Jan., 1957
ACTIVITYCOEFFICIENTS IN MIXEDELECTROLYTE SOLUTIONS
99
ACTIVITY COEFFICIENTS I N MIXED ELECTROLYTE SOLUTIONS BY WILLIAMJ. ARGERSINGER, JR.,AND DAVIDM. MOHILNER Contribution from the Chemical Laboratory of the University of Kansas, Lawrence, Kansas Received August 67,1966
In three-component aqueous electrolyte systems in which one solute is known to obey Harned's rule, the activity coefficient of the second electrolyte solute and the osmotic coefficient for the mixture may be directly computed without further assumptions by the integration of fundamental cross differentiation relations. The method is applied to four hydrochloric acid-metal chloride systems. In every case, Harned's rule is approximately followed by the second electrolyte solute, but the deviations are significant.
Introduction In many aqueous solutions of two electrolytes with a common ion, it is found that a t constant total ionic strength, the logarithm of the activity coefficient of one solute is a linear function of the ionic strength of the second solute. This behavior has been most extensively studied by Harnedlus and his co-workers for aqueous solutions of hydrochloric acid and a metallic chloride; it has been observed as well in solutions of two different metallic chlorides by Owen and C ~ o k e ,Robinson5 ~ and McCoy and Wallace.6 Although such behavior is not ~ n i v e r s a l , ~it- ~is sufficiently common in a t least the simpler systems to justify further study and use. In most systems thus far studied i t has been possible to determine the activity coefficient of but one electrolyte solute. Thus, Harned has very carefully determined the activity coefficient of hydrochloric acid in the mixed solutions from electromotive force measurements. Although additional measurements of solvent water activity would permit the calculation of the activity coefficient of the second solute by means of the GibbsDuhem equation, it has rather been customary to assume a linear variation for the second solute and to compute the rate of variation with ionic strength fraction. Glueckauf lo has proposed a sensitive test of the hypothesis which indicates for several systems that the assumption although invalid a t low ionic strength becomes increasingly valid a t high ionic strength. There is a need, therefore, for means of determining directIy the activity coefficient of the second solute from measurements of the activity of one other component. This has been accomplished by McKayl' and McKay and Perting12 through the use of cross differentiation rela(1) H. S. Harned and B. B. Owen, "The Physical Chemistcry of Electrolytic Solutions," Reinhold Publ. Corp., New York, N. Y., 1950, p. 459. (2) H.9. Harned, THISJOURNAL, 68,683 (1954). (3) H.S. Harned and R. Gary, J . A m . Chem. Soc., 7 6 , 5924 (1954); 77, 1995, 4695 (1955). (4) B. B. Owen and T.F. Cooke, ibzd., 69, 2273 (1937). ( 5 ) R. A. Robinson, ibkd., 74, 6035 (1952); R. A. Robinson and C. K. Liin, Trans. Faraday Soc., 49, 1144 (1953); R. A. Robinson, ibid., 49, 1147, 1411 (1953). (6) W. H. McCoy and W. E. Wallace, J . A m . Chem. Soc., 7 8 , 1830 (1956). (7) H. S. Harned and M. A. Cook, ibid.,69, 1890 (1937). 61, 704 (8) R. A. Robinson and R. 0. Farrelly, THISJOURNAL, (1947). (9) 0.D.Bonner and V. F. Holland, J . A m . Chem. Soe., 7 7 , 5828 (1955). (10) E. Glueckituf, H. A. C. McKay and A. R. Mathieson, J . Chem. 'SOC., 8 299 (1949); TTans. Faraday SOC.. 47, 428 (1951). (11) H.A. C. McKay, Nature, 169, 464 (1952). (12) H.A. C. McKay and J. K. Perring, Trans. Paradag Soc., 49, 163 (1053).
tions applied to vapor pressure measurements. It also may be a,ccomplished if sufficiently extensive electromotive force measurements are available to define the activity coefficient of the first solute over t'he entire range of interest of both composition variables. Derivation of Relations The mean activity coefficients of two electrolyte solutes with a common ion are related by the fundamental cross differentiation equation in which m represents molality and v and y possess their usual significance.l 3 The total ionic strength p may be written as fi = PI
and kl
+
-
PZ; PI = klml; PZ = k2m2
W I + ~ I -
2
,. ke
-
w2+22-
2
(2)
(3)
where z represents the absolute value of the valence of the indicated ion. If electrolyte solute 1 obeys Harned's rule, then log Y1 = log
YI(0)
- Ly12P2
(4)
in which. yl(o)is the activity coefficient of electrolyte solute 1 in its own pure solution a t the same total ionic strength p as the mixed solution; y1(0)and 0112 are functions of p only. This relation has been established a t several different ionic strengths in each of a number of systems; in what follows it is supposed that the relation has been established over the entire range of ionic strength of interest, and 0112 and yl(0) are therefore known functions of p over this range. Equation 4 may be substituted in eq. 1 to yield (5)
This is now integrated from zero to ml a t constant m2, under which restriction dml = dp/lcl. The result is U2 & [log YZ
- log
YZCO)
(rdl =
2 {log
Yl(0)
G) -
In eq. 6 , y~ is the activity coefficient of electrolyte solute 2 in the mixed solution containing ionic strengths pl and p2, respectively, of the two solutes, and y2(o)(p2) is that in a pure solution of solute 2 (13) Brackets or braces will enclose quantities to be multiplied; parentheses will be reserved to indicate functional dependence on ionic strength.
WILLIAM 5. ARGERSINGER, JR.,AND DAVIDM. MOHILNER
100
at ionic strength p2. The quantities ~ ~ ( ~ )and ( p ~ I ( o ) ( ~ Irefer ) to pure solutions of solute 1 a t the indicated ionic strengths. Equation 6 may be rearranged with the help of eq. 3 to yield
)
Vol. 6 i
I n eq. 13 and eq. 14 to follow, +1, 42, a12and DLZI are all to be evaluated a t ionic strength p. If eq. 11 and 13 are to be identical a t all values of p and p ~ it may be s h o p that 5 all ha
VI
kl
(y,2
-2;P [ ;
[q51
- 11 - --k2 VZ
1+2
-
,
lI1 (14)
and Thus, if y1(o),yzc0)and a12are known as functions of p , it is possible by means of eq. 7 to compute 7 2 , the activity coefficient of solute 2 in the mixed sol~tion.’~Within the limitations imposed by the experimental knowledge of yl(0), yZ(o)and LYIZ, the results are as valid and a8 direct as those computed for y2(0),for example, by means of the Gibbs-Duhem equation from measurements of vapor pressure or of fueezing point depression. It is often convenient to introduce as a composition variable the ionic strength fraction of solute 1; thus one may write x = P I / P = k t m l / p ; 1 - x = ~ Z / P= k z m d p (8) With this substitution eq. 7 becomes
The relationship between these expressions and those derived by Harneda is most clearly seen by considering the solvent activity in the mixed solution. The appropriate cross differentiation relation may be written in the form
Here & is the practical osmotic coefficient for the mixed solution. After the substitution of eq. 4 and Bjerrum’s relation between the activity and osmotic coefficients for electrolyte solute 1 in its own pure solution, this equation may be integrated to yield
+g
Y2
VI
&d
P )
cr2lb)
{r+i(r)
- P Z ~ ( P Z-) 2.3 P Z [ P w A-~
i
~ 2 ~ 1 2 G 2 ) I
(11)
in which 41 and $12 are the osmotic coefficients of solutes 1 and 2, respectively, in their pure solutions at the indicated ionic strength. Now if Harned’s rule holds as well for solute 2, that is, if log
72
=
log
W(0)
- CrZlP1
(12)
then Harned3 has shown that
(14) The advantage of vapor pressure measurementa over electromotive force measurements aa a meam of determining all the activities in a mixed electrolyte solution lies in the fact that the integration of the cross differentiation relation may be carried from the reference state for the solutes (infinite dilution), and so independent data for yqo) are not necessary.
2
L
+ 6 a21(fi2) v2
adu2)
#fG) (15) The first of these two results is the usual expression for computing 0121 from a12and the osmotic coefficients; the second is the test of the linearity relations proposed by Glueckauf. Thus, eq. 7 and 11 permit the direct calculation of yz and dX,and eq. 12 and 13 with 14 the approximate calculation by means of Harned’s rule. Applications The wealth of careful experimental work done by Harned and his co-workers provides the best material for exploiting the relations here derived. Even so, in most instances the linear rule has been established at only a few different values of the ionic strength, and so a12cannot be said to be very well known as a function of p . It is necessary to use a suitable empirical expression for a12and to limit the computations to the ionic strength range over which 0114 is known. Thus if in a system a12 has been measured for pa. f y f p b , at any particular ionic strength p’ in this range the accessible values of x are given by 0 6 2 f 1 - p a / p ’ . The upper limit on x may be increased to unity only if it is permissible to extrapolate the empirical expression for an to infinite dilution. The equations have been applied to the data of Harned and Gary3 at 25” for hydrochloric acid in solutions containing barium, strontium, aluminum or cerium chloride. The requisite values of ylco), YZ(O),41 and $2 have been obtained by interpolation from values in the literature,16 and the observed values of alzhave been incorporated in expressions of the form 0112
2
=
=
A +B 4%
+ C v’G
(16)
which is suggested by the observed variation of 0112 with p in a number of systems.16 Other empirical expressions which were tested and found to give negligibly different results over the allowed composition range were considered unsuitable for possible extrapolation. The results of the calculations are given as the unprimed quantities in Tables I-IV for the four systems a t the accessible compositions in each case. The primed quantities in the tables are the corresponding values of log yzor dx computed on the assumption that Harned’s rule holds for the second electrolyte solute. Discussion As the tables indicate, the values of the activity and osmotic coefficients calculated by means of the, (15) R. A. Robinaon and R. H. Stokes, “Electrolyte Solutions,” Academic Press, Inc., New York. N. Y., 1955, pp. 488-487. (16) Ref. 1, pp. 468-469.
101
ACTIVITYCOEFFICIENTS IN MIXEDELECTROLYTE SOLUTIONS
Jan., 1957
TABLE I ACTIVITYAND OSMOTIC COEFFICIENTS IN HCl-BaCh SOLUTIONS a18
+ 0.0332 + 0.0135 4,ii
= 0'0184 -
dF
0 2 z
0
0.05 .1 .15 .2 .25 .3 .35 .4 .45 .5 .55 .6 .65
log
Ya
-0.408 .400 .392 .383 .373 .365 .356 .347 .338 .330 .322
log Y l :
Ox
CP
-0.408 .400 .392 .383 .375 .367 .359 .350 .342 ,334 .326
0.886 .911 .933 .954 .975 .994 1.012 1.027 1.044 1.058 1.073
0.886 .910 .932 .952 .972 .990 1.008 1.024 1.040 1,054 1.069
log Y l
-0.403 ,391 .378 .365 .352 .339 .326 .313 .300 .286 .272 .258 .244 .232
log Y? =
C
C'
-0.403 .390 .377 .364 .351 .338 .326 .312 .300 .287 .274 .260 .248 .234
0.934 .966 .996 1.023 1,050 1.075 1.098 1.121 1.143 1.164 1.184 1.203 1.222 1.240
0.934 .966 .996 1.024 1.051 1.075 1.098 1.121 1.142 1.163 1.183 1.202 1.220 1.238
'
TABLEI1 ACTIVITY AND OSMOTICCOEFFICIENTS IN HC1-SrCla SOLUTIONS al*= log
0 0.05 .1 .15 .2 .25 .3 .35 .4 .45 .5 .55 .6 .65 .7 .75
Ya
-0.336 .324 .311 .297 .283 .271 .258 .244 .230 .218 .206 .194 .182 .171
log 7,p
-0.336 .324 .311 .299
,286 .274 .261 .249 .236 .223 .211 .198 .186 .173
1 3
-z-+ 0.0541 f 0.0013 z/F
0*0027
C
C'
1.009 1,038 1.065 1.090 1.112 1.133 1.153 1.172 1.190 1.207 1.223 1.237 1.252 1.265
1.009 1.036 1.062 1.085 1.108 1.128 1.148 1.166 1.184 1.201 1.217 1.232 1.247 1.262
log
YX
-0.236 .213
.189 .168 .147 ,124
.loo
+
.8
.078 .056 ,034 .011 ,010 ,032 .055 .075 .096 .115
log
r - 5 71'
-0.236 .214 .193 .171 .149
.128 .lo6 .084 .062 .041 .019
+ .003
.024 ,046 .068 .089
.111
C
+a
1.193 1.231 1.265 1.298 1.331 1.362 1.392 1.420 1.446 1.471 1.494 1.516 1.537 1.558 1.577 1.596 1.613
1.193 1.232 1.268 1.302 1.333 1.363 1.391 1,418 1.443 1.467 1.490 1.512 1.534 1.554 1.574 1.594 1.612
C 1.246 1.289 1.327 1.361 1.394 1.422 1.450 1.475 1.496 1.518 1.536 1,554 1.569 1.585 1.599 1.014 1.628
C' 1.246 1,288 1.326 1.358
TABLE I11 ACTIVITY AND OSMOTIC COEFFICIENTS IN HCl-AlClo SOLUTIONS 0.0052 f 0.0674 0.0009 4,ii alt = T - 3 log ,xp +a 6' log Y I log YIP -0.480 1.008 1.008 -0.352 -0 352 .464 1.048 1.043 .318 .321 .447 1.084 1.073 ,284 .291 .430 1.111 1.100 .253 .260 ,413 1 136 1.123 .222 .229 .397 1.158 1.145 .188 199 .380 1.177 1.164 .153 ,168 .363 1.194 1.182 .121 .138 .347 1.210 1.199 .088 .lo7 .330 1.225 1.215 .057 .076 .313 1.240 1.230 .025 .046 .297 1.253 1.244 f .003 .015 .280 1.265 1.257 .032 4- .015 .263 1.277 1.270 .061 .046 .087 .077 .115 .lo7 .139 .138
-
z
0 0.05 .1 .15 .2 .25 .3 .35 .4 .45 .5 .55 .6 65 .7 I75 .8
log Y I
-0.480 .462 .443 .424 .406 .389 .372 .355 .337 .321 .306 .289 .272 .259
-
-
.
1.388 1.415 1.440 1.462 1.484 1.504 1.523 1.542 1.559 1.576 1.592 1.608 1,623
WILLIAM J. ARGERSINGER, JR.,AND DAVIDM. MOHILNER
102
Vol. 61
TABLE IV ACTIVITY AND OSMOTIC COEFFICIENTS IN HCl-CeC13 SOLUTIONS < X
0
0.05 .1 .15 .2 .25 .3 .35 .4 .45 .5 .55 .6 .65
og 7 2
-0.585 .580 .575 .571 .566 .563 ,558 ,554 .549 .546 .543
log r /
-0.585 .581 ,577 .573 .568 .564 .560 .555 .551 .547 .543
a12
3 2
=
-o’0202 di + 0.1122 - 0.0051 4,ii ,!J
c
C’
log ya
0,847 .874 .895 .916 .937 .956 974 .990 1.007 1.023 1.038
0.847 .871 .893 ,914 ,934 .953 .971 .988 1.006 1.022 1.039
-0.578 .569 .558 .549 ,540 .532 525 .517 .509 ,502 .495
cross differentiation relations are but slightly different from those calculated on the assumption of Harned’s rule for the metallic chloride. The differences, however, are considerably larger than the variations from linear behavior for the hydrochloric acid, for example, or the experimental uncertainties in the independent values for the separate pure solutions, or finally, the quite small variations arising from the use of different reasonable empirical expressions for a12. That the particular choice of expression used for a12 is not itself responsible for the deviations from Harned’s rule may be inferred from comparisons of the osmotic coefficients at compositions for which a12is specifically known from experiment; for example, & = 1.496 and &’ = 1.484 in the hydrochloric acid-aluminum chloride system a t p = 5, x = 0.4,and in the hydrochloric acid-barium chloride system at p = 2, x = 0.5 the values are 1.073 and 1.069, respectively. The general consistency in the trend of calculated values in a given system a t a specified total ionic strength further argues against random errors in the interpolations used to obtain y1(0),y2(0),41and $2 as the source of the devia.tionsfrom Harned’s rule. Glueckauf’s test was applied by Harned and Gary to their results in all four systems and it showed that Harned’s rule does not hold precisely. This test, however, in its very nature is extremely sensitive; for these systems the left member of eq. 15 varies over the appropriate ionic strength
.488 ,481 .476
log
ya
-0.578 ,571 .563 .555 .548 ,540 .532 .525 .517 ,510 .502 .494 .487 ,479
= 3 9r
0.914 .945 ,974 ,998 1.022 1,045 1.067 1.088 1.109 1.129 1.150 1.170 1.190 1.209
d.‘
0.914 .940 ,964 .988
1.011 1.034 1.056 1.078 1.099 1.121 1.142 1.163 1,184 1.204
ranges by from 7.7 to 44.4%, instead of remaining constant as required by the test, but still Harned’s rule may be said to hold with only slight deviations. As Robinson17has pointed out, the effect of deviations from Harned’s rule is much greater with respect to solute activity coefficients than with respect to solvent activity; the calculated vapor pressures of the solutions are not here presented because they differ so slightly from those computed by the approximate method. The computed values of log y2 may be fitted to quadratic expressions as suggested by Robinson17; the coefficients of the quadratic terms are generally quite small. The deviations from linearity are in most cases rather less than the difference between the straight line determined by the computed values of log y 2 and that required by the usual calculation of a 1 2 through eq. 14. It may be concluded that the activity coefficients of the second electrolyte solute satisfy quite well a linear relation such as eq. 12, but the slope of the line is given less accurately by eq. 14, because the latter is based on the premise that Harned’s rule holds precisely. Acknowledgment.-This work is part of a project supported by a grant from the General Research Fund of the University of Kansas, and was also aided by a summer grant from the E. I. du Pont de Nemours Company. (17) R. A. Robinson, in “Electrochemical Constants,” National Bureau of Standards Circular 524, Washington, D. C., 1953,p. 171.
a
*
