J. V. LEYENDEKKERS
946
Thermodynamics of Mixed Electrolyte Solutions. Ionic Entropy Correlations and Volume Fraction Statistics
Divirhn of Fisheries and Oceanography, C S I R O , Cronulla, N.S. W . 2280, Australia (Received March 26, 1970) Publication costs borne completely by The Journal of Physical Chemistry
Linear correlations are illustrated between the ionic entropy and various thermodynamic properties of mixing of two electrolytes with a common ion, vix. Harned coefficients, excess free energy, and enthalpy of mixing. The Harned coefficients are resolved into pairwise-interaction components and correlations between the ionic and structural entropy shown. Volume fraction statistics are used to predict the activity coefficients of electrolytes in 1: 1/1: 1 mixtures. A comparison between the experimental and predicted results indicates that the theory needs modification to account for changes in the hydration parameters and long-range interactions. If such changes are negligible the agreement is good.
Introduction A survey of the theory of multicomponent electrolyte solutions was published in 1968. Recently,2 equations were given which enable the free energy and related properties of a mixed electrolyte solution t o be calculated from data on single electrolyte solutions and common-ion mixtures. Equations based on the ion-component treatment of Scatchard enable the mean activity coefficient of any electrolyte in any mixture of ions to be calculated in a related way.3 A recent report* gives the parameters of 1:hese equations for 22 aqueous mixtures of two electrolytes. The effect of temperature on some of thcse parameters has also been ~ t u d i e d . These ~ references constitute much practically useful information 011 thie theory of mixed electrolyte solutions. I n the present paper an attempt is made to extend this information in a somewhat different direction, more emphasis being given to ion-solvent interactions and solution structure. A number of developments6 take account of effects of ion-solvent interactions, or hydration, on the activity coefficients in single electrolyte solutions but these theories have been somewhat neglected in cleaiing with mixed electrolyte solutions. The S been to the calciumonly application S E ~ to~ have strontium chqoride system7 where volume fraction and mole fraction sta1,istics were used. I n this paper, volume fraction s i atistics (VFS) are investigated for univalent eleotrolyl e mixtures. I n addition, the correlation betvvem Earned coefficients and ionic entropyS is examined in more detail.
Entropy Correlations H Lines. For an aqueous mixture of two electrolytes (say A and 13) a t constant ionic strength, I (equal to iT* I B ) ,the simple relationship
+
log
YA/YA’
=
-~ABIB
T h e Journal of Physical Chemistry, Val. 76, N o . 7 , 1971
(1)
often applies, where y represents the activity coefficient and CY the Harned coefficient. Generally, the dsta have been correlated in terms of the system A-B with I as the variable. However, by keeping A and I fixed and varying B it has been shown that the corresponding Harned coefficients can be simply related to a thermodynamic property of the ions, vix., the ionic entropy. For a large number of chloride systems CXAB is a linear function of the standard entropies of the ions, suitably weighted t o normalize charge asymmetric ~ y s t e m s . ~ For example, for a 2: 1/1: 1 chloride system such as CaC12-NaC1 the entropy function is
and for 2 : 1 mixtures, such as CaCI2-MgCJ2
(1) H. S. Harned and R. A. Robinson, “Multicomponent Electrolyte Solutions,” Topic 15, Vol. 2, The International Encyclopedia of Physical Chemistry and Chemical Physics, Pergamon Press, Edinburgh, 1968. (2) (a) P. J. Reilly and R. H. Wood, J . Phys. Chem., 73,4292 (1969); (b) R. H. Wood, M, Ghamkhar, and J. D. Patton, ibid., 73, 4298 (1969). (3) Y . C. Wn, R. M. Rush, and G. Scatchard, ibid., 7 3 , 2047 (1969).
(4) R. M. Rush, Oak Ridge National Laboratory Report, ORNL4402, UC-4-Chemistry (1969). (5) M. H. Lietzke, H. B. Hupf, and It. M ’ Stoughton, ~ J . Phys. Chem., 69, 2395 (1965). (6) (a) R. H. Stokes and R. A. Robinson, Trans. Faraday Sac., 53, 301 (1957); (b) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions,” 2nd ed, Butterworths, London, 1959; (e) E. Glueckauf, Trans. Faraday Soc., 51, 1235 (1955); (d) E. Glueckauf, “Structure of Electrolyte Solutions,” W. J. Hamer, Ed., Wiley, New York, N. Y., 1959, p 106; (e) R. H. Stokes and R. A. Robinson, J. A m e r . Chem. Soc., 70, 1870 (1948); (f) D. G. Milter, J . Pfiys. Chem., 60, 1296 (1956); (9)T. Ilceda, Rep. Liberal Arts Fac., Shizuoka Univ. J a p a n , Natur. Sci., 1 , 25 (1960). (7) S. Wu, Ph.D. Thesis, University of Kansas, University Microfilms, Inc., Ann Arbor, Mich., 1965. (8) J. Leyendekkers, J . Phys. Chem., 74, 2225 (1970).
THERMODYNAMICS OF MIXEDELECTROLYTE SOLUTIONS
-*ire'
#-.&
' - ; J
a
-b
-; -'
'
0
947
4 I
'
e
'
'
i~
'
/ Ib
'
'
20
;':e Figure 1. H lzne9 of CaC12, SrC12, and UOz(N03)t a t various ionic strengths. S A B O in cal deg-1 mol-' (at 25" and computed relative .to & t o := O),Le LYAB in reciprocal ionic strength: 0, X experimental; 0, predicted. Code: 1, H+; 2 Li+; 3, Ma+; 4,K + ; 5, Cs+, 6, Mg"'; 7 , CaZ+; 8, Sr2+; 9, Ba2+; 10, U 0 2 + ; 11, Cu2+; 12. NH4+.
The charactt:ristic line for a given electrolyte at a given ionic strength has been called an H line. Figure 1 shows some of these lines for the chlorides of calcium and strontium and for uranyl nitrate. The data used ale given in ref 7' and 9. For convenience the weights for chloride systems were represented as functions of the ionic charges; however, the anion weighting function was not suitable for the nitrate ion, the entropy function for asymmetric uranyl nitrate systems being
The entropy of ,zn oxyanion is correlated with the charge in a different way from that for a monatomic ionlo and complexing is more marked in nitrate systems; however, no satisfactory theoretical basis for the weighting values is available yet. Thermodynamic quantities such as the ionic entropies and the structural entropy ASst l1 represent potentially very useful reference quantities so that experimental data can be ustd Lo characterize structural effects in a quantitative witp. A number of other linear correlations have been noted between ionic entropies and parameters relatetl io ion-solvent and ion-ion interactions such as the vih>cositgcoefficient B of the Jones-Dole equation'l and for dilute solutjons of alkali halides, the b coefficient in the well known equation13
where' D = 0.5107 mol-'/2kg'/2 for water a t 25", p is a parameter characteristic of the electrolyte, Y represents the number of moles of ions formed from one mole of electrolyte, and x is the charge number of an ion. The structural entropy ASst or a quantity proportional t o it has been correlated successfully with DzO/ H20 transfer enthalpies and entropies, and has been shown t o be useful in interpreting trends in the data.14 The transfer entropies are believed to result primarily from long-range ion-solvent interactions while solution (9) (a) H. S. Harned and B. B. Owen, "The Physical Chemistry of Electrolytic Solutions," 3rd ed, ACS Monograph 137, Reinhold, New York, N. Y., 1958, Chapter 14; (b) R. A. Robinson and V. E. Bower, J. Res. Nut. Bur. Stand., 70A, 306 (1966); (c) R. A . Robinson and A. IC Covington, ibid., 72A, 239 (1968); (d) R . D. Lanier, J . Phys. Chem., 69,3992 (1965) ; (e) L. Jenkins and II. A. C. McKay, Trans. Faraday Soc., 50, 107 (1954); (f) E. Glueckauf, H. A. C. McKay, and A. R. Mathieson, J . Chem. Soc., Suppl., 2, 5299 (1949). (10) J. W. Cobble, J. Chem. Phys., 21, 1443 (1953). (11) H. S. Frank and M. W. Evans, ibid., 13, 507 (1945). (12) R. W. Gurney, "Ionic Processes in Solutions," McGraw-Hill, New York, N. Y., 1953. (13) D. T. Burns, Electrochim. Acta, 11, 1545 (1964). (14) (a) E. AM.Arnett and D. It. McKelvey in "Solute-Solvent Interactions," J. F. Coetzee and C. D. Ritchio, Ed., Interscience, New York, N. Y., 1969; (b) C. V. Krishnan and H. L. Friedman, J . Phys. Chem., 74, 2356 (1970). The Journal of Physical Chemistry, VoE. 76, N o . 7 . l07i
J. V. LEYENDEKKERS
948 viscosity variiLtions result from a balance of both shortand long-range interactions. Consequently it is assumed that the ionic entropy is an indicator of both types of interactions while ASwtdistinguishes the longrange type. A breakdown of the Harned coefficients into compclnents on the basis of simple ion-ion interaction theory and a comparison with these entropy term8 should therefore be interesting. Such an analysis might also give information on the general applicability of the W lines. IOUInteracizon Equations. Following the treatment given in ref 1 (p 11) and taking account of interactions between caticms it can be shown that for the electrolyte system A (cation 1 and anion 2) and B (cation 3, and anion 4 identied with 2), and assuming eq 1holds -(2.303/2‘)aa~
UcB12
- ( 2 . 3 0 3 / 2 ) ~ =~=~~3 4~ 1
WdB23
+ 2
+ +
~ d B 2 3
We613
-
ZC&
- ~ d a a (4)
Wr8i1
(3)
where
-
wc
zzc
(VlV4/VAVB~3~4
Zvd
I
(Y2&/VAVgx@4)
toe
E
( V I V3/VAVB&Z4)
201
5
(2P12/va2x122)
’&
VlV4/VAVBzlX2
%Id
(9V2/VAVBg1&
zk!
=2:
ZCf
2VlVZ/VA2x1x2)
-
-1/x12(21
ug = - 1/zs2(X3
+
XZY
+ &)*
These weights are equivalent t o those given by Reilly and Wood2 for the pairwise interaction terms (the numbering here is different and all x values are positive). Cation-Anion Interactions. The Bl2 and Bzaterms in eq 3 and 4 are characteristic of solutions A and B, respectively. It is assumed that these terms may be represented by the B ‘ values listed by Pitzer and BreweP (i.e., equivalent to b in eq 2 when p = 1). These values have been weighted on the basis of ionic strength and plotted against the entropy of the corresponding cation (Figure 2). For electrolytes with negligible complexing, B ’ fits the line (2VlV2/VAXlZZ)B12
= a
-
(b’/zlz2)S01
(6)
to within kO.01. The coefficients a and b’ are functions of the ionic strength but a t higher concentrations, following the trend of B ‘ , become effectively constant. At 1 rn and above b’ has a value around -0.004. Similar correlations are found for other electrolytes such as the nitrates and chlorates of uni- and divalent cations.
&‘I[
-Mp
2V3V4/VB2x&)
VlV,,/VAYBXlXZ 2Va2/~~2X324
Here, x repregents %heabsolute charge value of the ion; note that xz I= 24, while v2 Z v4 may apply. The B terms represent interactions between ions of opposite charge and 6 terms represent the interaction between ions of like charge. For charge symmetric systems when 613 = (635 611) or the 6 terms are negligible, then CYAB = - C Y B A , obviously R special case. The excesfi free energy of mixing when Harned’s rule applies, and I* -- I B = l/ZI, is given by
+
+
P / R T 1 2 == - (2.303/4) ( C Y A B / X ~ Z ~ -=
yf =
CYBA/z3&)
YeBi2 --k y d B 2 3 -k ye613 -k
+
Yf6ii
20
16
G
t
(5)
where ‘/~(WC/~I% Yd
+ ‘&/xaZr) f
1/Z(Wd/z122
ud/%x4)
and so on, from eq 3 and 4, or, in terms of the charge values YC
(ZI -
23)/X122(xI
- x3)/&xP(xI
yd
=
-(z1
Ye
=
i/(Zl .f- &.)(x3
+
+
+
ZZ)~(Z~
+
xZ)(x3
Figure 2. Chloride interaction parameters for the single electrolyte solutions us. the conventional entropy of the cation, at various ionic strengths. Units as in Figure 1. rB ’ = 1 / % v (log y& z+z-D.H.)/I (where T = ~ V + V - / V Z + Z - ) .
+
Z2)
-k
22)’
z2)zlX3
The Journal of Physical Chemistry, Vol. 76, N o . 7 , 1071
(15) J. Greyson and H. Snell, J. Phys. Chem., 73, 3208 (1969). (16) G. N. Lewis and M. Randall, “Thermodynamics,” revised by K. S. Pitzer and L. Brewer, 2nd ed, McGraw-Hill, New York, N. V., 1961.
949
THERMODYNAMICS 01' !XIXED ELECTROLYTE SOLUTIONS
\ x \ \
U
1
'
1
5: Figure 3. The cation-cation interaction terms of chloride systems us. the conventional ionic entropy function: charge symmetrical systems, ,9Ao := - $ 0 . , asymmetric, SA"= (l/&' $. 2.99') - Jlo.Units as in Figure 1: A, 0.5 M ;0, 2 M ; e, 3 M ; X, 6 M
Single electrolyte solutions that exhibit complexing are in reaiity multicomponent systems. The B ' value therefore does ]not represent B12but the resultant of a number of interaction terms. I n addition, associated changes in the ionic strength and charge symmetry introduce furtlier complications. However, for some of these systems it might be useful to correlate B ' with weighted sums of t h e entropies of the constituent ions, or alternatively, simply correct for the entropy change due to complex formation. A reasonable amount of data are avzilable on entropies of formation of comp1exesl7 and an equation for predicting the entropies of complex ions has been derived.ls For example, the
deviations from eq 6 for the copper and zinc chloride systems are of the order of the expected AS, the entropy change due to complex formation.'' For the more complicated system of cadmium chloride the deviation is quite spectacular (the experimental rB is -0.28) but is consistent with the A S associated Kith the release of five or more water molecules. Extension to mixed solutions will introduce further complications since different complex species might dominate; e.g.] in mix'
(17) (a) G. iM.Nancollas, "Interactions in Eloctrolyte Solutions," Elsevier Publishing Co., Amsterdam, 1966; (b) C. W. Davies, "Ion Association," Butterworths, London, 1962. (18) J. W. Cobble, J. Chem. Phys., 21, 1446 (1953). The Journal of Physical Chemistry, Vol. 76, XQ.7, 1.971
J. 67. LEYENDEKKERS
950
r
r
-I:
co
c c
a 4
t5
c
c 0
- 500
-300
-400
(A?- ns:‘)(s,-
-100
-200
0
’3)
+
Figure 4. Excess free energy of alkali halide systems (- (OIAB (YBA) = 4GE/2.303RTZ2for equimolal mixtures) as a function of the ionic entropy and ritructural entropy ALPt. Coding and units as in Figure 1. Inset: slopes of lines in Figure 3 us. the structural entropy ( 2 m).
tures of cadmium chloride and sodium chloride the species Cd(& beconies much more important.lg Cafion-Cation Interactions. An estimate of the cation interaclion terms can be obtained from eq 3 using the experimental Harned coefficients and B ’ values, uix.
Values of A A B have been calculated for the alkali halide systems and plotted against a simple function of the ionic entropies (Figure 3). This was taken as S30 SIo for symmetric systems and ( l / & O - 1/2S10 - ,Silo for asymmetric systems to conform to the H line weights whiich were derived empirically. The results may be summarized by the equation
+
AAB
=T
d
+ b”S,O
(8)
where SAois the appropriate entropy function and d = (we - w i ) 6 ~ for. 1he case where cation 3 is equivalent to car-ion 1 i.e,*B mixture of the single electrolyte with AA(HCI) itself. The quantiiy d is also equal to bl’S1 for the symmetric systems with the acid as a component. The H line can therefore be interpreted in terms of eq 3, 6, and (3, For a simple 1:1/1:1 system the line is given by
+
The Journal of Fhzlsiccll Ch.ernistry, Vol. 76, No. 7 , 1971
(YAB
= -d/1.15 =
a.4
+
+ BSio -- B&O
(9)
bASS0
+
where b A = -B = -(I/&’ b”)/1.15 and U A = -d/1.15 BSl0. I n summary, the H lines should apply generally to strong-electrolyte mixtures with a common ion but for charge asymmetric systems the weights will depend on the common ion. Free Energy and Enthalpy Correlations. The above analysis indicates that the slopes of the H lines are related to the cation interactions. I n addition, b” was found to be linearly related to the structure entropy ASst (inset, Figure 4). This suggests that a combined entropy function of the type
+
- ASist)(& - Xs) would be proportional to the free energy for symmetric systems where only cation interaction terms apply (eq 5). The difference in the 611 terms will cause some scatter but the correlation is quite good (Figure 4). These free energy estimates cannot be expected to be very accurate and the data are limited. Accurate enthalpy data are readily available, however, and these have been used to extend the analysis.20 The experi(19) P. J. Reilly and R. H. Stokes, Aust. J. Chem., 23, 1397 (1970).
THERMODYNAMICS OF MIXEDELECTROLYTE SOLUTIONS
r 160
-
110
-
uo IO0
9
44-
SH
Figure 6. The parameter ho (asymetric charge systems) as a function of the ionic entropy. Coding as in Figure 1, e.g., 237 represents a LiCl-NaCi-CaClz mixture. Units as in Figure 5 . Figure 5 . The parameter ho for cation-cation interactions at an ionic strength of I m, at 25' (derived from enthalpy data%), as a function of the ionic and structural entropies: 0, SH values corrected for A88'. Coding (and units of s")as in Figure 1. A dashed number indicates bromide salt. Units of ho: cal/kg of solvent ,/ional2.
mental data for exaess enthalpy of mixing, H E are usually expressed jn the forrn20
BE = yA!/BI2RT(hO
+
hl(YA
- YB))
(10)
where y is the ionic Eltrength fraction of the electrolyte, ho is a measure of the ion interactions, and hl a measure of skew. The correlation indicated in Figure 4 did not apply for enthalpy data. However, when ho (for YA = Y B = 0.5) values are plotted against the sum of the cation entropim (Figure 5 ) , the points fall into three groups which c m be correlated with the water structure concepts; viz. elecirdytes are classified, according to their effect on t h water ~ structure, as makers (14) or breakers iB) . Mxtures of structure promoting electrolytes (M f AI) fall on line 11, (M B) mixtures fall B) on line B. On this basis on line MB, and (B sodium chloTide is ;2 B, contrary to its usual classification. However, the sodium ion seems to have only a small net effect on solvent structure.l5pz1 These classiiicnLiom~are based on a combination of short- and long-range structure effects, whereas classifications for d>lute :+oluLionsare based on long-range
+
+
effects,16 e.g., all the alkaline earths are considered &!I types here but in dilute solutions this only applies to the magnesium ion. For charge asymmetric systems (alkali chloride/alkaline earth) the contribution of the cation interactions to hohas been estirnated,20calthough the accuracy is uncertain.20d The values fall close to the MB line when plotted as a function of the entropy of the monovalent cation; apparently the contributions of the alkaline earth cations are all small or approximately the same value. This result is consistent with the very low RThovalues for alkaline earth mixtures.2oe The lines M, MB, and B can be made to closely coincide (line E) if the entropy function is corrected for the long range structural entropy.l 1 This correction is given by a weighted sum of the AXst values of the component ions for the B and M points and the weighted difference for the MB points. The sum is E+
+ z2~3)(fiASP 4f2AXzSt 4- fdd3Pt)
= (z~zg
(Ila)
where f represents the ionic strength fraction of the ion, e.g., for the point 34 (NaCl-IICI) E+
'/2(A.Sxacst
+- D.8~1.'~ Jr ~ A X C I - ~ ' )
(20) (a) T. F. Young, Y. C. Wu, and A. A. Krawetz, Wkscuss. Faraday SOC.,24, 37, 77, 80 (1957); (b) Y. C. Wu, M. B. Smith, and T. F. Young, J . Phys. Chem., 69, 1868 (1965); ( c ) R. N. Wood, J. D. Patton, and M . Ghamkhar, ibid., 73, 346 (1969); (d) R. H. Wood and M. Ghamkhar, ibid., 73, 3959 (1969); (e) R. H. Wood and H. L. Anderson, ibid., 70, 992 (1966). (21) M. Karninsky, Discuss. Foraday Soc., 24, 171 (1'357).
The Journal of Physical Chemistru, Vol. 75, N o . 7 , 1971
J. V. LEYENDEKKERS
952 and the diflerence i,s e-.
log
+ X Z X ~ ) ( ~ -~ AfiASzst) SI~~
(~1x3
(llb)
as the anion contributions cancel out. The open circles in Figure 5 indicate the corrected entropies. A mixture Gf the single electrolyte solution with itself represents either an (ILI 34) or a (B B) mixture and therefore should fall on line At or line B. The where SAA’ is entropy should correspond to ~SAAO, the value of the entropy function of the H line equation when the Nariied coefficient is zero. This is not equivalent to SIoas can be seen from eq 9. The points for these single electrolyte “mixtures” were therefore estimated by using StlLdovalues from the H lines and eq 11 for the structure correction. The resultant ho values correspond to the 61, interaction term. These enthalpy correlations should prove useful for filling in gaps in the data (the structural corrections, which are only approximate, are not necessary for this). Similar plots are obtained for asymmetric systems2OC when all the interaction terms are included (Figure 6). For the system NC1-n4Cl2 the entropy function is (&-O For more complicated mixtures the lines can be combined, e.fl., for the chloride system of structure breaking (B) and structure making (14) cations M2+/B*/AI~+the heat of mixing 50-50 mol% of B +/M + chloride mixtures with the divalent chloride is given by23
+
+
+
AH,
=
(‘/812)(RThMztg tc‘ -t
- ‘/zRThl\l .+Bec’)
AiSThh12+ip1+~’
=
2I”(cmstant
+ S B +o +
5/g&
+o
(12)
+ 2SM2+o)
the entropy Icoeficients have been derived from the lines MB (Figure 5 ) and MB and 14 (Figure 6). Experim ental vahes of AH, for alkali chloride mixtures with magnesium rind barium chlorides have been plotted against this entropy function (dashed line, Figure 6) ; predictions for other systems of this type are indicated for interest.
Volume Fraction Statistics (VSF) Method of Predicting Activity Coefficients in Mixed Electrolyte Solutions
.-y& =
log
ye’
+ log
ye
+ X(W))
(14)
-0.5091”2/(1
+- PI’”)
(14a)
where, at 25”
D.H.
=
The symbol p represents an empirical parameter estimated by an iterative method, X ( m >is an empirically derived deviation function which for a 1: 1 electrolyte (at molality m) has the form6d
X ( m ) = -0.06m2/(1
(13)
+ 1/2m)2
(14b)
A similar function has not yet been derived for 2 : 1 electrolytes but the following equation fitted both calcium and strontium chloride down t o I = 0 . 3 ~ ~ 2X(m)
=
+
0.0079 - 0.0195~1 0 . 0 0 7 3 ~ ’ ( 1 4 ~ )
This is a less satisfactory form than (14b) since at I = 0.1 m the value is still relatively large. The following equations, based on those statistically derived by Kirkwood, have recently been given by GlueckaufZ2and apply up to values of PI”* = 20 for all types of aqueous electrolyte solutions. At 25” and p P 2
+
where p = (”4 1/4(c/dom)) and c trolyte. Equation 14 may be written log ye’ = lX+z-IID.H. -
=
molarity of elec-
0.511p21a’2
+
so that X ( m ) = -0.511p213”/4(l PI^'')^ up to about = 2. While it would be preferable to work with the general eq 14d and 14e, values of the parameters p and h (see eq 15) have been derived on the basis of eq 14b and 14c so that it is convenient to use these equations here. The entropy term is given by
I
+
where, on the basis of Glueckauf’s treatment of 1: 1 electrolytes)6dthe electrostatic term is given by The Journal of Phhysieal Chemistry, Vol. 76, No. 7 , 1971
Ix-x+l(D.H.
log 7’ = f ~ r ( r h - v)/2.303(1
This method, despite its present shortcomings, is not bounded by linearity restrictions, and Wu7 has shown that I t can be usefully applied to ternary systems and has certair advantages over the mole-fraction-statistics approach; e.g., the hydration parameters are additive. The activity of EL strong electrolyte may be represented by log
ye’ =
( ( h - v , / v ) log (1
+
+
~ W Y )
+ fmr>
-
( h / v ) 1% (1 - Smh) (15)
where f = 0.018, r = apparent molatr volume of electrolyte (at m = 1for 1: 1 electrolytes, m = 0.7 for 1:2 electrolytes) divided by the molar volume of water, m = molality of electrolyte, v = number of moles of ions produced by the dissociation of 1. mol of electrolyte, and h = hydration parameter, derived in conjunction with p . (22) E. Glueckauf, Proc. Roy. Soc., Ser. A , 310, 449 (1969).
OF' R ~ I X E DELECTROLYTE SOLUTIONS THERMODYNAMICS
953
.c
Figure 7. A survey of the comparison of VFS theory with experiment over the range of I = 0.1 to 5 m (Table 11, linear fit). The first two numbers represent the electrolyte system (Figure 1) and the last number (or number in brackets) represents I ; e.g., 352 represents NaCl-CsC1 ai. 2 m; X represents LiNOsLiCl system. When the points for a given system are too crowded only key identificatisn numbers are shown.
When the electrolytes are of the same charge type and the ionic strength is constant the electrostatic terms should not change appreciably. This was found to hold for the system CaCl2-SrClz.' For such systems the activity coefficient ratio reduces to h,log ys. In systems where the electrolytes are of different charge type, variation in the electrosttihic term might be appreciable. An approximate eet'mate of this variation could be made on the following basis. In the derivation of the deviation functions, which characterize the charge type, the distance of closest approach has a fixed value for a particular electrolyte. I n the mixed electrolyte the D.H. term for either electrolyte may be considered constant and the deviation functions will be some combination of the individual deviation functions suitably weighted according to the charge type; e.g., for a system such as NaCI-CaCh it could be assumed that as mA -+0 log
YAB~' = (VA/V>X(~)B'
O.~Z(W)B' and as mB -+0 log
yBAel
-
.(m)Ao
x ( ~ ) A ' ,since Y = V A
=
+
1.2x(m)A0 - 22(m)BQ
YB
(14f) (14g)
T'hs Journal of Physical Chemistry, Vol. 76, N o . 7, 1971
J. V, LEYENDEKKERS
954
Table I: Hariied Coefficients Calculated on the Basis of VFS and Compared with Experimental Values
1 :1 Electrolytes aABC
I
U
aABb
EXP
PAB
a
aBAb
EXP
aBAC P5A
LiC1-HCldpe
0.025 0.05 0.075
-0,0068
-0.0101
- 0.0068
0,0071
0.0106
0.0071
- 0.0000
-0.0000
-0.0068
-0.0102
-0.0068
-0.0102
0.1
--0.0068
-0.0102
0.5
- 0.0071
-0.0104
1
- 0.0074
-0.0107
2
-0.O081
-0.0113
3
- 0.0090
- 0.0122
4
-0.0103
-0.0133
5
-0.0120
-0.0150
6
- 0.O145
-0.0176
0 025
- 0,0359
-0.0599
- 0.0068 - 0.0000 -0.0068 -0.0000 -0.0068 -0.0000 -0,0071 -0.0000 -0.0074 - 0.0000 - 0.0081 -0.0000 -0.0089 -0.0000 -0.0101 -0.0000 -0.0118 -0.0001 -0.0141 -0.0001
0.0071
0.0106
0.0711
- 0.0000 0.0071
0.0106
0.0713
- 0.0000 0.0071
0.0106
0.0074
0.0108
0.007 0.0013 0.006
0.0077
0.0111
0.005
0,0085
0,0118
0.005
0.0094
0.0127
0.004
0.0107
0.0139
-0.0025
0,0125
0.0156
0,0152
0 0183
0.0514
0.0759
I
- 0.0086
0.0071 0 0000 0.0074 - 0.0000 0.0076 - 0.0000 0.0085 - 0.0000 0.0095 - 0.0000 0.0109 - 0,0000 0.0128 - 0.0001 0.0158 - 0.0001 I
NaCl-HCld I
0.05 0.1
- 0.0360 - 0.0361
- 0.0600 -0.0600
0.5
- 0.0372
-0.0609
1
-0.(I384
- 0.0620
2
-0.0411’
- 0.0640
3
-0.0446’
-0.0677
4
-0.0489’
-0,0719
5
-0 .0544’
-0.0775
6
- 0.0619’
-0.0855
0.025
-0.0438
- 0.0872
0.05
- 0.0439
- 0.0872
0.1
--0.0400
- 0.0873
0.5
-0.0450
- 0.0880
1
- 0.0461’
-0.0889
2 3
-0.0486’ - 0.0516’
-0.0910 -0.0937
_____._I__ T h e Journal of Phusicai! Chemistry, VoE. 76, N o . 7 , 1971
-0.0359 -0.0005 -0.0359 -0.0005 - 0.0360 - 0.0005 -0.0369 - 0.0005 -0.0378 - 0.0006 -0.0399 -0.0007 - 0 0423 - 0.0009 - 0.0450 - 0.0011 - 0.0482 - 0.0014 -0.0518’ -0,0019 ~
KCI-HCI~ -0.0438 - 0.0009 - 0.0439 - 0.0009 - 0.0439 - 0.0009 - 0.0445 -0.0010 -0.0451 - 0.0010 - 0.0464 -0.0012 - 0.0478 -0.0014
0 0514 ~
- 0.0008
0,0515
0.0760
0.0515 - 0,0008
0.0517
0.0761
0.043
0.0518
-0 0.0534
0.0777
0.037
0.0553
0.0794
0.032
0.0597’
0.0836
0,031
0 . 0651f
0.0888
0.031
0. 071Sf
0.0956
0.030
0. 08Osf
0.1046
0,030
0.O92lf
0.1172
0.029
0.0712
0.1036
I
0008
0.0540 - 0.0008 0.0564 - 0.0009 0.0623 - 0,0011 0.0698 -0.0014 0.0798 -0.0017 0.0933 - 0.0022 0.1129 - 0.0030 0.0712 - 0.0018
0.0713
0.1038
0.0715
0.1040
0.077
0.0738
0.106
0.062
0.0764
0.109
0.056
0.0714 -0.0018 0.0717 -0.0018 0.0752
- 0.0020
0,0790 - 0.0021
0 . 0821f
0.114
0.057
0,0880 - 0.0025
0. 088gf
0.121
0.062
0,0993 - 0.0029
THERMODYNAMICS OF ~ I I X E ELECTROLYTE D SOLUTIONS
955
-~ Table I (Continued) UAB‘
I
a
4
-- 0.01552’
- 0.0973
5
--0.05918’
- 0.1021
6
.- 0.0667’
-0.1087
0.025
_0,0464
-0.1017
0.05
-- 0.0465
-0.1018
0.1
.-O 0466
- 0.:1019
0.5
--0.0417
-0.1034
1
-- 0 . 0488’
-- 0.1048
2
--0.0514’
-0.1079
3
-- 0.0543’
-0.1115
4
--0.0578’
-0.1156
5
~-0
-0.1207
6
.-0. 0615’
aBA
a
PAB
EXP
aABb
- 0,0493
CZEA~
EXP
PEA
0.1138 - 0.0035 0.1330 - 0.0044 0.1596 -0.0056
0.0972’
0.130
0.066
0.1074’
0.1417
0.072
0 . 1206’
0.1570
- 0.0464 - 0.0013
0.0629
0.0882
- 0,0464 - 0,0013 - 0.0460 - 0.0013 - 0.0470 -0.0013 - 0.0475 -0.0014 - 0.0486 -0,0016 -0.0496 -0.0018 - 0.0507 - 0.0020 -0.0516 - 0.0024 - 0.0522 - 0.0029’
0.0631
0,0884
0.0634
0.0886
0.140
0.0662
0.0917
0.105
0.0691
0.0950
0.100
0 . 0757’
0.1022
0.099
0 . 0833’
0.1107
0.098
0 . 0923’
0.1209
0 . 1032’
0.1336
0.1188’
0.1502
-0.0017 - 0.0507 - 0.0021 -0.0520’ - 0.0026
CsCI-HCld
~
I
Oi521’
- 0.1273
Table 11: Const,mts for Statistical Term of Equation 13 Electrolyte
He1 LiCl
NaCl KC1 CSCl LiNOs KOH HBr LiBr NaBr KBr
h
r
Ref
5.57 5.31 3.60 2.58 2.27 4.54 5.23 5.80 5.29 3.94 2.66
1.05 1.03 1.03 1.40 2.30 1.68 0.40 1.42 1.40 1.40 1.98
6d
_l________l__
The variations of the ratios of (13a) with solution composition were computed for a number of mixed electrolyte solutions. As an example Table I gives the results for alkali halide-hydrochloric acid systems (the rest of the tables are available on request). The values of h and r used in the calculations are listed (Table 11).
Results Figure 7 givt:s a survey for 1: 1 electrolytes over the range 0.1 t u 5 ’m ionic strength. The most notable feature is the large deviations for systems of cations
~
~
a Least-squares linear fit. Value of ail when component i at trace concentration. Poor fit. Source of experimental values.
’
0.0630 -0.0021 0 0632 - 0.0021 0.0636 - 0.0022 0.0677 - 0.0023 0.0722 - 0.0025 0.0826 - 0.0029 0.0954 - 0.0034 0.1116 -0.0041 0.1327 - 0 0051 0.1614 - 0.0064
Least-squares quadratic fit.
Reference 9a.
with opposing structural influences on mater (e.g., H-Cs, Li-Cs, H-Na, H-K). Quite accurate predictions are possible if the cations are similar, e.g., CaSr7and Li-H. A detailed analysis of Ihe error pattern is not attempted in view of the difficulty in sorting out the contributions from the various types of interactions. Instead, a few general points are noted. The main errors in prediction arise from errors in the two parameters p and h. The problem is that these quantities do not represent physical realities. If this were ignored then Iz. values for chayge symmetrical mixtures could be derived empirically. These parameters would then be forced to absorb all the ion-ion and ion-solvent interaction effects; e”g.,for the CaC12SrClz system,’ for which interactions between the cations are small, the h parameters in the mixture showed little variation above 2 m, the values being approximately those of the single electrolytes. At 0.5 m the values were about two-thirds lower and increased to a maximum around 1.5 m. For the 1: 1systems the values in the mixtures were generally lower than those in the single electrolyte with the exception of sodium chloride at molalities below 5 , and the chlorides of lithium and hydrogen, in mixtures with cesium chloride, above 3 m. A better approach would be to look for an unequivThe Journal of Physicai Chemistry, Vol. 76, N o . 7 , 1071
956 oca1 way t o define the hydration number of an electrolyte (e.g., on the basis of e n t r ~ p y ~and ~ , treat ~ ~ ) it as a function of the co'ncentration (e.y., by reassessing the single electrolyte data on the basis of eq 14d and 14e). This would provide a firmer theoretical framework and simplify the Treatment of mixed solutions. Recently, some very in teresting near-infrared studies of the hydration of alkali halides were reported.24 The hydration number of sodium chloride was calculated t o vary from 4.8 at a concentration of 0.5 M to 3.5 at 5.0 M , while the vahes for cesium and potassium chlorides (at 3 M ) were about 1.0higher than those in Table 11. The excess thermodynamic functions of symmetrical svstems give a complete description of the formation of a solution from its components since they include effects of solvation and solute-solute interaction. A check on the accuracy of VFS in predicting these effects can be made by comparing the experimental H coefficient 3ums (eq 5) with the corresponding sums of the calculated values. The results are poor for many systems, notably 25 (Li-Cs) and t o a lesser extent 15 (a-Cs), and 35 (Na-Cs), the magnitude of the predicteld excesE) free energy being much too low. 11; seems likely that changes in the log ye' term are superimposed on changee$in h for these systems. The quantity f c u B - a g ~ )approximately characterizes the end ,solutions but cont>ainscontributions from the mixture in the form of the term@ m(daBA/dm) ~ ~ P B A where ,h' is the coefficient of the I B A 2 term in the extended form of eq 1,
The Journal of I'hysicai' Chemistry, Vol. 76, No. 7, 1971
J. V. LEYENDEKKERS For one of the electrolytes in the pair the error in prediction of this quantity partly cancels the error in the prediction of the excess free energy. This usually favors the more structure breaking cation of the pair. For systems containing cesium, the errors decrease with increasing ionic strength, whereas they increase for the other systems. This merely summarizes the trends to be observed from Figure 7, notably the lack of symmetry in the prediction errors for the components of the system. The VFS method of prediction is, in general, limited now by failure to treat the h values as variables and to account for variations in the electrostatic term. However, generally the correct sign and order of magnitude is predicted and the accuracy is good for systems containing cations with similar properties. The form of the VFS equations indicate that Harned's rule cannot be exact but the expansion conveiges quite rapidly. The linear fit is poor for many systems, yet the /3 coefficients are small and would be difficult to measure. The VFS method is therefore also useful for predicting the relative complexity of various systems and provides a reasonable estimate of the P coefficient. Acknowledgments. I thank Dr. h4. Whitfield and Dr. R. Hunter for helpful discussions.
(23) S. L. Bertha and G. R. Choppin, Inorg. Chem., 8 , 613 (1969). (24) W. C. McCabe and Harvey F. Fisher, J. Phys. Chem., 74, 2990 (1970).
