167APPENDIX VIII Determination of the kinetic equation for the Pd(II)/PADA system Given the reaction scheme in water

APPENDIX VIII

Determination of the kinetic equation for the Pd(II)/PADA system Given the reaction scheme in water

PdCl₃H₂O -PdCl₄ 2-K Cl- + H2L++ k₁ k₂ PdCl2L + 2H+ + Cl -PdCl₂L + 2H+ + 2Cl -PdCl3H2O -PdCl4 2-K Cl- + HL+ k3 k4 PdCl₂L + H+ + Cl -PdCl2L + H+ + 2Cl -PdCl₃H₂O -PdCl₄ 2-K Cl- + L k₅ k₆ PdCl2L + H+ + Cl -PdCl₂L + H+ + 2Cl -H+ K_A1 K_A2 H+ (I) (II) (III) (IV) (V) (VI)

and assuming the vertical proton transfer and Cl- _{complexation steps to equilibrate} rapidly, the kinetic law can be written as

] ][ [ ] ][ [ ] ][ [ ] ][ [ ] ][ [ ] ][ [ ] [ 2 4 6 2 3 5 2 4 4 2 3 3 2 2 4 2 2 2 3 1 2 L PdCl k L O H PdCl k HL PdCl k HL O H PdCl k L H PdCl k L H O H PdCl k dt L PdCl d − − + − + − + + − + + − + + + − + + = (VIII.1)

Under conditions of metal excess (CM>>CL) the mass conservation for the metal is CM =[PdCl42-]+[PdCl3H2O-]+[PdCl2L] ≈[PdCl42-]+[PdCl3H2O-]

If we now define ] [ ] [ ] [ 2 3 2 4 − − − × = Cl O H PdCl PdCl K (VIII.2) ) ] [ ] ([ ] [ ] [ ] [ 2 1 1 2 2 2 2 A A A f L H K K K H H H L L H + + = = ++ _{+ +} + α _(VIII.3) ) ] [ ] ([ ] [ ] [ ] [ 2 1 1 2 1 A A A A f HL K K K H H H K L HL + + = = + _{+ +} + α _(VIII.4) ) ] [ ] ([ ] [ ] [ 2 1 1 2 2 1 A A A A A f L K K K H H K K L L + + = = _{+ +} α _(VIII.5)

Equation (VIII.2) can be rearranged to obtain

] [ 1 ] [ ] [ ₄2 ₋ − − + = Cl K C Cl K PdCl M _(VIII.6) ] [ 1 ] [ _{3 2} − ₋ + = Cl K C O H PdCl M _(VIII.7)

Introducion of equations (VIII.3)−(VIII.7) into equation (VIII. 1) yields

M f HL A AI f L M f HL M f L H M C L H K Cl K Cl K k k Cl K Cl K k k K H Cl K Cl K k k L Cl K C Cl K k k L Cl K C Cl K k k L Cl K C Cl K k k dt L PdCl d ] [ ] [ ] [ 1 ]) [ ( ] [ 1 ]) [ ( ] [ ] [ 1 ]) [ ( ] [ ] [ 1 ]) [ ( ] [ ] [ 1 ]) [ ( ] [ ] [ 1 ]) [ ( ] [

)

(

2 6 5 4 3 2 1 6 5 4 3 2 1 2 2 α α α α + − − − − + − − − − − − − − + + + + + + + + = + + + + + + + + = (VIII.8)

Since Lf will be entirely in the complexed form PdCl2L, it turns out that d[PdCl2L]/dt = -dLf/dt. Therefore, equation (VIII.8) can be integrated obtaining equation (VIII.9)

[Lf] = [Lf]o e (t/τ) (VIII.9)

where 1/τ (s-1_{) is the time constant of the system and depends on the reactant} concentrations according to equation (VIII.10)

M HL A AI C H K Cl K Cl K k k Cl K Cl K k k K H Cl K Cl K k k _α τ

(

1 [ ] [ ]

)

]) [ ( ] [ 1 ]) [ ( ] [ ] [ 1 ]) [ ( 1 _{1 2 3 4 5 6 2} + − − − − + − − + + + + + + + + = (VIII.10) If we now define kI = (k1+k2K[Cl-])/(1+K[Cl-]) (VIII.9) kII = (k3+k4K[Cl-])/(1+K[Cl-]) (VIII.10) kIII = (k5+k6K[Cl-])/(1+K[Cl-]) (VIII.11)

taking into account the apparent reaction (VI.1) and equation (VI.13) (see Appendix VI) it turns out that

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + + = ] [ ] [ ₂ 1 H K k k K H k k _{III A} II A I HL f α (VIII.12)

which corresponds to equation (6.3) of chapter 6. Considering now the reaction scheme in DTAC:

PdCl₄ 2-K Cl HL+ k₇ PdCl₂L PdCl3H2O -K_H H+ + L k5 k_{6 PdCl} 2L + 2Cl -PdCl2L + Cl -KA2 H+ (VII) (V) (VI) PdCl3OH 2-+ L + L H2L++ KA1 H+ + Cl- + OH

-Assuming the vertical equilibrium steps be fast, the kinetic law can be written as

] ][ [ ] ][ [ ] ][ [ ] [ ₂ 3 7 2 3 5 2 4 6 PdCl L k PdCl H O L k PdCl OH L k dt L d f ₌ − _{+ − + −} − (VIII.13)

Under conditions of metal excess (CM>>CL), in the pH range employed in the presence of DTAC, the following mass conservation applies

CM=[PdCl42-]+[PdCl3H2O-]+[PdCl3HO-]+[PdCl2L]~[PdCl42]+[PdCl3H2O]+[PdCl3HO-]

where CM isthe overall concentration of the metal. If we now define ] [ ] ][ [ 2 3 2 3 − + − = O H PdCl H HO PdCl K_H (VIII.14)

equation (VIII.2) and (VIII.14) can be rearranged to obtain ]) ][ [ ] ([ ] ][ [ ] [ ₄2 _{+ − +} − − + + + = H Cl K K H C H Cl K PdCl H M _(VIII.15) ]) ][ [ ] ([ ] [ ] [ _{3 2 + − +} + − + + = H Cl K K H C H O H PdCl H M _(VIII.16) ]) ][ [ ] ([ ] [ 2 3 + − + − + + = H Cl K K H C K HO PdCl H M H _(VIII.17)

By introducing equations equations (VIII.5), (VIII.15), (VIII.16),(VIII.17) into equation (VIII. 13) yields

f M L H H H H f L C H Cl K K H K k H Cl K K H H k H Cl K K H H Cl K k dt L d α

)

(

]) ][ [ ] ([ ]) ][ [ ] ([ ] [ ]) ][ [ ] ([ ] ][ [ ] [ 7 5 6 + − + + − + + + − + − + + + + + + + + = − (VIII.18)

Integrating equation (VIII.18) one gets

[Lf] = [Lf]o e (t/τ) (VIII.19)

where 1/τ (s-1_{) is the time constant of the system equal to}

M L H H M L H H H H C H Cl K K H H K k k Cl K k C H Cl K K H K k H Cl K K H H k H Cl K K H H Cl K k α α τ

)

(

)

(

)

(

] ][ [ ] [ ] [ ] [ ]) ][ [ ] ([ ]) ][ [ ] ([ ] [ ]) ][ [ ] ([ ] ][ [ 1 7 5 6 7 5 6 + − + + − + − + + − + + + − + − + + + + = + + + + + + + + = (VIII.20)

taking into account the apparent reaction (VI.1) and equation (VI.13) (see Appendix VI) it turns out that

M L H H f C H Cl K K H H K k k Cl K k k

(

)(

)

α ] ][ [ ] [ ] [ ] [ _{5 7} 6 + − + + − + + + + = (VIII.21) If we now define kIII = (k5+ k6K [Cl-])/(1+K [Cl-]) (VIII.22) KHapp = (KH/(1+K [Cl-]) (VIII.23)

Rearranging equation (VIII.21) we obtain

⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ + + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + = Happ Happ III L f K H H H K k k k ] [ ] [ ] [ 7 α (VIII.24)