4292
P. J. REILLYAND R. H. WOOD
The Prediction of the Properties of Mixed Electrolytes from Measurements on Common Ion Mixtures
by P. J. Reilly and R. H. Wood Department of Chemistry, University of Delaware, Newark, Delaware
19711
(Received J u n e 6 , 1969)
An equation predicting the free energy enthalpy and volume of a general mixture of electroljrtes has been derived. The equation, developed from the work of Friedman, predicts the properties of the mixed electrolyte solution from a knowledge of the properties of pure electrolytes and common ion mixtures. The equation allows for all possible pairwise interactions and will be most accurate when all of the electrolytes are of the type normally considered to be fully dissociated. The equation has been used to derive a general form of the cross-square rule. This general form applies to any mixture of salts whether the mixture is symmetric or asymmetric.
Introduction I n a recent paper Wood and Anderson1 using Friedman’s2 approach for mixed electrolytes showed that the properties of charge-symmetric mixtures of electrolytes could be predicted from measurements on pure electrolytes and on common ion mixtures. The equation was based on pair and triplet interactions and neglected only like-charged triplets. The equation has an advantage over the equation proposed earlier by Scatchard3 because, for heats and volumes of mixing, it is expressed more directly in terms of the experimental measurements. The equations of Wood and Anderson have been tested on heats of mixing and have proved a ~ c u r a t e . ~Guggenheim6 has proposed an equation which includes only pairwise interactions. Recently, Scatchard6 has extended his treatment to cover chargeasymmetric mixtures. The present paper reports the extension of the Wood and Anderson equations to include charge-asymmetric mixtures.
Theory The treatment developed in this paper allows the free energy, or any related property, of a mixed electrolyte solution to be calculated if the properties of the pure electrolytes and of the common ion mixtures are known. The excess free energy of mixing (AmGex)for a common ion mixture containing 1 kg of solvent, formed at constant ionic strength by mixing a solution of ions 1 and 3 in y kg of solvent with a solution of ions 2 and 3 in (1 - y) kg of solvent, is found to obey an equation of the form2 AmGex = RT12y(l - y)[go
+ (1 - 2y)glI
(1)
where go and gl are constants characteristic of the mixture. Friedman2 suggests that the excess free energy is governed by the change in the concentrations of ion pairs and triplets in the mixing process. The excess The Journal of Physical Chemistry
free energy then depends on Bu-a constant characteristic of the species u, and A,Cu-the change in the concentration product of the species u when the mixture is formed. The excess free energy must therefore obey an equation of the form A m G x = RTCBuAmCU
(2) (Note that for the purposes of this derivation we are neglecting the difference between molal and molar concentration scales. I n addition, we are neglecting the difference between mixing a t constant solvent activity and mixing a t constant pressure. The approximations involved are discussed in Friedman’s book.) A species formed by a pair of ions of type 2 is represented by u = 020 while an ion pair formed by an ion of type 1 with an ion of type 3 means that u = 101. For the species u, the concentration product in each solution is equal to the product of the concentration of the ions in the species multiplied by the mass of solvent in the solution. To evaluate AmCU,the concentration product in each solution is calculated and the initial values are subtracted from the final value. If the mixing is performed a t constant ionic strength, I , by mixing y kg of solvent containing ions 1 and 3 with (1 y) kg of solvent containing ions 2 and 3 to form 1 lig of solvent containing ions 1, 2, and 3, then the compositions of each of the three solutions are as shown in Table I. To calculate A,nCoo2, one squares the concentration of ion 3 in the final solution and multiplies by the mass of solvent. From this one subtracts y (1) R. H. Wood and H. L. Anderson, J . Phus. Chem., 70, 992 (1966) (2) H. L. Friedman, “Ionic Solution Theory,” Interscience Publishers, New York, N. Y . ,1962; H. L. Friedman, J . Chem. Phys., 32, 1134, 1351 (1960). (3) G. Scatchard and S. S. Prentiss, J . Amer. Chem. SOC.,56, 2314, 2320 (1934); G. Scatchard, ibid., 83, 2636 (1961). (4) R. H. Wood and H. L. Anderson, J . Phys. Chem., 70, 1877 (1966).
(5) E. A. Guggenheim, Trans. Faraduy Soc., 62, 3446 (1966). (6) G. Scatchard, J . Amer. Chem. SOC.,90, 3124 (1968).
PREDICTION OF ELECTROLYTE PROPERTIES FROM COMMON IONPROPERTIES
Table I Conon, ion 1
Initial
solution 1
Final solution
Conon, ion 3
0
ka(zi - 2 3 )
Y
21 -21 -~ Zz(zz - 23) ZdZa - 2 3 )
1 - Y
-21 zl(zl - za)
Initi a1 solution 2
0
Maas of solvent,
Concn, ion 2
kg
- 21
zl(zl - 2,) -(1
Thus, pairs contribute only to the symmetrical term in eq 1. The effect of triplet formation is more complicated since, in general, triplets generate extra y dependencei.e., contribute to gl as well as to go. The values of for the various triplets are AmC300
= -8y(l
- y)13[l
+ Y ] / Z ~ ~-( Z2~ ~
) ~
AmCoS0= -8y(l - y)13[[2 - y]/z23(z -22 3 ) 3
Amcoos = 8y(l - y)13[2Z1+
- 323 (21- z2)Y](z1 - -%)2/z33(z1 - z3)3(zz - 23)' AmCZ1O= 8 y ( l - y)13[y]/Z12Z2(Z1 - Z S ) ~ ( Z-Z 2 3 )
1
Y21
4293
- y)21
multiplied by the square of the concentration of ion 3 in solution 1 and (1 - y) multiplied by the square of the concentration of ion 3 in solution 2, i.e. AmCOOz =
8y(l - y)13[l - y]/Z1ZzZ(Z -1Z3)(Z2 - 2 3 ) '
A mC ln o
AmCZo1 = 8y(l AmCozl
2 2
=
- y)13[Z2- Z3 (21 - Z~)~]/ZI~Z~(ZI - za)'(Zz - Z,)
8y(I - y)13[2Z1- Zz - 2 3 (21 - z2)y]/z22z3( -z~ ZS)(& - Za)3
- y)PIZ1 - ZS(21- Z~)~]/ZIZ~Z~(ZI - z3)2(zz - 23)' AmC102= 8y(l - y)13[Z1 + 2 2 - 223 Z2)Yl(z1 - Z~)/ZIZ~~(ZI - ZS)'(Z-Z 25)' (21 AmCO"= -8y(l - y)13[2Z1- 223 (21 - z z)Y](z1- Zz)/ZzZa2(Z1 - z3)'(&- Z L I ) ~ A m C1ll= -8y(l
By this process, the following values of AmCufor various ion pairs are obtained
AmCZo0= -4y(l - Y ) P / Z ~ ~-( Z2 3~) ' AmCoz0 = -4y(l - Y ) I ~ / Z ~ ~-( Z Z3)2 Z AmCoo2 -4y(l - y)12(Z1 - Z2)2/Z32(Z -12 3 ) ' X (22
- 23)'
AmC1l0= 4 y ( l - y)12/Z1Z2(Z1 -
A,C"J' AmCol'
z3)(& - 23) = -4y(l - y ) 1 2 ( z ~- Zz)/ZiZa(Z1 - 2 3 ) ' x (22 - 2 3) = 4y(l - y)12(Z1 - ZZ)/ZZZS(ZI - 23) X (zz - 2s)'
It seems likely that triplets formed from two anions and one cation and vice versa will be more important than triplets formed from three ions of the fiame charge type. As a first approximation, one need only consider the influence of AmCZo1,AmCoZ1, AmCl1lrAmC102, and A,C0l2, which contribute to gl as well as go unless 21 and ZZhave the same value. The approximate values of go and g1 are given by eq 4 and 5.
where 21, 22, and 2 3 are the charges of ions 1, 2, and 3, respectively (ion 3 is the ion common to both salts). Applying eq 2 to the AmCU above, one finds that the excess free energy due to pairs has the form2 AmGeXpairs
= RT12y(l
- 9)gprirs
where
-- gpairs
4
-B 200 212(21
- 23)'
(4)
BOZO
-
- 28)'
(21 - Zz)2B002
- z3)'(.&- 2 3 ) ' (2, - Z2)B'O' ZiZa(Zi - ZS)~(& - 2,) 23'(21
-
z22((z2
BlIO +
ziZz(zi - &)(&
-
- 23)
(2, 22)BO" + 222a(Z1 - Za)(Z2 -
-
23)'
(3) Volume 76,Number 19 December 1089
4294
P. J. REILLY AND R. H. WOOD
Experimentally it is found that ion pairs are often much more important than triplets in mixed electrolyte solution^.^^^^^ If triplets are neglected, it becomes possible to calculate the excess free energy-and hence, the total free energy-of any solution. Thus for a solution containing mihf moles of cation M iwith charge ZiMjrnjx moles of anion X j with charge Zjx, etc., in each kilogram - of solvent it will be shown that the free energy is given by
Ek M E "EmXZkmZ p pm
RT
1=3
E 4E
Z=Z
4Bpp
-
(Zpnl)'(Zprn)
m=(Z-1)
c
m=l
4(zlx- Z m X ) ' B p p
E p " E ~ X E mZX Z pm
(ZpM)2(Zpl)2(Zprn)*
where The excess free energy of mixing may also be calculated from the actual difference between interactions in the initial and final solutions. Since the interaction in a solution is given by the product of the kilogram. 3f solvent times the molalities of the interacting ions and RTB' we have the interaction in the final solution equal to
R TBPP (m M G'M~X, is the free energy of pure MIXmwhile gMkM2Xm and gxix,"k are the values of go, from eq 4, for the mixing of J I k and hLIl in the presence of X,, and the mixing of X 1and X, in the presence of &Ik, respectively. Equation 6 is equivalent to forming the mixed electrolyte solution by mixing quantities of solutions of each of the salts M I X l , RIlX2, MixJsuch that the final soIution contains mlM moles of 311, m2Mmoles of M 2 , etc. The quantity of M i x l solution to be taken is (E1bfE,XZi,/2EI) kg of solvent and the con) (-,TI/ centrations of 31, and X, are ( Z I / Z t M Z i l and ZIxZt,) molal, respectively. The excess free energy of mixing is given by the second and third terms of eq 6 and it may be shown that the equation correctly predicts the excess free energy, due to pairwise interactions, as follows. 1. The excess free energy of mixing generated by an ion 31, interacting with another ion of the same type may be calculated by substituting the relevant portion of eq 4 into eq 6 and simplifying by use of eq 6a to 6d, i.e.
2
The initial solutions include solutions of AI, in combination with each anion. The concentration of M, in each of these solutions is (2I/ZPMZpm) while the quantity of solution taken is such that the mass of solvent in each is E,ME,XZ,,/2EI. The interaction of M p with itself in each of these solutions is
RTBpp
(-) 21
Zp'Zpm
E,MEmXZpm 2EI
The excess free energy of mixing due to the interaction is obtained by subtracting from the value of the interaction in the final solution, the sum of the interactions in the initial solutions. Hence
EpMEmXZ,, 2EI
A,Gex
E
hiE 1 M E,xZpmZg m
4E
4Bpp (zPM)2(Z,m>2
k-(p+l)
The Journal of Physical Chemistry
m=l
This result agrees with the prediction of the equation and, from the symmetry of eq 6, it is obvious that the same result will apply to the interaction between two X , ions. (7) R. H. Wood and R. W. Smith, J. Phye. Chem., 69,2974 (1965).
PREDICTION OF ELECTROLYTE PROPERTIES FROM COMMON IONPROPERTIES
4295
2. The second type of interaction is that of an ion, A I p , with another ion of the same charge type, M,. The excess free energy due to this interaction may be estimated from eq 4 and 6 as before, ;.e. m=j
A RPq
Direct calculation of the effect of this interaction, from the compositions of the solutions, gives
Am@x = RTm,"mqMBPQ
This result agrees with that given by eq 6 and from the symmetry of eq 6 it is obvious that the interaction of two anions X,and X , will be correctly predicted. 3. The third and final type of pairwise interaction is between a cation, & I p , and an anion, X,. Proceeding as before, we find that eq 4 and 6 yield
Hence, in all possible situations the equation correctly predicts the ion-pair interactions. If the cations all have the same charge, Z', and the anions all have the same charge, Zx, the equation will correctly predict triplet interactions (except for triplets composed of three cations or three anions). In this case, gl vanishes and go of eq 4 becomes
(since Z1 = Zz). In this special case, E is equal to 21/(ZM - ZX). Equation 6 may then be written as
RT 4E
(ZM- Z X ) 3 g ~ 1 ~ m M (8)k The triplet interactions may be divided into two classes. The first class of triplets contains three different ions, e.g., M p , &I,, and X,. Combination of the restricted equations (7 and 8) gives the excess free energy of mixing due to this triplet as
RT 4E
A,Gex =
RT 4E
-RT
81
E,ME,ME,X(ZX- zx)3
x 81BPqr
(ZM)2ZX(ZM - ZX)3
RT E
=-
1-j
\
The actual interaction in the solution gives
AmGex = RTmpYmqMmrXBP~'
- RT E
The magnitude of the actual interaction in the system may also be calculated as follows.
The second class of triplet interactions are those between two ions of the same type, e.g., XIv, and an ion Volume 73, Number 12 December 1969
4296
P. J. REILLYAND R. H. WOOD
of the opposite charge type, e . g . , X,. The excess free energy of mixing due to this triplet is given by eq 7 and 8 as
mixtures (the Q&I~D.I? terms of eq 6) one should obtain the present equation. The advantage of the present equation is that for heats and volumes of mixing it is more directly expressed in terms of experimental quantities. To show this, the analog of eq 6 for heats of mixing is needed. This is
DW~ k
The actual interaction indicated from the composition of the solutions gives the following amount of excess free energy of mixing. A,Qx
=
RTBppr
Thus in this special case the equation correctly predicts the effect of both pairs and these triplets. It can be shown that eq 8 is identical with the equation of Wood and Anderson for the case where all the electrolytes are of the same charge type.
Discussion Equation 6 correctly predicts all pairwise interactions for any mixture of electrolytes. For mixtures containing only a single class of electrolytes (e.g., 1-1 or 2-1 salts) all triplets, except those formed by three ions of the same charge type, are correctly predicted. For other mixtures, the triplet terms are only partially accounted for. This means that the equation will not be as accurate for mixtures containing several classes of electrolytes but since pairwise interactions are often predominant,'J it should be a good first approximation. The present equations differ from Scatchard's equations in that the properties of the pure electrolytes are not represented by power series and the components of the mixed electrolyte solution are specified by the first term in eq 6. This term specifies the amount of each component from which the mixture is formed. By collecting terms in Scatchard's equation into terms representing the properties of the pure electrolytes of Our component terms in eq 6, and terms representing measurements on two salt common ion The Journal of Phyeical Chemistry
= i Z=j ( m = l - l )
where RTh, is the heat of mixing analog of RTgo and ~ J ' M ~ X ,=
~M~x,'FL(~I~X~)
In this equation HOM,X,,is obtained by measuring the heat of dilution of the pure electrolyte. Similarly, hi\^^^? is measured by one heat of mixing ( M k X m with X I Z X ~at) a molal ionic strength equal to that of the multicomponent mixture. The final equations do not show it but in deriving the present equation the concept that any mixture of electrolytes can be formed by a series of common-ion mixings was very important. The mixing scheme involves picking a cation and, with it as a common ion, mixing all the anions to get the final anion mixture. This process is repeated for each cation and the final mixture is then made by mixing the cations in the presence of the common mixture of anions. This concept makes it clear that it is only necessary to predict heats of mixing ions in the presence of a common mixture of ions in order to predict the properties of any mixed electrolyte. It was a consideration of this mixing scheme that led to the component rule expressed in the first term of eq 6. The general equations may be used to investigate the cross-square rules and to predict the limitations of the conventional form of the rule. The cross-square rule has usually been applied to mixtures of 1-1 electrolytes. If the four salts formed from two cations and two anions are mixed in pairs, it is found that the sum of the excess free energy of mixing of the four common ion rnixings is equal to the sum of the two mixings in which there is no common ion. If one represents the mixings as the sides and diagonals of a square as below, then the sum of the sides (square) mixings is equal to the sum of the diagonal (cross) mixings.
""m:: MY
(8) T.F. Young, Y. C. Wu, and A. A. Krawetz, Discussions Faraday sot., 24,37,77,80 (1957).
PREDICTION OF ELECTROLYTE PROPERTIES FROM COMMON IONPROPERTIES I n the general case, the influence of the charge of each ion must be taken into account. If one takes quantities of solution containing equal masses of solvent (e.g., 0.5 lig) and if each solution has the same ionic strength (I) the excess free energy of mixing generated in the pairwise mixings is as follows. 1. The excess free energy of mixing for the common anion mixture lIX-NX is
A m p X = 1/&T12gh$NX 2. The excess free energy generated by mixing M and N in the presence of the common anion Y is
A m p x = 1/4RT12g,NY 3. The first common cation mixture, MX-MY,
1
RTP
The final common-ion mixture is NX-NY which gives the following quantity of excess free energy
4.
+
(ZM- ZX)gx,M (ZN - 2Y)gxyN 4(ZM ZN - Z X - Z Y )
+
These two results show how eq 6 predicts the excess free energy of the cross mixings from a knowledge of the common ion (square) mixings. The sum of the cross-mixings therefore gives the excess free energy of mixing as
1
+ = '/4RT12 X {(zM+ ZN- 2zx)gMNx + (zM+
[
AmGexMX-NY
generates excess free energy as below AmQx = '/4RT12gXyM
4297
AmGeX~~Y-NX
ZN - 2zy)gMNy f (2ZM(2ZN
zx- zy)g X y M +
- zx - Z Y ) gxyN)/( ( Z M
zx - 2'))
AmGex = '/4RT12gXyN (In the above the effect of the small RTgl term is neglected.) The excess free energy of mixing involved in the two cross mixings may be calculated from the general equation (eq 6). The total free energy of the mixture of MX and NY is given by
+
+
(ZN - ZY)G0Mx (ZM- Z Y ) G o ~ y ( Z N - ZX)GoNX (ZM- Z X ) G o ~ y G = 2(P Z N - zx - Z Y )
+
+
(ZN -
+ (ZM-
zx)gMNx
+
+
zy)gMNy
+
(ZM - ZY)gxyM ( Z N - ZX)gxyN 4(ZM ZN - Zx - Z y )
RTP
+
1
The excess free energy of mixing is the difference between the total free energy of tJhe mixture and the free energy of the components. In this case, the components are hIX and NY so the excess free energy is Ampx
G - '/~G'YX - '/~G'NY = -(ZM ZX)GoMx (ZM - Z Y ) G o M y (ZN - Z X ) G o N X - (ZN- Z Y ) G o N y 2(ZM Z N - zx - ZY)
+
-
+
+
+
(ZN - z x ) g M N x (ZM- z y ) g M N y (ZM- ZY)gxyM (ZN - ZX)gx,N RT12 ____ 4(ZM ZN - Zx - Z y )
[
+
+
+ZN -
+
I
Similarly, the excess free energy of mixing for the salt pair MY-NX is
(9)
The cross-square rule is now seen to apply in its original form if the four salts are from the same class of electrolytes (1-1, 2-1, etc.). If the salts are from different classes, then the cross-square rule must be modified to include the charge factors shown in eq 9. The general relation between the cross and square mixings is that the sum of the excess free energy in the cross mixings is equal to a weighted sum of the excess free energy in the square mixings, i.e.
+
(ZM ZN - Zx - ZY)(AmGeXMX-NY
+ A m G e X ~ y - ~ x=)
+ ZN - 2ZX)AmGeX~x-~x(ZMf ZN -
(ZM
2 Z Y ) A m G B X ~ * y - ~ y (2ZM - Zx
- ZY)AmGexMX-My f (2ZN - Zx- Z Y ) A m G e ~ N X - N Y
Conclusion The equations presented in this paper allow the free energy and related properties of a mixed electrolyte solution to be calculated from the properties of single electrolyte solutions combined with knowledge of common-ion mixtures. The cross-square rule has been extended to charge-asymmetric mixtures. The following paper gives an example of the use of these equations to predict the heat of formation of chargeasymmetric mixture of three salts. Acknowledgment. The authors gratefully acknowledge the support of this work by the Office of Saline Water, U. S. Department of the Interior.
Volume 73, Number 12 December 1969
