Oscillations, Waves, and Interactions - GWDG

Multistep association of cations and anions 345
Figure 10. Graph of a Debye relaxation function (αλ)exc = Aωτ/(1 + ω 2 τ 2 ), full line and
of a Debye function with small differential term ∆Bν added: ∆B = 0.02πτA, dashed line;
∆B = 0.04πτA, dotted line.
complexes. The high-frequency relaxation region, however, is discussed in terms of
a distribution function theory in which no distinct ion complexes are assumed a
priori [32]. Another approach that does not proceed from the existence of stoichiometrically
defined complex structures but from long-range concentration fluctuations
is briefly presented in the discussion of zinc chloride solution spectra.
Figure 11 shows ultrasonic excess attenuation spectra for solutions of scandium
sulfate in water. These spectra obviously reveal three relaxation regions within
the frequency range of measurements. The three Debye relaxation terms are indicated
by dashed curves. They have been assigned to the coupled scheme [17]
� Sc 3+ �
aq + � SO 2−
4
k23
⇀↽
k32
�
aq
k12
⇀↽
k21
� Sc 3+ (H2O) SO 2−
4
� 3+
Sc (H2O) 2 SO 2−�
4 aq
�
aq
k34
⇀↽ (ScSO4)
k43
+
aq ,
corresponding with Eq. (1). Evaluation of the relaxation parameters of Sc2(SO4)3solution
spectra at salt concentrations between 0.0033 and 0.1 mol/l enabled a complete
characterization of the stepwise association scheme. As the pH of the solutions
was adjusted at 2.4, at which the sulfate ion is partly protonated, the protolysis
reaction
� Sc 3+ �
aq + � SO 2−
4
�
aq + H3O + ⇀↽ � Sc 3+�
aq + � HSO − 4
�
(11)
aq + H2O , (12)
coupled to the stepwise association mechanism, has been also taken into account [17].
The forward and reverse rate constants kij and kji, respectively, following from the
evaluation of the ultrasonic spectra are listed in Table 1 where also the equilibrium
constants
Ki = kij/kji , (13)
the association constant
Ka = K1[1 + K2(1 + K3)] (14)