3890
NOTES
A Further Examination of the Additivity Rule by Jacob A. Marinskyl Department of Chemistry, State University of N e w York at Buffalo, Buffalo, N e w York 14114 (Receised A p r i l 6, 1971) Publication costa borne completely by The Journal of Physical Chemistry
Isopiestic vapor pressure measurements of the ternary system, sodium polyacrylate (NaPAA)-NaCI-H20 were reported in a recent paper by Okubo, Ise, and lIatsui.2 They employed a transform of the Gibbs-Duhem equation, previously derived by IlcKay and Perringj3for the computation of the mean activity coefficient of polyelectrolyte and simple electrolyte in the experimental mixtures. To express the mean activity coefficient of the simple electrolyte in these mixtures the isopiestic ratio, R, was assumed to be a function of the following type
R
= 1
- ax - bx2
(1)
otherwise thermodynamically rigorous treatment thus seems inappropriate. The quantities R and x have consequently been redefined by the elimination of p to facilitate the analysis that follows. When this is done, in the region where R is essentially linear ( R = 1 - ax) interaction between polyelectrolyte and salt is apparently absent. (The average deviation in the value of a was only very slightly improved by employment of the quadratic equation.) If this is indeed the case the colligative properties of the NaC1-NaPAA mixtures in these experimental situations should be additive, each component retaining its characteristic osmotic coefficient at the experimental water activity. Also, since in these isopiestic studies the chemical potential of the solvent in the various mixtures is equal 42m2
+ 24ama = 24aMa +
By dividing both sides of eq 4 by (m2 rn3)~$~ and subtracting unity from each side of the resultant equation we obtain
I n this equation
R
=
2Ma
2M3/[2ma
+ + l)(m~/x)l (a
- 1 (5)
(2)
Eventually
and
x
(4)
= polymer ionic fraction =
(a
+ l)(mz/x)/Pm3 + ( a + l ) ( m & ) l
(3)
where m2 = concentration of polyelectrolyte in polyelectrolyte-KaC1 solution in equivalents per 1000 g of water, ma = molality of NaCl in polyelectrolyte-NaC1 solution, M3 = molality of NaC1 with same solvent vapor pressure as ternary solutions of total ionic con(a l ) m z / x , a = net valency of centration 2ms macroion (=px), and z = stoichiometric valency of macroion. To calculate R and x, the net valency of the polyelectrolyte, a, \vas defined as the fraction, p, of x ionizable groups that are ionized. The value of p was assigned from the results of transference experiments. Such an assignment is based on the conclusion that the fraction of counterions that are not free to move in solution (1 - p) is rigidly attached to the polyion and moves with it forming an integral part of a new “complex” macroion. This association concept has been criticizedS6 Indeed the appropriate interpretation of the electrochemical properties of polyelectrolytes has not been resolved. Incorporation of a p term obtained in this manner in an
+
+
T h e Journal of Physical Chemistry, Vol. 76, No. 16, 197’1
By substituting the definitions of R and x eq 6 takes the form 1
+ (& - 1)x
=
R
(7)
to lead to the identification of the a parameter of eq 1 with (1 - 4 2 / 4 3 ) when b = 0. The validity of this assignment is shown in Table I given below by the excellent agreement between a, computed from the isopiestic data, in eq 1 ( b = 0) and (1 - 42/48), calculated by (1). Visiting Professor, 1970-1971, Chemistry Department, McGill University, Montreal, Quebec, Canada. (2) T. Okubo, N. Ise, and F. Matsui, J . A m e r . Chem. SOC.,89, 3697 (1967). (3) H. A. C. McKay and J. K . Perring, Trans. Faraday SOC.,49, 163 (1953).
(4) (a) T. Okubo, Y. Nishizaki, and K.Ise, J . P h y s . Chem., 69, 3690 (1965); (b) F. T. Wall and J. J. Eitel, J . A m e r . Chem. Soc., 79, 1556 (1957). (5) A. Katchalsky, 2. Alexandrowicz, and 0. Kedem, “Chemical Physics of Ionic Solutions,” B. E. Conway and R. G. Barradas, Ed., Wiley, New York, N. Y., 1966, Chapter 15.
NOTES
3891
using osmotic coefficient values corresponding to the pure components at the solvent vapor pressure employed to obtain a given set of data.
expressed by the incorporation of an additional term whose characteristic property is such that it will be of negligible magnitude in those systems where the ratio (62/43is not crowding unity. The addition of the empirical term 1 / 3 ( 4 2 / + 8 ) satisfies this requirement
Table I : Comparison of a Parameter Computationa for NaPAA-NaCl-H20
-~ ( R -5 1)
Set
(1
= ab
0,666 0.638 0.616
1 2 3
(9)
with the b parameter corresponding to
-:)-a
0,658 0.640 0.621
-('")3
3 43
a The same net result is obtained with u'se of a term. This equivalence is due to the fact that p is assigned constancy in a simplifying approximation of the transference results. However, strict interpretation of transference data according to the complexation model would not yield a constant p, and this resemblance of both approaches is significant only in that it provides support for the analysis that is recommended. b The average deviation in the value of a was only very slightly improved by employment of the quadratic equation.
On the basis of this analysis it seems appropriate to suggest that the earlier empirical additivity rule6-' that has been employed to correlate the colligative properties of dilute polyelectrolyte-salt mixtures be modified by using in eq 4 the practical osmotic coefficient values corresponding to the pure components at the solvent vapor pressure of the mixture rather than to their values at the experimental polyelectrolyte and salt concentrations of the mixtures as in the earlier application of this empirical relationship. I n the more concentrated NaPAA-NaCl-N20 systems (sets 4 and 5 ) R is best described by the quadratic in IC to indicate the failure of any simple additivity relationship. It is interesting to note, however, that in these systems
This result, demonstrated in Table I1 presented below,
Table I1 : A Comparison between the Sums of a and b Evaluated with Eq 1 and 8 a
b
(eq 1)
(8s 1)
a f b (es 1)
a-tb
Set
4 5
0.492 0.354
-0.168 -0.305
0.234 0 049
0.233 0.052
I
(eq 8)
may be Of importance- It suggests that the a parameter defined by (1 - 42/43) may be better
With eq 9a the value of b is indeed small when the is small and increases rapidly in value when ratio ~$~/43 the ratio approaches a value of unity. A comparison of b computed with eq 1 and 9a is presented in Table 111. The correlation is acceptable t o support this kind of interpretation of the observed results.
Table 111: Comparsion of Computed b Parameters for NaPAA-SaCl-H20
-0.002 +0.014 -0.026 -0.168 -0.305
-0.013 -0.016 -0.018 -0.150 -0.284
It is believed that an important deficiency of the empirical additivity rule previously proposed to describe the colligative properties of polyelectrolyte, simple salt mixture^^-^ has been identified. Modification of this relationship that is facilitated by the excellent data of Ise and coworkers has yielded a more quantitatively correct expression. The real utility limits of this modification of the additivity rule may have been defined as well. When the ratio +2/+3 exceeds 0.5 t o 0.6, the simple expression (eq 4) cannot be employed and resort to the use of eq 9 and 9a in eq 1 may provide a more generally applicable relationship for anticipating the colligative properties of polyelectrolyte-salt mixtures over a much extended concentration range. Acknowledgment. The author wishes to express his appreciation t o the U. S. Atomic Energy Commission for financial support through Contract No. AT(30-1)2269. (6) Z. Alexandrowice, J . Polym. Sci., 43, 337 (1960). (7) z. Alexandrowicz, ibid., 56, 97, 116 (1962).
The Journal of Physical Chemistry, Vole76, N o . 36?1971
