1
544 Bull. Chem. Soc. Jpn. Vol. 80, No. 8 (2007)
Determination of Stability Constant by RAFA
The size of matrix A is c ꢃ t, where c denotes the number of
1.55
L concentrations and t is the number of times in which the
absorbances were recorded (c is smaller than t). Obviously,
1
1
.35
.15
a
T
the size of matrices C and Y are c ꢃ 2 and 2 ꢃ t, respectively.
The pure absorbance–time of X(yX), can be readily measured
in the absence of L, whereas yXL, is usually unknown. The val-
0.95
0
ue k can be simply calculated by fitting Eq. 12 the data of the
kinetic profile in the absence of L by a least-squares curve-
fitting method using MATLAB. Let
0
0
0
0
.75
.55
.35
.15
b
T
X
R ¼ A ꢂ AQ ¼ A ꢂ CX;0y :
ð20Þ
Here, the concentration profile CX;0 is calculated by optimizing
the value of Kf using Eq. 8. Thus, if Kf can be resolved, then
CX;0 can be optimized. When the rank of R is one less than the
rank of original matrix A, then the CX;0 is the optimum concen-
tration profile and the obtained Kf is the optimum formation
constant.
c
-0.05
0
50
100
150
200
250
300
Time / s
Fig. 1. Simulated absorbance–time plots for a system
CXL;0 can then be obtained by Eq. 4. By using RAFA on R,
ꢂ1
00
ꢂ1
0
ꢂ1
with Kf ¼ 500 L mol , k ¼ 0:01 s , k ¼ 0:07 s at
CX;0 ¼ 5:0 ꢃ 10 mol L and different L concentrations
a), after subtraction of X (b) and after subtraction of
0
0
00
k can be obtained. By using a suitable k value, all of the
components of residual matrix become zero (noise level).
ꢂ3
ꢂ1
(
T
XL
X and XL kinetic profiles (c) by using RAFA.
E ¼ R ꢂ AQ
0
¼ R ꢂ CXL;0y
:
ð21Þ
Table 1. PCA Results on Simulated Data
Experimental
Synthetic Data. To evaluate the performance of the method, a
set of kinetic profiles were created. The use of simulated data
makes it possible to know how each problem affects the perfor-
mance of the proposed method.
i
gi
gi=giþ1
R.S.D.
1
2
3
4
5
46.2076
2.7757
0.0181
0.0172
0.0168
16.6471
153.0082
1.0541
0.0435
0.0068
0.0058
0.0047
0
ꢂ1
00
Parameters setting was as follows: k ¼ 0:07 s , k ¼ 0:01
1.0259
ꢂ1
ꢂ3
ꢂ1
s
, CX;0 ¼ 5:0 ꢃ 10 mol L . Measurements were taken from 0
to 300 s with 1 s intervals (301 measuring time in all). The kinetic
profiles of all components were produced by a first-order rate
equation (Eq. 12 or 18). Vectors of X and XL forms were calcu-
nents.24,25
ꢂ1
The R.S.D. is a measure of the lack of fit of a principal com-
6
lated based on Eq. 8, by considering Kf ¼ 500 L mol and L con-
2
ꢂ1
ꢂ1
ponent model to a data set. The R.S.D. is defined as
centrations in the range 0.001–0.016 mol L with 0.003 mol L
intervals (6 concentration in all). The kinetic profiles were simu-
lated, and random noise was added to the set of artificial data
generated to more rigorously test the method. The error was a set
of noise in agreement with the Gaussian distribution with mean
zero and standard deviation equal to 0.2% of absorbance value.
Real Data. All experiments were performed with analytical
reagent grade chemicals purchased from E. Merck. These chemi-
0
1
1
2
c
X
gi
B
@
C
A
i¼nþ1
ðc ꢂ nÞðt ꢂ nÞ
R.S.D.(n) ¼
;
ð22Þ
where gi is the eigenvalue and n the number of considered
principal components. Table 1 presents the eigenvalues, ratios
of consecutive eigenvalues, and R.S.D. of matrix A by PCA. It
should be noted that R.S.D. (2) reaches 0.0068, thus satisfying
the noise level of the simulative experiment. Also, it should be
noted that the ratio of consecutive eigenvalues, when n ¼ 2, is
the maximum at the same time. It also shows that there exist
two kinetic profiles in the system, which coincides with the
assumption of the simulative experiment.
The value of k , for the simulated data in the absence of L
was obtained by least-squares curve fitting of the experimental
rate constants using MATLAB. The best k is the value that
gives the best fit of Eq. 12 through the simulated data.
Simulated data matrix A was processed by RAFA method.
The best estimation of Kf reduced the rank of the system by
one (Fig. 1b). The kinetic profiles in Fig. 1b belonged only
to the XL form. The obtained Kf was then used to estimate
cals were used without further purification. The stock solutions of
ꢂ
4
pared by dissolving the compounds in distilled water. Samples
-tert-butylcatechol, 3-methylcatechol, ꢀ-CD, and IO3 were pre-
were diluted by taking the appropriate aliquots from the stock
solutions followed by dilution with phosphate buffer (pH 3.60).
ꢂ1
The total concentrations of prepared buffers were 0.15 mol L
.
Absorption spectra were obtained with a Perkin-Elmer Lambda
5 UV–vis spectrophotometer. In each experiment, the sample
placed in a 1 cm path length quartz cells, and the measurements
4
0
ꢅ
were performed at 25 ꢄ 0:1 C.
0
Calculations. All calculations were performed in MATLAB
.5 (Math Works, Cochituate Place, MA).
6
Results and Discussion
Synthetic Data. Figure 1a shows created kinetic profiles
0
0
00
for the system described above. Based on principal component
analysis (PCA), the relative standard deviation (R.S.D.) meth-
od is widely used to determine the number of principal compo-
of k . The best estimation of k reduced the rank of the system
to zero (noise level) (Fig. 1c). The relationship between R.S.D.
of matrices R and E and the estimaties for Kf and k are shown
0
0