APPENDIX III

A. Derivation of the relaxation time equation for a reaction at the equilibrium The reaction between M and L is represented by the first order reaction respect to each reagents

kf

Mf + Lf ⇄ MLT (IIIA.1)

kd

whose kinetic law can be written as

€

−d[Lf]

dt = kf[Mf][Lf] − kd[MLT] (IIIA.3)

δ[L]=[Lf ]eq −[Lf ] (IIIA.4)

δ[ML] = [MLT ]eq − [MLT ] (IIIA.5)

and the (IIIA.2) becomes

€ d

(

_{[ ]}

[ ]

)

(

[ ]

[ ]

)

(

[ ]

[ ]

)

(

[

]

[

]

)

_{[ ]}

[ ]

[ ]

(

)

[ ]

The mass conservations can be written in differential form as

δ[M] + δ[ML] = 0 (IIIA.8)

Introduction of equations (IIIA.9) in (IIIA.7) reasonably assuming that product δ[M]δ[L] is negligible and taking into account that kf[Mf]eq[Lf]eq = kd[MLT]eq equation (IIIA.7) reduces to

€

−dδ L

[ ]

δ L

_{[ ]}

{

(

)

}

Integration of equation (III.10) yields

€

δL =δL0e−tτ (IIIA.11)

where 1/τ (s-1

) is the time constant of reaction (IIIA.1) and is expressed by equation (IIIA.12)

€

1

τ = kf

(

)

For CM ≥ 10CL (pseudo first order conditions), equation (IIIA.12) becomes

€

1

τ = kfCM + kd (IIIA.13)

which corresponds to equations (3.13) of chapter 3, (4.16) of chapter 4 and (5.24) of chapter 5.

  B. Derivation of the relaxation time equation for a dimerization reaction

The dimerization reaction for a complex ML can be described by the reaction kf

2ML ⇄M2L2 (IIIB.1)

kd

where ML is the complex formed starting from M and L. In this case the kinetic law corresponds to € d[M₂L₂] dt = kf[ML]eq 2 − kd[M2L2]eq (IIIB.2)

By using the same procedure described in eqs (IIIA.3)-(IIIA.7), relationship (IIIB.3) is obtained

€

dδ[M2L2]

dt = 2kf[ML]eqδ[ML] − kdδ[M2L2] (IIIB.3)

The mass conservation equation is

CML=[ML]eq+[M2L2]eq (IIIB.4)

whose differential form is given by eq (IIIB.5)

δML + 2δM2L2 = 0 (IIIB.5)

Introduction of eq (IIIB.5) in eq (IIIB.3) reduces to

€

dδ[M2L2] δ[M2L2]

= 4k

{

}

that integrated yields

€

δ[M2L2] = δ[M2L2]0e−tτ (IIIB.7)

where 1/τ (s-1

) is the time constant of reaction (IIIB.1) and is expressed by equation (IIIB.8)

€

1

τ = 4kf[ML]eq+ kd (IIIB.8)

that gives a straight line, whose slope and intercept correspond to 4kf and kd, respectively and that corresponds to eq (5.11) of chapter 5.

As the exact value of [ML]eq is not known, evaluation of 1/τ must be performed by iteration: at the first step [ML]eq in eq (IIIB.8) is replaced by CML. Thus, dependence of 1/τ on CML gives a first approximation of kf and kd, whose rate, kf/kd, corresponds to the apparent constant Kapp.

Introduction of Kapp in eq (IIIB.9) allows to determine the equilibrium concetration of the dimer, [M2L2]eq. € [M₂L₂]_eq= 2C_ML+ Kapp −1

(

)

(

)

By using eq (IIIB.4) is possible to determine [ML]eq , that can be reintroduced in eq (IIIB.8). This process can be iterated until convergence.

  C. Derivation of the relaxation time equation for coupled reactions at the equilibrium

Consider reactions (IIIC.1) and (IIIC.2) (charges omitted) coupled by means of MH2L: kf1 M + H2L ⇄MH2L (IIIC.1) kd1 kf2 MH2L ⇄MHL + H (IIIC.2) kd2

Mass conservation equations are:

CM = M + MHL + MH2L (IIIC.3)

CL = H2L + MHL + MH2L (IIIC.4)

that in their differential form reduce to

δM + δMHL + δMH2L = 0 (IIIC.5)

δH2L + δMHL + δMH2L = 0 (IIIC.6)

The reaction system (IIIC.1)-(IIIC.2) is characterized by two kinetic equations:

= k’f1 [M] [H2L] – kd1 [MH2L] (IIIC.7) = kf2 [MH2L] – k ’ d2 [MHL] (IIIC.8) where k’ f1 = kf1CM e k ’ d2 = k d2 [H + ].

The displacement of the concentration of each reactant from its equilibrium concentration is:

δ H2L = [H2L]eq – [H2L] (IIIC.9) δ MH2L = [MH2L]eq – [MH2L] (IIIC.10) δ MHL = [MHL]eq – [MHL] (IIIC.11)

Introduction of equation (IIIC.9), (IIIC.10) and (IIIC.11) in (IIIC.7) and (IIIC.8) gives:

€ −d([H2L]eq −δH2L dt = k ’ f1([H2L]eq – δH2L) – k d1([MH2L]eq - δMH2L) (IIIC.12) € d([MHL]_eq −δMHL) dt = k f2([MH2L]eq – δMH2L) – k ’ d2([MHL]eq - δMHL) (IIIC.13)

Equilibrium relationships can be written as in eqs (IIIC.14) and (IIIC.15) k’

f1 [H2L]eq = k d1[MH2L] (IIIC.14)

k f2 [MH2L]eq = k ’

d2 [MHL]eq (IIIC.15)

Thus, equation (IIIC.12) and (IIIC.13) reduce to

= k’

f1 δH2L – k d1 δMH2L (IIIC.16)

= -k f2 δMH2L + k ’

d2 δMHL (IIIC.17)

= k f2 δH2L + (k f2 + k’ d2 ) δMHL (IIIC.19)

By defining the following parameters: a 11 = k ’ f1 + k d1 a 12 = k d1 a 21 = k f2 a 22 = k ’ d2 + k f2

equations (IIIC.18) and (IIIC.19) turn to

= a 11 δH2L + a 12 δMHL (IIIC.20)

= a 21 δH2L + a 22 δMHL (IIIC.21)

Such a two-equation system can be solved by considering the eigen-values, λk, of the matrix having the coefficients a j k, that is by setting the matrix determinat to zero:

€

a₁₁−λk a12

a₂₁ a₂₂ −λk

= 0 (IIIC.22)

Equation (IIIC.22) gives the solutions:

€

λ

=

Eqs (IIIC.1) and (IIIC.2) constitute a system that can theoretically display two different relaxation effects, each one being characterized by its relaxation time 1/τk = λk.

We put k=1 for the fast step (IIIC.1) and k=2 for the slow step (IIIC.2). For the investigated Ni(II)/SHA system (Chapter 3) it results λ 1>>λ 2 . Thus, the equilibrium of reaction (IIIC.1) is not influenced by reaction (IIIC.2).

We can deduce from equation (IIIC.22) that

a 11 + a 22 = λ 1 + λ 2 (IIIC.24) a 11 a 22 - a 12 a 21 = λ 1 λ 2 (IIIC.25)

In the case of the Ni(II)/SHA system, being λ 1>>λ 2 and a 11 >> a 22, equation (IIIC.24) turns to

λ 1 = a 11 (IIIC.26)

that is

λ 1 = kf1 CM + kd1 (IIIC.27)

that describes the linear trend of 1//τfast (= 1//τ1 = λ1)with CM (Inset of Figure 3.10A of Chapter 3) . On the other hand, by applying the conditions λ 1>>λ 2 and a 11 >> a 22, equation (IIIC.25) reduces to

€ λ2 = a11⋅ a22− a12⋅ a21 a11 = a22− a12⋅ a21 a11 (IIIC.28)

  that can be written as

€ λ2 = k ' d 2+ (k' f 1/ kd1)⋅kf 2 1+k'f 1/ kd1 (IIIC.30)

By putting the definitions of k’f1 and k’d2 in equation (IIIC.30) and defining K1= kf1/kd1, equation (IIIC.31) is obtained

€

λ

=

− k

that describes the dependence of 1//τslow (= 1//τ2 = λ2) on CM (Inset of Figure 3.10B of chapter 3 and Figure 5.17 of chapter 5).