In the Laboratory
A Practical Integrated Approach to Supramolecular Chemistry. II. Kinetics of Inclusion Phenomena Jesús Hernández-Benito, Samuel González-Mancebo, Emilio Calle, M. Pilar García-Santos, and Julio Casado* Departamento de Química física, Facultad de Química, Universidad, E-37008, Salamanca, Spain
In a previous paper (1) we proposed an experiment for detecting the inclusion of an azo-dye with α-cyclodextrin and measuring the equilibrium constant of the formation of the corresponding inclusion complex. In this article we study the kinetics of the formation of that inclusion complex of the guest molecule, mordant yellow 7 (MY7), 3-methyl-5-(4-sulfophenylazo)salicylic acid, disodium salt, in the host molecule, α-cyclodextrin. The kinetic parameters involved in the formation of the inclusion complex were determined using the stopped-flow technique. As is known (cf., e.g., 2), with this method a steady flow is first established and then brought rapidly to a halt, after which the subsequent changes in concentration are monitored in real time. The stopped-flow technique is now standardized and is in fact one of the fast-reaction techniques embodied in commercial instruments.
With the stopped-flow technique, we followed the formation of the inclusion complex until equilibrium was reached. If S is used to designate the guest molecule and C is used to refer to α -cyclodextrin, the formation equilibrium of the inclusion complex, CS, assuming a 1:1 type mechanism, can be written thus: CS
If the term ∆[CS] is used to refer to the difference in concentrations of the complex at time t and at equilibrium, the relaxation time τ is defined as the time corresponding to the situation in which (∆[CS])0/∆[CS] = e) (3). The relaxation time is related to the rate constants, equilibrium constants, and concentrations, depending on the reaction mechanism. For the proposed mechanism, the reaction rate equation should be:
{
d [S] = k 1[S][C] – k {1 [CS] dt
(1)
If [S]0 and [C]0 are the initial concentrations of substrate and cyclodextrin, respectively, and x is the reaction variable (i.e., the decrease in reagent concentration), one can then write [S] = [S]0 – x; [C] = [C]0 – x; and [CS] = x. Therefore:
d x = k [S] – x [C] – x – k x 1 0 0 {1 dt
(2)
At equilibrium,
d x = 0 = k [S] – x [C] – x – k x 1 0 e 0 e {1 e dt *Corresponding author. Email:
[email protected].
422
(4)
Combining eq 2 and eq 4, one easily obtains
d x = k [C] + [S] + k 1 0 0 {1 x e – x dt
(5)
If we set ∆x = x – x e ,
d ∆x = { k [C] + [S] + k ∆x 1 0 0 {1 dt
(6)
Integrating eq 6:
∆x 0 = k 1 [C]0 + [S]0 + k {1 t ∆x
(7)
Since the relaxation time τ is that for which (∆x)0/∆x = e, one can write eq 7 in such a way that
1 = k [C] + [S] + k 1 0 0 {1 τ
(8)
Working in conditions where [C]0 >> [S]0, eq 8 can be rewritten
k1 k -1
k 1[S]0[C]0 = k 1[C]0 x e + k 1[S]0 x e + k {1 x e – k 1 x e2
ln
Kinetics of Inclusion Phenomena
S + C
which means that
(3)
1 = k [C] + k 1 0 {1 τ
(9)
On plotting the values of 1/τ against [C]0, a straight line should be obtained, with a slope of k1 and the intercept k {1. Table 1 shows the results obtained at 25 °C. The values of 1/τ are the means of five replicates. When 1/ τ is plotted against [C]o, the curve shown in Figure 1a is obtained. Its nonlinear profile is not consistent with a single-step mechanism such as that proposed. In fact, it has been shown that some guest molecules are buried in the cavity of the α-cyclodextrin through a two-step mechanism: the dye reacts with the α-cyclodextrin in a rapid equilibrium to form the intermediate (CS)*, which than gradually forms the final complex CS (4, 5): S+C
k1 k{1
(CS)*
K 1, fast
k2 k{2
CS
K2 , slow
When two equilibria are involved, two relaxation signals will be obtained. As before, the relaxation time corresponding to the first equilibrium, τ1, is such that
1 τ 1 = k 1 [S] 0 + [C]0 + k {1
(10)
To calculate the second relaxation time, τ2, corresponding to
Journal of Chemical Education • Vol. 76 No. 3 March 1999 • JChemEd.chem.wisc.edu
In the Laboratory
the assumedly slow equilibrium, one can write
Table 1. Relaxation Times for Different Initial Concentrations of a-Cyclodextrin 1/τ /s
[C]0/M
{1
k2/s{1 K1/M
{1
K2 KT/M {1
(12)
∆[S] + [(CS)*] + ∆[CS] = 0
(13) (14)
0.5708 ± 0.0007
8.754 × 10 { 4
0.6868 ± 0.0008
1.253 × 10 {3
0.8102 ± 0.0009
1.751 × 10
{3
0.9412 ± 0.0011
K1 = [(CS)*]/[S][C]
2.506 × 10
{3
1.088 ± 0.002
3.502 × 10 {3
1.207 ± 0.002
treatment analogous to that used to obtain eq 8 will readily allow one to deduce
5.011 × 10 {3
1.389 ± 0.003
7.003 × 10 {3
1.468 ± 0.004
1.002 × 10 {2
1.610 ± 0.005
Since
1 = K 1k 2 [S] + [C] + k {2 τ 2 1 + K [S] + [C] 1
Table 2. Constants for Inclusion Reaction of a-Cyclodextrin with Azo-dye Guest MY7
k{2/s
∆[S] = ∆[C]
0.5035 ± 0.0007
{4
N OTE: Values obtained from stopped-flow data. Solvent, phosphate buffer; pH = 11.0; T = 298 K; [S]0 = 2.503 × 10 {5 M.
{1
(11)
4.377 × 10 { 4 6.264 × 10
Constant/ Units
d [CS] = k 2 (CS)* – k {2[CS] dt
Plot Nonlinear (eq 16)
Lineweaver– Burk (eq 17)
Eadie–Hofstee (eq 18)
Hanes (eq 19)
0.26 ± 0.02
0.26
0.26
0.26
1.69 ± 0.18
1.71 ± 0.05
1.66 ± 0.09
1.69 ± 0.04
382 ± 29
374 ± 12
387 ± 28
381 ± 10
6.5 ± 0.9
6.6 ± 0.2
6.4 ± 0.3
6.5 ± 0.2
2870 ± 400
2850 ± 100
2850 ± 200
2850 ± 100
(15)
If one of the equilibria is reached much faster than the other, only the relaxation signal corresponding to the slower one will be seen. In this way, if one knows the values of τ2 at different reagent concentrations, it is possible to calculate the values of the two rate constants involved. When one works in conditions such that [C]0>> [S]0, eq 15 can be written
K 1k 2[C] 0 1 τ 2 = k obs = 1 + K [C] + k {2 1
(16)
0
At high values of [C]0, K1[C]0 >> 1 and kobs = k 2 + k{ 2. At low values of [C]0, K1[C]0
