1680
Hiroshi Maeda
Thermodynamic Analysis of the Potentiometric Titration of Aggregation Systems Hiroshl Maeda Department of Chemistry, Faculty of Science, Nagoya University, Nagoya, Japan (Received May 29, 1974; Revised Manuscript Received February 7, 1975)
A thermodynamic analysis of the potentiometric titration of aggregation systems with or without phase separation is given. The effect of aggregation on the titration properties is shown to be composed of essentially two termw, representing the effects on the chemical potentials of polyion and solvent. The nonelectric free energy change associated with aggregation is shown to be unaffected even if the intrinsic dissociation constant of dissociable sites changes as a result of aggregation. For certain types of phase separation, the nonelectric free energy change can be obtained from the titration data in just the same manner as in the case without phase separation.
Introduction Aggregation of polyions is considered to affect their titration properties profoundly. Accordingly, we can expect to obtain information about the nature of the interactions responsible for the aggregation from an analysis of titration data. An excellent example has been presented on a wellcharacterized aggregation system.l On the other hand, titrations of aggregation systems are sometimes encountered which are poorly characterized, for example, solutions of polypeptides in &coil equilibrium.2 Another example is the micellar solutions of ~ u r f a c t a n t salthough ,~ in the latter example the monomer is not a polymeric material carrying many charges. Characterizations of these systems by means of various methods are undoubtedly of primary importance. However, thermodynamic analysis of the titration. data of these systems will be also effective for their approximate characterizations. However, a theoretical basis to analyze the titration data of aggregation systems has not been satisfactorily presented. It is uncertain, for example, whether the titration properties of aggregation systems can be related to the chemical potential of the polyelectrolyte alone, as in the titration of ordinary polyelectrolyte solutions where no aggregation occurs, (See eq 3.) The largest effect due to aggregation of polyions will arise from the drastic change in charge density. Furthermore, the aggregation reduces the number of solute particles in the solution, and this will be expected to affect, at least conceptually, the chemical potentials of every species present in the solution. Also, aggregation sometimes affects the intrinsic dissociation constant of dissociable sites. The purpose of the present study is to examine how these effects of aggregation can be taken into consideration in the potentiometric equation and to estimate the extent of the effects on the titration behavior. Some aggregation is accompanied by phase separation and hence the effects of phase separation are also examined to some extent. The effects of polydispersity are not considered here, since the extension to polydisperse samples can be made in a formal way. The results derived from the present analysis will be applied to experimental data, which will exhibit the usefulness as well as the limitations of the present approach. Theoretical Description of the System. Consider a solution containThe Journal of Physical Chemistry, Vol. 79, No. 16, 1975
ing four components: water (w), poly(weak acid) (p) (e.g., polyacrylic acid), alkali (a) (e.g., NaOH), and simple salt (s) (e.g., NaCl). To discuss a solution in a state of ionization lower than the self-ionization, another four component system will be considered where an acid component (ac) (e.g., HC1) is introduced in place of an alkali component to supress self-ionization. Let us denote the numbers of moles of these components as n's with corresponding subscripts. Counterions from the alkali and those from the simple salt are assumed to be of the same species. A similar assumption is made about the coions from the acid and the salt. Furthermore, we assume that there are two different states with respect to the chain conformation of polymers in the solution: random coil conformation (denoted by C) and another conformation (denoted by A) which can form aggregates. C e A + 4 _L **e e4 9.0
The numbers of the respective species are denoted by n , and nt ( t = 1 , 2 , 3 , . . .) and their degrees of ionization by a, and at. The average degree of ionization of the whole solution a is given as follows:
Here denotes the average degree of ionization of the A conformation and f the fraction of the polymers in random coils, i.e., f = nc/np. The Gibbs-Duhem relations for the respective four component systems are ~p
dPp
+
(na
+ ns) dP+ -I-
np dPp + n, d ! ~ ++ (nS + a ),
dP- +
d ~ +.
dPH (2') Here wup,ww, w+, w-, and WH represent the chemical potentials of polymer component, water, counterion, coion, and hydrogen ion, respectively. The polymer component is here defined as a fully protonated uncharged specie^.^ The number of dissociable sites on a polymer molecule is denoted by nw dPL, =
Macroscopic Expressions for the Equation of Titration. Potentiometric titrations of ordinary polyelectrolyte solutions where neither conformational change nor aggregation occurs can be described by the following equation:6
Potentiometric Titration of Aggregation Systems app/apH
=
e-%i\p& 9
(3)
f f x
Combining eq 3 with eq 2 or 2', we have for a change at constant np, n,, and n,
Here n+ and n- denote the number of moles of counterions and coions. The number of excess hydrogen ions, nH, can be given by
nH = (n,ax - nJ or
(?Z,LYX
+
n,)
f nH
dPH
n+ dP+ YZ- d p - -t 12,
n,d'(€) =
[-n,X
