Aqueous Solutions of Two or More Strong Electrolytes Vapor Pressures and Solubilities Herman P. Meissner* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Mass. 02139
Charles L. Kusik Arthur D. Little Inc., Cambridge, Mass. 02140
A relationship is presented for determining the vapor pressure of water over aqueous solutions containing more than one strong electrolyte. Illustrations present use of this relationship in determining the solubility in such mixed solutions of electrolytes whose crystals are hydrated.
T h e activity coefficients of strong electrolytes in aqueous solutions can be predicted a t various t'emperatures and ionic strengths by recently published methods (Meissner and Kusik, 1972; lleissner and Tester, 1972: and Neissner e t al. 1972). Vapor pressure of water over such solutions, containing only one dissolved electrolyte, can also be predicted (Kusik and lleisener, 1973). The object here is to extend these procedures to the calculation of water vapor pressures over aqueous solutions of more than one electrolyte, and to illust'rate application of these methods to the estimation of the solubility of electrolytes whose crystals are hydrated. Mixed Solutions
solution of a n electrolyte 12 is here called "pure" or "mixed," depending on whether other electrolytes are present. Mixed solutions are described as containing cations 1, 3, 5, and so forth, and anions 2, 4, 6, and so forth. It is convenient t o characterize these solutions in terms of the individual ions, using the definition of ionic strength presented in the nomenclature table. To illustrate, consider a solution which is 2 m in NaCl and 2 ?n in n'a2S04.Designating Na+, C1-, and S042-as ions 1, 2, and 4,the ionic strengths p1, p2, and p4 are 0.5 (6 X 12), 0.5 (2 X 12),and 0.5 (2 X 2'9, or 3, 1, and 4, respectively. The anionic strength of the solution itself, namely pa equals (p? pq) or 5, the cationic strength is p c or 3, making pT, the solution's t'otal ionic strength, equal to (pa p c ) or 8. Finally, again using the definitions in the nomenclature table, the anionic strength fractions Y2 and Y4 in this system are, respectively, 0.2 and 0.8, while the cationic strength fraction SIis uniby. A \
+
+
Mixed Solution Vapor Pressure
The activit'y of water, designated as (a,")12 over a pure solution of electrolyte 12 and (a,Jrnix over a mixed solution is defined as the ratio p / p o , namely, the ratio of the partial pressure of water over the solution and over pure water a t the temperature in question. I n a mixed solution ( a w ) m i x can be ~ ~ , ( a w 0 ) 2 3 , and so forth, all taken calculated from ( u ~ " )(aWo)14, a t the total ionic strength and temperature of the mixed solution, as follows: log
(aw)niix
Lylj7S
log
=
log (am0)12 f X 1 Y 4 log . . . $- zy3y2 log (aw0)32
s1I72
(awo)16
(aw0)14
+
+
log
~y3y4
(1)
The derivation of this equation is outlined in later paragraphs. For a mixed aqueous solution of NaCl and KCl, here called electrolytes 12 and 32, respectively, Equation 1 obviously simplifies to (a,)
in i x
=
[(am")N aC lXLy'
1 [(a," 1KC lX3"1
(2)
Thus a solution which is 5.09 m in 9aC1 and 2.31 m in KC1 is saturated with both salts a t 3OoC, and has a total ionic strength of 7.4; consequently S S a + is 0.69, Ycl- is 1.0, and X K +is 0.31. At 30°C and a total ionic strength of 7.4, a solution containing only NaCl or only KC1 is supersaturated. The activities of electrolytes and the vapor pressures of water over these two pure supersaturated solutions are readily estimated by the methods previously discussed (Kusik and AIeissner, 1973; Meissner and Tester, 1972; Meissner et al., 1972). For example, starting with literature values for y o of 0.67 and 0.57, respectively, for a solution of pure NaC1 and pure KC1 a t 25°C a t a n ionic strength of 2, then by estrapolation to a n ionic strength of 7.4, ( u ~ " ) N ~ c and I ( a z c " ) ~ care, 1 respectively, 0.71 and 0.78. Substituting into Equation 2! (aw)rnix is predicted to be 0.73 vs. a n experimental value of 0.722 (Adams and Mert'z, 1929). Water activities over several other saturated aqueous solutions of tvio 1: 1 electrolytes having a common ion are tabulated in Table I. Inspection shows fair success again to be attained. A more interesting application of Equation 1 lies in calculating water vapor pressures over multi-ion solutions. Thus, a synthetic brine with a measured vapor pressure of 18.35 mm H g at 26OC has the following ion molalities: 2.36 ;\Ig2+, 1.13 Sod2-, 0.83 K+,5.75 C1-, and 2.46 ?;a+. When we designate these ions as 1, 2, 3, 4, and 5, respectively, Equation 1 becomes : (am)rn ix = (a,")IZX,Y? (a,")14xLy4 (a,")32x3yz (LzwO)34X3Y4 x (a,")s 2 ~ ~ Y ~ ( U z54XJd c0)
(3)
T o solve, i t is necessary first to determine ( a , " ) l ~ , ( a m o ) ~ 4 , and so forth, each a t the total ionic strength of the misture, 1.13 X 22 0.83 X l 2 5.75 X namely 0.5 (2.36 X 22 l 2 2.46 X 12), or 11.47. I n sequence for the sis terms as written on the right-hand side of Equation 3, the values of a , ' (calculated by the methods outlined previously) are 0.95 for MgSO1, 0.62 for 1IgCl2, 0.93 for K2S04,0.65 for KC1, 0.89 1'2, for NazS04, and 0.51 for NaC1. Similarly, values of SI, X3, Y4, and X , are, respectively, 0.742, 0.44, 0.065, 0.56, and
+
+
+
+
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 2, 1973
205
0.193. Substituting into Equation 3, (u&ix is 0.73, making the water vapor pressure equal t’o 18.4 mm Hg, very close to the experimental value of 18.35 as reported above.
Table 1. Activities of Water at 3 0 ° C in Solutions Saturated with T w o Salts Having a Common ton Water activity in mixed solution ExperiCalcumental lated (Adams from and Equation Mertz,
Solubility
The ionic activity product equation, which relates the solubilities of electrolyte 12 in pure and in mixed saturated solution, is usually written as follows: (m1”)Y’( r n * ” ) Y 2 (y120)”’~=
Jrnl)”’(rnd”* (Y12)”’?
The symbols used here are again defined in the nomenclature table. For low concentrations with very insoluble electrolytes, ylZoand y12are both unity, whereupon Equation 4 reduces to the familiar solubility product equation. For systems of higher concentration, these electrolyte activity coefficients can be predicted with reasonable success by the method of Meissner arid Tester (1972) and Meissner and Kusik (1972). However, Equation 4 applies only to electrolytes like NaCl and KC1 which as solids contain no waters of crystallization. This equation must be modified for application to saturated solutions in which the solid phases are hydrated, as is normally the case with electrolytes whose ion charges exceed uriity. For convenience, the moles of water per mole of electrolyte in the crystal are here called no when in contact with pure saturated solution, and n when in contact with mixed saturated solution. For such hydrated solids, when no and n are identical : ( ~ l l O ) ” ’ ( ? ~ * O ) ” ” ~ l * O )= ”~~~azoo)l~n~
(5)
(~l)Y1(~~2)Y2(~12)Y1?(~za)mixno
Occasionally, no and n are unequal. Thus, for pure sodium carbonate, the solid phase iii contact with pure saturated solution a t 25°C is Na2C03.1OH20, but in contact with a mixed saturated solution containing 11% KaC1 is NazCOa.7H,O (International Critical Tables, 1928 IV, p 301), making no and n equal to 10 and 7 , respectively. For such cases, the solubility relationship becomes:
where (see nomenclature table for definitions) : A F , = Pno -
F, - (no- n)F,
(7)
Values of the free energies in Equation 7 are presented in the standard references or can be calculated from water activities for saturated solutions in which both crystal species are present. The derivation of Equabions 4, 5 , and 6 is presented in later paragraphs. It is worth noting that when no and n are equal, then Equation 6 reduces to Equation 5 . I n mixed solutions containing only small amounts of other electrolytes, no and n are usually identical; consequently Equation 5 applies. When concentrations of both pure and mixed solutions are low because of low solubilities, then (a,”)12 and (a&ix are both substantially unity, and both Equations 5 and 6 reduce to Equation 4. Extension of the thoughts above indicate that Equation 6 applies also to situations where the left-hand side as well as the right-hand side represent mixed solutions. FeClz Solubility
Consider the problem of determining how much HC1 can be dissolved a t 25°C in a 1.4 rn solution of ferrous chloride (here called electrolyte 12) before solid FeClz.4H20starts to precipitate. From the literature, a pure saturated solution of this tetrahydrate a t 25°C has a molality of 5.07; hence p is 206
Ind. Eng. Chem. Process Des. Develop., Val. 12, No. 2, 1973
1
1929)
0.73 0.64 0.77 0.82
0.722 0,676 0.735 0.786
Electrolyte molalities
(4)
NaCl NaCl KCl KC1
5.09 4.11 2.05 4.74
KCl NaXOa h”4C1 Kxo3
2.31 8.55 5.74 2.70
Table I1 Molalities NQS04 NaCl
System
Solid phases
I I1
Na2S04* 1 OHzO Anhydrous NarSO4 and NaCl
1,97 0.69
0 5.51
15.21. Here 7 1 2 and (a,”) are, respectively, 3.8 and 0.55, found by extrapolation (Kusik and lleissner, 1973; Meissner and Tester, 1972), when we know that is 0.9 for a pure (unsaturated) FeCL solution at 25°C when p is 6 (Harned and Owen, 1958). Substituting into the left side of Equation 5, we get: (rnl”)”’(rnZ”)
”?
(y12”)”’2(a,”) 12no
=
(5.07)(2 X 5.07)*(3.8)3(0.55)4 The foregoing product, which is numerically 2617, must equal the right side of Equation 5, or 2617 = (1.4)(2 X 1.4
+ m 3 ~ ) ~ ( ~ ~ ~ ) ~ (8) ( u ~ ) ~ ~ i ~
Here r n 3 2 represents the moles of HC1 which must be present to form the desired saturated solutions of ferrous chloride. I n m 2 ) . By these solutions, by definition, pr equals (1.4 X 3 trial, various values of mas are assumed and corresponding values of 7 1 2 are obtained by the method of Neissner and Kusik (1972). Values of are calculated by Equation 1. Equation 8 is in balance when ma2is 6.6 a t which point 9 is, of course, 11.8, YF.&12 is 8.6, and (u&ix is 0.43. Agreement with the experimental value for rna2of 7 m (Seidell and Linke, 1965) is thus within 6%.
+
Errors
Errors may be encountered in applying Equations 5 and 6, not only because of inaccuracies in estimating values of activity coefficients and vapor pressures, but because these terms are raised to powers, magnifying such inaccuracies. For example in Table 11, systems at 25”C, solution I is pure while solution I1 is saturated with both sodium sulfate and sodium chloride (International Critical Tables, Vol. 4, p 288, 1926). There is some disagreement in the values for Na2S04molality in saturated solution 11, inasmuch as they are alternatively reported in the International Critical Tables as 0.82 on page 287 of Vol. 4, and as 0.595 on page 982 of Seidell and Linke. The moIality, mrI, of sodium sulfate in solution II will nom be calculated by trial from Equation 6 and compared with the experimental value in Table 11. The free energy of hydration of anhydrous sodium sulfate is reported (Rossini, 1952) as follows: Sa*SOd(c)
+- 10H20(1) = S a z S O ~ . l O I I ~ O ( c ) ,
A F , = -1250 ea1 a t 25°C
Values of no and n for the decahydrate and the anhydrous salt in this equation are, respectively, 10 and 0. From Robinson and Stokes (1959), ~ s ~ ~ for s ( pure - , ~ solution I is 0.155, while (amoj is 0.96 as estimated by the method of Kusik and Neissner (1973), making the left side of Equation 6 equal to (1.97 x 2j2(1.97j(0.55j3(0.96j100r0.076. Similarly, when we know Y N ~ ~ SforO solution ~ I, its value in a mixed solution which is 5.5 m in NaCl, as in solution 11, can be determined by the methods of Aleissner and Kusik (1972). Thus, when mII is assumed to be 0.46, then pT is 6.89 while y x a r ~ ois4 0.32. By trial, Equation 6 is in balance as follows when rn11 is 0.46:
0.076
=
(2
VLII
Derivation of Equations 4 and 5
-t5.51)2(nz~~)(0.32)3 esp
(-1250/1.99
X 298)
There appears t,o be a significant difference between this calculated value for m11 and the average of the experimental values reported earlier. Caution should obviously be used in applying Equations 5 and 6, especially when either n or no is large. Derivation of Equation 1
For a system of two 1: 1 electrolytes 12 and 32, having ion 2 as the common ion, the Gibbs-Duhem equation is:
-55.5 d
111
+
d 111 ml?yl2
(uW),,,iX=
1~112~12
m32V32
ci In mazy22 (9)
Separating and rearranging the right-hand side of Equation 9 give us : - 5 5 . 5 d In (aW),,]ix=
+
-/- ~ 3 2 d m 3 2 m 1 2 ~ 1 2d In 7 1 2
ulldm12
+
d
m32~32
S
+2
d log
p12
(10)
rI2+
2 s 1132 d log
I(32
(11)
B u t from Meissner and Kusik (1972), for a three-ion system: d log
r12=
0.5 (.YI d log
ri2'
+ Y2 d log riro + sad log
d log
r3?=
0.5 (SI d log
riz'
+ Yp d log
r32'
S 3
(12)
+
d log rho) (13)
From Kusik and lleissner (1973), the activity for water over the pure electrolyte solutions are, respectively,
-55.5 log ( a m l x O )=l ~E 2122
S
+2
PT
I n these derivations, the standard state for water is the pure liquid arid for the electrolyte is in its familiar pure hypothetical solution in which both the mean ionic molality arid activity are unity. For each equation, the derivation is based on the reversible isothermal transfer of electrolyte (and any associated maters of crystallization) by two alternative paths from the standard states just listed to the same solid crystalline form. Providing the crystals produced are identical, the free energy changes over these two paths must be equal: Path I. Starting with the standard states just nientioiied, transfer 1 mol of electrolyte plus no moles of water into the saturated pure solution, simultaneously allowing the hydrated crystals to precipitate. Obviously no is zero for a n anhydrous crystal. Since the free energy change of forming crystals from saturated solutions is always zero, the molal free energy change over this path is obviously RT[ln (mlo)v'(m20)Y2 (y120)v12 noIn ( a w o ) l 2 ] . Path 11. Again starting from the standard states mentioned above, transfer 1 mol of electrolyte plus n moles of water into the saturated mixed solution, simultaneously allowing the crystals to precipitate. Since this free energy change of crystal formation from saturated solution is again zero, the molal free energy change over this path is RT [hi ( m l ) y l ( ~ ~ ~ ~ ) yn! ( y ~ ~ ) y ' ~ In ( a v l m i x l . Equation 4 is derived by equating these free energy changes for paths I and I1 above, recognizing that no and n are both zero since the crystals precipitating are anhydrous. By neglecting second-order terms, the derivation of Equation 5 proceeds along similar lines for the case when the solid phases are hydrated, with noand n being equal. When no and n are not identical, as in Equation 6, then the crystals precipitated from the pure solution of path I above are not identical with those from the mixed solution of path 11. Before the free energy changes of these two paths can be equated, allowance must be made for rehydrating the crystals of path I1 from n to no.The free energy change of this crystal hydration step is of course given by Equation 7 . It is obvious t h a t Equation 6 results when the molal free energy changes of the tlvo completed paths are equated.
+
111 7 3 2
Hold the ratio rnl~,'ms?coilstant and Equation 10 is integrated from zero to the final electrolyte concentration. Katural logarithms are converted to base 10 logarithms, the quantities of the form m l ~ v 1 2 z l z eare replaced by 2 ~ 1 2 and , d log yI2is rewhereupon: placed by z1z2 d log rI2,
-55.5 log (awjmix= !K!LF 2 , 3 2122
Substituting Equations 12 and 13 into Equation 11, eliminating the ititegrals by using Equations 14 and 15, and combining with Equation 16, yield Equation 2 for a three-ion system. It is obvious that this derivation can be extended to multi-ion systems of 1: 1 electrolytes, yielding Equation 1. This derivation is rigorous for systems in which all cations have the same charge, and all anions have the same charge, as in a solution of NH4C1 and K S 0 3 , or a solution of CaC12 and lIgC1,. When cations and/or anions have unequal charges, as in a solution of HC1 and FeC12, or S a C l and Na2S04, then Equation 1 is still a good first approuimation as indicated by the foregoing examples.
d log
(14)
+
Nomenclature = activity of water, (uw0jl2being for a pure solution of a n electrolyte identified by subscript, (uwjmixbeing for mixed solution AFc = free energy change of crystal hydration, as in Equation 7 J' = molal free energy of formation from the elements, F,,, referring to crystals carrying no moles water of hydration per mole electrolyte, F,, referring to crystals carrying n moles water of hydration per mole electrolyte, and F ,
(a,)
By definition for this system of 1: 1 electrolytes: PI2 = X l P T ; G I 2
-tw 3 2 )
= Y Z P T ; P32 = X 3 P T
(16)
For this three-iori system, of course, Y pis unity, and S1I'r
+
S317L =
1.0
(17)
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 2 , 1973
207
being the free energy of liquid water, namely -56.69 Kcal/g-mol a t 25OC m = molality, in g-mo1/1000 g solvent; mo referring to pure solutions and m to mixed solutions. The ions or electrolyte referred to are indicated by subscript n = moles of water of crystallization per mole of anhydrous electrolyte in a solid, noand n being used to distinguish between solids in equilibrium with saturated pure and mixed solutioiis, respectively X = cationic fraction, as inpl/pc or p 3 / p c Y = anionic fraction, as in p2/pa or p 4 / p a z = number of unit charges on the ion, indicated by the subscript. rllhiis z is unity for Ka+, also for NOs-, but is 3 for A13+,also for IQi3= moles of ions formed upon complete dissociation of 1 Y mol of electrolyte. Thus, v1 is for the indicated cation, v 2 is for the indicated anion, while u12 is the total number of ions namely (u1 v2j. T o illustrate, v12 is 2 for NaCl and 5 for A12(S04), y = mean ionic activity coefficient for the electrolyte indicated by the subscript, ylz0 being for the pure solution and ylz for the mixed solution I’ = reduced activity coefficient, namely y121/z1z2for the electrolyte indicated by the subscript. The term, I’1zo, refers to pure solutions, 712 to mixed solutions 1.1 = ionic strengths indicat,ed by the subscript. The ionic
+
. . . , of the individual ions are 0.5 mlz12, 0.5 pT equals 0.5 (mlz12 m2.Q mgsZ . . .). For a pure system, pT equals 0.5 m12Y12z1zz. The cationic strength pC of a solution is (1.11 1.13 . . . ), the anionic strength 1.1, is ( p ~ p4 . . .) strengths 1.11, ~
1 ,
mz2, and so forth. The total ionic strength
+ + +
+
+
+ +
literature Cited
Adams, J. R., Mertz, A., Ind. Eng. Chem., 21, 305 (1929). Harned, H. S., Owen, B. B., “Physical Chemistry of Electrolyte Solutions,” 3rd ed., Iteinhold, New York, N. Y., 1958. International Critical Tables, McGraw-Hill, New York, N. Y., 1926.
Kusik, C. L., Meissner, H. P., “Vapor Pressyes of Water Over Aqueous Solutions of Strong Electrolyes, Ind. Eng. Chem. Process Des. Develop., 12, 112 (1973).
Lleissner, H. P., Kusik, C. L., “Activity Coefficients of Strong Electrolytes in Multicomponent Aqueous Solutions,” A.Z.Ch.E. J., 18, 2294 (1972). Meissner, H. P., Kusik, C. L., Tester, J. W., “Activity Coefficients of Strong Electrolytes in Aqueous Solutions-Eff ect of Temperature,” ibid., p 3661. Meissner, H. P., Tester, J. W., “Activity Coefficients of Strong Electrolytes in Aqueous Solutions,” Ind. Eng. Chem. Process Des. Develop., 11, 1128 (1972).
Robinson, It. A,, Stokes, R. M., “Electrolyte Solutions,” 2nd ed., Academic Press, New York, N. Y., 19.59. Rossini, F. I)., “Selected Values of Thermodynamic Properties,” C ~ T C500, . Natl. Bur. Stand., 1952. Seidell, S., Linke, W. F., “Solubilities,” Vol. 11, 4th ed., Amer. Chem. SOC.,Washington, D. C., 1965. RECEIVED for review September 11, 1972 ACCEPTEDDecember 11, 1972
Measurement of Fractionation in Analytical Distillation of Crude Oil R. J. O’Donnell Chevron Research Co., Richmond, Calif. 3@02
A procedure is outlined to characterize the fractionating ability of a column used for distilling wide-boiling range multicomponent mixtures such as crude oils. It can be used to predict boiling point curves of fractions distilled b y the column, The experimental procedure consists of obtaining gas-liquid partition chromatograms on a series of small (i.e., instantaneous) cuts from the distillation. These are converted to simulated true boiling distillation curves b y a computerized integration program. The slopes of the boiling curves thus determined provide a direct measure of the column’s fractionating ability. The distillation curve of any finite cut can be calculated by integrating the instantaneouscurves over the interval desired.
T h e fractioiiatiiig power of a laboratory batch distillation column commonly is expressed as a certain number of theoretical trays. This is a valid and useful concept for the separation of mixtures containing relatively few components, but i t has little meaning when applied to distillation of multicomponent wide-boiling range material such as crude oil. A more useful approach, particularly for the distillation of crude oil, is the determiriatioil of t,he true boiling point (TBPj range of instantaneous cuts. The value of this method is twofold. First, it can exlxess tiumerically the fractionation of a column while distilling crude oil. Second, it can be used to predict quite precisely the TI3P distillation of any finite fraction. 208
Ind. Eng. Chern. Process Des. Develop., Vol. 12, No. 2, 1973
Analytical distillations of crude oils often are referred t o as true boiling point (TBP) distillations. I n this paper, we will refer to them as analytical TBP distillations to distinguish them from the simulated TBP distillation derived from a chromatogram of a distillate fraction, either instantaneous or broad, obtained from the analytical TI31’ distillation of a crude oil. It is the degree of fractionation realized in the analytical TBP distillation which is to be measured. Boiling ranges of distillates, both laboratory-produced fractions and commercial products, sometimes are expressed as results of ASTM distillations of the fractions. Such distillations use little fractionation; and, in the case of the atmo-
