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Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979 391

Double Salt Solubilities H. P. Meissner Massachusetts Institute of Technology, Cambridge, Massachusetts

C. L. Kusik’ Arthur D. Little, Inc., Cambridge, Massachusetts 02140

Solubility products of double salts in aqueous solutions are predicted using previously published methods for estimating water activities and activity coefficients of the constituent simple strong electrolytes over the temperature range of 0 to 250 OC or higher. Anhydrous and hydrated double salts such as glaserite (Na2SO4.3K2SO4)and schoenite (MgS04.K2S04-6H20) are considered. Calculations at 25 OC are presented for the system Na,S04-K,S04-H20, determining saturated solution compositions and identifying the associated equilibrium solid phases.

Objective

Using previously published methods of predicting activity coefficients (Kusik and Meissner, 1978), solubility limits in multicomponent aqueous solutions have been calculated for simple strong electrolytes, such as represented by the formula A,,B,;nH,O, where UA and vB are stoichiometric coefficients and n represents the number of moles water of hydration, if any (Meissner and Kusik, 1973). Of equal interest are solubility limits of double salts formed from three or more ions in solution. The generalized formula and dissociation reaction of such a double salt formed from four ions is as follows

~(AuABuB).P(Cu,D,,).~HzO 9

avAAZA + avBBZB+ /3vcCzC + PvDDZ” + nH,O (1)

Here, A and C represent cations, B and D represent anions, and cy, P, and n are numerical constants, respectively representing the moles of simple salts AUABvB, CUCDYD, and water per mole of double salt. The term Z represents the absolute value of the charge on the ions identified by subscripts A, B, C, or D. Typical double salts containing 3, 4, and 5 ions are listed in Table I. . The object here is to present a method of calculating solubility limits of such salts. F o r m u l a Selection

In a double salt like glaserite, which is derived from two cations and one anion, the anions represented by B and D in eq 1 are identical. This double salt may have been written in various ways, three of which are as follows: (a) NaZSO4.3K2SO4;(b) 1/3Na2S04.K2S04; and (c) l/sNazS04.3/,K2S04. These three expressions are equally valid, being simple multiples of each other. In all three formulas, it is obvious that uA, vB, uc, and UD are respectively 2, 1, 2, and 1, with the ratio P / a equalling 3. Obviously, the calculated molality of glaserite in a particular solution depends on whether formula a, formula b, or formula c is chosen for reference. Thus, a given solution which is 0.1 m in glaserite by formula a is 0.3 m by formula b, and 0.8 m by formula c. Ion molalities in this solution, however, are the same with all three formulas, being 0.2 m in sodium ion, 0.6 m in K+ ion, and 0.4 m in S042-. Clearly, in defining double salt molalities, it is necessary to stipulate a, 0, and n of the salt under discussion as well as its molality, to avoid misunderstanding. In dealing with double salts, it is convenient to choose that formula which contains one gram-equivalent of material. In this discussion, therefore, formula c is chosen 0019-7882/79/1118-0391$01 .OO/O

for glaserite. Formulas per equivalent of contained material for other typical double salts are listed in the last column of Table I. An advantage of this choice of one equivalent is that it keeps a, 0, and n relatively small, and thus avoids the appearance of large exponents in the activity equations (below). Moreover, since one equivalent of material must contain one equivalent of cations and also one equivalent of anions to maintain electrical neutrality, the following always applies to a four ion solution CYZAYA + P Z C Y C = ~ZBVB+PZDUD= 1 (2) S o l u b i l i t y P r o d u c t . For strong electrolytes, the logarithm of the activity a, of a salt in solution equals the sum of the logarithms of its constituent ions (and water activity) suitably weighted by the stoichiometric coefficients. Per equivalent of double salt it follows directly from eq 1 that

a, =

mB)“”(mC)B”C( mD)B~(yAB)a(YA+YB)(yCD)B(uc+~)(a,)n (3) where m and y represent the molality and the mean ionic activity coefficient of the simple salt indicated by subscript while a and P are constants chosen so that eq 2 is satisfied. In saturated solutions, the activity of any electrolyte becomes identical with the so-called thermodynamic solubility product K , The solubility product is usually considered as applicaKle to sparingly soluble salts such as BaS04 since at low concentrations, activity coefficients of the ions in solution are unity. However, the concept is equally applicable to electrolytes of higher solubility, providing that appropriate values of activity coefficents are used (Denbeigh, 1955). Equation 3 appears formidable upon first inspection, but is easily applied to determine activities and solubility products. Thus, for one equivalent of glaserite (l/sNazS04.3/sKzS04), in which n is zero since this double salt carries no water of crystallization, (,A)”’”(

K,, =

~

~

~

~

2

~

8

~

~

6

~

8

~

s

~

~ sol. ~

(4) T o illustrate, consider a solution of 25 “C (D’Ans, 1933) containing only K+, Na+, and SO4’- ions, having respective molalities of 1.52, 0.94, and 1.23, which is saturated with both glaserite and arcanite (K2S04).The ionic strength O ~ Y K ~ S Oin ~ of this solution is 3.7, while values of Y N ~ ~ Sand this multicomponent solution are each 0.193 as calculated by methods previously published (Kusik and Meissner, 0 1979 American Chemical Society

s

~

3

~

s

392

Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979

Table I. Typical Double Salts common formula burkeite carnallite darapski te glaserite hanks i te kainite mendozite pol yhalite teepleite

formula per g-equiv

’

2Na,S0,~Na,C03 KC1.MgC1,*6H20 NaNO,.Na,SO, Na,S0,~3K,S04 9Na,SO,. 2Na,C03.KCl KCl.MgS0,.3H2O

Na SO .l/sNa,CO /3kCf I / 3hgC1,.2H2d I / 3 NaNO,.’/ Na,SO, 1/8Na2S04.3/8KZS04

9/~3Na,S0,~2/23Na,C03~1/~3KCI l/sKCl*l/~MgSO,~H,O”

1/~Na,S0,~1/8Al,(S0,)3~~~/4H,0 ‘ I S K , S O , ~*MgSO,.I/ ~/ CaSO,.’/ ,H,O

0.5Na,SO4~O.5A1,(SO,),~11H,O K,SO,*MgSO,. 2CaS0,. 2H,O Na,B,04~2NaC1~4H,0

‘ I 4Na,B,0,.

l/

,NaCl.H,O

As with all double salts containing 4 ions or more, kainite’s formula per equivalent could have been written in an alternative way such as: 1/6K2S04.1/6MgS04.1/6MgC12.Hz0. It follows that alternative ways of writing a, (eq 3) for kainite

include: a, = m K 1 i 3 m M g 1 i 3 m c 1mS0,’ 1 i 3 ’ 3(YKC1)2’ ’(7MgSO, )” ’(awl = K I ’ ’mMg I ’ %l” ’“SO, I ’ ’(7K,SO, )’ ’ ’(7MgSO, ) I ’ ’( y M g C 4 ) 1 / 2 ( a w ) .With either formula values of a, are equivalent if y and ow are obtained by equations for multicomponent solutions presented earlier (Kusik and Meissner, 1978). Table 11. Calculated Thermodynamic Solubility Product for Glaserite Saturated Solutions (25 “C)from Eq 4 glaserite : ionic calcd values molalitiesa solid phase(s) in strength of K , at 25 “ C addition t o elaserite“ K’ Na’ SO”,+ MCZ+ c1I (eauiv basis) K2S04

Na,SO,.lOH,O NaC1, Na,SO, KC1, Naci KCl, K,SO, ~

Na2S04,

Na,SO,~lOH,O KC1, NaCl, Scc Ast,b NaCl, Na,SO, KC1, K,SO,, Scc KC1, Scc Ast, NaC1, Scc

3.69 8.25 8.41 7.86 5.86 8.64

0.127 0.136 0.117 0.114 0.107 0.120

1.52 1.02 1.10 2.22 3.90 0.84

0.94 4.48 6.50 5.40 1.80 5.81

1.23 2.75 0.81 0.24 0.15 1.99

0 0 0 0 0 0

0 0 5.99 7.13 5.41 2.67

1.58 0.91

2.56 5.32

0.78 1.22

1.97 0.88

6.52 5.54

10.8 10.1

0.138 0.117

2.64 1.96 1.17

1.02 1.97 3.06

2.56 0.65 1.03

1.42 1.73 1.94

5.42 6.08 6.06

12.5 9.77 11.1

0.133 0.136 0.128 average 0.125

a

D’Ans (1933).

Sc = Schoenite, 1/4KZS04.1/4MgS04.3H20. Ast = Astrakanite, ]/4Na2S0,.1/4MgS04.2Hz0.

1978). Substituting into the above expression, a, and K,, for glaserite a t 25 “C is found to be 0.127. Inspection shows that the composition of the solids (dry basis) dissolved in the saturated solution just mentioned is not the same as that of glaserite itself. Indeed, at 25 “C it is found experimentally that the composition of a solution on a dry basis is never that of glaserite when glaserite is the equilibrium solid phase. As a result, glaserite is said to be “incongruently” soluble a t this temperature. This finding is not an uncommon one for double salts and in no way prevents the use of eq 4 for calculating saturated solution compositions from known values of K , . The fact t i a t the solubility product (and activity) of a double salt at a given temperature remains the same in all saturated solutions, regardless of the type and concentration of any other ions present, is illustrated in Table 11, where values of (K,,) for glaserite are listed in various solutions saturated a t 25 “ C with this double salt and other salts. The solubility products in each of these saturated solutions were calculated from eq 4 by the method just described. Inspection shows that despite the wide variation in both the compositions and ionic strengths of these saturated solutions, calculated values of K,, for glaserite a t 25 “C in all these systems remain relatively constant, varying within 15% of the average value of 0.125.

Phase Diagrams Whether saturated or not, solution compositions are conveniently expressed as gram-equivalents of each ion per 1000 g of water. Obviously, a t a fixed temperature, a solution containing more than two ions can be characterized by: the value of w , the grams of water per

gram-equivalent of total anions (or cations) in solution; the fraction “f,” of the total equivalents of cations present in solution which is represented by each cation; and the fraction “f,” of the total equivalents of anions present which is represented by each anion. The total equivalents of anions present, of course, equal the equivalents of the cations. Regardless of the number of ions present, such a solution at a given temperature having a known set of values of w, f,, and f, can be readily tested for saturation with a given compound. This involves calculation of the compound’s activity in this solution by substituting its appropriate ion molalities and activity coefficients into eq 3. If the value so obtained is smaller than this compound’s activity (solubility product) a t saturation, then the solution being tested is less than saturated. When a, and Kspare numerically equal, then the solution is saturated. When a, is greater than Ksp,then the solution is supersaturated, and a t equilibrium, the salt will precipitate. Various techniques can be used to present isothermal data on compositions of saturated solutions (Meissner, 1971). Solutions containing three ions, such as Na’, K’, and SO:-, are conveniently represented by a diagram such as Figure 1. Thus the experimentally determined compositions a t 25 “ C of saturated solutions for the K+Na+-SO?--water system are shown by the triangular and diamond shaped points of Figure 1 (D’Ans, 1933; Seidell and Linke, 1965) and contrasted with the (circular) points as calculated by the method outlined below. Glaserite is the stable phase along the dotted line (which is drawn through the calculated points) with K2S04, the stable phase on the solid line indicated on the left-hand side of the figure and Glauber’s salt on the solid line to the right

Ind. Eng. Chem. Process Des. Dev., Vol. 18,

3

Figure 1. Comparison of predicted and experimental solubilities at 25 "C for the Na2S04-K2S04system.

of the figure. The calculated solution compositions which can coexist with two solid phases (K2S04-glaserite, and glaserite-Glauber's salt) are shown by the two unshaded circles. General procedure in calculation is now illustrated with an arbitrarily chosen dilute solution at 25 "C, for which fNa+ and f F + are to be 0.75 and 0.25, respectively, while fso42of course is 1.0. Such a solution can be made by dissolving in water a small amount of salt mixture containing a ratio of 8 mol of Na2S04to 1 mol of glaserite (Na2S04.3K2S04). Upon evaporation, the object is to find the grams of water, w, when this solution first reaches saturation of 25 "C, and to identify the solid phase(s) which forms, recognizing that values of K,, for glaserite, arcanite, and Glauber's salt are 0.125, 0.14, and 0.38, respectively. As a first step, arbitrarily pick a value of w,such as 833 g of water per equivalent of cations (or anions) in solution. The sodium ion molality mNa is then 0.75 X 1000/833 or 0.9, while mK is 0.3, and mso, is 0.60. Using the methods described earlier, values of y for K2S04and Na2S04are estimated in this multicomponent solution as 0.247 and 0.248, respectively. Substituting into eq 3, a, is found to be 0.04 for glaserite. Similarly, after determining the water activity by earlier published methods (Kusik and Meissner, 1978) the values of a, for arcanite (K2S04)and Glauber's salt are 0.03 and 0.08, respectively, in this solution. Since in each case a, is far smaller than the corresponding Ksp,the solution is far from being saturated with any of these three salts. An arbitrary amount of water is therefore next removed, so that w is now 750 g of water per equivalent of cations making mNaequal to 0.999 and mK equal to 0.333 mol/ 1000 g of water. Estimating values of y for K2S04and Na2S04as before, determining a,, and calculating a, for all three salts, electrolyte activities a, for each salt are still found t o be smaller than the corresponding solubility products Ksp. Finally, when w is chosen as 500 g of water, the mNaand mKare respectively 1.5 and 0.5 and the value of a, for glaserite in this solution is found to be almost identical with its solubility product, namely 0.125. Calculations, therefore, indicate that this solution is saturated with glaserite, but not with K2S04or Glauber's salt, because values of a, for these latter two salts lie well below their respective solubility products ITsp,

No. 3, 1979 393

Figure 1was constructed by the procedure just outlined. Inspection of the calculated points shows these to be close to the experimental points for solutions saturated with one or two solid phases. Similar procedures can obviously be used for strong electrolyte systems containing more than three ions and at temDeratures other that 25 "C (Meissner et al., 1972, 1977). Solubility Product Expression Derivation. The following three-step isothermal reversible process is involved in showing that eq 3 is the thermodynamic solubility product of one gram-equivalent of a double salt in its saturated solutions, and in showing this product to be constant. Step 1. Transfer 1g-equiv of the double salt (eq 1) into the solution saturated with the double salt in question. This involves adding CY moles of the simple salt AvABQand P moles of simple salt C,,D, from their respective hypothetical standard state soyutions, in which their respective activities are, of course, unity. The Gibbs free energy change for the transfer of CY moles of A, B, is now: aRT In ((mA)YA(mB)Q(y)U~I and for P moles of &,D, is PRT In ( ( m c ) u ~ ( m D ) ~ ( y c ~The ) u ~quantities ~). TAB and YCD are the activity coefficients of the indicated electrolytes in this saturated multicomponent solution, as calculated by the methods mentioned earlier. Step 2. At the same time, transfer n moles of water from its standard state (pure liquid) into the saturated solution is question. The free energy change is nRT In (a,) where a, can be calculated by published methods (Kusik and Meissner, 1978). Obviously, if the solid double salt carries no water of hydration, then n is zero. Step 3. Simultaneously allow one equivalent of the double salt formula (1) to precipitate. The Gibbs free energy change of this step is zero. Upon combining terms, the sum of the free energy changes to steps 1, 2, and 3 becomes

.IG = R T In { m A a u ~ m B a v ~ m C ~ v ~ m D ~ ~( 5y) ~ ~ a u ~ ~ y For a fixed temperature and regardless of any other ions present, the free energy change of eq 5, per gram-equivalent of double salt, is always the same from the standard state solutions to all solutions saturated with the double salt in question. The quantity in brackets in eq 5 is, therefore, also a constant for all these saturated solutions, and is the solubility product Kspper equivalent of double salt. For these saturated solutions it is equal to the electrolyte activity a, as presented in eq 3. Note that this development applies or can be extended to all double salts, regardless of whether they involve three ions (carnallite and glaserite), four ions (kainite and POlyhalite), five ions (hanksite), etc., and regardless of whether waters of hydration are present or not. Precision. Errors in estimating water activities or activity coefficients of electrolytes in multicomponent solutions are generally within 20% with occasional values showing larger deviations (Meissner and Tester, 1972; Meissner and Kusik, 1972, 1973). Errors of this magnitude can significantly affect calculated values of (Ksp),especially for double salts, since these exist in equilibrium with mixed (multicomponent) solutions. Solid solutions could introduce another source of potential error. Thus, the relationships proposed here should be used only for orientation purposes when direct experimental evidence is not available. Nomenclature a, = activity of a dissolved electrolyte a, = activity of water, namely p l p " , where p is the water vapor pressure over the salt solution and p" is the water vapor

394

Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979

pressure over pure water at the same temperature fi

= fraction of total cations in solution represented by cation

v i = stoichiometric coefficient: see eq 1; also vij = v i

AG = Gibbs free energy change

+ vi

i

f, = fraction of total anions in solution represented by anion

I

j = ionic strength = ~ O . 5 m , Z I 2

K,, = solubility product for one equivalent of salt with examples shown in Table I m, = molality, g-mol (of the ion indicated by subscript)/1000 g of water n = number of moles of waters of hydration (crystallization)/g-equiv of solid double salt w = g of water per total equivalents of cations or anions in a saturated solution 2, = ionic charge on ion represented by subscript a = number of moles of simple salt A,B,/g-equiv of double salt (eq 2 must be satisfied) /3 = number of moles of simple salt C,,D,,/g-equiv of double salt (eq 2 must be satisfied) y = mean ionic activity coefficient of the ion pair, indicated by subscript, in the multicomponent solution under consideration

Literature Cited D'Ans, J., "Die Losungsgleichgewichthe der Systeme der Salzeozeanischer Salzablagerungen", Verlagsgesellschafl fur Ackerbau, Berlin, 1933. Denbeii, K. G., "Principles of Chemical Equilibrium", p 308, Cambridge Unhrersity Press, 1955. Kusik, C. L., Meissner, H. P., AIChE Symp. Ser. 173, 7 4 , 14-20 (1978). Meissner, H. P., "Processes and Systems in Industrial Chemistry", pp 46-61, Prentice-Hail. Englewood Cliffs, N.J., 1971. Meissner, H. P., Kusik, C. L., AIChf J., 18, 294 (1972). Meissner, H. P., Kusik, C. L., Ind. Eng. Chem. Process Des. Dev., 12, 205 (1973). Meissner, H. P., Tester, J. W., Ind. Eng. Chem. Process Des. Dev., 1 1 , 128 (1972). Meissner, H. P., et ai., AIChE J., 18, 661 (1972). Meissner, H. P., Kusik, C. L., Field, E. L., "Estimation of Phase Diagrams and Solubilities for Aqueous Multi-ion Systems", presented at 70th Annual AIChE Meeting, New York, N.Y., 1977. Seidell. S., Linke, W. F., "Solubilities", Voi. 11, 4th ed, pp 296-316, American Chemical Society, Washington, D.C., 1965.

Received for reuiew December 23, 1977 Accepted January 2, 1979

Use of the Vaporization Efficiency in Closed Form Solutions for Separation Columns Phillip C. Wankat" and Joseph Hubert School of Chemical Engineering, Purdue University, West Lafayette, Indiana 4 7907

Modified forms of the Winn, Fenske, and Kremser equations utilizing a constant vaporization efficiency are derived. The modified forms of the Smoker and Underwood equations are presented but not derived. These equations allow the rapid calculation of required number of stages without requiring the assumption of equilibrium stages. They also allow estimation of the average vaporization efficiencies from data on column performance, and then simulation of column performance under different operating conditions.

Introduction

yAj

Accurate solutions for staged separation systems can be obtained on computers by using various stage-by-stage or successive approximation methods. Although the modern computer techniques are quite rapid, they still require too much computer time for problems such as preliminary economic estimates or reactor recycle calculations where a large number of iterations through the separator are required. For these problems, for preliminary hand calculations, and for simulating existing installations closed form solutions such as the Kremser, Smoker, Fenske or Underwood equations are very useful. These equations are usually derived with the following assumptions: constant flow rates, constant relative volatility or linear equilibrium isotherms, and equilibrium stages. The last assumption can be relaxed if a stage efficiency can be incorporated. However, the usual Murphree efficiency is usually difficult to utilize (the Kremser equation is an exception (King, 1971)). An alternate efficiency which has frequently been used in multicomponent separation calculations is the vaporization efficiency. The vaporization efficiency, EAl, can be defined as (Holland, 1975) where the vapor is assumed to form an ideal solution. We will rewrite eq 1 as 0019-7882/79/1118-0394$01.00/0

=

EA]KAjXAj

(2)

where KAj = KAj*yAj includes the ideal solution K value and the activity coefficient evaluated using the mole fractions of the liquid leaving plate j . The vaporization efficiencies are bounded and can be estimated from pilot or plant tests, or from models of separation plates (Holland, 1975). The vaporization efficiencies also have the advantage that they are easy to incorporate in calculations. For binary systems the vaporization efficiencies are not equal unless the stage is an equilibrium stage. In this paper the vaporization efficiency will be used to derive modified forms of the Winn and Fenske equations for total reflux in distillation columns and the Kremser equation for linear equilibria. The modified forms of the Smoker and Underwood equations will be presented without derivation. The use of these equations to determine the vaporization efficiencies from operating data is also developed. A sample calculation using these equations will be presented. Total Reflux Equations

Equations for total reflux in distillation have been presented by Winn (1958) and Fenske (1932) (or see King, 1971; or Smith, 1963). Both of these equations are easily modified to include the vaporization efficiency. The derivation will be done for the Winn equation since the Fenske equation is a special case of the Winn equation. 0 1979 American Chemical Society