Appendix VII

Appendix VII

165 APPENDIX VII

Determination of the kinetic equation for the Ga(III)/HQ system Given the reaction scheme

M ³⁺ MH ₂ L ⁴⁺

MHL 3+

M(OH) ²⁺ +

HL ML ²⁺

H ₂ L ⁺

M(OH) ²⁺ +

H ₂ L ⁺ +

k ₂

k ₃ k _-3 k _-2 k _-1 k ₁

K _A1 K _C2

K _C1 K _H1

H ⁺

H ⁺

H ⁺ H ⁺

( I )

( II )

( III )

where the vertical equilibria (proton transfer) (complexation) are assumed to be much faster than the horizontal ones (complex formation), the kinetic law can be written as

( ^{[ ] [ ] [ ]} )

] ][

[ ] ][

[ ] ][

] [ [

2 3 3

2 4

2 1

2 3

2 2 2

2 3 1

− +

− +

− +

+ +

+ +

+

+ +

−

− +

+

=

−

ML k MHL k

L MH k

HL MOH

k L H MOH k

L H M dt k

L d _f

(VII.1)

Defining

) ] ([

] [ ]

[ ] [ ₂

2

f A L

H H K

H L

L H

= +

= ⁺ ₊ ⁺

α (VII.2)

) ]

([

] [ ]

[ ] [

1 3

f H

M H K

H M

M

= +

= ⁺ ₊ ⁺

β _(VII.3)

Appendix VII

166

2 1 1

2

2 1 2

] [ ]

] [ [

] [

C C C

C C T

ML H K H K K

K K ML

ML

+

= +

= ⁺ _{+ +}

γ (VII.4)

after introduction of equation (VII.2), (VII.3) and (VII.4) in equation (VII.1) and appropriate rearrangement of the latter equation one obtains

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ ⎟ ⎟ ×

⎠

⎞

⎜ ⎜

⎝

⎛ + +

⎥ −

⎥

⎦

⎤

⎢ ⎢

⎣

⎡ ⎟ ⎟ ×

⎠

⎞

⎜ ⎜

⎝

⎛ + +

=

− _{+ + −} ^{− + − +} _{T ML}

C C C

M L H f H f

A f H

K ML K

H k K

H k k

M H L

K K k H

K k k dt

L

d α β [ ] [ ] [ ] γ

] ][

] [ [ ] [ ]

[

2 1 2

2

2 1 3 2

2 1 1 3 1 1 2

(VII.5)

Taking into account the apparent reaction (VI.1) and its kinetic law (VI.2) (see Appendix VI) it turns out that

M L H H

H A

f H

K K k H

K k k

k α β

2

2

1 1 1 3

1 2

] [ ]

[ ⎟⎟ ⎠ ×

⎜⎜ ⎞

⎝

⎛ + +

= _{+ +} (VII.6)

ML C

C C

d K K

H k K

H k k

k ⎟⎟ × γ

⎠

⎜⎜ ⎞

⎝

⎛ + +

= ₋ ^{− + − +}

2 1

2 1 2

2 3

] [ ]

[ (VII.7)