculated from eq 16 using Rk*T,values from Table Iv, and our "best fit" Rk* values at other temperatures are shown as functions of temperature in Figures 3 and 4. At temperatures from 500 to 700 "C, which are not shown in these figures, "best fit" values of T generally fall along a linear extrapolation of their corresponding values at 300-450 "C. As can be seen in Figure 3, "best fit" T values are nearly electrolyte independent when electrolytes with a common anion are compared. It should be noted that the size of the filled circles in Figure 3 represents the complete range of "best fit" T values for LiCI, KC1, RbCl, and CsCl, while the diameter of the open circles for NaCl is just twice that of the filled circles. In contrast to the relatively electrolyte independent T values in Figure 3, the "best fit" T values for electrolytes with a common cation shown in Figure 4 reveal large and systematic differences with increasing temperature. Nevertheless, consideration of the effects of uncertainties in experimental values of AHjo and ASID (Table I) on "best fit" values of T indicates that T can be taken as electrolyte-independent within its uncertainty. For example, the differences between the filled and open squares representing T values for KOH (Figure 4) are solely the consequence of using AHl' and AS," data for OH- from ref 35 (filled squares) rather than the OH- data given in Table I (open squares). Note that the region between the filled and open squares defines a rather large uncertainty envelope. Similar calculation of the T uncertainty envelopes for all of the salts shown in Figures 3 and 4 indicated that the T function could be taken as electrolyte independent. However, these calculations do not permit unambiguous determination of a unique T function. Consequently, values of T used in our present calculations are obtained by regression of "best fit" (35) Meot-Ner, M.; Speller, C. V. J . Phys. Chem. 1986, 90, 6616.
3). The filled symbols represent "best fit" values obtained by using A H , O and AS,' for OH- from ref 35 and RKoH*Tr = 2.65 8, instead of the corresponding values given in Tables I and IV.
values for the electrolyte of interest. The simple, empirical function given by T
= 6,
+ 6zT + 63/P
(17a)
permits reasonably close description of "best fit" T values for any specified electrolyte from 0 to 700 "C. However, at temperatures below 175 "C we also used the relation T
= 64
+ bST + 6 6 P
(17b)
when a closer description of "best fit" T values was desired. This closer fit is needed to provide a more reasonable prediction of standard-state entropies and heat capacities at temperatures below 175 "C. Coefficients 61-66 obtained from "best fit" T values for NaCl using eq 17a,b are given in Table 111.
-
P-T Surfaces for Thermodynamic Properties One of the major objectives of this study is to provide correct predictions of the qualitative shapes for the P-T surfaces of the standard-state thermodynamic properties of aqueous electrolytes at H 2 0densities ranging from vaporlike to liquidlike. For a given thermodynamic property, the shapes of the P-T surfaces for 1 : l aqueous electrolytes are very similar at liquidlike H20 densitie~.~ Calculations using our present model indicate that this strong similarity also prevails in the critical region and at vaporlike HzO densities. Our model also suggests that the P-T surfaces for non- 1 : 1 electrolytes and ion complexes should generally resemble those for 1:l electrolytes. In the following discussion, we have chosen the P-T surfaces for NaCl to illustrate the general shapes of these surfaces. Differentiation of our model equations for AhG with respect to pressure and temperature provides explicit equations for the entropy, A$, heat capacity, AhcB and volume, AhV, of hydration. The equations for AhS and AhCp are given by eq A1 and A2 in the Appendix, and the equation for AhVis given by Ahv,
=
2
(n)(vHzO*
ions
- vHzOo)
+
( 2 s R / R k * ) ( d ( 1 / € ) / d P )+T VkRT@H,O (18) where VHzoois the molal volume of pure H20, @ = P(dV/dP), and ( n ) represents the mean hydration number for an aqueous ion as defined by n=6
(n)
En(xn/xg)/[l
n= 1
n=6
+ n=2 (Ix n / x J I
(19)
Our calculated values Of Ahc, Ahs, Ahv, and AhCp for NaCl are shown as functions of temperature at selected pressures in parts a, b, c, and d of Figure 5, respectively. We have extended our
4946
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989
Tanger and Pitzer 31-)
Temperature, "C
,
-z/ 200
0
600
400
Temperoture,
800
-5 -5
-15 -20
-25 -30 200
400
300
500
Temperature,
600
'C
0.50
-e
0
V
=
4
500
/..1
750 Temperature, ' C
IO00
present calculations from 700 to 1000 "C (e.g., Figure 5 ) because this involves only a linear extrapolation of VH20*(eq 15) and T (eq 17). It follows from the definition of A& (e.g., eq 13) that Ahvk = Vko, AhSk = - ~ i o n s V , ~ g o , and AhCp,k = Cp"p"k CionSviCpog, where Sgois given by the Sackur-Tetrode equation is 5R/2 for alkali-metal and halide ions. Since Ahvk and cpog = V k o and Cpogis small, Figure 5c, and Figure d also represent the P-T surfaces for the more familiar quantities pko and C p o k . The P-T regions where inner-shell and outer-shell contributions for NaCl predominate are denoted by C and D, respectively, in parts a, b, c, and d of Figure 6 for the Gibbs energy, the entropy, the volume, and the heat capacity, respectively. In regions A and B of Figure 6a-q standard-state contributions predominate, and in region A these contributions represent 95% or more of the total hydration property. In region A of Figure 6d, AhCpis 0.5R or less. The P-T isopleths for the mean hydration numbers of Na+ and C1- ions are shown in Figure 7. At 25 "C and 1 bar the mean hydration numbers for the ions considered in this study are (5.70)Li+, (5.81 ) N a f , (5.53)K+, (5.36)Rb+, (4.93)Csf, (5.89)F, (5.92)0H-, (5.66)C1-, (5.22)Br-, and (5.01)1-. Note that for a given charge these values of ( n ) generally decrease with increasing ion size. At a given P and T, inner-shell and outer-shell contributions also generally decrease in magnitude with increasing ion size, as can be deduced from eq 6, 7, and 11 and Tables I and IV. Nevertheless, figures corresponding to Figures 6 and 7 for other 1:l aqueous electrolytes would be quite similar to those for NaC1. Outer-shell contributions predominate in region D (Figure 6) and is obtained from the Born equation (eq 11). The equations for A h T , ,A h T . ,and Ahci,kare represented by the terms containing E in eq 18, A l , and A2, respectively. In accordance with the Born equation, AhGt'' becomes more highly negative as the dielectric constant increases. The increase in our Born radius term, Rk*,with temperature slightly modifies this
sko
I
CI
0
250
Figure 6. Pressure-temperature regions where outer-shell contributions (region D), inner-shell contributions (region C), and standard-state conversion contributions (region B) for NaCl predominate. Predominant contribution regions are shown for Gibbs free energies (a), entropies (b), volumes (c), and heat capacities (d). In region A, values of AhG, A$, and AhV are within 5%, and values of Cpo are within 10%of corresponding limiting values for unhydrated aqueous ions in an ideal gas solvent.
-3
5
0
Temperature. ' C
O C
-2
m
-3
1000
0.25
o -0.25
- 1.25 50
-1.50 100
1
'
300
500
'
'
700
Temperature,
'
900
C'
Figure 5. Standard-state properties for hydration of Na+ + CI- from the semicontinuum model as a function of temperature at selected pressures in bars. Calculated values are shown for Gibbs free energies (a), entropies (b), volumes (c), and heat capacities (d).
Thermodynamic Properties of Aqueous Electrolytes
Temperature, 'C
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989 4941 in the present study indicate that outer-shell contributions give the correct limits for our model because their magnitudes are much larger than the corresponding standard-state terms. Inner-shell contributions predominate in region C (Figure 6 ) , and Ahc"s'for an ion is given by eq 6. The expressions for & v i ' , A h S p ,and Ahcpkare given by the terms preceding the outer-shell contributions in eq 18, AI, and A2, respectively. The inner-shell contribution, Ah@,., becomes more highly negative and ( n ) approaches a maximum value of 6 in response to the increase in K j with decreasing temperature. The terms fH@ and &/+,, in eq 6 govern the pressure dependence of Ah@,.. At a given temperature, the increase in fHlo with pressure works to increase AhG',s,,but For low this effect is opposed by a concurrent decrease in c&/$,. temperatures at liquidlike densities, these opposing effects nearly cancel, so that Ah@. changes very slowly with increasing pressure. The P-T dependence of ( n ) in Figure 7 provides a good indication of the general P-T dependence of Ah(??.. , At liquidlike densities the values of AhG;;"'and Ah$s. are generally greater than 40% of the corresponding outer-shell values. The contributions &k$" and Ahc$ predominate at high liquid densities because the dielectric constant derivatives which govern the outer-shell contributions approach small values at these low temperatures. This predicted significance of inner-shell contributions at high liquid densities agrees with similar conclusions obtained from Monte Carlo calculations,1zs26molecular dynamics simulation^,^^ and neutron scattering experiments3*at 1 bar and 25 "C for aqueous ions. The dielectric constant of H 2 0 is very close to unity, and H 2 0 behaves ideally at pressures less than the vapor-liquid saturation pressure from 0 to 100 "C and increasing to 10 bar at 1000 OC. Consequently, in P-T regions A and B, A h G y is negligible and the standard-state contributions for ions are given by AG,, = R T In (MwP/lOOO), ASs, = -R In (MwP/lOOO), ACp- = 0, and AV,, = RT/P, where Mw denotes the mole weight of HzO. In region A, inner-shell contributions are also very small or negligible because mean hydration numbers for ions are generally less than -0.2 (Figure 7). It follows that in region A the above ideal gas equations for standard-state contributions permit close approximation of the total values for the respective hydration property. This indicates that in the limit of stable, unhydrated, aqueous ions in an ideal gas HzO solvent, = u,RT/P, and c p o k = ~ i o n s ~ i C pThe o g .corresponding limits of AhGkand Ahsk for 1:l electrolytes are obeyed by the 0.01 isobars for NaCl in Figure 5a,b at high temperatures. Also note that, although the ion-solvent interaction in region A is small or negligible, AhGk is still significant because of the standard-state conversion term. We now consider the changing nature of ionsolvent interactions indicated by our model for a path that starts in the A region for AhG (Figure 6a) and ends in the D region. Our path does not cross the vapor-liquid saturation curve, and the solvent density increases continually from beginning to end. In taking the ion from A into C, some HzO molecules are temporarily captured by the strong electrostatic field adjacent to the ion. Although a few additional H 2 0 molecules are close enough to be affected by the ion's electrostatic field without being captured, their total contribution to ion-solvent interactions is small. The predominant species for these physically separable, gaslike ion hydrates is given by the mean hydration number (Figure 7). These inner-shell ion hydrates are the species detected in the gas-phase mass spectrometric ~ t u d i e s . ' ~ - ' ~ - ~ ~ As the ion is taken through region C and into region D, the increasing concentration of the solvent brings many more H 2 0 molecules within the electrostatic field of the ion. The total charge-dipole interaction for molecules not immediately adjacent to the ion becomes large. These more distant molecules are assigned to the outer-shell region of our model. These long-range electrostatic ion-solvent interactions are significant for AhG throughout the liquid range and at pressures ranging from 10
-
-3111' 0
' '.
1
I
I
250 500 750 Temperature, 'C
1000
Figure 7. Isopleths of the mean hydration number of the inner shell region, ( n ) , (eq 19) for Na+ (a) and CI-(b) as functions of pressure and temperature.
effect but does not change the qualitative shape of the P-T surface obtained by using a constant radius. In the vicinity for Ah? of the critical point, AhSk, AhVk, and Ahcp,k closely correspond to outer-shell contributions because the pressure and temperature derivatives o f t strongly diverge at the critical point. The Born equation predicts that Vk", Sk", and mkoapproach --OD at the solvent critical point. In addition, predicted critical point divergences for Cp"k, as well as (dVko/dP)Tand ( d p k o / d r ) P are , --m for T T, from T < T, and +m for T T, from T > T,, where T, denotes the critical temperature, 374 "C. These predicted divergences as well as their signs are consistent with qualitative geometrical constraints at the solvent critical point derived from thermodynamic reasoning as well as calculations from a lattice gas model.36 The signs for these divergences depend on whether the solute-solvent interactions are repulsive versus sufficiently attractive compared to solvent-solvent interactions. Ionic solutes fall in the latter category. The nature of the geometrical-thermodynamic arguments36can be easily grasped from consideration of the P-Tsurface for AhGk(Figure Sa). Note that the critical isobar (P,= 221 bar; not shown) would be tangential to the vapor-liquid "hoop" at the critical point and would have a slope of +-ma This geometry predicts that Afik and Sk" approach --m at the critical point. Similar analysis of critical isotherms for AhGk and Ahvk as Well as Critical isobars for AhSk and Ahvk permits prediction of the critical point divergences given above for other thermodynamic properties. It should be noted that predicted critical point limits for standard-state contributions are also infinite, but they have opposite signs from those given by the Born equation. Calculations
-
-+
(36) (a) Griffiths, R. B.; Wheeler, J. C. Phys. Reu. A 1970, 2, 1047. (b) Wheeler, J. C. Ber. Bunsen-Ges. Phys. Chem. 1972, 76, 308.
vko
-
(37) Impey, R. W.; Madden, P. A.; McDonald, I. R. J. Phys. Chem. 1983, 87, 5071. (38) Hahn, R. L. J . Phys. Chem. 1988, 92, 1668.
4948
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989 I
I
Tanger and Pitzer I
I
35
45
k 5. 5
-
2 5
15
35
O5
2 5
I
15
T e m p e r a t u r e , "C
Figure 8. Isopleths of the uncertainty in the outer-shellcontribution (eq 20) for NaCl expressed as 10-3(6A&/R), in K, as functions of pressure
and temperature.
0
bar in the saturated vapor to -50 bar at 1000 "C.
Uncertainties in Model Calculations We have evaluated the major known sources of uncertainties in our model as functions of pressure and temperature in order to estimate the reliability of AhGkvalues in region IV of Figure 1. If we assume that all uncertainties in A h G y values arise from uncertainties in the dielectric constant, then we can express this assumption as 6AhG$"/R = (27/Rk*)(St/[(t
+ st)(€ - 6t)])
(39) Sunner, J.; Kebarle, P. J . Phys. Chem. 1981, 85, 327.
1000
Figure 9. Isopleths of the uncertainty in the inner-shell contribution (model I, see text) for NaCl expressed as 10-'(6AhC/R), in K, as functions of pressure and temperature. L4,
I
3-
35 3
(20)
where 6 e is the uncertainty in t. The estimated uncertainty in t is 2.5% or less in the P-T region, where we use the t equation of Uematsu and FranckS2*Estimated uncertainties in our Kirkwood-type equation29 are about 3% from 325 to 575 "C, are 5% from 575 to 725 "C, and increase from about 5% to 10% as temperatures approach 1000 "C. These estimated uncertainties are given for t, but they more strictly apply to t - 1 because the Kirkwood equation should become more accurate as E approaches unity (Le., 6t -,0 as E -, 1). In our calculations using eq 20, we took the uncertainty in (e - 1) as 2.5% at T I 325 "C and we interpolated linearly over three higher temperature intervals using uncertainty values for (t - 1) of 3.5% at 575 "C, 6.5% at 725 "C, and 10% at 1000 "C. Isopleths of lO-%AhGy/R in kelvins for NaCl are shown in Figure 8. Note in this figure that 10-36AhGy/Ris generally less than f 0 . 2 5 at temperatures below -400 "C or at pressures less can reach values of than 100 bar. In contrast, 10-36AhGo~s~/R f 1 in the critical region. Nevertheless, these uncertainties are relatively small and they do not permit any change in the qualitative shape of the P-T surface for (e.g., Figure sa). Values of 6AhGy for other 1:1 electrolytes are similar to those for NaCl because the ratios (RNa~I*T,/Rk*T,) are near unity (Table IV). We estimated uncertainties in inner-shell contributions, bAhGp, by providing for the uncertainties in Kj (eq 7) arising from the experimental uncertainties in AHj" and ASj" (Table I). The standard deviation of van't Hoff plots and the precision of replicate measurements from one or several laboratories suggest that the experimental uncertainties are typically f 1 kcal mol-' for AHj" and f 2 cal mol-' K-' for ASj0.3s In our calculations, we used two different models for the distribution of these uncertainties between AHj" and ASj" values. However, we neglected the additional uncertainties from the estimation of some AHj" and ASj" values in Table I and from the possible dissociation of ion hydrates in the vacuum region of the mass ~ p e c t r o m e t e r . ~The ~ combined consequences of these additional uncertainties are expected to have a relatively small effect on dAhGp. Model I for 6AhGp accounts for the covariance uncertainties in the AHj" and ASj" values obtained from van't Hoff plots. All
-
250 500 750 T e m p e r a t u r e , "C
-3
0
250 500 750 Temperature, "C
1000
Figure 10. Isopleths of the uncertainty in the inner-shell contribution (model 11, see text) for NaCl expressed as 10-)(6AhC/R), in K, as functions of pressure and temperature.
values of AHj" in Table I are adjusted by +1 or -1 kcal mol-' and then corresponding ASj" values are obtained using eq 7, the adjusted AHj", and the constraint that for eq 7 In Kj (adjusted) = In Kj (Table I) at T = Ti*, where (l/Tj*) is the average (1/7') for the experimental Kj data. Using these two sets of adjusted values for anj"and AS.", we calculated maximum and minimum values of Kiand AhG$ (eq 6 ) . The value of was then obtained from ( 1/2)lAhGy',,,,, - AhGp,,i,,j. Isopleths of bAhci.s, shown in Figure 9 represent our calculated values from model I for NaCl. Model 11 for 6AhGp accounts for temperature-independent, systematic uncertainties in experimental In K j values. For this model, all AS," values in Table I are adjusted by +1 or -1 cal mol-' K-I (or f0.5 in In Kj),but AHj" values are unchanged. Calculated maximum and minimum values of A h @ ' are then used to determine 6AhGp,as in model I. Isopleths of 6AhGy/R from model I1 for NaCl are ,shown in Figure 10. Comparison of 6AhGp values for NaCl from models I and I1 with corresponding values for other electrolytes indicates that 6AhG;;"'values for most 1:1 electrolytes are within 25% of those for NaCl (Figures 9 and IO)., With respect to P-Tregions I, 11, and I11 (Figure l ) , the 6AhGP values from models I and I1 are generally larger than corresponding differences between AhGkfrom
Thermodynamic Properties of Aqueous Electrolytes 3.51
'I
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989 4949
'W
-2 321 bar
-.I /
/
(
/
300 I .ol
200
I
1
I
I
400 600 Temperature, "C
I
)Calculated using Pitzer and Pabalan (1986)
Ld
500
I
800
Figure 11. Isopleths for 2, 5, and 10% increases in local dielectric constant with respect to the bulk H20value obtained from compressible electrostatic continuum calculation^^^ for a lo-A radius ion (see text). The dashed portions of the isopleths represent our graphical extrapolations. The P-T boundary curves for regions I, 11, and I11 of Figure 1 are also shown. The sinuous curve at 321 bar correspondsto the temperature range where provision for compressibility effects may be required for accurate description of heat capacity data using the semicontinuum a p proach (see Figure 14 and text).
our model and AhGk from eq 13. Clearly the empirical methods parameters have permitted we used to evaluate the Rk* and pH@* some compensation for the uncertainties in AhGp. Nevertheless, in P-T regions I and I1 values of AhGkfrom eq 13 should be more accurate than 'our values because Vk0and Cpokvalues from eq 13 are closely consistent with. experimental data. Provision for the combined effects of 6AhG'** and 6 A h G p on predicted values from our model indicates that these uncertainties would not permit any change in the qualitative shapes of the P-T surfaces for the thermodynamic properties (e.g. Figure 5 ) . In addition to the uncertainties considered above, solvent compressibility effects in the outer-shell region should increase our model's uncertainties as pressures and temperatures approach those of the critical p ~ i n t .The ~ ~calculations ~ ~ ~ of Quint and WoodZ5 using a compressible electrostatic continuum model provide some guidance for delineation of this higher uncertainty region. They calculated the increase in the local dielectric constant of the solvent with respect to the bulk solvent value using a hard-sphere ion with a 10-A radius. The curves labeled 2,5, and 10 in Figure 11 are P-T isopleths representing the reported percent increase in the local dielectric constant. The boundary curves for P-T regions I, 11, and 111 in Figure 1 are shown also in Figure 11. In these three regions, our "best fit" Rk* values are pressure independent, which is consistent with insignificant compressibility effects. This suggests that the 2% isopleth (Figure 11) could roughly encompass the P-T region where compressibility effects are significant with respect to our AhGoaS* model. The experimental apparent molal heat capacities for NaCl reported by White et al."O also provide a means for investigating these compressibility effects in the critical region. However, comparison of these apparent molal heat capacities with standard-state heat capacities from our model requires provision for the effects of ion-pair formation, as discussed in the next section of our paper.
Comparison with Critical Region Data Apparent molal heat capacities for a 0.015 m NaCl solution have been measured from 331 to 445 "C at 321 bar.40 At these experimental conditions, the predominant species for aqueous NaCl changes from dissociated ions at 331 "C to associated ion pairs at 445 "C. Provision for ion pairing in the calculation of solution heat capacities has been treated p r e v i ~ u s l yand ~ ~ the
700
900
Temperature, "C
Figure 12. Graphic representation of eq 23 (curve) for the logarithm of the dissociation constant of the ion pair, NaClO, as a function of temperature. Circles and hexagons denote experimental and calculated log Kd(,,,) estimates, respectively.
corresponding equation for the apparent molal heat capacity is given by
where 8, m,, and A denote the fraction ionized, the stoichiometric molality, and the ion-pair-dissociation reaction, respectively, and AJ and AH represent molal Debye-Huckel parameter^.^^ In derivation of eq 21, we assumed that the activity coefficient of the ion pair, NaCl", is unity and that the excess heat capacity and enthalpy of the aqueous electrolyte, Na+ + Cl-, can be closely approximated by the Debye-Huckel limiting laws because our calculations will be restricted to m, = 0.015. It follows from the definition of 8 that 0 = I(c)/C,where C denotes molarity and I(c)represents the true ionic strength on a molarity basis. For the present case of a dilute solution, we take pWlnN tHzO and approximate C as pHzom,. In our calculations, the equilibrium constant for the dissociation reaction on a molarity basis is expressed as log Kd(c) = 1% [ I ( c ? / ( c -
I(c))] - ~ A T ( C ) [ I ( C ) " ~ +/ (I(C)"~)] ~
(22) where the term containing AT(c)represents the expression for the logarithm of the mean activity coefficient as used by Quist and Marshall.43 Consequently, a known value for log &(c) permits calculation of 0 by iterative solution of eq 22 for the value of I(c). Calculations in the present study indicate that the relation log &cm)
= 3.311
- 2.928 x iO4/T +
6
~a,JJ@j) (23)
n= 1
affords close approximation of experimental and theoretical estimates of log &cm) at 321 bar (see below), where anrepresents an empirical constant. Equation 23 permits derivation of the equations for AH and AC, and in combination with eq 22, it also allows derivation of the equation for ( d 8 / d 7 ) p f i . It follows that predicted values for @Cpcan be calculated from eq 21 by using Cpokfrom our model and eq 22 and 23. However, before presenting the results of our @Cpcalculation, we will first discuss the origin of eq 23. Experimental log &(c) values for NaCl obtained from conductance studies43are given at specified constant densities for selected temperatures from 400 to 800 "C. For a constant density, we used graphical fit lines to describe the log &(c) data as a function of temperature over the range 400-550 "C. Calculation of the temperature for pure H 2 0 at 321 bar, which corresponds
(40) White, D. E.; Wood, R. H.; Biggerstaff, D. R. J. Chem. Thermodyn. 1988. 20. 159. (41) Peiper, J. C.; Pitzer, K. S. J. Chem. Thermodyn. 1982, 14, 613.
(42) Bradlev. D. J.: Pitzer. K. S. J. Phvs. Chem. 1979. 83. 1599. (43j Quist, A. S.; Marshail, W. L. J. h y s . Chem. 1988, 72, 684.
4950
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989
'
0O 8
r
-
7
-
Tanger and Pitzer
7
321 bar
0.61
0
\
V+L
-
0,4t NaCl
0.015 m
\ i
i
-12
350
400
450
500
350
320
360
400
400
450
500
Temperature, *C
Temperature, 'C
440
T e m p e r a t u r e , "C
Figure 13. Fraction ionized, 0, for a 0.015 m NaCl solution as a function of temperature at 321 bar obtained by using eq 22 and 23.
to the experimental densities, permitted interpolation and extrapolation of the log &(@ data. From the known density we obtained log &(m) at 321 bar using log &(m) log &(c) - log pHIO. The circles in Figure 12 represent our interpolated and extrapolated values for log Theoretical estimates of log &(m) were calculated by using AhGNaCIO (ref 7; Pitzer-Pabalan model), AhGNa+,Cr(present study), and the ionization constant for pure ideal gas NaCl" (ref 31). At 321 bar, the Pitzer-Pabalan model is expected to be valid at temperatures above 480 "C where ( n ) for NaCl" is less than 12. The open hexagons in Figure 12 represent our more reliable estimates for log KdCm).The stippled hexagons (Figure 12) represent estimates obtained by extrapolation of the Pitzer-Pabalan model, and these estimates become more uncertain as temperature decreases from 480 "C. Consideration of our model's implications for log &(m) values at conditions where Na+, CI-, and NaCl" are unhydrated species should approach log in ideal gas HzO indicated that log &cm) K(ionization) - [log (M,/lOOO) + log PI in this limit. We derived the first two terms in eq 23 by using the van't Hoff equation to approximate log K(ionization) in the above limit expression. The summation term in eq 23 seemed reasonable because the density of HzO at 321 bar has the same qualitative temperature dependence as the difference between our complete log &(m) values and their corresponding limit values. The solid curve in Figure 12 representing eq 23 affords close description of the experimental and theoretical estimates for log Kdcm,.Calculation of 0 by using eq 22 and 23 indicates that the fraction ionized is 0.94 at 350 OC and 321 bar but decreases rapidly and approaches zero with increasing temperature as shown in Figure 13. Calculated values of and AhG for NaCl from the semicontinuum model at 321 bar are denoted by the solid curves in Figure 14a,d. Corresponding values of @ecp - Cpo for 0.015 m NaCl computed from eq 21-23 are represented by the solid curve in Figure 14b. Addition of the solid curves in Figure 14a,b gives the solid curve for @ecp in Figure 14c. Comparison of this calculated curve for @Cpwith the measured values for NaCl reveals significant discrepancies in the range 395-435 "C. This temperature range at 321 bar is indicated by the sinuous curve in Figure 11. Note the position of this curve in relation to the isopleths for the percent increase in the local dielectric constant. This observation suggests that solvent compressibility effects omitted from our model could be a significant source of error in our predicted @Cpvalues. The increase in the local dielectric constant produced by compression of the local solvent increases the magnitude of AhG0.'. relative to its Born equation value. This effect approaches a maximum at the critical point of HzO. As pointed out by Wood et al.,24 the omission of compressibility effects from the Born equation could represent a very small percent error in predicted free energies, but the percent error will increase to large values with successive differentiation of the free energy with P and T . Their calculations at 327 "C and 177 bar indicated that solvent compressibility effects for a 3-A radius ion would increase predicted values from the Born equation by less than 1% in the free energy but would reduce the corresponding predicted value of IcPp"I by -40%. Note that reducing the magnitude of values
cpo
cpo
Y Y
'O
a 7
/
-60-601
i;
-65-
-70-
-75
I/'
-Colculoted
- - - Pasluloted
-80 350
400
450
! '0
Temperature, 'C
Temperature, "C
Figure 14. Standard-state heat capacity (a), apparent molal (0.015 m) minus standard-state heat capacity (b), apparent molal (0.015 m) heat capacity (c), and Gibbs free energy of hydration (d) for NaCl at 321 bar as functions of temperature. Calculated curves (solid) obtained by using
the semicontinuum model and eq 21-23. Postulated curves (dashed) explained in text. Experimental apparent molal heat capacity data in (c) (symbols) from ref 40. relative to our calculated solid curve in Figure 14a would allow better agreement with the experimental @Cpdata (Figure 14c). The dashed curve extending from 370 to 500 "C in Figure 14a represents a general reduction in the magnitude of our calculated cpovalues such as might be produced by solvent compressibility effects. These revised values are constrained to return the values of hhG and A& obtained from our semicontinuum model as temperatures approach 370 or 500 OC (e.g., dashed curve in - Cpovalues (dashed Figure 14d). Corresponding revisions of @Cp curve; Figure 14b) within their expected uncertainties gives good agreement with @Cpdata (dashed curve; Figure 14c). The differences between the dashed and solid curves in Figure 14d yield rough estimates for the uncertainties in hhG arising from solvent compressibility effects, 6AhGS.'.. values of 10-36AhGs~c~/R at 32 1 bar maximize at about 1 K from 415 to 425 OC. Comparison of these estimates for ?JA,G~.~./R with corresponding values of ?JAhGi."/R(Figures 9 and 10) suggests that, in general, 6AhGs.c. values in the critical region could be well within our expected uncertainties for inner-shell contributions. We also have evaluated the uncertainties in our @CPcalculations arising from the uncertainties in log Kd (eq 23) and 6 (ref 29). This analysis suggests that combined revision o f t and log Kd values within their respective uncertainties would allow reconciliation of calculated and experimental @Cpvalues. Consequently, these results also support the proposition that, in general, 6 A h P c .could be much smaller than 8AhGi,s.in the critical region. In principle, replacing our Born equation treatment of AhCO.s. with a compressible electrostatic continuum model should improve the accuracy of our semicontinuum approach in the critical region. However, the accuracy of currently available models of this typez5 must be improved before explicit theoretical modification of our outer-shell contribution is worthwhile. In addition, improved accuracy in values o f t and log Kd at critical region conditions will be needed before measured solution properties can be used for meaningful comparisons with compressible, electrostatic continuum models. Nevertheless, solvent compressibility effects in the critical region are not expected to change the qualitative
cpo
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989 4951
Thermodynamic Properties of Aqueous Electrolytes shapes of our predicted P-T surfaces for the thermodynamic properties.
Conclusions In principle, the semicontinuum model can be extended to permit estimation of standard-state thermodynamic properties for any individual ionic species or complexed ionic species at any temperature and H20density of practical interest. In this paper, we have demonstrated that the qualitative shapes of predicted P-T surfaces for properties of selected 1:1 aqueous electrolytes are well-characterized from 0 to 1000 OC and any pressure up to 5 kbar. Our model is consistent with theoretical limits for standard-state properties at the solvent critical point and at the gasphase conditions where the ion is unhydrated and H20behaves ideally. We have forced our model to give exact agreement with the best available experimental values of AhG at 1 bar and 25 OC, but it also provides a reasonable description of experimental AhG values over a wide range of liquidlike densities. Finally, we have provided estimates of the uncertainties in our predicted values to facilitate meaningful calculations on geochemical and industrial processes at supercritical temperatures.
It follows from eq A4 that
Acknowledgment. This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Division of Engineering and Geosciences of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.
Appendix It follows from eq 12 that the standard-state entropy, A,,$ and heat capacity, AhCp,of hydration for the kth aqueous electrolyte are given by
It follows from eq 7 that and
and
(A2) where
ss = RTp/1000 (A31 Rk* is given by eq 16 and 17 and, taking note of eq 6, we define as
Evaluation of +*/&and its first and second temperature derivatives is straightforward using eq 10. Registry No. Li', 17341-24-1; Na', 17341-25-2;, 'K 24203-36-9; Rb', 22537-38-8; Cs', 18459-37-5; F, 16984-48-8; CI-, 16887-00-6; Br-, 24959-67-9; I-, 20461-54-5; OH-, 14280-30-9; NaCI, 7647-14-5.
