4425
C O M M U N I C A T I O N S TO T H E E D I T O R
The Spectrophotometric Determination of Association Constants for CuSO4 and Cu(en)zSzOa
S i r : Some years ago we’ studied the formation in aqueous solution of the ion pairs CuSO4 and Cu(en)zSZO3(en = ethylenediamine) by measuring the optical densities of the appropriate constant ionic strength mixtures. For each ion pair, a range of values of K was obtained from these optical densities,2 the precise value depending on the activity coefficient assumption employed. However, Hemmes and Petrucci, who recently made similar measurements, argue3 that spectrophotometry can yield unambiguous (although not very precise) values of these association constants. Their arguments are that, for each ion pair, plots of log K’ 8 A g / r / l + BdZ/i us. 1 for different values of d converge as I + 0 and that, for CuSO4, these plots give a value of K which agrees with “the conductance value.” The first argument undoubtedly shows the calculation of K from K’ to be insensitive to the activity coefficient assumption used. However, to prove that spectrophotometry gives an unambiguous value of K one must also show that values of K‘ derived by this method do not depend on one’s activity coefficient assumption. Thus, consideration must be given to the role of the activity coefficient in the calculation of K’ from optical densities, a point not discussed by Hemmes and Petrucci. To calculate K’ from optical densities one needs a figure for Ae, the difference between the extinction coefficients of the ion pair and the free ions. Since this quantity is generally unknown, it must be determined simultaneously with K’ by finding a value of Ae which gives figures for K‘ such that K is constant. Since K = K ’ / ( Y * ) ~an ) activity coefficient assumption is required in this operation. Moreover, for species like of Ae which is CuSO4 and C ~ ( e n ) ~ S ~ the 0 ~value , obtained from the optical densities of a given set of solutions can be shown to depend on the assumption made about the variation of -y+ within the set. With different assumptions about this variation, different values of Ae and different sets of values of K’ may be obtained f r o m the same set of optical densities. For example, in our studies of CuSO4 and Cu(en)zSzOl,we considered the possibility that, at constant I
+
log
y+ =
log Y*O
+ ab
(1)
with r*OJa! constants (cf. Harned’s rule) and b the concentration of S02-/S2032-. We found that the value of Ae which was consistent with the optical densities of one
of our sets of solutions depended on the figure assumed for a. Unless this quantity was arbitrarily fixed, a range of values of AE was consistent with such a set of optical densities. Further, with this range of values of Ae a range of values of K’ was obtained for a given solution. Hemmes and Petrucci obtained K’ by a graphical method4 which assumes y+ to be constant at constant I and thus fixes a at zero. It is therefore not surprising that they obtained a unique value of K’ for each of their solutions. However, since activity coefficients are not necessarily strictly constant at constant 116 a may well not be zero. Table I
A B C
D
a
108/Aa
KP
3 . 4 M-1 0 M-‘
8 . 1 cm M 6.3“ cm M 5 . S b cm M
42.8 4 1.OM-1 35.3 4 0.9 M-1 31.3 rt 0.8 M-l 26.6 f 0 . 5 M-1
0
M-1
-1.25 M-l
4.9 cm M
’The mean of the figures in Table 111, ref 2. of the slope of the plot of ab/D - D‘ us. (a b deviations indicated by &.
+
Our estimate
- 2).
Mean
Table I1 108 a / M
108 b / M
2.63 2.63 2.63 2.63 2.63 2.63 2.63
8.00 12.00 16.00 20.00 24.00 28.00
4.00
-
a ( A = (D Table 11, ref 3.
D’)caiod
10aA(A)a
-2 +1 0 $2 +3 -1
-1
- (D -
108A(B)a
0
+3 +1 $1 +l -5 -4 D’)obsd).
108A(C)a
10*A(D)a
-1 $2 +1 $2 $3 -2 0
0 $3 0 +1 +2
(D -
-4
-3 D’)obsd
from
We have analyzed Hemmes and Petrucci’s optical data6 for Cu(en)zSzOl assuming eq 1 to apply at constant I and find that the values of Ae and K’ which (1) R.A.Matheson, J . Phys. Chem., 69,1537 (1965);(b) R.A. Matheson, ibid., 71, 1302 (1967). (2) K = (nCuSO1)/(~Cu2*~5042-) or (“Cu(en)zSzOs)/(aCu(en)zz+ %Oa2 -). The corresponding concentration quotients will be denoted by K’. (3) P. Hemmes and S. Petrucci, J . Phys. Chem., 72, 3986 (1968). (4) Discussed in ref lb. (6) See, e.g., R. A. Robinson and It. H. Stokes, “Electrolyte Solutions,” Butterworth and Co., Ltd., London, 1965,Chapter 15. (6) Table 11,ref 2. Volume 76,Number 18 December 1960
4426
COMMUKICATIONS TO THE EDITOR l
L-O,
u I
2.6
LL
2.4
o
-/ ” 1
f
d=5%
2.2 1
d
1
0
0.05
0.10
0.05
0.10
Ib
0
2.6
2.2
i
O
l
lL i--
= d
\
‘ 0.05
0.10
I. Figure 1.
are consistent with the results for any one of their sets of solutions depend on the figure assumed for a and, unless this is zero, differ from the values originally obtained by these authors. Table I lists some values of l / A e ’ LY and KO’ (defined below) which are consistent with their data for solutions of I = 0.0875 M . Although the values of l / A e corresponding to the three figures for a are very different, each gives rise to a set of K’ values such that log KO’ = log K’ - 2ab is constant and therefore (cf. eq 1) K ’ / ( Y & )is~ constant. Of course if LY # 0, K’ is not constant and cannot be expected to be constant. We have also used each of the sets of KO)’ LY and Ae in Table I to calculate the concentrations of C ~ ( e n ) ~ and S ~ the 0 ~ values of D - D’ for all the solutions with I = 0.0875 M . The results (see Table 11) confirm that these four sets of parameters differ little in their consistency with the experimental optical densities of these solutions. It seems that unique values of l / A e and K’ cannot be obtained from these optical densities unless some arbitrary restriction The Journal of Physical Chemistry
’
+
+
(7)
I
0
is placed on yi, e.g., if a is fixed at zero. The position can be shown to be similar in the case of Hemmes and Petrucci’s other solutions for which different values of a again given different figures for l / A e and K’. Thus, if a is not arbitrarily fixed, Hemmes and Petrucci’s optical densities produce a range of values of KO‘ at each I . However KO’ varies with I and there remains the possibility that the extrapolation procedure used by Hemmes and Petrucci may produce a definite figure for K despite the ambiguity in KO1. We have therefore constructed plots of F(K0’) = lo$ KO’ 8Adj/ 1 B d d j vs,I ford = 0,5, and 10 A using values of KO’ calculated from Hemmes and Petrucci’s optical densities for their solutions of I = 0.0335,0.0465, 0.0664, and 0.0875 M8 assuming (a) a = 3.4 103/Ae = 8.1 cm M ; (b) a = 0, 103/Ae = 6.3 cm M; and (c) a = -1.26 M-’, 103/Ae = 4.9 cm M . In each case (see Figures la, lb, and IC) the plots for the three values of d converge satisfactorily as I + 0. However, the limits to which they extrapolate correspond to different values of K (275 to 347 M-l for Figure la, 195 to 246 M-I for lb, and 141 to 178 M-’ for IC). Evidently, this extrapolFtion does not eliminate the ambiguity which results from the dependence of K‘ and Ae upon the activity coeficient assumption used to calculate these quantities from the experimental data. In regard to the agreement between Hemmes and Petrucci’s figure for Gus04 and “the conductance value,” it is in our view misleading to speak of “the conductance value” when, for electrolytes like CuSO4, the value of K which is obtained from conductances depends on the assumptions made in the calculation^.^ KO,’ = lim K’ for the constant ionic strength series in question. b-+O
(8) The set of solutions with I = 0.109 was not considered because the points for this set diverged from the original graphs (Figure 2, ref 2). (9) J. E. Prue, “Ionic Equilibria,” Pergamon Press, Ltd., London 1966, Chapter 3. D E P A R T M E N T O F CHEMISTRY,
THEU N I V E R S I T Y
R. A. MATHESON
O F OTAQO
I)UNEDIN, N E W ZEALAND
RECEIVED APRIL28, 1969
On the Reliability of the Spectrophotometric Determination of Association Constants. The Case of CuSO4 and Cu(en)&Oi
Xir: We wish to start our answer from the concluding point of Dr. Matheson’s statements,’&namely from the claimed unsuitability of the conductance method to determine a unique value of association for electrolytes like CuSOa. (1) (a) R. A . Matheson, J . Phys. Chem., 4425 (1969) ; (b) J. E. Prue, “Ionic Equilibria,” Pergamon Press, Ltd., London, 1966.
