Petgrmon

r. INTN,ODUCTION

Most chemical industries comprise pro@sses that in-volve multicomponent electrolyte solutions. Damageprevention of industrial equipment requires accurateknowledgc of the concentration of electrolytesthroughout a process. Recent environmental concernsrequire that the conccntration of electrolytes in finalproducts or in waste streams be controlled with pre-cision. Prediction of the thermodynamic properties ofmulticomponent electrolyte solutions therefore posesan important challenge for engineers.

In practice, the equilibrium thermodynamic prop-erties of multicomponent solutions are often deter-mined within the framework provided by Pitzer (1991)or that provided by Chen et al.(1982). Pitzer's equa-tions are bascd on a virial expansion for the osmoticpressure. The coefficients appearing in such an expao-sion are determined by rcgression of experimeutaldata. The model ofChen er al.(1982\ is based on anextension of the NRTL model of Renon and Prausnitz(1968), which was originally conceived for non-elec-trolyte solutions. The parameters appearing in sucha model are also determined by regrcssion of cxperi-menial data Both formalisms lead to results of com-parablc quality, and are generally included in com-mcrcial process simulators (e.g. ASPEN PROCESS).

For multicomponent solutions, both formalisms re-quire binary and ternary cxpcrimcntal data for allcomponents in the mixture. Unfortunately, such datamay not be available for a number of systems ofengineering importane. Clearly, new methods thatrequire only limited experimcntal information butthat lead to reasonable predictions of the equilibriumthermodynamic properties of electrolyte solutionswould be of great usc in chemical engineering prac-tice.

Chnical Engircqing Sciqce, Yol 50, No. 12, pp. 1953- 1959, 1995Copyrigbt,O l99j Elsvicr Scictre Ltd

Printcd itr Gr€r Brirain. All dghrs rcerycdfln9_2i09/95 $9.50 + 0.@

fit(xr-2s09(9s)fixx7-x

Recently, we have proposed a novel interpretationof the NRTL model which appears to provide goodresults for phase equilibrium in binary electrolytesolutions (Kolker and de Pablo, 1995). Encouraged bythe success of this new method, in this paper weextend it to multicomponent solutions and apply it toseveral systems of engineering importance. Our exten-sion is based on an integration of the Gibbs-Duhemequation. Such an integration is described in Section2. In Section 3 we review briefly the model and themethods described in our previous paper (Kolker andde Pablo, 1995), which we refer to as (I) throughoutthe remainder of this work. Section 4 discusses thecalculation of solubilities in aqueous solutions con-taining both two electrolytes and an electrolyte anda non-elcctrolyte, and Section 5 presents our calcu-lations of solubility of a volatile solute (carbon diox-ide) in several aqueous electrolyte solutions.

Z TTIERMODYNAMIC BACKGROUIYD

We propose to calculate activity coefficients foreach salt by integrating the Gibbs-Duhem equationfor a multicomponent system. The general form of thisequation (temperature and pressure are constant) is

m1dp1* in,dp,:g ( l )

where m1 = const i, ttl 1noUtity of pure water(: 55.508), m; is the molality of solute i, and n is thenumber of components (1 solvent and n - I solutes).Chemical potentials (p and p) are related to activ-ities (in differential form) by

dtti : RTdln ai(m2, ,.., mn). Q\

The activities are functions ofn - I independent vari-ables (molalities). We may therefore write the follow-

THERMODYNAMIC MODELING OF CONCENTRATEDMULTICOMPONENT AQUEOUS ELECTROLYTE AND

NON-ELECTROLYTE SO LUTIONS

ALEXANDER KOLKER and JUAN J. DE PABLODepartment of Chemical Engineering, U""ri:ilr%Xisconsin-Madison, 1415 Johnson Drive, Madison,

(Receioed 12 October 1994; accepted in reuised.form 9 January 1995)

Ahtnct-A ncw approach is presented for calculation of activity cocfficients in multicomponent aqueouselcctrolyte and non+lectrolyte solutions. The approach is bascd on an integration of thj Gibbs-Iiuhemequation. It is shown that using standard thermodynamic properties for the lure components and for thecomponents at infnite dilution in water, it is possible to predict joint solubiiities of both electrolytes andnon'€lectrolytes. To illustrate thc usefulness ofthc method proposed in this work, calculations ofsblubilityare prescnted for the two-electrolyte systems Hro-Nacl-KCl, HrG-NaNo.-KNo3, HrG4aclr-Naciand for a system containing both an electrolyte ind a non+lectrotyte Hro-sucrose-l.iacl. The soluiility oicarbon dioxide at different pressurcs in water containing various elecirolytes [Ca(NOr)r, NaCl, Catlr,NarSOnl is also calculated.

cEs 50-t2-H 1953

ALsxeNorn KoLKER and Julx J. DE PABLo

ing general expressions:

" /d ln ai\dlna,: t t-=----j I dmr. (3)

172\ omt '/^1 "

Substituting expressions (2) and (3) into eq. (l), we find

3 /d ln at \mt ) | ---:--l Qttt*

y72\ 0m* / ^ , , *

: 3 /d lnar\+)nr) . { - | dn*:g (4)

i4 *72\ omy 1n,. ,

and, after some rearrangements, we arrive at

: f /d lnar\ 3 /d lnar\ lf ln, l - l +In, l := l ldrq:s.

, " rL - \ dmr /^, ,* i7z \omx/^t . , )

(5)

Equation (5) is satisfied if all the coemcients in par-entheses vanish, thereby leading to the following n - |equations:

/d ln c ' \ + /d ln ar\mt l - - - : - l +Lmtl

- - l :u '

' \dmx /^, ,* i -=2 \om*,/^t . .

k:2, . . . ,n. (6)

Next, we proceed to convert eqs (6) into a form inwhich each ofthem contains only the activity ofone ofthe salts, a2, and the activity of water, c1. Thermodyn-amic consistency requires that the partial derivativesbe related by (McKay, 1953)

(0ln a;lAml)^,, , = (?ln al6ftii)^1, ,' (7)

Substituting eqs (7) into eqs (6), we have

n

L ^t@lnayf 1mi)^, , , * m1(0laa1f 0m1)^, , * : O,

k - 2, . . . , n (8)

To integrate these n - I equations, we introducea new set of independent variables, namely the molefraction of the salt i in the mixture of solutes (ona water-free basis), zr : mtl[i=zmt (with !i=rzr : l),and the mole fraction of water, xr: mrl{mr *

Li=r^t). We replace molalities by these new indepen-dent variables to get

(0ln a1l0m1)^, , , : (0ln allAx ),,......-,(0xrl1mi)^t , ,n- |

+ I (a hapl0zl),,(oz/0mi)^,,,l=z

(e)where the partial derivatives of x1 and zi can bewritten explisitly as

(0x1f 0m1)^,.,: - x?lmr

(0zi/0m11^,,, = xr(1 - z)lft l tr)

(az1l1m)^,,, = - zfi tl(x"m)

and where x" denotes the mole fraction of all solutescombined, i.e. x": I - xr.

Substituting expressions (9) into eqs (8), we arrive ata much simpler set of equations, namely

(| lnallax),, .-, : ft,rU t" or,Untr)^,. *,

k:2, . . . ,n. (10)

Note that eqs (10) contain a derivative of the activityof water and, in contrast to eqs (8), only one sol-ute-activity.derivative. In order to facilitate the sub-sequent integration ofeqs (10), we express the deriva-tives of the activity of water with respect to molalitiesin terms of derivatives with respect to x1 and z;. Aftersome algebraic manipulations we arrive at:

(0ln aylAx )",

k = 2, . . . ,n. (11)

Activities, d1, and activity coefficients, y1, are relatedby

ar = x*!r , k = l , . . . , n. ( l2 l

Note that the mole fraction of solute x* is related tox" through x1 : zzxs. Substituting this definition intoeqs (11) and integrating them by parts from x1 to I atzr : const * l, we arrive at expressions for the activ-ity coefficients of each solute.

For components k:2, . . , ,n - 1 we f ind

r l

ln7, : lntP - x1/x"ln7, - | lnyl/x"2dx1JXr

For the "last" solute, component n, we get

l rlnTn - fnTo - x1fx" lny1- | lnTt/x"2dx1

r - l t l

+ L ,i I t0tnvtl1z),,1x? dx, (14)j=2

"rr

where ln yi (and ln ff ) denotes the activity coemcientof component k at infinite dilution (at x1 - 1) in purewater.

Equations (13) and (14) are exact thermodynamicrelations. They do not depend on any molecularmodel but are merely an integral form of theGibbs-Duhem equation. Their usefulness resides inthe fact that the activity coefficient of each solute canbe calculated from the activity coefficient of water andits dependence on z1 $ :2,..., n * l). The activitycoefficient of water in a multicomponent solution,lny1, and its derivatives with respect to a solute's molefraction in the anhydrous mixture (0lny110z),,can&,obtained either from experiment or from a model. In

.- , = - (0 lna1lax)"r . . . . . " - ,xr /x,

+ (0lna1l1z),,(l - z)lx:

/ " - r \ /- ( I zphal l /z l l fx ! ,

\ ;=2, ; ' . t / l

- t r - rdJ ' ( otnyl lazf lx!dxy

*,='j', * rr, [',{ant

rloz,),,1x! dx, (13)

ll *3'

n. ( l l )

related

arrive at

n. (10)

r activityone sol-the sub-: deriva-rolalitiesz;. After

t lx",r?

(r2)iated toon intotolat

e activ-

( l3)

(14)

ificientn pure

.'namtcIecularof therdes inlte caner andctivitylution,s molecan be.1e1. In

the latt€r case, a solution theory is required; ulti-mately" the accuracy of the calculations depends onthe accuracy of such a theory.

3. CALCULATION OF THE ACTIVITY COEFFICIENT OF

WATER

To calculate In 71 in a multicomponent solution weapply a method originally developed for binary solu-tions. While a detailed description can be found in (I),for convenience Appendix A gives a condensed ver-sion ofthe necessary equations. In this work we treata multicomponent system as pseudobinary, i.e. weconsider the system to consist of a water (component1) and a mixture of all dissolved salts, which is treatedas one unique complex pseudocomponent 2.

The calculation of lnyl in the framework of thismethod requires estimation of several molecularcharacteristics of the mixture (see I and Appendix A).As initial dat\ functions nl2Qr, ..., zn_) and022Q2,..., zo-r) for the mixture of solutes are neces-sary. We define a function n9zQz,...,z,_,), whichcorresponds to the average number of nearest neigh-bors ofcomplex pseudocomponent 2 in the absence ofwater, as a linear combination of nfi:

nl, = izrnfl*i=2

where nf1 denotes the average number of nearestneighbors of pure component /<. We define a functiongzz(22,..., zo-,), which represents a Gibbs free energydifference between pure complex pseudocomponent 2,and the same pseudocomponent at infinite dilution inthe solvent as

nn

9zz: lzr4l- LtrptQr)k=2 h=2

(15)

where p! is the standard chemical potential of com-ponent k in the anhydrous mixture in a "supercooledfused salt" reference state, and pf is the standardchemical potential of the same component in a refer-ence state corresponding to a "hypothetical ideal solu-tion of unit concentration" (Robinson and Stokes,1965). The former can be written as

pPQ*) : pl,tzt = l) + RTln ys"z* (16)

where ln y1." are the activity coefficients of the salts intheir anhydrous mixture (xz

- l). Their values de-

pend on the interaction energies ofthe salts with eachother. An ideal mixture of salts is formed when allIny1," are equal to zero.

Equation (15) can be rewritten in the followingform:

1955

and where gft are determined as described in Appen-dix A (eq. (A4)1. Note that our definitions of n!2 and922 for a complex pseudocomponent follow naturallyfrom our interpretation of these functions for a binarysystem (I).

4 RESULTS OF CALCULATIONS

The model presented in this work has been used topredict solubilities in ternary systems. While the equa-tions obtained in previous sections are applicable forcalculating the activity coefficients of a wide class ofsystems, for simplicity, we restrict our attention to thecalculation of branches of solubility for componentswhich do not form crystal hydrates or double salts inthe corresponding solid phases. Generalization to sys-tems that form crystal hydrates or double salts in solidphases will be presented in a forthcoming paper.

We give examples for the following two-electrolytesystems: HrO( l)* KCI(2|-NaCI(3), HrO(1FKNO3 (2)-NaNOr(3), and H2O(l)-CaClz(2)-NaCl(3). Wealso present results for a system containing both anelectrolyte and a non-electrolyre: H2O(llsucrose(2)-NaCl(3). Note that we use the same equations forthese different types of systems; most conventionalmodels cannot be applied directly to both of them.

Equations (13) and (14) for the activity coefficientsof the components of a ternary system (n : 3) take thefollowing form:

lnyr : lny? -x1lx" ln71 - l ' nyr l t l ar ,

ln l. - ln yf - xrf x"lny1

To calculate gzz wa need to determine the excessGibbs energy of a supcrcooled fused anhydrous mix-ture of salts, gf. Unfortunately, direct experimentaldata for this quantity are not available. There is,however, experimental information about the heat ofmixing, hf, of fused alkali-metal nitrates and halides(Lumsden, 1966). In most cases, such fused mixturesare described well by regular solution theory. In sucha theory, the excess Gibbs free energy of mixing, 9f, isgiven by

S!=h!:A2sz2z3.

Extrapolation with respect to temp€rature of the ac-tivity coefficients given by Lumsden (1966) (we as-sume that lny" a. llT) leads to the value ofAzt

- - 720 callmol for the mixture KNO3-NaNO3

and, As t - 100 cal/mol for the mixture KCI-NaCI.These numbcrs reflect the fact that these anhvdrousmixtures of salts are not ideal but exhibit smali nega-tive deviations from idealitv.

Thermodynamic modeling

(17)

-,t - 'r l:,(Ltny,lLz),,/xl dx, (le)

- J'h7'7'"'d"'

* , , [ ' , (otnyl loz), ,1x!

dxr. (20)

gzz: lz*9**9!k=2

where gf is the excess Gibbs energy of the system ona solvent-free basis {anhvdrous mixture)

n

g"E : RT I z1 lny1.,,L=2

(18)

, : i l ;. i l

t : : t '

1956 Arrxexorn Konrn and JUAN J. oe Plnto

For the system CaClz(2)-NaCl(3), the activity coef-ficients extrapolated to the standard temperature 5

25"C have the form (Lumsden, 1966)

lo yz." : - 7.6z2tl(2 - zt)2, 4

lnYr," : -15'2zl lQ-zt)z i ,o-

To calculate gf we use eq. (18). ?To obtain gf for the anhydrous system NaCl(2)-

sucrose(3), we used data for the standard Gibbs en-ergy of transfer of NaCl from water to an aqueoussucrose solution (Wang et al., 1993). These authorsmeasured the dependence of this quantity on the con-centration of sucrose in solution up to 30% wt; theyfound it to be linear. Assuming that a linear extrapola-tion up to pure 1007o sucrose can be made, we ob-tained the standard Gibbs free energy of transfer ofNaCI to pure sucrose. From this quantity, we thendetermined the activity coefficient of NaCl at infinitedilution in sucrose and, consequently, the valueAzt x - 6 kcal/mol (assuming that regular solutiontheory is valid for this binary system).

The individual component activities for the aque-ous ternary system on the branches ofsolubility,lna2and Ina3, must be constant and satisfy

lnar: - {AI Io.r( l -TlToi+IAHr*(1 -TIT+)

- Lco,r,lTo.* - T(l + lnTs*lT))\lRT (21)

for components 2 and 3, respectively (I). Here, A116.1and fo.r are the solute's enthalpy and temperature offusion and Aco,1 is the change of heat capacity uponfusion. The second term in the right-hand side of eq.(21) has to be added if i phase transitions occur in thecrystal structure ofthe salt, each with enthalpy changeLHi at temperature ?";. Enforcing these criteria per-mits calculation of the joint solubility.

Figures l-4 show experimental data and the resultsofour solubility predictions. In general, the predictedresults are in reasonable agreement with experiment.We should point out, however, that the description of

Nat'|O3, m3

Fig. 2. Solubility isotherrr for the system water-KNO, (21NaNOr(3) at 25'C: (l) shows experimental data (Linke andSeidell, 1965); (-) gives the results ofour calculations.

€oz

-3

Fig. l. Solubility isotherm for the system water-KCI(2FNaCI(3) at 25'C: (l) shows experimental data(Linke and Seidell, 1965); (-) gives the results of our

calculations.

m2,CaCl2

Fig. 3. Solubility of NaCl in aqueous solutions of CaCl, at25"C: (I) shows experimental data (Linke and Seidell, 1965)(-) gives the results ofourcalculations; (- - - -) gives

the calculated solubility assuming 9f : 0.

m3 , grJctos€

Fig. 4. Solubility isotherm for the system water-sucrose(2FNaCl(3) at 25"C: (l) shows erperimental data (Linkcand Seidell, 1965I (-) gives the results of our catcu-

lations.

m,, KCL

Thermodynamic modeling 1957

) r (2Y(e andrtlons.

a sup€rcooled fused anhydrous mixture of salts byregular solution theory is only a first approximationwhich happens to work for the systems studied here.For other mixtures, a more refined treatment of sol-utFsolute interactions is likely to be necessary.

To illustrate how sensitive our results could be tothe value of9f, we have also calculated the solubilityof NaCl in aqueous solutions of CaClz, assuming thatg! :0. This calculated solubility corresponds to thedashed line in Fig. 3; these results indicate that ne-glecting solute-solute interactions can in some casesintroduce appreciablc errors into the predictions ofthc model.

5. SOLUBILTTY OF CO2 IN AQUEOUS SOLUTIONS

CONTAINING AN ELECTROLYTE

Vapor-liquid equilibrium is determined by the con-dition that the chemical potential of each componentbe equal in the vapor and in the liquid phases, i.e.

p9., + RTInPlzez: #9,riq * RTlny2x2 (for CO2)(22\

p?,, + RflnP(l - yz)rpt: p?,uo * Rllnal

(for water)

where P is the total pressure, rp1 and E2 arc thefugacity coefficients of water and CO2, 12 and y2 arethe mole fractions of carbon dioxide in water and inthe vapor, respectively, y2 is its activity coefficient inwater, and c1 is the activity of water. The values ofpl and 1^rflq are the standard chemical potentials ofcarbon dioxide and water in the vapor phase and inthe pure liquid state. Following the approach that wehave taken in our previous work (I), we have chosenthe standard state of carbon dioxide in water to behypothetical superheated pure liquid CO2 at the tem-perature and pressure of the system. The advantage ofthis standard state compared to a standard state atinfinite dilution is the independence of the former onthe presence of other solutes.

Equations (22) lead to the following equation forthe solubility of the gas:

lny2x2: fuo2' - F\,i lRT + ln Pq2y2 e3)

where

lz : | - exp (p?,tq - pl.,)lRT'l atlPa L

r 1 - exp [(AG?,rio - LGI)lRr]lP.

The mole fraction of water in the vapor phase is verysmall at elevated pressures of carbon dioxide. Theactivity of water in the binary system COz-HzO isclose to unity. For simplicity, we assume that the ratioar/er x l. (At elevated pressures, rp1 could deviatesignificantly from unity. However, the exponent in thesecond term in the expression for y2 is much smallerthan unity. At elevated pressures, this term, which isinversely propoftional to P, b€comes even smallermaking y2 close to unity. Therefore, our assumptionthat the ratio a1fq1 rv I at all pressures does notintroduce an appreciable error in y2).

The difference between the chcmical potentials ofgnseous and superheated liquid CO2 on the right-hand of the eq. (23) is equal to

F9.u - F9..riq: LG|, - AGg,riq

: AIIo(1 - fTo) - LH/l - TlTo)

- LcolTs - T(l + ln TslT))

- u9v - P,) (?r4\

Here AG!., and AG!,6 are the Gibbs energies offormation in the gaseous phase and in the superheatedliquid state, A.H6 and A.flo are the heats of sublimationand melting of solid carbon dioxide, Ac, is the differ-ence between the heat capacity of liquid and gaseouscarbon dioxide, ?"6 and I are sublimation andmelting temperatures, u! is the molar volume of liquidcarbon dioxide and P, is the reference pressure (usu-ally I atm).

The activity coefficient of carbon dioxide is given byeq. (20). At elevated pressures, a Poynting correctionis employed according to

Iny2 : lny2(P,, * (oz-u$

e - P,)RT

where u2 is the partial molar volume of carbon diox-ide in water.

To calculate the fugacity coefficient ofcarbon diox-ide we use the equation of state of (Nakamura er cl.,1976). Values of AiIq, LH, To, To, Lco for CO2 aretaken from the literature (Angus et c/., 1976;Vukalovich and Altunin, 1968).

Note that in calculating the solubility of CO2 wehave not taken into account the formation of ions dueto the reaction

CO2+H2OeH++HCOt.

Its equilibrium constant is very small and for a molal-ity greater than l0-a the effect of dissociation disap-pears (Edwards er a/., 1978).

We have applied eqs (19), (23) and (24) to calculatethe solubility of carbon dioxide in water containingelectrolytes, Ca(NO3)2, CaCl2, Na2SOa and NaCl.The last two systems have been studied extensively byCorti et af. (1990) using Pitzer's equations. Within theframework of our approach, we need to determine theexoess Gibbs energy of carbon dioxide-salt anhydrousmixtures. Unfortunately, in contrast to mixtures ofsalts, even indirect thermodynamic information forsuch a quantity is not available. If we apply regularsolution theory we need to determine only one adjust-able parameter, Asf RT, for each of these mixtures.To do so, we have no other alternative than to use atleast one experimental value for the solubility of car-bon dioxide. The values of the parameter A6f RT arcgiven in Table l. (Note that this parameter does notdepend on pressure.) Figures 5 and 6 show the resultsof our calculations for the solubility of carbon dioxideat different concentrations of electrolyte along withthe corresponding experimental data. Again, cal-culated results are in reasonable asreement with ex-periment.

t95t AreiixDBn.Kotree and Juex J' oe Plslo

Table l. Values of the parameter Arrl\T at 25'C. for an-' iidlous carbon-dioxide--supercoolcd fusod salt mixtures

Salt NaCl Ca(NOr)z CaCl2 Na2SOa

lytes in aqueous solutions ano for the.solubility of

a gas (carbon dioxide) in water contalnlng several

electrolytes at different pressures'

In alicases, our predictions are in reasonable agree'

ment with experiment. Note that our model relies on

exact thermoiynamic equations for the activity coeffi-

cients for multicomponent systems' These equations

are not dependent on any molecular model' Further-

torr, ou. model does not require empirical para-

meters but it requires instead the excess Gibbs energy

of anhydrous soiute mixtures. In this work, for simpli-

city, we use regular solution theory to describe such

.i*trr.r; a more elaborate model could in principle

be used.The main practical difficulty faced by our method is

the determination of the excess Gibbs energy for a hy-

pothetical anhydrous liquid mixture of solutes'.Such

iutu "r.

not always available. However. gf has a

clear physical significance (as opposed to empirical

p"tutn"tit. required by conventional models)' This

iaises the possibility of calculating it either from a mo-

lecular theory or, as it has been shown in this paper'

from available independent experimental data'

In practice, it may prove useful to solv€ the revers€

problem of using the data for joint solubility in.water

io determine an expression for the excess Gibbs en-

ergy of the corresponding anhydrous solute mixture'

Tiis procedure may be particularly.useful when a spe-

an" "ppfi""tion

requiris information regarding sol-

ute-solute interactions.

Acknowleilgements-The authors are grateful to. the Nation-

;ffi;;;; ffi;;tion for financial sqpporl'JJdP is grateful

," ti" b"-iff. and Henry Dreyfus Foundation for a New

Faculty Award.

A2!lRT - 0.2 - 6.1 - 9.89 10.5

g ,..g 2.4

; 2.2

; 20

€' 'E 16

't.0

0.9

0.8

E 0.7

o 0.6og 0.56E o.r

0.3

molality ol tfa el€cfol$e' m3

Fic. 5. Solubility of carbon dioxide in water containing

lli:,,ilvi"Jlrl r.iact (r), (2) cacl' (o), (3) Na,son (A)' (4)Ca{NO.), (V); the pressure is l atm, the temperature ts

;iC]ii .rtJ*t "*lpa.ental

data (onda er al" l97o\;- -

i '

) givei thc rcsuits of our calculations'

tnoldity of ttto sl€cbolYto' ml

Fir" 6. Solubility of carbon dioxide in water contarning

ffiilyGiiit i.raq (rI. (Q-^c^ac!..(o); ttre q1111c is47.3 atr., thc temperaturc tt' j5'C' (i) slows cxpcrimental

d;l'il"fiit;;h Kvrovskava, 1e75); (-) sivcs theresults of our calculations'

CONCLUSIONS

We have developed a new method for the calcu-

lation of the thermbdynamic properties of multicom-

poi"n, aqueous electrolyte and non-electrolyte solu-

iions. To-illustrate the usefulness of the method' we

h""" pr"r"ttr"d results of calculations for the joint

,otuUiiity ofa variety ofelectrolytes and non-electro-

REFERENCF.s

Ancus. S., Armstrong B. and de Reuck' K" 1976' lnterna-''"i;;;i;;iv"i)i, rabtes of the Ftuid

^state .carbonDioxide. IUPAC Vol. 3' Pergamon Press' uiloro'

s"";;;. "id-di"artv,

D., 1985, Extension of the- sPecific-l"f,.".,i*rnodel to include gas solubilities rn hi-q! pm-

;;;;;-ft;. . Geokhim' Coimochim' Acta 49' -r95-2ro'

cri"i, di---s;it' H., Boston, J' and Evans' L" 19E2'.Local-';;;;;Ji"t model for excess Gibbs energv of electrolvte

svstems. A.I.Ch.E. J. 2& 588-596'c;il:ri;;F;blo, J. and Prausnita J', lee0' Phase equilib-

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J. phvi. Clwm. 94, 7876-7880'ei*i.it r" r"r""*l' l.' Newman, J' and Prausnita J" 1978'

Vapour-liquid equilibria in multicomponcnt aqueous

.Jitions of volaiile weak electrolytes' A'I'Ch'E' J' A'

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trolvte solutions' A.I.Ch.E, J' (in press)'

d;;w.;ils.ii;i A" resl, iotibttities of-tnorganic att-

Urio't Organic Compoutds' 4th Edition' Vols I and 2'

Washington, DC'L";;;"j" igeo, tn,,*avnanics of Molten salts Mix'

ttlres, Acadcmic Press, London'Mil;i;; S;JKurovskav4 N', 1975, Investigation.of co'"-*i"liriiv

in solutions of chlorides at elcvated temper-

"i"*t "i,a Preslrure$ Geokhimiya 4'. 547-550'.

f"f.I("v. A., 1b53' Activities and activity coefrcicnts tn ter-'--i"iltyrt*..

Trans. Faradav Soc' 49'217-246.'

Thermodynamic modeling 1959

bility of: several

le agree-relies on.y coem-luationsFurther-.rl para-j energyr simpli-ibe suchrrinciple

hod ist hy-SuchasairicalThis

. mo-paper,

reversen waterbbs en-

n a sPe-ng sol-

Nation-gratefula New

I nterna-Carbon

specificgh tem-

210.I, Local

equilib-diox-

uttons.

., 1978,queousJ, A,

odeling

s Mix-

Co.emper-

wherc

Rllnyf :

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APPENDD( AHere we give equations for the activity coefficients of

a binary system in the framework of our approach (I). A de-tailed description of their use has already been pres€nted (I)and need not be given here again.

These equations are:

ln l t :1111 * lnTtot

lnTr:11t1 * lnY2ot

- Ltn(A,,2 * A\2x6)l(x1Arz * xz)z

+ t2{Alt + xlA'2)l@2A2r + xl)2lx:

RTlnyl : - ltr2(Al2 - A\zxtr)l@ltz * xz)2(Al)

+ t2{Ar, - x62A'2)/(x2A2, + xr\2)xl

and where lnyro" and lny2es are the Debye-Huckcl contri-butions to the activity coemcient of water and that of theelectrolyte, respectively. [n these equations

421 :exp(t21fa2tRT\ tzr : 9zr - Qry dzr : t rzr

At2 = exp(rr2fdrzRT), t n : grz - |tz, ,r, = nr, + (i;\

The gr; are Gibbs free energies of interaction b€tween speciesof type d and a central species of type j. The numbers ni; arethe average numbers of nearest neighbors of type i whichinteract with a central species of type.l (t, j: 1,2). Theseequations differ from the commonly used ones (Prausnitzet al,, 19861in three rcsp€cts: (l) gr, * 921, (11) u12 { d21 and(iii) a;; are not regarded as adjustable parameters becausethey can be expressed in terms of n;i (and the latter can becalculated from molecular information).

To calculate g z t g r z, g t r, and nf 2 in terms of g 22, n! 1, andn!, we use the following set of four equations:

t21exp(t21f nl lRT) * tp: g22

t pexp (t 21ln!2RT) + r 21 : g 1 1

no12fnor, : expf(gpln!2 - gtrln!)lRT] (A3)

n! 1 [exp (r2 r/n!, Rr\ - 2) + n?z [exp ( - t r t/not"RT) - 2] : 0

where

g22 : ( c9- Aci.) - R?'lnm1 + AIIo(l - f/To)

- AcolTa - f(l + lnTolT\f

+ IArrr(l - TlTrl + 2RTA,lpln(r + pkfz) (A4)t

Here AG!- is the Gibbs free energy of formation of the salt atinfinite dilution in the molal scale at the temperature of thesystem, AG! is the Gibbs free energy of formation of the purccrystal salt at the temperature of the system, rn1 is themolality of pure water ( : 55.51), Affe and fo are the salt'senthalpy and tempcrature of fusion and Aco is the change ofhcat capacity upon fusion. The term IrAfIl(l - I/Tj has tobe added if & phase transitions occur in the crystal structureof the salt, each with enthalpy change AI[1 at temperatureI.. Parameters 1, and p in the expression for theDebye-Huckel contribution have been given by Pitzer andSimonson (1986). The value ofn!2 is the average number ofmolecules of type 1 interacting with a molecule of type 2 atinfinit€ dilution (i.e. it is considered to be a hydration num-ber), and the value of nlt is the hydration number for purewater (n!1 x 4.6 at 25"9.

ic andand 2.

ln ter-