APPENDIX IV
Separation of contributions in aqueous and micellar media for two coupled reaction at the equilibrium.
The reaction between a metal, M, and a ligand, L, can be described by the two step
mechanism kf1 M + L ⇄ (ML)I (IV.1) kd1 kf2 (ML)I ⇄ (ML)II (IV.2) kd2
Reactions (IV.1) and (IV.2) (charges omitted) are coupled by means of ML.
A surfactant containing solution is formed by two pseudo-phases (aqueous and micellar
phases) that are homogeneously mixed. In such a solution, depending on the reactants
affinity for each phase, a complexation reaction can take place only in water, in micelle or
in both media. In the last case the overall concentrations can be expressed as
CL T = CL W + CL S (IV.3) CM T = CM W + CM S (IV.4) CML T = CML W + CML S (IV.5)
where the superscripts “T”, “W” and “S” indicate the concentrations in the whole solution,
in water and in micelle, respectively.
Differential forms of equilibrium concentrations for each species are given by eqs
(IV.6)-(IV.8). δ[L]T= δ[L]W + δ[L]S (IV.6) δ[M]T = δ[M]W + δ[M]S (IV.7) δ[ML]T = δ[ML]W + δ[ML]S (IV.8)
The relationship between the reactants concentrations in water and in micelle are expressed
by the retention coefficients
RL = CL W /CL S (IV.9) RM = CM W /CM S (IV.10) RML = CML W/C ML S (IV.11)
that were already defined in chapter 3.
Retention coefficients can be expressed also by using the equilibrium concetrations
differential forms: RL = δ[L] W /δ[L]S (IV.12) RM = δ[M] W /δ[M]S (IV.13) RML = δ[ML] W /δ[ML]S (IV.14)
When using pseudo-first order conditions (CM ≥ 10CL), the kinetic law for step (IV.1),
corresponding to the fast kinetic effect, can be expressed for both the aqueous and micellar
medium as reported in eqs (IV.15)-(IV.16).
€ −dδL W dt = kf WC MWδLW− kdWδ[ML]W (IV.15) € −dδL S dt = kf SC M S δLS− kd S δ[ML]S (IV.16)
The mass conservations can be written in differential form, for both phases, as
δ[M] + δ[ML] = 0 (IV.17)
δ[L] + δ[ML] = 0 (IV.18)
Introduction of eq (IV.18) in eqs (IV.15) and (IV.16) gives
€ −dδL W dt = kf WC M W δLW+ kd W δ[L]W (IV.19) € −dδL S dt = kf SC M S δLS+ kd S δ[L]S (IV.20)
expressed as € 1 τ fast W = kfWCMW+ kdW (IV.21) € 1 τ fast S = kf SC M S + kd S (IV.22)
Combination of eqs (IV.19)-(IV.22) gives
€ −d δ[L] W +δ[L]S
(
)
dt = 1 τ fast W δ[L]W+1 τ fast S δ[L]S (IV.23)Introduction of eqs (IV.6) and (IV.12) in eq (IV.23) leads to
€ −dδ[L] T dt = 1 τ fast W RL (1+ RL) δ[L]T+1 τ fast S 1 (1+ RL) δ[L]T (IV.24)
Therefore the rate constant for the reaction in the overall solution corresponds to
€ 1 τ fast T =1 τ fast W RL (1+ RL) +1 τ fast S 1 (1+ RL) (IV.25)
Insertion of eqs (IV.21) and (IV.22) in eq (IV.25) allows to express
€
1 τfast T
as a function of
the metal concentration in each phase.
€ 1 τ fast T = kf W CMW + kd W
(
)
RL (1+ RL) + kf S CMS + kd S(
)
(1+ R1 L) (IV.26)By introducing eqs (IV.7) and (IV.10) into eq (IV.26), one can obtain the 1/τT dependence
on the total metal concentration in solution, CM T . € 1 τ fast T = kf W RM (1+ RM) RL (1+ RL)+ kf S 1 (1+ RM) 1 (1+ RL) ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ CM T + kd WR L+ kd S
{
}
(1+ R1 L) (IV.27)that corresponds to eq 3.14 of chapter 3.
Eq (IV.27) represents a straight line, whose slope and intercept depend on the forward and
reverse kinetic constants both in water and in micellar solution. Being the kinetic constants
in water (kf W
and kd W
) known from literature, it is possible to determine kf S
and kd S
Reactive step (IV.2) corresponds to the slow kinetic effect of the whole mechanism. As
demonstrated in Appendix IIIC, the rate constant of the slower of two coupled reactions
can be expressed as € 1 τslow T =K1kf 2CM T 1+ K1CMT + kd 2 (IV.28)
where K1 represents the equilibrium constant of reaction (IV.1) and
€
1 τslowT is the rate
constant of the slow step in the overall solution. The same equation can be used to describe
the rate constants in aqueous and micellar phases:
€ 1 τslow W =K1 Wk f 2 WC MW 1+ K1WCMW + kd 2 W (IV.29) € 1 τslow S =K1 Sk f 2 S C M S 1+ K1SCMS + kd 2 S (IV.30)
The overall kinetic law for step (IV.2) can be expressed as the sum of the contributions in
both the phases, in analogy with eq (IV.23).
€ −d δ[MLI] W +δ[MLI]S
(
)
dt = 1 τslow W δ[MLI] W +1 τslow S δ[MLI] S (IV.31)Introduction of eqs (IV.8) and (IV.14) in eq (IV.31) gives
€ −dδ[ML] T dt = 1 τslow W RML (1+ RML) δ[ML]T+1 τslow S 1 (1+ RML) δ[ML]T (IV.32)
and the rate constant for the slower reactive step in the overall solution is provided by eq
(IV.33). € 1 τslow T =1 τslow W RML (1+ RML) +1 τslow S 1 (1+ RML) (IV.33)
By introducing eqs (IV.29) and (IV.30) in eq (IV.33) it is possible to express
€
1 τfast T
as a
function of the metal concentration in water (CM W
) and in micellar phase (CM S
€ 1 τslow T = K1 Wk f 2 WC MWRML 1+ K1WCMW + kd 2 WR ML+ K1Skf 2S CMS 1+ K1SCMS + kd 2 S ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 (1+ RML) (IV.34)
Insertion of eqs (IV.7) and (IV.10) in eq (IV.34) gives eq (IV.35), where
€
1 τslowT depends on
the total metal concentration.
€ 1 τslow T = K1 W kf 2 W RMRML 1+ RM+ K1 W RMCM T + K1 S kf 2 S 1+ RM+ K1 S CM T ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ CM T (1+ RML) + kd 2 W RML+ kd 2 S
[
]
(1+ R1 ML) (IV.35)Eq (IV.35) corresponds to eq (3.20) of chapter 3. All parameters in water are known from
literature. Once known K1 S
, for instance from static experiments, it is possible to derive the
kinetic constants of the slow effect in the micellar phase (kf2 S
and kd2 S
), without the
