To determine the rate constants for the second order consecutive reactions, a number of chemometrics and hard kinetic based methods are described. The absorption spectroscopic data from the

reaction was utilized for performing the analysis. Concentrations and extinctions of components were comparable, and all of them were absorbing species. The number of steps in the reaction was less than the number of absorbing species, which resulted into a rank-deficient response matrix. This can cause

difficulties for some of the methods described in the literature. The available knowledge about the system determines the approaches described in this work. The knowledge includes the spectra of reactants and

product, the initial concentrations, and the exact kinetics. Some of this information is sometimes not available or hard to be estimated. Multiple linear regression for fitting the kinetic parameters to the

obtained concentration profiles, rank augmentation using multiple batch runs, mixed spectral approach which treat the reaction with pseudo species concept, and principal components regression are the four

groups of discussed methods in this study. In one of the simulated datasets the spectra are quite different, and in the other one the spectrum of one reactant and the product share a high degree of

overlap. Instrumental noise, sampling error are the considered sources of error. The aim was investigation of relative merits of each method.

augC:

augX:

References:

1 T. J. Thurston and R.G. Brereton, Analyst 2002, 127, 659.2 A. R. Carvalo and R.G. Brereton, T. J. Thurston, R. E. A. Escott, Chemom. Intell. Lab. Syst. 2004, 71, 47-60.3 T. J. Thurston and R.G. Brereton, D. J. Foord, R. E. A. Escott, J.Chemom, 2003,17, 313-322.4 R.Tauler, Chemom. Intell. Lab. Syst. 1995, 30, 133.5 S. Wold, K. H. Esbensen and P. Geladi, Chemom.Intell. Lab. Syst. 1987, 2, 37.

augmentationaugmentation

PCR:PCR:

Conclusion:Conclusion:When the pure spectra of each component are available, MLR is the best choice, and gives accurate estimates of rate constants in this catalytic system, without requiring any knowledge of initial concentrations.

When pure spectra are not available, and data from three or more reactions are available, rank augmentation can be used to obtain estimates for the pure spectra of all species and to calculate the more accurate estimates of the rate constants than mixed spectra and PCR methods.

When the pure spectra of each component are not available and the data from three or more reactions is not available, PCR and mixed spectra are suggested. The accuracy of the estimated rate

constant is similar. The choice between mixX and mixD or pcrT and pcrC or pcrD, depends on the type of error and level of noise or error present in response matrix. In presence of instrumental noise, mixX

and pcrT is better than mixD and pcrC or pcrD. In contrast, in presence of sampling error, it is better to use mixD and pcrC or pcrD.

To estimate the rate constants of this system, it is better to use two or more of these proposed methods and compare the obtained results to give the most accurate rate constants, as possible.

When the pure spectra of each component are available, MLR is the best choice, and gives accurate estimates of rate constants in this catalytic system, without requiring any knowledge of initial concentrations.

When pure spectra are not available, and data from three or more reactions are available, rank augmentation can be used to obtain estimates for the pure spectra of all species and to calculate the more accurate estimates of the rate constants than mixed spectra and PCR methods.

When the pure spectra of each component are not available and the data from three or more reactions is not available, PCR and mixed spectra are suggested. The accuracy of the estimated rate

constant is similar. The choice between mixX and mixD or pcrT and pcrC or pcrD, depends on the type of error and level of noise or error present in response matrix. In presence of instrumental noise, mixX

and pcrT is better than mixD and pcrC or pcrD. In contrast, in presence of sampling error, it is better to use mixD and pcrC or pcrD.

To estimate the rate constants of this system, it is better to use two or more of these proposed methods and compare the obtained results to give the most accurate rate constants, as possible.

pcrT is less sensitive to noise than pcrC and pcrD. pcrC and pcrD are less sensitive to sampling error.

pcrT is less sensitive to noise than pcrC and pcrD. pcrC and pcrD are less sensitive to sampling error.

Underestimation of k2 and

Overestimation of k1

At high levels of sampling error

Underestimation of k2 and

Overestimation of k1

At high levels of sampling error

Maryam Khoshkam and Mohsen Kompany Zareh * Institute for advanced studies in basic sciences (IASBS), Zanjan

Maryam Khoshkam and Mohsen Kompany Zareh * Institute for advanced studies in basic sciences (IASBS), Zanjan

12

111

6.0

..2

21

Sk

litmolSk

PWVU kk

Dataset2: high overlapDataset2: high overlap

Sampling error:Sampling error:

iisi n xhx ~1

2.ˆ

1. ˆˆ11 TTTRC

RSSRCTCTRk k

k )(estimated

2

1. CCRTC PCRkRSSPCR

2

22. .2 DDRTDRD PCRkPCRCTRk k

pcrT C into TpcrT C into T

pcrC C into TpcrC C into T

pcrD D into T, completely

pcrD D into T, completely

augX shows higher tolerance limit to noise, compared to augC and mixX.

augX is sensitive to sampling error, similar to mixX. Specially for highly overlapped data.

augC has accurate results in presence of sampling error, similar to mixD.

augC has accurate results in presence of sampling error, similar to mixD.

Instrumental noise:Instrumental noise:

GXX Nn~

Data matrix

0.00

0.50

1.00

1.50

400 410 420 430 440 450 460 470 480 490 500

wavelength

abso

rban

ce

Dataset1: low overlapDataset1: low overlap

Second order consecutive reaction:Second order consecutive reaction:

Pure spectra

0.00

0.50

1.00

1.50

400 410 420 430 440 450 460 470 480 490 500

wavelength

abso

rban

ce

Data matrix

0.000.200.400.600.801.001.201.401.601.802.00

400 410 420 430 440 450 460 470 480 490 500

wavelength

abso

rban

ce

Concentration profile

0.000.200.400.600.801.001.201.401.60

0 2 4 6 8 10 12 14 16 18 20

Time

Con

cent

ratio

n

Concentration profile obtained from runge kutta algorithm by solving ordinary differential equations of component.

Concentration profile obtained from runge kutta algorithm by solving ordinary differential equations of component.

Noise level%

AverageRelative Standard deviation (RSD)%

Accuracy %

k1 k2 k1 k2 k1 k2

Dataset 1Instrumental

noise

0.1 1.999 0.60 0.16 0.06 -0.02 0.05

1 2.005 0.60 0.61 0.28 0.27 0.06

2 2.02 0.60 1.27 0.59 0.89 0.05

Sampling error

0.1 1.998 0.60 0.13 0.06 -0.06 0.02

1 1.996 0.601 1.26 0.71 -0.17 0.13

2 1.991 0.602 2.17 1.28 -0.42 0.37

Dataset 2Instrumental

noise

0.1 2.00 0.60 0.18 0.06 0.02 0.02

1 1.999 0.601 0.75 0.29 -0.01 0.11

2 2.00 0.601 1.82 0.60 0.02 0.16

Sampling error

0.1 1.999 0.60 0.29 0.09 -0.02 0.01

1 2.01 0.599 2.61 0.79 0.66 -0.06

2 1.98 0.60 5.37 1.72 -0.84 0.63

augX

augC

2ˆ.ˆ.ˆ

321ˆˆˆ),,( XXXSCCCC RSSSCXXCSkkkk k

2ˆ.ˆ ˆˆ CCC kRSSSXC

)(estimatedk

Noise level%

AverageRelative Standard deviation (RSD)%

Accuracy %

k1 k2 k1 k2 k1 k2

Dataset 1

Instrumental noise

0.1 1.996 0.60 0.43 0.14 -0.18 0.13

1 1.97 0.61 1.31 0.54 -1.57 1.16

2 1.91 0.62 1.88 1.08 -4.59 3.36

Sampling error

0.1 2.00 0.599 0.10 0.08 0.01 -0.01

1 2.01 0.60 0.80 0.78 0.31 0.01

2 2.02 0.597 1.32 1.36 0.75 -0.35

Dataset 2Instrumental

noise

0.1 1.99 0.60 0.42 0.12 -0.19 0.24

1 1.93 0.61 2.14 0.65 -3.58 2.24

2 1.72 0.64 5.16 2.22 -14.01 6.97

Sampling error

0.1 1.999 0.600 0.10 0.06 -0.02 0.02

1 2.001 0.601 0.94 0.54 0.08 0.09

2 1.99 0.601 2.17 1.22 -0.26 0.24

0.40

0.50

0.60

0.70

0.80

0.90

1.001.10

1.50

1.90

2.30

2.70

0.00

0.02

0.04

0.06

0.080.10

0.12

0.14

0.160.18

0.20

RSS

k2k1

Noise level

%

AverageRelative Standard

deviation (RSD) %Accuracy %  

k1 k2 k1 k2 k1 k2

Dataset 1

Instrumental noise

0.1 2.00 0.60 0.27 0.09 0.01 0.00  

1 2.004 0.60 2.45 0.78 0.21 0.02  

2 2.01 0.601 5.15 1.70 0.59 0.16  

Sampling error

0.1 1.999 0.60 0.23 0.13 -0.03 0.02  

1 2.001 0.60 2.68 1.45 0.05 0.01  

2 1.99 0.602 4.24 2.43 -0.53 0.37  

Dataset 2

Instrumental noise

0.1 1.999 0.60 0.23 0.09 -0.04 0.01  

1 2.004 0.60 2.08 0.82 0.22 0.003  

2 2.00 0.599 4.35 1.57 0.02 -0.02  

Sampling error

0.1 2.00 0.599 0.59 0.22 0.02 -0.01  

1 1.98 0.60 6.16 2.33 -0.88 0.46  

2 1.97 0.61 14.51 8.69 -1.48 2.05  

Mixed SpectraMixed Spectra

mixXmixX

mixDmixD

Pure spectra

0.00

0.50

1.00

1.50

400 410 420 430 440 450 460 470 480 490 500

wavelength

abso

rban

ce

Application of chemometrics methods with kinetic constraints for estimation of rate constants of second order consecutive reactionsApplication of chemometrics methods with kinetic constraints for estimation of rate constants of second order consecutive reactions

Concentration profilf of pseudo species

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2 4 6 8 10 12 14 16 18 20

Time

Con

cent

rati

on

2ˆ.~

.ˆ ˆˆˆ)( XXXFDC RSSFDXXDFkk k

k estimated

2ˆ.ˆ ˆˆ DDD kRSSkFXD

mixXmixX

mixDmixD

Pseudo speciesPseudo species

CBA Concentration matrix of pseudo speciesConcentration matrix of pseudo species

o

tU

P0U

U t

1

pcrDpcrDpcrTpcrT

Noise

level%

Average

Relative Standard

deviation (RSD) %

Accuracy %  

k1 k2 k1 k2 k1 k2

Dataset 1

Instrumental noise

0.1 1.999 0.60 0.26 0.08 -0.005 0.02  

1 1.995 0.60 2.51 0.83 -0.23 0.04  

2 1.993 0.60 4.92 1.70 -0.32 0.08  

Sampling error

0.1 2.00 0.599 0.26 0.14 0.02 -0.008  

1 2.01 0.598 2.60 1.42 0.39 -0.18  

2 2.003 0.60 4.85 2.58 0.17 0.09  

Dataset 2

Instrumental noise

0.1 1.999 0.60 0.22 0.09 -0.01 0.00  

1 1.998 0.60 2.12 0.73 -0.10 0.05  

2 2.005 0.60 4.36 1.58 0.28 0.14  

Sampling error

0.1 2.001 0.599 0.60 0.21 0.05 -0.01  

1 1.98 0.604 7.17 5.46 -0.76 0.72  

2 2.06 0.598 13.27 4.33 3.12 -0.19  

Noise level%

AverageRelative Standard

deviation (RSD) %Accuracy %  

k1 k2 k1 k2 k1 k2

Dataset 1

Instrumental noise

0.1 1.998 0.60 0.60 0.21 -0.06 0.02  

0.5 1.996 0.60 1.20 0.46 -0.21 0.15  

1 1.97 0.604 2.74 1.09 -1.47 0.72  

Sampling error

0.1 1.999 0.60 0.24 0.14 -0.02 0.01  

1 2.001 0.599 2.36 1.42 0.06 -0.007  

2 1.98 0.605 7.77 7.13 -0.96 0.96  

Dataset 2

Instrumental noise

0.1 1.999 0.60 0.80 0.29 -0.04 0.02  

0.5 1.993 0.601 1.83 0.70 -0.34 0.23  

1 1.95 0.607 5.27 2.89 -2.40 1.17  

Sampling error

0.1 2.00 0.599 0.25 0.15 0.03 -0.02  

1 1.993 0.601 2.68 1.66 -0.34 0.26  

2 1.98 0.605 7.23 6.81 -1.25 0.97  

Noise level%

AverageRelative Standard deviation (RSD)

%Accuracy %  

k1 k2 k1 k2 k1 k2

Dataset 1Instrumental

noise

0.1 1.997 0.60 0.51 0.17 -0.12 0.04  

0.5 1.995 0.60 1.29 0.50 -0.22 0.10  

1 1.98 0.60 2.78 1.03 -1.08 0.55  

Sampling error

0.1 1.999 0.60 0.23 0.13 -0.02 0.02  

1 2.00 0.599 2.25 1.26 0.02 -0.07  

2 2.00 0.599 4.61 2.46 0.005 -0.02  

Dataset 2

Instrumental noise

0.1 1.999 0.599 0.58 0.20 -0.02 -0.003  

0.5 1.98 0.602 1.40 0.55 -0.74 0.35  

1 1.96 0.605 2.87 1.14 -2.14 0.97  

Sampling error

0.1 2.00 0.599 0.22 0.13 0.01 -0.01  

1 1.99 0.60 2.17 1.29 -0.31 0.04  

2 2.02 0.596 4.73 2.53 1.16 -0.63  

MLRMLRNoise level

%

AverageRelative Standard deviation

(RSD) %Accuracy %

k1 k2 k1 k2 k1 k2  

Dataset 1 Instrumental

noise

0.1 1.999 0.60 0.30 0.15 -0.04 0.02

1 2.00 0.599 1.37 0.73 0.02 -0.09

2 1.999 0.601 2.59 1.49 -0.02 0.15

Sampling error

0.1 200 0.599 0.09 0.05 0.00 -0.005

1 1.999 0.60 0.45 0.30 -0.005 0.05

2 1.999 0.60 0.93 0.59 -0.005 0.01

Dataset 2Instrumental

noise

0.1 2.001 0.599 0.27 0.15 0.03 -0.02

1 2.002 0.60 1.49 0.85 0.13 0.05

2 2.008 0.599 3.07 1.69 0.42 -0.18

Sampling error

0.1 1.999 0.60 0.09 0.06 -0.005 0.01

1 2.001 0.599 0.44 0.28 0.04 -0.01

2 2.003 0.599 1.01 0.65 0.16 -0.05

0.40

0.50

0.60

0.70

0.80

0.90

1.001.10

1.50

1.90

2.30

2.70

0.00

0.02

0.04

0.06

0.080.10

0.12

0.14

0.160.18

0.20

RSS

k2k1

pcrCpcrC

Noise level%

AverageRelative Standard

deviation (RSD)% Accuracy%

k1 k2 k1 k2 k1 k2

Dataset 1Instrumental

noise

0.1 2.00 0.599 0.39 0.14 0.02 -0.003

0.5 1.99 0.601 1.64 0.66 -0.47 0.20

1 1.96 0.605 3.56 1.35 -1.98 0.89

Sampling error

0.1 1.999 0.60 0.22 0.14 -0.04 0.02

1 1.999 0.60 2.49 1.42 -0.04 0.13

2 2.005 0.598 5.17 3.28 0.25 -0.18

Dataset 2Instrumental

noise

0.1 1.999 0.60 0.74 0.28 -0.005 0.02

0.5 1.988 0.601 1.65 0.64 -0.59 0.28

1 1.96 0.604 3.65 1.29 -1.75 0.77

Sampling error

0.1 2.00 0.599 0.26 0.15 0.00 -0.005

1 2.01 0.597 2.26 1.29 0.62 -0.41

2 1.998 0.60 7.45 6.81 -0.06 0.31

0.10

0.30

0.50

0.70

0.90

1.10

1.30

1

1.4

1.80

2.20

2.60

3.00

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

RSS

k2 k1

Abstract:Abstract:

CSXCS

X MLR. ˆ

2

)( CCCkkMLRRSSk

estimated

0.1

0.35 0.6

0.85 1.1

1.35 1.0

1.4

1.8

2.2

2.6

2.3 0.00

0.04

0.08

0.12

0.16

0.20

0.24

0.28

RSS

k2 k1