Supplementary Material (ESI) for New Journal of Chemistry
This journal is © The Royal Society of Chemistry and the Centre National de la Recherche Scientifique 2001
-1No Manuscript : B104573N
The pyrophosphate-assisted reduction
manganese(II) and its reverse reaction
(Revised)
of
chromium(VI)
Joaquin F. Perez-Benito and Conchita Arias
Electronic Supplementary Information (ESI)
by
Supplementary Material (ESI) for New Journal of Chemistry
This journal is © The Royal Society of Chemistry and the Centre National de la Recherche Scientifique 2001
-2Contents
Appendix:
Derivation of the theoretical rate laws.
Table 1S:
Apparent kinetic orders of Mn(II) at various initial concentrations of Cr(VI).
Table 2S:
Apparent kinetic orders of Mn(II) at various temperatures.
Table 3S:
Dependence of the initial rate on the concentration of sodium perchlorate.
Fig. 1S:
Dependencies of parameters a and b on temperature.
Fig. 2S:
Dependence of the initial rate on the temperature.
Supplementary Material (ESI) for New Journal of Chemistry
This journal is © The Royal Society of Chemistry and the Centre National de la Recherche Scientifique 2001
-3Appendix
Derivation of the theoretical rate laws
According to the mechanism proposed, the active forms of the oxidant and reductant are the
complexed forms of Cr(VI) and Mn(II) with pyrophosphate, respectively. The concentrations of
these two complexes can be obtained from a combination of a balance of matter and the
expression of the equilibrium constant for the formation of the corresponding complex.
The total concentration of Cr(VI) can be expressed as the sum of the concentrations of its
free and complexed forms:
[Cr(VI)]T = [Cr(VI)]f + [Cr(VI)]c
(1S)
and the equilibrium constant for the formation of the Cr(VI)-pyrophosphate complex [eqn. (7)] is:
Kc =
[Cr(VI)]c
[Cr(VI)]f [Py]
(2S)
where [Py] is the concentration of pyrophosphate. From eqn. (1S) and (2S), the concentration of
the Cr(VI)-pyrophosphate complex can be obtained as:
[Cr(VI)]c =
K c [Cr(VI)]T [Py]
1 + K c [Py]
(3S)
Similarly, the total concentration of Mn(II) can be expressed as:
[Mn2+]T = [Mn2+]f + [Mn2+]c
(4S)
and the equilibrium constant of formation of the Mn(II)-pyrophosphate complex [eqn. (8)] as:
Km =
[Mn2+]c [H+]
[Mn2+]f [Py]
(5S)
leading to the following expression for the concentration of that complex:
[Mn2+]c =
K m [Mn2+]T [Py]
[H+] + K m [Py]
(6S)
Supplementary Material (ESI) for New Journal of Chemistry
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-4-
On the other hand, from the mechanism proposed [eqn. (7)-(11)] the reaction rate can be
defined as:
v =
d[Cr(III)]
dt
= v II = k II [Cr(V)] [Mn2+]c
(7S)
where it has been considered that the intermediate Cr(IV) is in steady state.
To obtain the concentration of the intermediate Cr(V) we can consider two different
alternatives:
First alternative : Considering the forward reaction in eqn. (9) as the slow step
In this case, the intermediate Cr(V) would be in steady state:
d[Cr(V)]
dt
= v I - v -I - v II = 0
(8S)
and in terms of rate constants and concentrations:
kI [Cr(VI)]c [Mn2+]c - k-I [Cr(V)] [Mn(III)] - kII [Cr(V)] [Mn2+]c = 0
(9S)
from where it can be deduced that the concentration of the intermediate Cr(V) is:
[Cr(V)] =
k I [Cr(VI)]c [Mn2+]c
k II [Mn2+]c + k -I [Mn(III)]
(10S)
and the reaction rate [from eqn. (7S) and (10S)]:
v =
k I k II [Cr(VI)]c [Mn2+]2c
k II [Mn2+]c + k -I [Mn(III)]
(11S)
From eqn. (3S), (6S) and (11S), we obtain:
v =
(12S)
Supplementary Material (ESI) for New Journal of Chemistry
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-52 k k [Cr(VI)] [Mn2+]2 [Py]3
Kc Km
I II
T
T
+
2+
(1 + K c [Py]) ([H ] + K m [Py]) K m k II [Mn ]T [Py] + k -I ([H+] + K m [Py]) [Mn(III)]
Finally, since in this case Mn(III) is formed in a slow step, at time zero the only Mn(III)
present would be the one added to the reacting mixture (if any), and the expression obtained for
the initial rate is:
vo =
(13S)
[Cr(VI)]T,o [Mn2+]2T,o [Py]3
K m k II [Mn2+]T,o [Py] + k -I ([H+]
2 k k
Kc Km
I II
(1 + K c [Py]) ([H+] + K m [Py])
+ K m [Py]) [Mn(III)]ad
where [Cr(VI)]T,o and [Mn2+]T,o are the total initial concentrations of Cr(VI) and Mn(II),
whereas [Mn(III)]ad is the concentration of added Mn(III).
Second alternative : Considering the reaction in eqn. (10) as the slow step
In this case, the reaction given in eqn. (9) would be in quasiequilibrium. The initial
concentrations of the species involved in that reaction would then be related with the associated
equilibrium constant by the expression:
KI =
[Cr(V)]o [Mn(III)]o
[Cr(VI)]c,o [Mn2+]c,o
(14S)
Since in this case Mn(III) is formed in a fast step [eqn. (9)] before the slow step [eqn. (10)],
even if no Mn(III) is added to the reacting mixture some Mn(III) is present at time zero. Thus, the
initial concentration of Mn(III) would be given by the sum of two contributions: the initial
concentration of Mn(III) formed in the reaction, symbolized by [Mn(III)]f,o, and the concentration
of Mn(III) added to the reacting mixture (if any):
[Mn(III)]o = [Mn(III)]f,o + [Mn(III)]ad
Because of stoichiometric requirements, we can write:
(15S)
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-6-
[Cr(V)]o = [Mn(III)]f,o
(16S)
given that, before the slow step, Cr(V) and Mn(III) are formed [forward reaction in eqn. (9)] and
consumed [backward reaction in eqn. (9)] in equal proportions. Equation (16S) would be valid
only at time zero, since at higher times the concentration of Mn(III) formed in the reaction would
increase because of its accumulation as a reaction product.
From eqn. (14S)-(16S) we obtain:
[Mn(III)]f,o ([Mn(III)]f,o + [Mn(III)]ad)
KI =
[Cr(VI)]c,o [Mn2+]c,o
(17S)
and from eqn. (17S):
[Mn(III)]f,o = 1
2
[Mn(III)]2ad + 4 K I [Cr(VI)]c,o [Mn2+]c,o - [Mn(III)]ad
(18S)
and by substitution of eqn. (3S) and (6S):
[Mn(III)]f,o =
(19S)
1
2
[Mn(III)]2ad +
[Py]2
4 K c K m K I [Cr(VI)]T,o [Mn2+]T,o
(1 + K c [Py]) ([H+] + K m [Py])
- [Mn(III)]ad
Finally, from eqn. (6S), (7S), (16S) and (19S) we obtain the following law for the initial rate:
vo =
K m k II [Mn2+]T,o [Py]
2 ([H+] + K m [Py])
(20S)
4 K c K m K I [Cr(VI)]T,o [Mn2+]T,o [Py]2
[Mn(III)]2ad +
- [Mn(III)]ad
(1 + K c [Py]) ([H+] + K m [Py])
Supplementary Material (ESI) for New Journal of Chemistry
This journal is © The Royal Society of Chemistry and the Centre National de la Recherche Scientifique 2001
-7Table 1S Apparent kinetic orders of Mn(II) at various initial concentrations of Cr(VI)a
[Cr(VI)]o / 10-4 M
order (Mn2+)
1.28
1.26 ± 0.02
2.56
1.25 ± 0.02
3.84
1.21 ± 0.01
5.12
1.31 ± 0.02
6.40
1.18 ± 0.02
8.20
1.18 ± 0.02
10.2
1.25 ± 0.01
12.3
1.22 ± 0.02
14.3
1.12 ± 0.03
a [Mn2+] = (1.09 - 10.9)  10-4 M, [Na P O ] = 5.47  10-2 M, [HClO ] = 0.731 M, 25.0 oC.
o
4 2 7
4
Supplementary Material (ESI) for New Journal of Chemistry
This journal is © The Royal Society of Chemistry and the Centre National de la Recherche Scientifique 2001
-8Table 2S Apparent kinetic orders of Mn(II) at various temperaturesa
T / oC
order (Mn2+)
15.0
1.07 ± 0.02
20.0
1.02 ± 0.01
25.0
1.26 ± 0.02
30.0
1.24 ± 0.03
35.0
1.16 ± 0.02
a [Cr(VI)] = 1.28  10-4 M, [MnSO ] = (1.09 - 10.9)  10-4 M, [Na P O ] = 5.47  10-2 M,
o
4 o
4 2 7
[HClO4] = 0.731 M.
Supplementary Material (ESI) for New Journal of Chemistry
This journal is © The Royal Society of Chemistry and the Centre National de la Recherche Scientifique 2001
-9Table 3S Dependence of the initial rate on the concentration of sodium perchloratea
[NaClO4] / M vo / 10-8 M s-1
0.000
0.91 ± 0.04
0.073
0.92 ± 0.01
0.146
0.83 ± 0.03
0.219
0.90 ± 0.02
0.292
0.88 ± 0.03
0.366
0.86 ± 0.01
0.439
0.96 ± 0.11
0.585
0.98 ± 0.04
0.731
1.02 ± 0.04
0.877
1.06 ± 0.01
1.023
1.14 ± 0.02
a [Cr(VI)] = 5.12  10-4 M, [MnSO ] = 2.19  10-3 M, [Na P O ] = 9.94  10-3 M, [HClO ]
o
4 o
4 2 7
4
= 0.366 M, 25.0 oC.
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This journal is © The Royal Society of Chemistry and the Centre National de la Recherche Scientifique 2001
-10-
1.9
-16.9
1.1
a
-17.7
3.24
3.36
-1
ln ( b / M )
-1
)
-16.1
0.3
3.48
T -1 / 10-3 K-1
Fig. 1S
Dependencies of parameters a (empty circles) and b (filled circles) on temperature.
[Cr(VI)]o = 5.12  10-4 M, [MnSO4]o = 2.19  10-3 M, [Na4P2O7] = 9.94  10-3 M and ionic
strength 1.39 M.
Supplementary Material (ESI) for New Journal of Chemistry
This journal is © The Royal Society of Chemistry and the Centre National de la Recherche Scientifique 2001
-11-
vo
-1
)
-17.2
-19.0
-20.8
3.23
3.36
T
-1
3.49
-3
-1
/ 10 K
Dependence of the initial rate on the temperature at [Cr(VI)] o = 1.28  10-4 M,
[Na4P2O7] = 5.47  10-2 M and [HClO4] = 0.731 M. [MnSO4]o = 1.09, 2.19, 3.28, 4.37, 5.47,
Fig. 2S
6.56, 7.65, 8.74, 9.84 and 10.9  10-4 M (from bottom to top).