1305
PREDICTION OF OSMOTIC AND ACTIVITYCOEFFICIENTS are not fast compared with H, CH,, etc.I6 Furthermore, the hot substitutions of recoil T for heavier groups (such as CHB, F, C1, etc.) also proceed in good and the escape of the replaced heavy atom must take place on a time scale of almost sec, long enough for several vibrations of each C-H bond in the reacting molecule. We postulate that the failure to observe an inversion mechanism is more involved with the difficulties in getting several groups into simultaneous rapid motion than it is in the great rapidity of the T / H time scale, In this view, the reaction is fast, ( ~ 1 O - lsec), ~ but some heavy groups do move appreciably while the substitution is occurring-the dynam-
ical requirements for simultaneous rapid acceleration of several heavy substituents are severe, apparently severe enough to reduce the probability of such a reaction to the negligible level. The substitution of T-forH with retention certainly also requires the acceleration of one or more additional groups to account for the high vibrational energies but is enough less restrictive to permit substitution to occur. This conceptual approach suggests that the substitution of tritium for CH8, F, or other heavy group will probably also be found to proceed with retention of c o n f i g ~ r a t i o n , ~ ~ even though this time scale cannot approach the 10-14sec time scale conceivable for T-for-H substitution.
The Prediction of Osmotic and Activity Coefficients in Mixed-Electrolyte Solutions by P. J. Reilly,* R. H. Wood, Department of Chemistry, University of Delaware, Newark, Delaware 19711
and R. A. Robinson Department of Chemistry, State University of N e w York, Binghamton, N e w York 18901
(Received August $1, 1970)
Publication costs assisted by Ofice of Saline Water, U.S. Department of the Interior
The equations of Reilly and Wood for the prediction of free energy have been used to derive expressions for the osmotic and activity coefficients of many-component charge-asymmetric mixtures of electrolytes. Two levels of approximation are available with the equations. I n the first level of approximation only data on single-salt solutions are used. I n the full equations data on common ion mixtures of electrolytes are used. The first level of approximation is used to predict activity coefficients in mixtures of hydrochloric acid with alkaline earth perchlorates. -4comparison with the experimental results of Stokes and Stokes and of Weeks show t h a t the predictions are superior to those of the ionic strength principle. T h e full equations are applied to mixtures of three cations with a common anion. The results are compared with experimental measurements on the following systems: LiC1-NaC1-KC1, LiC1-NaC1-CsC1, and LiC1-NaC1-BaClz reported in this paper; NaC1-KC1-BaC12 reported by Robinson and Bower; and HC1-CsC1-BaC12 reported by Lietzke, Hupf, and Stoughton. I n all cases the predictions agree with the experimental measurements t o within the experimental error.
Introduction I n a recent paper’ the authors presented an equation for the prediction of the excess free energy of any mixedelectrolyte solution. The equation used Friedman’s2 approach to mixed electrolytes and predicts the free energy, heat content, volume, etc., of any mixture of electrolytes from measurements on pure solutions and common ion mixtures. The equation correctly accounts for all pairwise interactions. The purpose of this paper is to derive the corresponding equations for the osmotic and activity coefi-
cients and to show how these equations can be used to predict the properties of mixtures with more accuracy than has previously been possible. As an example of their use the equations are used to predict the trace activity coefficient of any electrolyte in a solution of another electrolyte. If the trace elec-
* Correspondence to be addressed t o University of Lethbridge. Lethbridge, Alberta, Canada. (1) P. J. Reilly and R. H. Wood, J. Phys. Chem., 73, 4292 (1969). (2) H. L. Friedman, “Ionic Solution Theory,” Interscience, New York, N. Y . , 1962. The Journal of Physical Chemistry, Vol. 76,N o , 9,1971
P. J. REILLY, R. H. WOOD,AND R. A. ROBINSON
1306 trolyte and the supporting electrolyte have a common ion, the situation is the same as that investigated by Harned3 and the equation is of the same form as the Harned equation. Stokes and Stokes4 and also Weekss have obtained experimental data on the more general case with no common ion. Even using only the terms involving single-salt solutions the equation predicts the experimental results much more accurately than the principle of ionic strength. As a further test the full equations (including terms involving tu-0-salt mixtures) were used to predict the properties of mixtures containing three cations and a common anion. The difference between the predictions and the measured values was less than the experimental error in the measurements.
Substitution into eq A-9 gives In
yrtnrX
= In y%x
+ (1 - y) X
= In y0sx
+
and In
yrtNX
Results and Discussion The derivations of the equations for the osmotic and activity coefficients of many-component electrolyte solutions are given in the Appendix. Given the equations of Reilly and Wood, the derivation is straightforward, although tedious. Equation A-6 givestheosmotic coefficient and A-9 the activity coefficient for a general mixture in terms of the osmotic and activity coefficients of pure electrolyte solutions at the same ionic strength (4O and yo) and interactions between pairs of electrolytes (gaINX). The symbols in these equations are defined in eq A-la to A-le. Ionic strength (I) and equivalents per kilogram of solvent ( E ) are the basic concentration scales. Equations A-6 and A-9 can be used for any mixture of electrolytes. In this paper the equations will be applied to two types of mixtures: (1) mixtures of two salts without a common ion (LIX with NY); (2) mixtures of three salts with a common anion (LX, NX, and NX). Specific equations will be derived for these cases but first it is useful to derive the equations for the simplest possible case : two salts with a common ion (IIX and NX). If the mixture is formed at constant ionic strength by adding y kg of solvent containing the salt ilTX to (1 - y) kg of solvent containing the salt N X then, the final solution contains a mixture of the three ions in 1 kg of solvent. The concentrations of the three ions are
If these values are used to calculate the quantities required for eq A-6, then eq 1 results.
The Journal of Physical Chemistry, Vol. 76, N o . 9, 2971
Equations 2a and 3a can be used to describe the trace activity coefficient of one salt in a solution of the other. If the value of y becomes zero, eq 2a gives (In
YrtblIX)traoe
= In yonrx
+
while a value of 1 for y converts eq 3a to (In
y*sX)traoe
+ + 1-
= In yoxx
[-g(l-
4OMX)
4OSX
1
- '/2Z"ZX~g,,sX (3b)
These equations are of the same form as the Harned relations. It is interesting that in this case the trace activity coefficient depends on the activity coefficient of the pure electrolyte in a solution of the same ionic strength. This result is due to the common ion and does not apply to a trace activity if there is no common ion. This is because when both ions are present in trace amounts (no common ion), they will never interact with each other because of their very small concentrations. Equations for ihe M X , N Y System. The trace activity coefficient of the salt N X in a solution of the salt NY is obtained by applying eq A-9 to a mixture of the two salts and then reducing the concentration of 31X to zero. The result is (3) See for examule H. 9. Harned and B. B. Owen, "The Physical Chemistry of Electrolytic Solutions," 3rd ed, Reinhold, Iiew York, N. Y . , 1958. (4) J. M. Stokes and R. H. Stokes, J. Phys. Chem., 67, 2442 (1963). (5) I. A. Weeks, Aust. J. Chem., 20, 2367 (1967).
PREDICTION OF OSMOTIC AND ACTIVITY COEFFICIENTS
1307 constant molal ionic strength for common ion mixtures specified by eq 4. Tables I-VI11 give the approximate values of the Table I: Trace Activity Coefficient of Hydrochloric Acid in Magnesium Perchlorate Solutions
-
(ZN
ZY)
ZNZY
I
2 -(@"
-
(1 -
~ O N Y
ZY)gmY
+ In
+
(ZN
Ionic
-
Y'XY)
- ZX)gxyN}
(4)
While equations similar t o eq 2 and 3 have been applied to common ion mixtures, very little work has been done on systems of the type described by eq 4. The literature contains some data on metal perchloratehydrochloric acid ~ystems4,~ and it is of interest to see how the predictions of eq 4 compare with the experimental data. The data provide a severe test of the equation since the mixture is not symmetric-the cations involved being magnesium, calcium, strontium, and barium. The trace activity coefficient of hydrochloric acid in an alkaline earth perchlorate solution is given by (In
= (1 -
YfHC1)traoe
-
$'N(ClOa)z '/2(1
I
+ In
+ In + In
- 4'NClz
3/4(1 3/4(1
$J'HCIOa
yoN(CIOd)z)
+ +
4/3(1
=
(1 -
- 4'HClOn
4/d1 2(1 -
~OHCI
+ In
~ ~ O N C I ~
+ 1n + In
y0Ncl2)
(4a)
YOHClOa)
Y'HCI)
+ j ( 3 g ~ , r +~ '
+
-
-
I
~OHCI)
Calcd 2'
0.01 0.04 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 1.5 2.0 2.5 3.0
0.904 0.835 0.782 0.742 0.721 0.706 0.693 0.689 0.684 0.681 0.677 0.677 0.689 0.706 0.724
0.910 0.841 0.790 0.754 0.732 0.721 0.715 0.712 0.713 0.712 0.718 0.747 0.789 0.850 0,926
0.905 0.842 0.796 0.767 0.756 0.755 0.757 0.763 0.772 0.783 0.809 0.897 1.009 1.147 1.316
a The trace activity coefficient as reported by week^.^ Calculated from eq 4a using only the osmotic and activity coefficients of the single-salt solutions, i e . , the first 4 terms of eq 4a. Activity coefficient of the pure electrolyte at the same ionic strength. (Except where noted the properties of pure electrolyte solutions, required for these computations, were taken from R. A. Robinson and R. H. Stokes, "Electrolyte Solutions," 2nd ed (revised), Butterworths, London, 1970.)
Table I1 : Trace Activity Coefficients of Hydrochloric Acid in Calcium Perchlorate Solution4
while the trace activity coefficient of the alkaline earth perchlorate in hydrochloric acid is given by (In y*N(C'oa)z)traoe
7
Calcd I b
-
- { 2 ~ ~ , ~ ~ 1 03 g4C I , C 1 0 1 ~ }
4
Activity caeff----
Exptla
-
Y'NClz)
@N(ClOa)a)
+
Y'HClOr)
T-
strength
2gci,cioaH}
(4b)
Weeks has measured the activity coefficient of HC1 in HC104 so that gC1,CIOaH can be calculated. Unfortunately the properties of the other common ion mixtures are not known. It is still of interest to see how accurate the equations are when the term involving measurements on mixtures (the last term) is dropped from the equation and only the properties of singlesalt solutions are used to predict the activity coefficients of the mixtures. This is equivalent to assuming the excess free energy of mixing is zero (Young's rule)6 at
Ionic strength
0.1 0.3 0.6 1.0 1.4 2.0 3.0 3.3
7
Activity coeff-------
Exptla
Calcdb
CalodC
0.782 0.726 0.704 0.712 0.734 0.788 0.913 0.956
0.789 0.731 0.712 0.717 0.737 0.786 0.911 0.960
0.796 0.756 0.763 0.809 0.876 1.009 1.316 1.434
a The trace activity coefficients as reported by Stokes and Stokes.4 * Calculated from eq 4a using only the osmotic and activity coefficients of the single salt solutions, i.e., the first 4 terms of eq 4a. Activity coefficient of the pure electrolyte a t the same ionic strength.
trace activity coefficients from eq 4a and 4b (without the last term) together with the actual trace activity coefficients. It is possible t,o use the equations in this paper to give quite an accurate first approximation for the activity coefficient of a salt in a mixture even if the (6) T.F. Young and M. B. Smith, J . Phys. Chem., 58, 716 (1954). The Journal of Physical Chemistry, Vol. 76,N o . 9,1971
P. J. REILLY,R. H. WOOD,AND R. A. ROBINSON Table I11 : Trace Activity Coefficient of Hydrochloric Acid in Strontium Perchlorate Solution6 Ionic strength
,-----Activity Exptla
coeff-----Calcdb
0.01 0.04 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 1.5 2.0 2.5 3.0 3.5
0.904 0.835 0.782 0.747 0.728 0.718 0.712 0.710 0.709 0.710 0.714 0 748 0.792 0.845 0.912 0.989
0.908 0.842 0.794 0.759 0,742 0.734 0,729 0.727 0.729 0.729 0.737 0.766 0.809 0.864 0.912 1.042
I
CalcdC
Ionic strength
----Activity Exptla
coef-Calcdb
CalcdC
0.905 0.842 0.796 0.767 0.756 0.755 0.757 0.763 0.772 0.783 0.809 0.897 1.009 1.147 1.316 1.518
0.01 0.04 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.0 2.5 3.0
0.814 0.700 0.620 0.567 0,544 0.533 0.529 0.530 0.534 0.540 0.549 0.560 0,634 0,737 0 867 1.020
0.815 0.709 0.632 0.582 0.558 0,553 0.550 0.548 0.551 0.553 0.567 0,577 0.647 0,752 0.888 1.076
0.819 0.714 0.640 0.595 0.577 0.573 0.569 0.565 0.566 0.567 0.576 0.583 0,633 0.706 0.802 0.925
5 The trace activity coefficient as reported by Weeks.6 b Calculated from eq 4a using only the osmotic and activity coefficients of the single-salt solutions, is.,the first 4 terms of eq 4a. c Activity coefficient of the pure electrolyte at the same ionic strength,
Table IV : Trace Activity Coefficient of Hydrochloric Acid in Barium Perchlorate Solution6 Ionic strength
0.01 0.04 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 1.5 2.0 2.5 3.0 3.5 4.0
----Activity Exptla
0.904 0.835 0.782 0.747 0.723 0.718 0.712 0.708 0.705 0.705 0,709 0.735 0.778 0.828 0,880 0.941 1.000
ooeff-----Calcdb
0.909 0.840 0.793 0.754 0.736 0.725 0.720 0.716 0.712 0.716 0.721 0.744 0.784 0.848 0.893 0.965 1.047
0,906 0.842 0,796 0.767 0.756 0.755 0.757 0.763 0.772 0.783 0.809 0.897 1.009 1.147 1.316 1.518 1.762
only information available is in the standard tables of osmotic and activity coefficients of single-electrolyte solutions. The equations also show that for more accurate predictions only the common ion mixtures of electrolytes need be measured, There are a number of other approximations available for estimating the activity coefficients in a mixed-elec-
I
a The trace activity coefficient as reported by Weeksn6 b Calculated from eq 4b using only the osmotic and activity coefficients of the single-salt solutions, L e . , the first 4 terms of eq 4b. c Activity coefficient of the pure electrolyte at the same ionic strength (ionic strength principle).
Table VI: Trace Activity Coefficient of Calcium Perchlorate in Hydrochloric Acid Solution4
Calcdc
a The trace activity coefficient as reported by Weeks.6 b Calculated from eq 4a using only the osmotic and activity coefficients of the single-salt solutions, ie., the first 4 terms of eq 4a. c Activity coefficient of the pure electrolyte at the same ionic strength.
The Journal of Physical Chemistry, Vol. 76, N o . 9 , 1971
Table V : Trace Activity Coefficient of hhgnesium Perchlorate in Hydrochloric Acid Solution6
Ionic strength
-----Activity Exptla
coeff-Calcdb
CalcdC
0.01 0.04 0.1 0.25 0.5 0.7 1.0 2.0 3.0 4.0
0.817 0.705 0.627 0.564 0,543 0.547 0.572 0.740 1.040 1.553
0.814 0.708 0.629 0.560 0,542 0.539 0.561 0.716 1.006 1.497
0,819 0.711 0.634 0.566 0,540 0.530 0.535 0.606 0.743 0 944
7
I
The trace activity coefficient as reported by Weeks.6 Calculated from eq 4b using only the osmotic and activity coefficients of the single-salt solutions, Le., the first 4 terms of eq 4b. Activity coefficient of the pure electrolyte at the same ionic strength. Q
trolyte solution. The simplest approximation is the ionic strength principle’ which gives the trace activity coefficient the same value as the activity coefficient of the pure electrolyte at the same ionic strength. Tables I-VI11 give the values of the activity coefficient indicated by the ionic strength principle. Using only osmotic slnd activity coefficients from single-salt solutions eq 4a and 4b are consistently better than the ionic strength principle. (7) G. N. Lewis and M. Rnncfpll, “Thermodynamics and Free Energies of Chemical Substances, McGraw-Hill, New York, N. Y., 1923.
1309
PREDICTION OF OSMOTIC AND ACTIVITY COEFFICIENTS Table VII: Trace Activity Coefficient of Strontium Perchlorate in Hydrochloric Acid Solution5 Ionic strength
0.01 0.04 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.0 2.5 3.0
----
.--.
Exptla
Activity coeff Calcdb
Calcd‘
0.810 0.689 0.602 0.547 0.522 0.510 0.504 0,502 0.504 0,506 0.514 0.521 0.577 0.661 0,772 0.912
0.814 0.708 0.628 0.577 0.551 0.544 0.540 0.535 0.536 0.536 0 548 0.556 0.612 0.700 0.814 0.971
0.817 0.705 0.620 0.559 0.528 0.815 0.504 0 494 0.489 0.485 0.488 0.489 0.507 0.537 0.582 0.638
I
I
a The trace activity coefficient as reported by Weeks.5 * Calculated from eq 4b using only the osmotic and activity coefficients of the single-salt solutions, i.e., the first 4 terms of eq 4b. c Activity coefficient of the pure electrolyte a t the same ionic strength.
pairs formed by two ions of the same charge type but this equation is only valid for mixtures of univalent electrolytes and in addition it approximates the properties of pure electrolytes with a linear deviation function at high concentrations. Scatchardlo proposed equations for mixtures of electrolytes but the amounts of pure electrolytes to be taken to make the mixture are not specified. The prediction of the properties of the mixture will depend on the amounts of pure electrolytes chosen. A later equation of Scatchardl’ defines the components exactly. Before these equations can be tested, the same approximations will have to be made and the appropriate expressions for activity and osmotic coefficients derived. lla Equations for the L X , M X , N X System. This section will be concerned with the case of a mixture of three cations and one anion. The simplest case is a mixture of univalent ions. Equation A-6 applied to the osmotic coefficient of a mixture of LiC1, NaC1, and CsCl becomes
Table VIII: Trace Activity Coefficient of Barium Perchlorate in Hydrochloric Acid SolutionS Ionic strength
0.01 0.04 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5
r -
Exptln
0.809 0,690 0.555 0.535 0.507 0.491 0.482 0.477 0.475 0.475 0.477 0.480 0.519 0.884 0.674 0.793 0.946
Activity coef-Calcdb
0.813 0.707 0.627 0.572 0.547 0.538 0.533 0.526 0.526 0.523 0.534 0.539 0.587 0.665 0.767 0.903 1.072
CalcdC
0.817 0.705 0.619 0.556 0.524 0.508 0 494 0.481 0.473 0.465 0.464 0.462 0.462 0.472 0,480 0.513 0.539 I
a The trace activity coefficient as reported by Weeks.8 b Calculated from eq 4b using only the osmotic and activity coefficients of the single-salt solutions, ;.e., the first 4 terms of eq 4b. c Activity coefficientof the pure electrolyte at the same ionic strength.
Another approximation is that of Guggenheim.8 Since this equation treats only cation-anion pairs and ignores cation-cation and anion-anion pairs, it is not capable of precision at high ionic strengths. The restrictions imposed by this equation on the ion-size parameter are also unnecessary and are absent from the equations presented in this paper. More recently GuggenheimO has proposed an equation which includes
This equation shows how the osmotic coefficient of a mixture containing three cations and one anion can be predicted from the osmotic coefficients of the pure electrolytes and measurements on two-salt mixtures. In the case of a mixture of equal amounts of all three cations eq 5 can be reformulated in terms of the osmotic coefficients of the pure electrolytes ( ~ O L ~ C I , etc.) and the osmotic coefficients of 50:50 mixtures of two salts (4LiC1-NaC1, etc.). The result is 9~LiCI-NaCl-CsCl
4[4LiCl--NaCl
$‘LiCl-CsCl
+
t
4NaCI-CaCII
4OLiCl
-
-~ O X ~ CI +OC~CI
(6)
In order t o test the prediction for univalent cations, the isopiestic method already used for mixtures of two salts12 was easily extended to mixtures of three salts. (8) E. A. Guggenheim, “Thermodynamics,” 3rd ed, North-Holland, Amsterdam, 1957; Phil. Mag., 19, 588 (1935). (9) E. A. Guggenheim, Trans. Faraday SOC., 62, 3446 (1966). (10) G. Scatchard, J. Amer. Chem. Soc., 83, 2636 (1961). (11) G. Scatchard, ibid., 90, 3124 (1968). ( l l a ) NOTEADDEDIN PROOF.This has been done. See G. Scatchard, R. M. Rush, and J. s. Johnson, J . Phys. Chem., 74, 3786 (1970). (12) R. A. Robinson, ibid., 65, 662 (1961).
The Journal of Physical Chemistry, Vol. 75, N o . 9,1071
P. J. REILLY, R. H. WOOD,AND R. A. ROBINSON
1310
Table IX : Isopiestic Measurements on Binary and Ternary Salt Mixtures at 25' (A) The binary system: NaC1-CsCl. ~ N ~ = C rncscl I = m/2 mefa
(2,7093 2.7221 2.7671
(D) The ternary system: LiC1-NaC1-KC1. mLicl = mNscl = mxcl = m/3
m
Q
mefa
m
c
3.0000 3.0140 3.0620
0.9268) 0.9277 0,9308
2.9349 (3.0534 3.0903 3.1660
2.8950 3.0000 3.0342 3,0963
1,0553 1.0677) 1,0709 1,0802
A t m = 3, V'LiC1 = 1.2847,' ( D O N ~ G I = 1.0453,' q'xcl = 0.9367." ((O'Liol f 'P'N~CI f ' P O K C I ) / ~ = 1.0889. P L I C I - N ~ C I = 1.1683,d 'PLiCI-KCI = 1.0659, (oNaC1-KD1 0.9783."" Hence, qLiCI-N&]-KCI calculated by eq 6 = 1.0648. (B) The ternary system: LiC1-NaC1-CsC1. ~ L ~ = C ImNaCl = mcscl = m/3 mreP
m
2,9003 (2.9250 3.0242
2.9751 3.0000 3.1011
(E) The ternary system: LiC1-NaCl-CsCI. 0.3256 m, rnXvacl= 0.3244 m, mCsCl = 0.3500 m
rnL,Cl = Q
1,0126 1.0143) 1.0209
At m = 3, P'L~CI = 1.2847,b'P'N~CI = 1.0453,' ' P ~ C S C I = 0.8809.b ('P'LICI f 'P'XaC1 'P°CsC1)/3 = 1.0703. 'PLiCI-NaC1 = 1.1683,d ' P L ~ C I - C ~ C I= 0.9917,b PNaCI-CsCI = 0.9268. Hence, 'PL,CI-~,-~CI-C~CI calculated by eq 6 = 1.0151.
+
(C) The binary system: LiC1-KCl. ~ L , C=I mKcl = m/2 mrefa
m
2.9868 (3.0495 3.0816
2.9360 3,0000 3.0328
Q
1.0636 1.0659) 1,0677
c, mrefa
m
Q
calcd 1
c, calcd 2
4.5088 4.7102 1,1045 1.2117 1.1073 4.9910 5.2325 1.1339 1.2586 1.1397 V(ca1cd 1) = 0.3526 &icl 0.3244 ~ O N ~ C I 0.3500 qocsc~. pp(ca1cd2) by means of eq 5, using data for the binary systems LiCl-NaCl,d LiCl-CsCl,h and NaC1-CsCl.'
+
+
(F) The ternary system: LiC1-NaC1-BaClz, 1 ~ i c i= 0.33391, 1 ~ = 0.33281, ~ ~ I1 B ~=c0.33331 ~ ~ c,
Q,
mrera
I
v
calcd 1
calcd 2
1.8776 2.8743 2.9834
2.1371 3.1888 3.2973
1,0295 1.1215 1.1337
1.0383 1.1383 1.1494
1.0308 1.1217 1.1315
V(ca1cd 1) by eq 7 using only the osmotic coefficients of the single-salt solutions. q(ca1cd 2) by eq 7 using osmotic coefficients of the single-salt solutions and also the terms for the binary mixtures LiC1-NaCl,d LiC1-BaC12,bNaCl-BaCIQ.'
'
a In all cases, sodium chloride was the reference salt. Values in parentheses are interpolated. R. A. Robinson, unpublished. R. A. Robinson and R. H. Stokes, "Electrolyte Solutions," 2nd ed (revised), Butterworths, London, 1970. R. A. Robinson, R. H. Wood, and P. J. Reilly, J . Chem. Thermodyn., in press. e R. A. Robinson, J. Phys. Chsm. 65, 662 (1961). R. &I. Rush and R. A. Robinson, J . Tenn. Acad. Xci., 43, 22 (1968). R. 1vI. Rush, Oak Ridge National Laboratory Report 4402. See Table X.
'
All salts were recrystallized twice from water. Isopiestic vapor pressure measurements using sodium chloride as the reference salt gave values of the osmotic coefficient of the mixed-salt solutions. The results of these measurements are given in Table IX. For the ternary system LiC1-NaC1-CsC1 at m = 3.000, I$ = 1.0143 in good agreement with the prediction of eq 6, I$(calcd) = 1.0151 (Table IXB). Similar results are obtained for this system using eq 5 at m = 4.7102 and 5.2325 (Table IXE). Equation 6 also predicts the LiC1-NaC1-KC1 system accurately a t m = 3.000 (Table IXD). The choice of eq 5 or 6 for prediction of these mixtures is a matter of convenience. Robinson and BoweP have reported measurements of the osmotic coefficients of NaC1-KC1-BaCL mixtures using the isopiestic method. For this mixture eq A-6 becomes The Journal of Physical Chemistry, Val. 76, N o . 0 , 1071
m 3 ENaEBa 2 m
3 EE[gK,Bac' 2 m
+ I-((gK,Bacl)] b bl
(7)
I n order to test this equation it is necessary to have values of the bracketed terms in eq 7. These were derived from the measurements of Robinson and (13) R. A. Robinson and V. E. Bower, J . Res. Nat. Bur. Stand., 69A, 365 (1965).
PREDICTION OF OSMOTIC AND ACTIVITYCOEFFICIENTS Table XI : Isopiestic Equilibria in KC1-BaCl2 Solutions
Table X : Isopiestic Equilibrium in NaCl-BaC12 Solutions Concn of NaC1,
Concn of BaClx,
m
0.3591 0.2213 0.0794 0,4033 0.2797 0.1453 0.7382 0.4910 0.2610 0,9479 0.6389 0.3356 1.1653 0.7442 0,2721 1.4141 0,9784 0.5259 1.3930 I , 0035 0.4308 1.2466 0.7037 0.3073 1.5471 1.0496 0.5733 1.8885 1.3509 0.7850 0.3184 1.0177 0.7602, 0.4557 0.3008 1.9192 1.3574 0.8234 0.3348 1.7350 1.1799 0.5922 1.7263 1.0073 0.2911 2.0250 1.3969 0.6356
QNaBaOl
b
+
m
Ionic strength
Osmotic coeff
I-
0.1046 0.2037 0.3060 0.0736 0.1628 0.2597 0.1644 0.3378 0.4999 0.1963 0.4092 0.6193 0.2730 0.5586 0.8790 0.1421 0.4356 0,7414 0.3219 0.5821 0,9647 0.4906 0.8501 1.1124 0.3006 0,6303 0.9452 0.1878 0.5407 0.9136 1.2206 0.7670 0.9375 1,1377 1.2396 0.1845 0.5542 0.9060 1,2274 0.4108 0.7744 1.1591 0.5781 1.0448 1.5106 0.4451 0.8514 1.3458
0.6726 0,8324 0.9974 0.6241 0.7681 0.9244 1.2314 1,5044 1.7607 1.5368 1.8665 2.1935 1.9843 2.4200 2.9091 1.8404 2,2852 2.7501 2.3587 2.7498 3.3249 2.7181 3,2540 3.6445 2.4489 2.9405 3.4089 2.4519 2,9730 3.5258 3.9802 3.3187 3.5727 3.8688 4.0196 2.4727 3,0200 3.5414 4.0170 2.9674 3.5031 4.0695 3.4606 4.1417 4.8229 3.3603 3.9511 4.6730
0.9006 0.8818 0.8629 0.9078 0.8901 0.8719 0.9232 0.9112 0.8994 0.9366 0.9289 0.9202 0.9557 0.9813 0.9462 0.9599 0.9572 0.9536 0,9758 0.9754 0.9748 0.9850 0.9867 0.9882 0.9853 0.9868 0.9887 0.9941 0,9979 1.0010 1.0038 1.0010 1.0019 1.0038 1,0048 0.9956 0.9989 1,0018 1.0048 1.0088 1.0130 1.0176 1.0302 1.0378 1,0449 1.0349 1.0422 1.0499
+0.01894 +O. 02637 0,05489 0.03194 f0.02581 +0.03680 0.00966 4- 0.00551 -0.00118 f0.00178 0,00044 0.00270 - 0,00444 0,00557 0.00924 0.01046 0.00530 0.00620 - 0.00612 - 0.00725 -0.01014 0,00421 0.00409 -0.00113 - 0.00305 - 0.00467 - 0.00315 - 0.00970 - 0.00798 - 0,01013 - 0.01487 - 0,00844 -0,01140 -0.01414 -0.01662 -0.01046 - 0.01045 - 0.01350 -0.02127 -0. 01158 0,01360 - 0.02035 -0,01278 -0.01749 - 0.02542 -0.01212 0.01588 -0,02247
bI
1311
(QN*BSC')
+ + +
+
-
-
-
-
Bower14 on the NaCl-BaC12 and KC1-BaC12 systems> and the measurements of Robinson12 on NaC1-KC1. The results of the calculation for the NaCl-BaC12 and KC1-BaC12 systems are given in Tables X and XI. The values of
were calculated from eq 1. Note that this quantity is a function of the ionic strength but not a function of the
Conon of KCl,
Conon of BaClz,
m
0.6309 0 4178 0 1967 0.3099 0.2308 0.1411 0.5789 0.4185 0.2048 0.6644 0.5072 0.3186 0.7074 0.6163 0.5063 0,4067 0,8081 0.5648 0,3064 0.6330 0.4063 0 2358 1.3606 1,1475 1,0259 0.8270 1.9274 1.4671 0.8579 0,3443 1,3394 0.6887 0.2503 1.8961 1.1888 0,5955 1,6329 1.2541 0.3002 1.8855 0.5746 0,2539 2.5365 2.2111 1.7890 1.1054 2.3901 1,5011 0.4287 2.3016 1.3772 0.9462 I
I
I
QI
