P. J. REILLY AND R, €3. WOOD
3474 ion pairs, arid the resultant effectively large cation might not be able to form stable aggregates larger than pairs, a conclusion supported by the observations of Kubas and Shriver of the molecular weight of sodium tetraphenylborate in THF.Ie Although the d u e s reported here cannot be considered to be quantitative because of the difficulties of working with alinninum alkyls as well as the presence of some ion-ion, ion dipole, etc., interactions,20 these should be relatively small. For the latter of these, this is indicated by the iow ionic conductances observed in thest. systems.71 The results are in agreement with
Heats of Mixing Aqueous Electrolytes. 21,
Na+IICl-, Br-
previous studies of this system,16and it is thus proposed that the general trends are correct.
Acknowledgment. Support of this work by National Science Foundation Grant No. G P 11427 is gratefully acknowledged. (19) G. J. Kubas and D. F. Shriver, J . Amer. Chem. Soc., 92, 1949 (1970). (20) H. S. Harned and B. B. Owen, “The Physical Chemistry of Electrolytic Solutions,” 3rd ed, Reinhold, New York, N. Y., 1958, p 460. (21) 6.N. Hammonds and M . C. Day, J. Phys. Chem., 73, 1151 (1969).
IX. The Reciprocal
1
P,J. Reilly and R. H. Wood* Department of Chemistry, University of Delaware, Newark, Delaware
19711
(Received M a y 8, 197%)
The heats of mixing aqueous solutions of all combinations of the reciprocal salt pair Mg2+,Na+/lCI-, Br- have been measured a t 25”. I n one set of experiments the initial solutions had the same molal ionic strength ( I = 4, 3, or 6). I n another set of experiments the initial solutions had the same number of equivalents per kilogram of solvent ( E = 1, 3, or 6). For charge-asymmetric mixtures the magnitude of constant E mixings is less than that of the constant I mixings. Young’s cross-square rule holds quite accurately for both constant I and consbant E mixings even a t the highest concentrations. The results indicate that predictions of the properties of multicomponent charge-asymmetric mixtures based on constant E mixings may be very useful and equations for these predictions are derived.
Introduction Previous measurements on heats of mixing aqueous electrolytes2-6 have shown that Young’s cross-square r ~ l e .is~obeyed # ~ quite accurately for charge-symmetric mixtures (mixture8 of salts of the same charge type). The present measurements were undertaken in order to test this rule a t very high concentrations for a reciprocal salt pair containing salts of different charge type. Previous measurements on charge-asymmetric mixturese indicated that a concentration scale based on equivalents per kilogram of solvent ( E ) might be more useful than thr: mold ionic strength ( I ) . For this reason, measurerri~ents were made on the heats of mixing all of the pnskible combinations in the reciprocal salt pair 31gz+, 2 K a f //2C1-, 2Br-. The experiments were performed both at constant molal ionic strength and a t constant equivalents per kilogram of solvent. The reaction a t constant ionic strength is given by The Journal of Physical C%emistry, Vol. 76, No. 23, 1972
bAnwJ,O)
+ ( Y B ~ w P , ~ ( ~ w , Y A ~ , Y B(1)~ ) --j-
where a solution is characterized by amount of water, ionic strength of salt A, and ionic strength of salt B. The change in enthalpy for this process (AmH)is represented by the equation
+
AmH/nwMw = yAyB12(RThc? (yB where
yA
- YA)RThl’
+ . . . (2)
is the ionic strength fraction of salt A,
YB
is
(1) Presented in part at the 158th National Meeting of the American Chemical Society, New York, N. Y., Sept 1969. (2) T . F. Young, Y. C. Wu, and A. A. Krawcta, Discuss. Faraday SOC.,24,27, 77,80 (1957). (3) Y . C. Wu, M . B. Smith, and T. F. Young, J. Phys. Chem., 69, 1868, 1873 (1965). (4) R. H. Wood a,ndR . W. Smith, ibid., 69,2974 (1965). (5) R . H. Wood and H. L. Anderson, ibid., YO, 992 (1966). (6) R. H. Wood and M. Ghamkhar, ibid., 73,3959 (1969).
3475 the ionic strength fraction of salt €3, 1 is the molal ionic strength (1 =- L,'pZimlZtz), A,H is the change in e ~ ~ t h a(in~ ~c y~ ~ ~ arid } ~nwMw e ~ is ~ the , weight of ~ ~ ~ ~ a RThor n t i t y is the measure of heat of mixing a t y = 0.5, and the measure of the skew in the heat of mixing. Ne~tethat nomenclature has been changed Eram l~revious~ u ~ l In ~this~ series ~ ~so that ~ othe~ atjty uf solveiit ou tho left hand side of the equation vcii explicj tly a d the coefKcients ho and hr have a ~ ~ ~ ~ 1'~ th3e nndicate ~ s ~ that r ~the~ mixing ) t process is a t constant ionic 3trengtb. A mixing at constant equivalents per kilogram of sslvernt ( E ) 1s given by the equation ( Y ~ ~ ~ w-t- , (ye ~ '%w,Q,z) ? ~ ) + ( n w , y ~ ' E , y ~ E )(3)
where E i s the Concentration in equivalents per kilogram of solvenl ( E =- l/lzZ,m,\Zil);YA' and YB' are the equivalent fractions of salts A and B, respectively, in the final mixturc. 'The change in enthalpy in this process i~ given by the equation A
~
+-
~ =- YArYEi'E2(RThoE / ~ w ~
~
~
(YB - YA)RThiE) (4) When two electrolytes of the same charge type are mixed, the conutant I mixing is also a constant E mixing so that the two processes are the same. The difference between 4- RTho'(MgBr2-NaBr) ( 4/5)R Tho'(NaBr-NaC1)
+
("5)
+
+ RTho"(KaBr-MgCiz) RThoE(MgClZ-NaC1)+ RThoE(MgBr2-NaBr) + RThoE(NaBr-NaC1) + RThoE(MgBrz-MgC12) (7)
RTho"(MgBr2-NaC1)
=
should hold just as it does for charge-symmetric mixtures. According to the theory, the interaction between pairs of ions and triplets (except triplets containing ions all of the same charge) do not contribute to deviations from the cross-square rule. The experimental results in Table IIB show that the cross-square rule at constant E holds very accurately. It is within experimental error a t E 1 and 3 and just slightly beyond the expected experimental error a t E = 6. This result indicates that the equations in Appendix A for the prediction of multicomponent, mixtures based on constant E mixings may be very useful. I n particular, the fact that constant E mixes containing a common ion are generally smaller than constant I mixes indicates that if RTho terms are unknown the equation a t constant E will give more accurate results because the neglected RTho terms will be smaller. Since this equation depends on the neglect of changes in ionic strength it would be expected to be a poor approximation at low concentrations (perhaps below 0.1 m or so). More results will be needed for charge-asymmetric mixtures at high concentrations before the relative merits of the equations a t constant I and constant E can be determined. 2=
Acknowledgment. The support of the Office of Saline Water, U. S. Department of the Interior, is gratefully acknowledged.
Appendix A The derivation of an equation based on constant I mixtures has been given by Reilly and W00d.l~ The
R Tho'( MgBrz-MgClz) (6)
The results in Table IIA show that this equation is obeyed well within the experimental error, even a t a molal ionic btrangth 0f 6. The low values of the constant E mixings compared with the constant I mixings prompted us to explore the conditions under which a cross-square rule for constant E mixings should hold. This involved a new derivation of the general equations for the prediction of a multicomponent mixture from common-ion mixings given by Reilly and Wood. I n the new derivation constant El mixes were used instead of constant I mixings and the influence of the changes in ionic strcngth on the interactions between the ions mas neglectedaZ3With this approximation, the results turn out to be identical with those derived for charge-symmetric mixtures. The details of the changes necessary to perform this derivation are given in Appendix A. The resulting equations can be used to show that within thc approximations inherent in the derivation, the unweighted cross-square rule The Journal of Physical Chemistry, Vol. 76, N o . 8.!4 1978
(16) Y . C. Wu, R . M. Rush, and G. Scatchard, J . Phys. Chem., 72, 4048 (1968) ; 73,2047,4434 (1969). (17) A. K. Covington, T. H. Lilley, and R. A. Robinson, ibid., 72, 2759 (1968). (18) R. A. Robinson, A. K. Covington, and C. P. Bezboruah, J . Chem. Thermodyn., 2,431 (1970). (19) R . F . Platford, ibid., 3, 319 (1971). (20) H. E. Wirth, R. Lindstrom, and J. Johnson, J . Phys. Chem., 67, 2339 (1963). (21) H. E. Wirth and W. L. Mills, J . Chem. Eng. Data, 13, 102 (1968). (22) H. E. Wirth and A . LoSurdo, ibid., 13,226 (1968). (23) There is some evidence (besides the present results) that this may not be too serious an approximation. The following observations are consistent with the idea that at moderate to high concentrations (above1 = 0.1 or so) changes in ionic strength do not greatly affect the activity coefficients of the ions. (1) Statistical mechanical calculations (P. N. Vorontsov-Veliaminov, A. M. Eliashevich, J. C . Rasaiah, and H. L. Freedman, J . Chem. Phys., 52, 1013 (1970), J. C. Rasaiah, D. N. Card, and J. P. Valleau, ibid., 56, 248 (1972)) show that the energy difference between a system of charged spheres and the uncharged system is roughly constant except a t very low concentrations. Thus the energy of charging the particles does not vary strongly as the concentration changes in the more concentrated region. (2) The values of RTho for mixing many 1-1 electrolytes vary slowly with concentration at moderate concentrations (see ref 4). One of the things these coefficients depend on is the activities of the ions in the solution. (3)The concept of ionic strength is neglected quite successfully for fused salts.
Replacement in constant E equation
syrnhol aud definition in constant I equation
1 =
('/e)2i7&Zi2
== ~iioialionic strength
m,,= molality ion x charge on eetion k ZIX = charge on iznion 1 g ~ ~ ~= ~ goT xformconstant I mixing of MkX, with M,X,, ZkM
=I
E
tot,al ooncentr8tion in equiv per kg of solvent E, = m,jZ/ = equiv of x per kg of solvent 1 -1 ~MNX = go8 for constant E mixing of MX with NX = I/?Zirn,/Zil =
Bnmo
=
Oex
cluster integral for n ions of type 1, m ions of type 2, o ions of type 3
= T o t d e x ~ f i free ~ s energy per kg of solvent
GoB1zxm=: excess free energy per kg of solvent of a solution of pure M ~ X M a t ionic strength I yx = ionic strength fraction of X
derivation for constant E mixtures follows the same path except that all of the mixings refer to constant E mixings. The transformation scheme (Table 111) converts the equ:&ions for constant I into ones for constant E. The common ion mixing shown by Table I of ref 15 becomes the mixing of a solution of MX containing y kg water and ,E equivalents per kg water of both ions 1 (M) and 3 (X) with a solution of NX containing (1 y) lcg of water and E equivalents per kilogram of water of both ions 2 (N) and 3 (X). The A,Cooz etc. are calculated in the same way and lead to the analog of eq 3 and 4 of ref 15 lor the contributions of cluster integrals to a t,mo salt comnmn ion mixing.
Similarly the equation €or the excess free energy of a multicomponent mixture (eq 6 of ref 15) becomes
- cluster integral reduced by the product of
_ I _ -
Z1nzzmz80
the charges on all of the ions in the cluster Gex/n,,.M,v = total excess free energy per kg of solvent. The weight of solvent i s explicit for clarity G0Mx/nwMlv= excess free energy per kg solvent of a solution of pure MX a& equivalent concentration E yx' = equivalent fraction of X
+ R T E ' Z M ~,xYnfYNYxgim" N +
Gex/nwlKw= ~ M , , X Y M Y X ( G M X O / ~ ~ M ~ )
RTE2Zx
