J. Phys. Chem. 1986, 90, 3005-3009 but in the present work, in which the CS2pressure is 13 Pa, a partial sulfide formation is observed in the case of SrTi03. In SrZr03, Zr3S4is formed in addition to SrZrS3. Therefore, it might be possible to say that SO2-is formed by the reaction s20 2 S02”
+
s022-
+0 2
-- so*-+
02-
However, an in situ measurement will be necessary to clarify this reaction mechanism. The SO2-ion can be formed in CS2-02treated polycrystalline SrTi03, but not in a single crystal of SrTi03 which is treated in the same way. This fact will indicate that the SO2-is formed as a result of a surface reaction between the atmosphere gas and the surface species. Until now SO2-formation has been found only in S r T i 0 3 and S r Z r 0 3 . It will be a subject of interest to know which substances can form SO2- by the procedures reported in this work. The band gap of SrTi03 is 3.2 eV.I2 Ordinary SrTi0, exhibits photocatalytic properties only when irradiated by ultraviolet light. However, SrTi0, polycrystals in which SO2-ions are formed have certain photocatalytic properties when illuminated by visible light with wavelengths longer than 480 nm (480-730, 2.6-1.7 eV).9 One possible explanation is that SO2- ions create an electronic band which works as the valence band from which the electron is excited to the conduction band. If this explanation is true, the SO, band may be located 1.7-2.6 eV below the conduction band. In the present case, the SOT electrons are in a deep impurity level and so they are confined rather closely to the SO2molecule.13 (1 1) Clearfield, A. Acta Crystallogr. 1963, 16, 135. (12) Kung, H. H.; Jarett, H. S.;Sleight, A. W.; Ferretti, A. J. Appl. Phys. 1911,48, 2463.
3005
The SO2- ESR spectrum of the photocatalyst, prepared from SrTi03 which has been processed by CS2-02 cycles and used for photocatalytic reactions, Figure 2, has a AHm, value twice as large as those of the spectra in Figure 1 . This fact suggests that the SO2-dopant electrons form a narrow band in the oxide crystal lattices when activated by the photocatalytic reactions.14 The shift of the g factor in the x direction may also be related to this band formation. Many oxide semiconductors have been tested for use as raw materials for the photocatalytic decomposition of water. Some of the results have been reviewed in chart form by Kung et al.lS and by Scaife.I6 Their theory seems to predict that a semiconductor which has both an ideal flatband potential value and an ideal band gap does not exist. Such difficulties might be overcome by bringing in a new electronic energy band into the semiconductor (doping).” If the newly created band works as a new valence band, the original wide band gap does not have to be considered. The introduction of SO, into SrTiO,, SrZr03, or other crystals may be useful to add visible light response to wide band gap oxide semiconductors. Registry No. SrTi03, 12060-59-2; SOT, 12143-17-8; SrZr03, 12036-39-4.
(1 3) Ayscough, P. B.Electron Spin Resonance in Chemistry; Methuen: London, 1967, p 391. (14) We thank the referee of their manuscript who pointed out this pos-
sibility.
(15) Kung, H.H.; Jarett, A. S.;Sleigh, A. W.; Ferretti, A. J. Appl. Phys. 1971, 48, 2466. (16) Scaife, D. E.; Solar Energy, 1980, 25, 41. (17) Sekine, T.; Yonemura, M.;Ueda, H. Book of Abstracts, The Fifth
International Conference on Photochemical Conversion and Storage of Solar Energy; Osaka, Japan, 1984; p 169.
Thermodynamics of Multicomponent, Miscible, Ionic Systems: Theory and Equations Kenneth S. Pitzer* and John M. Simonsont Department of Chemistry and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720 (Received: September 17, 1985; In Final Form: January 24, 1986)
Equations of a type which has proven successful for binary and pseudobinary ionic systems miscible from dilute solution in polar solvent to the fused salt are extended to an indefinite number of components. The underlying theory is discussed as well as the optimum form for empirical terms. In brief, the equations comprise an ideal term, an extended Debye-Huckel term, and a Margules expansion. The latter is carried out to the three-suffix level but could be extended. Relationships between molality and mole-fraction standard states are given. Examples are given showing the extent to which parameters measured for binary systems determine the properties of more complex systems.
Introduction Multicomponent ionic systems of limited concentration have been successfully treated by semiempirical equations’ based upon a virial series plus a Debye-Hiickel term derived from rigorous statistical m e ~ h a n i c s . ~The , ~ short-range forces or specific solvent effects are treated empirically through the virial coefficients while the form of the equations and the Debye-Hiickel term remain theoretical. The more convenient and temperature-independent composition measure, molality, replaces concentration. These equations have been very successful for multicomponent systems of great complexity such as the seawater evaporate solutions studied by Harvie and Weare.4 Any virial expansion becomes unsatisfactory at sufficiently high concentration and for a pure fused salt the molality is infinite. Present address: Chemistry Division, Oak Ridge National Laboratory, P. 0. Box X,Oak Ridge, TN 37830.
Thus an alternate framework is required for systems miscible to the fused salt and such an approach may be advantageous for other systems of very high but limited solubility. Equations appropriate for binary of pseudobinary, miscible, ionic systems have been presented.5*6 They were successful in treating several aqueous systems with metal nitrates melting in the range 373-473 K and the NaCl-H20 system’ in range 373-823 K. In this paper we present equations appropriate for miscible, ionic systems with an unlimited number of components of neutral species or singly charged ions. They are based on the two developments Pitzer, K. S.;Kim, J. J. J . Am. Chem. SOC.1914, 96, 5701. Pitzer, K. S.J . Phys. Chem. 1913, 77, 268. Mayer, J. E.J. Chem. Phys. 1950, 18, 1426. Harvie. C. E.; Weare, J. H. Geochim. Cosmochim. Acta 1980,44,98 1. ( 5 ) Pitzer, K. S . Ber. Bunsenges. Phys. Chem. 1981, 85, 952. (6) Pitzer, K. S.J. Am. Chem. SOC.1980, 102, 2902. (7) Pitzer, K. S.; Li, Yi-gui. Proc. Natl. Acad. Sci. U.S.A. 1983, 80, 7689.
0022-3654/86/2090-3005$01.50/00 1986 American Chemical Society
3006
The Journal of Physical Chemistry, Vol, 90, No. 13, 1986
described above. The relationship to treatments of multicomponent, nonelectrolyte systems is also considered. Experimental measurements for several systems including LiN03-KN03-H20 are treated with these equations in two accompanying papers.
Definitions: Ideal Mixing Terms For a miscible system the appropriate measure of composition is a mole fraction, but for an ionic system one may either recognize or ignore the ionization of the salt. While much of the literature uses nonionized mole fractions for salts, it is both theoretically and empirically preferable to recognize the ionization. Indeed this is necessary for mixtures without common ions. Thus, the mole fraction of the j t h species (ion or neutral) is xi = n j / C n i i
(1)
where ni is the number of moles of the ith species with cations and anions included as separate species. Electrical neutrality must also be maintained, of course. The ideal entropy and Gibbs energy of mixing are
Pitzer and Simonson For systems with all pure components liquid at the conditions of interest, the normal standard states are the pure liquid components. This yields y j = 1 and aj = 1 for the pure liquid in eq 6. Even if a component melts at a somewhat higher temperature, it may be useful to take the supercooled liquid as the reference state; this procedure was successful for the system NaCl-H20 a t temperatures well below the melting point of NaCl. Where there are more than one species of both cation and anion, the reference basis of pure fused salts becomes ambiguous. Thus an equimolal system of Na+, K+,C1-, NO3- can be referenced to NaCl and K N 0 3 or to KC1 and NaN03. In some cases there is a preference. Thus in a system dominated by NaCl the reference should be to pure NaCl and to the Na+ salts of other anions and the C1- salts of other cations. But in other cases an arbitrary choice may be required. There are various reasons for wishing to define an infinitely dilute reference state on a mole fraction basis and for relating it to the standard state on a molality basis. We follow PrausnitzIo and others in using the symbol yj* for an activity coefficient based on the infinitely dilute reference state; thus on a mole fraction basis yj
-- -1 as xj
yj*
where xi0 is the mole fraction of the ith species before mixing; for each ion in a pure salt MX and xo = 1 for a i.e., xo = neutral species. This simple result is readily derived for the case that all particles mix randomly without respect to charge but that charge neutrality must be maintained for each pure component and for the mixture. Such completely random mixing is, of course, a poor structural model for fused salts where there is a strong pattern of alternation of charge. But, for the mixing of salts where the ions have the same magnitude of charge, the assumption that there is no cation-anion mixing while cations mix randomly and anions mix randomly (Temkin model) yields the same entropy of mixing as is given by eq 2. This was demonstrated by Laitys and by BlanderS9 Various models have been proposed for mixing of unsymmetrical fused salts but none have been sufficiently justified to justify more complicated definitions for our purposes. Thus we take eq 2 and 3 as definitions of ideal mixing and limit the present treatment to singly charged ions and neutral molecules. The excess Gibbs energy for any amount of material is GE and, per mole of particles, is 2.Thus
AmG = AmG1+ GE
(4)
2 = G E / Ei n i
xj”
1 as xj
(9)
where xj” is the value for the pure fused salt. In the limit of very small fraction of species j in a single molecular solvent, the mole fraction becomes a mole ratio. xj
-
n j / n l as xj
-
mi = (nj/nl)(lOOO/Ml)
m j / x j = 1000/MI
gE = Cxi~iE= RTCX,In yi i
i
+ R T In (x,yj*) (1 2a)
+ R T In (mjyim)
pj(molality) = $(molality)
(6a)
(12b)
+ R T In (1000/M,) ( 12c)
(6b)
Differentiation of CEwith respect to nj at constant T,P, and other ni yields the excess chemical potential p y ; also
R T In y j
(1 1)
with M I replaced by a mean solvent molecular weight when required. The chemical potential must be the same for a given composition on either basis. Thus, in the very dilute range and with yjmthe activity coefficient on a molality basis
Mj”(mo1e fraction*) = Mj”(mo1ality)
pjE =
(10)
where M I is the molecular weight of the solvent. If one has a mixed solvent n, is replaced by the total number of moles of solvent and M I by the mean molecular weight. The ratio of mj to x j is
Mj(mole fraction) = Fj”(mo1e fraction*)
and for a salt MX In a M X= In ( a M a X )
0
Under these conditions the molality is
The activities aj and activity coefficients y, are related by In aj = In (xjyj/x,O)
0
(7)
(8)
For electrolytes the activities and activity coefficients can ordinarily be measured for neutral combinations of ions. Thus for a salt mixture containing M+, N+, X-, Y- one can determine ( U M U X ) , (aMaY),(aNax), and (aNaY)but not any of the individual a’s. (8) Laity, R. W. In Reference Electrodes; Ives, D. J. G.;Janz, G. J., as.; Academic: New York, 1961; pp 544-6. (9) Blander, M. Molren Salt Chemistry; Interscience: New York, 1964; pp 130-5.
where the last equation gives the conversion from the conventional standard Gibbs energy on a molality basis to that for the mole fraction basis retaining the infinitely dilute reference state. Again these fi’s are ordinarily measurable only for neutral combinations of ions. The chemical potential for the pure liquid standard state may also be related to that for the infinitely dilute reference state. Take a salt MX as an example PMX
cLO(MX,liq)
+ R T In ( X M X X Y M Y X / X M ~ X X ~ )
= fiMxo(molefraction*)
+ R T In (
(13a)
x M x ~ ~ ~ * ~ (13b) ~ * )
pMxO(molefraction*) = po(MX,liq) - R T In
( X M ~ X X ~ ~ M * ~ Y(1 X 3c) *~)
(10) Prausnitz, J. M. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice-Hall: Englewood Cliffs, NJ, 1969.
Multicomponent, Miscible, Ionic Systems
The Journal of Physical Chemistry, Vol. 90,No. 13, 1986 3007
where yj*O is the value of yj* in the pure liquid MX. For a neutral molecular liquid these equations simplify for a single particle and xi” is then unity. We turn now to the excess Gibbs energy which arises from inequalities in interparticle forces. If there is a substantial ionic concentration, interionic forces are effectively screened from the R2long range to short range. Then interionic forces can be combined with all other interparticle forces on the same basis and one expects the same type of expression for excess Gibbs energy as was found for nonelectrolytes. At very low ionic concentration, however, the alternating charge pattern and its accompanying screening effect is lost and the long-range nature of ionic forces must be considered. This is the effect described by the DebyeHuckel treatment. For the full range of composition one expects the excess Gibbs energy to comprise two terms: a short-range-force term GS and a Debye-Huckel term GDH;thus GE
= GS + GDH
(14a)
gE=gS+gDH (14b) where the last equation refers to one mole of particles. The logarithms of the activity coefficients are similarly sums of terms for short-range forces and for the Debye-Huckel effect.
terms involving w ~remain , ~ for a pure fused salt MX. A notation used by WohlI2 and ~ t h e r s ’ is ~ Jrelated ~ by Ai/ = w.. IJ - uIJ. . and Ai/ = wjj uii. We prefer a form in which one parameter (uJ will become zero in a symmetrical case rather than two parameters becoming equal. Our third parameter, Cijk, is equivalent, except for sign, to the c* or Q’used by others. Here we prefer a positive sign in eq 18a. The expression for the short-range-force contribution to an activity coefficient is obtained by differentiation of eq 18a.
+
In y; = C’xj[(l - xi)wij + (2xi - 2x;
+ 2xixj - xj)uij] -
j
~ . ’ C ’ X ~ k [ w+ j k 2(xj - xk)ujk - (1 - 2xj)Cjjkl k>J
C’C’C’2XjxkxICjkj - In Tiso (19a) I>k>j
The prime on the summations is a reminder that terms with any two indices equal are omitted and that neither j nor k nor I may equal i in the multiple sums. For ions the last term is a reference-state correction which is best considered for complete salts; it is zero for neutral species. For a salt MX one wishes the product y M y X In (YMYX)’ = In 7 2 + In yxS - In (YMYS)O
(19b)
Excess Gibbs Energy from Short-Range Forces There is an extensive literature concerning semiempirical or purely empirical expressions for the excess Gibbs energy of nonelectrolytes. Prausnitz’O discusses the merits of many of these equations. Among the simplest and most generally effective are the van Laar equation, which involves two parameters for a binary system, and the various extensions of the Margules equation, which comprise series expansions. A “three-suffix” Margules equation also involves two parameters for a binary system. Either equation can be generalized to multiple-component systems, but the Margules system, as generalized by Wohl,”J2 is more flexible and has been found to be more successful in various tests on nonelectrolyte~.’~ A general Margules expansion can be written ? / R T = Z C U j j X j X j F y y U j j k X j X j X k + ... (15)
where the last term is the value for the pure fused salt, which may not be zero, and must be subtracted. An appropriate expression for gS0/RT would eliminate the need for the correction in eq 19b, but it may be easier to handle this problem for the activity coefficient. A shift to the infinitely dilute reference state merely changes In yiso. Before concluding this section it is interesting to consider a few examples which illustrate the extent to which the parameters measurable from simple systems are able to determine the properties of more complex systems. Consider first the common ion mixture of two salts M X and N X with a cation fraction F of ion M. Then the mole fractions are XM = F/2, xN = (1 - F)/2, and and xx =
where n’s with all suffixes equal are zero. If there are only two components, one has
For pure liquid MX, F = 1, ykx = 1, and In is convenient to define
i l
i
l
k
In (YMyX)’ = (1/4)(2wM,X + (1 - F)2[2wM,N+ c M , N , X UM,X - uN,X - ( l - 4F)UM,NI)- In 7% (20)
gS/RT = x+2(2ai2 + %a112 + 3x20122) = xlX2[wI2 +
U 1 2 h
- x2)l
wMX8NX
(16)
with
= (2WM,N+ CM,N,X uM,N
=
uM,X
= wM,X/2. It - uN,X)/4
(21a) (2 1b)
UMUI,N/4
whereupon w12 = 20.12 + (3/2)(a112 + a1221
(1 7a)
u12 = (3/2)(a112 - a1221 (17b) where the second line for eq 16 shows that only two of the three parameters in the first line are independent. If one generalizes the definitions of eq 17 and defines a third quantity related to a,23, one can write for the general case ?/RT = c c x j x j [ w j j + ujj(xj - xj)] + ccxxjxjxkcjjk - gSo/RT j> I
k> j>l
wij = 2aij
+ (3/2)(aiij + nijj)
uij = (3 /2)(aiij - aijj) cijk
= 6Ujjk - (3/2)(aiij
(18a) (18b) (18c)
+ ajjj + ajjk + ajkk + ajjk + njkk)
( 18d) Note that interchange of subscripts leaves wij or c i j k unchanged but changes the sign of uij. Also gS0 is the value for the same material in the reference states of the various components. The values are zero for neutral species but not for electrolytes where (11) Wohl, K. Trans. Am. Inst. Chem. Eng. 1946, 42, 215. (12) Wohl, K. Chem. Eng. Prog. 1953, 46, 218. (13) Adler, S.B.;Friend, L.; Pigford, R. L. AIChE J . 1966, ZZ, 629.
gS/RT = F(1 - F ) [ ~ M x , N-x (1 - 2F)uM,N1/2 (22a) In &X
= ( l - F)’[WMX,NX - ( l - 4F)uM,N1
(22b)
Measurements at a variety of values of F allow the two parameters WMx,Nx and UMvN to be determined. Consider next the binary mixture of a neutral species, 1, with an ionic component MX. Now xM= xx = (1 - x1)/2 and
gS/RT = ( 1 / 4 h ( 1 - x i ) P w i , ~+ W , x - wM,x + (3x1 l)(U1,M + #I,X) + (1 - xl)Cl,M,XI (23a) In 7’‘ = (1/4)(1 - xl)2[2wl,M + 2w1,X - WM,X + (6x1 - 1) x (U~,M + U i , x ) + (1 - ~ X I ) C ~ , M ,(23b) XI In
= xi2[wi,M + wi,x - W M , X / ~ + (3x1 - 2) x (#l,M + ul,X) + ( l - xl)CI,M,XI ( 2 3 ~ )
(YMYX)’
Now define W ~ , M= X (2wl,M + 2 ~ 1-, WM,X ~
+ 2 1 1 1 ,+~ ~ u I , x ) / ~ (24a)
UI,MX = -(3/4)(Ui,~ + UI,X)+ C I , M , X / ~ (24b)
whereupon
$ / R T = xi(1 - xi)[wi,Mx
(1 - xl)Ul,MXI
(25a)
3008 The Journal of Physical Chemistry, Vol. 90, No. 13, 1986
+
(25b)
+ 4(1 - XI)UI.MXI
(25C)
In yIs = (1 - X , ) ~ [ W , , ~(1~- 2XI)UI,MX] In (YMYx)’ = Xl2[2wi,,x
Again we note in eq 24a,b the combinations of the original parameters that can be measured with this system. Next, consider the three-component system with a neutral species (1) and two salts with a common ion MX, N X and with cation fraction F of M. Also define a total mole fraction of ions XI = (1 - x,) whereupon XM = FxI/2, XN = (1 - F)x1/2, and xx = x1/2 and use the definitions of eq 21a,b and 24a,b.
?/RT = XIXI(Fw1,MX + (1 - F ) w I , N X + xIIFUl.MX + ( l F)ul,NX F(1 - F)Ql,MX.NXI) X12F(1 - F)[wMX,NX + xI(2F - ~ ) U M , N ] (26a) ’/~ TIs = X?IFw1,MX + ( l - q w l , N X - F(l - F ) w M X , N X / 2 + (XI - ~ I ) [ F U I , M X(1 +- F)~I,N +xF(1 - F)&I,Mx,Nx]+ x F ( 1 - F)(1 - ~ F ) ~ M(26b) .N)
In (YMYX)’ = 2x,((1 - xlF)wI,MX + xI(1 - F + 2FXI)Ul,MX - F)[wI,NX + - X1)U1.NX (1 - ~ ~ I F ) Q I . M x + , Nx1(1 x I ) - F) 1 + 2x1F(1 - ~ ~ U M , N (26C) ]) ((1 - ~ I F ) ~ M x , x1[3FNx QI.MX.NX
--
(uI,M + U1,N + C1,M.N)/4 + (uM,X + UN,X - cM,N,X)/8 (27) Thus we note that, for this ternary system, all but one of the parameters of eq 26 can be determined from the binary systems. The additional parameter Ql,MX,NX enters only with rather small coefficients and its effect may be rather small, but that remains to be determined from measurements on real systems. Another important type of three-component systems involves two neutral species numbered 1 and 2, and one ionic species MX. Now xI= 1 - x, - x2 = 2XM = 2xx. We give only the excess Gibbs energy in this case, but the activity coefficients can be derived by differentiation. ? / R T = XIXZ[WI,Z u1,2(x1- X2) + Zl,2.MXXII XlxI[wl,MX + (2/3)(l - XI + xI/2)Ul,MXI + X2XI[w2,MX + (2/3)(1 - x2 + XI/2)U2,MXI (28a) ZI.2,MX
=
(C1,2,M
+ c1,2,X)/2 - (CI,M,X + c2,M,X)/6
(28b)
Again, only one new ternary parameter appears in addition to those for the binaries in eq 16 and 25. While it would be desirable to present an equation for an indefinitiely complex system, this becomes very involved when the ternary terms are included. Thus we give only the general equation for binary terms. Now the mole fractions of neutral species are x,, xn,, ... and the total mole fraction of ions x I = 1 - Ex,. For the ions the cation fractions are Fc, F,, ... and the anion fractions are F,, F,,, .... Then $/RT = CZx,tx,,w,tn + ~ ~ ~ : X , , . X I F ~+’ ~ W , , , ~ ~ n3nl
n c a
(x12/2) [CCCF&FcFnw&a.cn+ CCCFnlFnFcwcn,,cnI (29) r‘>c n ,
a h c
It is evident that all of the parameters in eq 29 can be determined from two-component systems. Presumably all ternary parameters for the general system could be determined from the full array of three-component systems. For practical work with specific systems of four (or possibly a few more) components, the expression for the excess Gibbs energy with ternary terms can be derived more easily than the completely general expression. The types of terms will be those appearing in eq 26 and 28 together with a few others if there are more than two neutral species or more than two salts. Debye-Hiickel Effect In dilute ionic solutions the distribution changes from a random pattern a t extremely low concentration to one of alternating positive and negative charges at moderately higher concentration.
Pitzer and Simonson Debye and Hiickel gave a simple treatment which describes this effect. The limiting term at low concentration is given rigorously; the effect at higher concentration is given approximately. There are various alternate forms with respect to the higher concentration range. We choose a, form which has its origin in the pressure equation of statistical mechanics and has been most satisfactory in empirical tests.2,6 The basic variable is the ionic concentration weighted by the square of the charge. Concentration is replaced by molality for most work at moderate dilution but is replaced by mole fraction for our present problem of systems continuously miscible to the fused salt. Thus, ionic strength on a mole fraction basis is defined as 1, = (1/2)cx,z,2 i
(30)
where zi is the charge on the ith species of ions. Thus for singly charged ions Z, = xI/2. Then the activity coefficient of a single uncharged solvent is5,6 (In yl)DH= 2Ax12/2/(1
+
(31)
The excess Gibbs energy is gDH/RT = -(4A,Z,/p)
In [(l
+ pZx1l2)/(l + p(Z2)1/2)] (32)
represents Z, for pure fused salt which is where for singly charged ions. The Debye-Huckel parameter A, is A , = (1 / 3 ) ( 2 * N A d l / M 1 ) 1 / 2 ( e 2 / 4 * € ~ D k T ) 3 /(33) 2 where d , , M I , and D are the density, molecular weight, and dielectric constant (relative permittivity) of the solvent while NA is Avogadro’s number, e is the electronic charge, k is Boltzmann’s constant, and to is the permittivity of free space. Most of the literature on electrolytes is now written for esu’s; in that case eo = (4r)-l. The parameter p in eq 31 and 32 is related to a hard-core collision diameter a in the approximate theory. p =
a(2e2NAdl/M1eoDkT)1/2
(34)
In practice, however, the hard core is not an accurate model and is not independently known; hence, p is treated empirically. In multicomponent systems it complicates the equations greatly to make p dependent on the composition of ionic components. It has been found to take a standard value for p and let the short-range-force terms accommodate any composition dependency of p . It is less clear whether p should be made dependent on the variables d,, D, and T which also appear in eq 34 and are known. For the NaCl-H20 system over an extremely wide range of temperature and density, p was defined7to recognize these variables as follows p
= 2150(dl/DT)‘/2
(35)
with dl in g - ~ m and - ~ T i n K. In calculations for several metal nitrate^^,^ in water near 373 K, p was given a fixed value of 14.9. While the numerical parameter 2150 in eq 35 was selected to best fit data for NaCl-H,O over the wide range 373-573 K, it yields at 373 K a value (14.6) very close to that chosen in the other research. There may be cases where it is best to make a new choice of p, but for a wide variety of aqueous systems it seems desirable to use either the definition of eq 35 or, for a narrow range of temperature, a fixed value consistent therewith. Then the short-range-force parameters determined for various salts in different investigations can be combined to make predictions for new multicomponent systems for which there are few or no measurements. A Gibbs-Duhem transformation of eq 31 or differentiation of eq 32 yields the mean activity coefficient of an ionic component of charge z, for the infinitely dilute reference state
J . Phys. Chem. 1986, 90, 3009-3013 or for the pure fused salt reference state (In yJ)DH= -z,~ A,((2/p) In [(I z,q1
+ ~ Z , l / ~ ) / ( +l p(z,0)1/2)] + - 2Z,/Z?)/(l + pZ,'/2)] (37)
Here Z," is the ionic strength of the pure fused salt which is 2;/2 for the MX type. One may note that when z; reduces to zero, eq 36 and 37 reduce to eq 3 1. W e have written these equations for ions of charge zJ for general interest even though our applications will involve only singly charged ions. For a single solvent of known properties and a defined value for p, the various Debye-Huckel terms are fully determined and the only disposable parameters are those in the short-range-force function. Also, as noted above, there is a relationship between p and these last parameters. Consequently, the value of p and the form of the extended Debye-Huckel expression should be stated clearly in all cases. The situation is more complex for systems with more than one neutral species, Le., for mixed solvents. If one accepts the macroscopic dielectric constant of the mixed solvent, as seems best, and takes M a s the mean molecular weight of the solvent, then A, is determined by eq 33. But A, is now a function of the solvent composition. Thus, differentiation of the excess Gibbs energy,
3009
eq 32, will introduce derivatives of A, into the expression for the activity coefficient of a particular solvent species. The physical picture related to these composition derivatives of D and of A , is a preferential solvation of ions by one species of solvent. This is primarily a short-range effect which is also represented by terms for short-range forces as given in the previous section. If the excess Gibbs energy is taken as the basic expression, the results for the various activities will be consistent, but the equations become complicated. The best procedures should be explored for real systems involving electrolytes in mixed solvents. It should also be mentioned that this entire treatment is designed for liquid systems well removed from the critical region. Near the critical point or critical curve the compressibility becomes very large and special effects arise. Also the Debye-Hiickel derivation basically involves the Helmholtz energy and the ionic concentrations. The simple conversion to Gibbs energy and mole fraction is a satisfactory approximation only for relatively incompressible fluids.
Acknowledgment. This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Division of Engineering and Geosciences of the US. Department of Energy under Contract No. DE-AC03-76SF00098,
Thermodynarnlcs of Multicomponent, Miscible, Ionlc Systems: The System LiNO3-KNO3-H,O John M. Simonson* Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
and Kenneth S . Pitzer Department of Chemistry and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720 (Received: September 17, 1985; In Final Form: January 24, 1986)
Vapor pressures of water over KN03-H20 and LiN03-KN03-H20 (50.34 cation 7% Li) are reported in the temperature range 373 C T/K C 436. Water activities calculated from these vapor pressures and other available results are fitted to an equation appropriate for multicomponent electrolyte solutions which are miscible to fused salts. The resulting equation gives the excess Gibbs energy over the complete composition range of the three-component system. Parameters for the binary fused salt mixture are determined from aqueous solution data and compared with approximate values calculated from phase diagrams. Excess thermodynamic properties, including solute activity coefficients and excess enthalpies, are calculated from the model.
Introduction Electrolyte-molecular solvent systems which are miscible at moderate temperatures from dilute electrolyte to the fused salt are relatively uncommon, but are of interest for both theoretical and practical reasons. There are electrolytes of industrial or geological interest which become extremely soluble in water at high temperature and pressure. Any new type of system is of theoretical interest. Thus we chose to investigate a prototype system. Mixtures of metal nitrates, with relatively low melting temperatures and high solubilities in water, have been the most extensively studied systems of this type. Solvent activities in these solutions have been determined by direct vapor pressure,'** isopiestic: differential transpiration," and dew point methods.' A (1) Trudelle, M.-C.; Abraham, M.; Sangster, J. Can. J. Chem. 1977, 55, 1713. (2) Campbell, A. N.; Fishman, J. B.; Rutherford, G.; Schaefer, T. P.;Ross, L. Can. J. Chem. 1956, 34, 151. (3) Braunstein, H.; Braunstein, J. J . Chem. Thermodyn. 1971, 3, 419. (4) Tripp, T. B.; Braunstein, J. J. Phys. Chem. 1969, 73, 1984. (5) Tripp, T. B. J . Chem. Thermodyn. 1975, 7, 263.
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complete, consistent description of the measured thermodynamic properties of very concentrated systems has not been available due to limitations in the composition range of the experimental results and the lack of a comprehensive set of modeling equations appropriate for multicomponent mixtures of electrolytes in molecular solvents. In this work we report the results of water vapor pressure measurements over K N 0 3 solutions near 393 and 423 K,and over the mixed electrolyte system LiN03-KN03-H20 near 373, 393, and 436 K. These results are combined with other available data and treated with equations developed in an accompanying paper to give a consistent and comprehensive representation of the thermodynamic properties of the LiN03KN03-H20 system.
Experimental Section Lithium nitrate was prepared by neutralizing a hot-filtered solution of lithium hydroxide (Fisher Scientific) in distilled water (6) Tripp, T. B. Proc. In?. Symp. Molten Salts, Princeton, NJ 1976, 560. (7) Sacchetto, G. A.; Bombi, G. G.; Ma-, C. J . Chem. Thermodyn. 1981, 13. 31.
0 1986 American Chemical Society
