127 -Appendix IV-
APPENDIX IV
Derivation of the rate equation for the two kinetic effect observed in the reaction between gold(III) and PADA in SDS
The slow effect
Between pH 4 and 6 the Au(III) species prevailing on Au(OH)Cl3− and
Au(OH)2Cl2− . These species are repelled by the negatively charged SDS
micelles. So, one can reasonably suppose that the species distribution in SDS is the same as in pure water. Therefore, the reactions taking place are
Au(OH)2Cl2− + Cl− K Au(OH)Cl3− + OH− (IV.1)
Au(OH)Cl3− + L L Au(OH)Cl+ + 2ClL − L k1 k-1 (IV.2) Au(OH)2Cl2− + L L Au(OH)2+ L + Cl− + OH− L k2 k-2 (IV.3)
Note that, differently from water, here the experiments indicate that the reverse steps are operative as well.
Under conditions of pseudo first-order, i.e. CAu ≥ 10 CPADA, the concentrations
of the Au(III) species can be expressed as a function of the total gold(III) concentration, CAu, using the relationships
] Cl ) OH ( Au [ ] Cl ) OH ( Au [ CAu = 3− + 2 2− (IV.4) and ] OH [ ] Cl [ K ] Cl ) OH ( Au [ ] Cl ) OH ( Au [ 2 2 3 − − − − = (IV.5)
128
-Appendix IV-
Introduction of equation (IV.5) to equation (IV.4) yields:
(
)
] OH [ ] Cl [ K C ] Cl [ K ] Cl OH Au [ 3 − Au− − − + = (IV.6)(
)
] OH [ ] Cl [ K C ] OH [ ] Cl OH Au [ 2 2 − Au − − − + = (IV.7)The relationship between the free ligand and the bound ligand is obtained using the mass conservation equation of the ligand is differential form.
(
OH
)
ClL
]
0
Au
[
]
L
L
[
−
+
δ
2=
δ
+ (IV.8)where δ[i] represents the departure of the ith species from equilibrium. Express
the rate of reaction as the rate of ligand disappearance:
(
)
(
)
{
}
{
k [Cl ] k [OH ]}
[Cl ] [Au(
OH)
ClL ] ] L L [ ] Cl OH Au [ k ] Cl OH Au [ k dt ] L L [ d 2 2 1 2 2 2 3 1 + − − − − − − − δ × + − − − δ + = − δ − (IV.9)Introduction of equations (IV.4) and (IV.7) in to equation (IV.9) yields:
{
k 1[Cl ] k 2[OH]}
[Cl ] [L L] Au C ] OH [ ] Cl [ K ] OH [ 2 k ] OH [ ] Cl [ K ] Cl [ K 1 k dt ] L L [ d − − δ − − + − − + − + − − + − + − − = − δ −
(IV.10)Integration of equation (IV.10), after variable reparation yields the expression for the relaxation time of the slow process
(
k [Cl ] k [OH ])
[Cl ] C ] OH [ ] Cl [ K ] OH [ k ] OH [ ] Cl [ K ] Cl [ K k 1 2 1 Au 2 1 s − − − − − − − − − − − + + + + + = τ (IV.11)129 -Appendix IV-
A plot of log kd as pH of the data of Table 1 yields a straight line with a slope of
0.52 which indicate that kd contains a term independent of [OH−] and a tern
proportional to [OH−], thus confirming equation (IV.11) as the most suitable to represent the experimental results.
The fast effect
Between pH 8.0 and 9.0 the main reacting species is Au(OH)3Cl− in equilibrium
with a small amount if Au(OH)3(H2O). Under these circumstance the must
probable reactions are
Au(OH)3(H2O) + Cl− K' Au(OH)3Cl− + H2O (IV.12)
Au(OH)3(H2O) + L L Au(OH)2+ + OHL − L
k1
k-1 (IV.13)
The mass conservation equation with respect to gold(III) is
CAu = [Au(OH)3(H2O)] + [Au(OH)3Cl−] + [Au(OH)2+ ] L L
(IV.14)
where the last term can be neglected because CAu ≥ 10 CPADA. The equilibrium
constant of step (IV.12) can be written as
] Cl [ ' K )] O H ( ) OH ( Au [ ] Cl ) OH ( Au [ 2 3 3 − = − (IV.15)
Introduction of equation (IV.15) into equation (IV.14) yields:
] Cl [ K 1 C )] O H ( ) OH ( Au [ 3 2 Au − ′ + = (IV.16)
The mass conversation equation with respect to ligand, written in differential form
130 -Appendix IV- 0 ] L ) OH ( Au [ ] L L [ − +δ 2 2 = δ (IV.17) The reaction rate can be written as
] 2 L 2 ) OH ( Au [ ] OH [ 1 k ] L L [ )] O 2 H ( 3 ) OH ( Au [ 1 k dt ]] L L [ d − δ − − − δ = − δ − (IV.18)
Integration of equation (IV.18), after variable separation, yields the reciprocal time constant: ] OH [ k ] Cl [ K 1 C k 1 1 Au 1 s − − − + + = τ (IV.19)