Singular Value Decomposition is used as a Chemo metric Tool for Kinetic Investigations

Mushini Venkata Subba Rao¹*, Ananta Ramam V.², Muralidhara Rao V.² and Sambasiva Rao R.²

¹Dept. of Chemistry, G M R Institute of Technology, Rajam 532 127, Andhra Pradesh, India.

²School of Chemistry, Andhra University, Visakhapatnam 530 003, Andhra Pradesh, India

*Corresponding Author Email: subbarao.mv@gmrit.org, srmushini@rediffmail.com

ABSTRACT:

Singular Value Decomposition (SVD) is a matrix based chemo metric tool, working under windows environment, and is used to test several data sets namely “simulated”,” literature reported” and “experimental”. Singular Value Decomposition employs the use of eigen value analysis for identifying the number of species present in a reaction.

KEY WORDS: Chemometrics, SVD, Kinetics, species.

INTRODUCTION:

Chemometrics^1-4 comprises Chemistry, advanced Mathematics, Statistics and Information theory. It is an interdisciplinary area emerged in late 1970's. With the advent of chemometrics, multi-variate and multi response data acquisition from statistically designed experiments, an expert system inferences⁵ results in reliable chemical information even in noisy environment.

Kinetometrics, a subfield of chemo metrics deals with the knowledge of rates of chemical reaction and mechanistic details. The scope of chemo metrics in kinetic investigations is already reported^6-8 by a few workers. A few popular subfields are envirometrics^9,10, pharmacometrics, synthometrics, qualimetrics and speciometrics¹¹.

Singular Value Decomposition (SVD) of a matrix or a second order tensor is a linear mathematical decomposition into the influence of row designees (U), the importance of each influence (diagonal element (S)) and the corresponding influence of column designees (V). U and V are orthogonal matrices. The power of SVD is best used by selecting the statistically significant positive singular values.

Truncated SVD obtained by deleting all zeros and insignificant singular values together with their corresponding vectors is useful to reproduce data matrix. SVD finds extensive application in multivariate quantitative analysis using UV-Visible, NIR (Near infrared) and excitation - emission spectra.

This technique can also be applied in kinetic studies for identifying the number of reactant and product species and to determine the number of independent reaction paths contributing to the rate law.

Data reduction technique:

A linear or non-linear model can represent the variation of response or its function with the magnitude of the influencing factors. For instance, in a linear model y = a₀ + a₁x, all the data points are reproducible by the parameters a₀ and a₁. This is a data reduction technique as the number of parameters is far less than the number of data points. Variation of absorbance(Y) with concentration (Beer’s law) or logk(Y) with substituent (X) (Hammet equation) is a well-known linear model. The slopes and intercepts of these models bear chemical significance. In the case of non-linear models and polynomial models, the number of parameters increases with order of the polynomial.

Dimension reduction technique:

The other situation is the response or its function (Y) dependence on a number of influencing factors like X₁, X₂, and X₃. The following are to be considered for analyzing and modeling.

· Are the variables are to be correlated

· Which variables are to be selected with the constraint that each is independent of the other?

For example, estimation of number of species from absorbance data over a range of wavelengths is a dimension reduction technique. Singular Value Decomposition (SVD), Principle Component Analysis (PCA) and Factor Analysis (FA) are used in dimension reduction of chemical data.

Eigen values and Eigen vectors:

Eigen values, also called hidden roots, find extensive applications in quantum chemistry, multivariate quantitative analysis and in solution processes. Many physico-chemical phenomena in multi-dimensional space can be expressed as eigen problems. Similarly, the number of independent reaction paths contributing to the rate law is equal to the number of eigen values of kinetic data matrix. The use of eigen value analysis for identifying the number of species is presented with data sets which are simulated, literature reported and experimental.

In this paper, SVD technique is used for identifying the number of species involved in a kinetic investigation. For identifying the number of species a few simulated data sets and reported data sets with Methyl red, chloranilic acid and Hf-chloranilic acid have been chosen. The experimental data obtained “on the spectrophotometric studies. The spectra of hexaaquochromium(III) absorbance values at different pH conditions are taken and are analyzed by SVD technique. And also the other experimental data obtained from the substitution reaction of hexaaquochromium(III) with EDTA has been analyzed by SVD technique.

Hardware and software:

An IBM Pentium II Computer is used for analyzing data. MATLAB (Version 4.2c.1 for Windows environment) from Math Works Inc. is employed to develop programs and graphic outputs. MVATOB is developed from the subset of method based in matrix notation. MVATOB consists of SVD among others like MLR, PCA2, PLSC1 m-files. This package was thoroughly tested with several data sets (simulated, literature reported and experimental data).

EXPERIMENTAL AND RESULT ANALYSIS:

Simulated data sets:

Example 1:

The angle between column vectors of the data set is 30⁰. Eigen value analysis shows that the percentage explainabilities in the two column spaces are 78.8 and 21.1. A large amount of variance in the data is explained by the first eigen vector [-0.7071 -0.4472; 0.7071 -0.8944] direction. The residuals with one component are high and they are completely accounted by the second one

Example 2:

The vectors of the data set are orthonormal (i.e. angle between them is 90⁰). The eigen value and eigen vector matrices are the same and numerically equal to identity matrix. The Percentage Explainability (PE) of each eigen value is 50. Hence, the eigen vectors in 2D space reduce to a circle.

Literature reported data sets:

Example 1: Methyl red

The spectra of Methyl red¹² (8´8 matrix) in the pH range (2.20-6.60) have one maximum, but the isobestic point and eigen vector analysis indicate the presence of two species. The absorbance matrix in Table 1A is highly correlated with the matrix in Table 1B. Representing the residuals in absorbance after considering eigen values upto 1, 2 and 3, it is evident that the residuals are high when eigen value is one, where as the residuals are within the range ± 0.03 with two eigen values. This indicates the existence of two species. Model with three eigen values appears to be over ambitious (over fitting). Similar conclusions are drawn with a data matrix of 4´4 size, but with inferior quality. When drawn in scree plot, the result unequivocally establishes the presence of two species within the experimental pH range (2.20-6.60). The first two Eigen values explain 74.40 % and 23.82 % (Table 1C).

Table 1A. Spectra of Methyl red at different pH values

pH	Absorbance at the wavelengths (nm)
	400	425	450	475	500	525	550	575
2.20	0.017	0.045	0.062	0.110	0.197	0.278	0.332	0.377
3.00	0.058	0.080	0.100	0.152	0.238	0.320	0.372	0.417
3.40	0.167	0.180	0.187	0.222	0.282	0.330	0.368	0.400
3.80	0.420	0.402	0.395	0.365	0.342	0.310	0.300	0.292
4.60	0.770	0.735	0.690	0.565	0.437	0.278	0.192	0.131
5.40	1.015	0.990	0.922	0.742	0.528	0.282	0.147	0.051
6.20	0.935	0.940	0.875	0.702	0.488	0.255	0.118	0.032
6.60	0.443	0.480	0.462	0.342	0.235	0.108	0.048	0.015

Table 1B. Correlation matrix of Methyl red

	X1	X2	X3	X4	X5	X6	X7	X8
X1	1.00
X2	1.00	1.00
X3	1.00	1.00	1.00
X4	1.00	0.99	0.99	1.00
X5	0.94	0.93	0.93	0.96	1.00
X6	-0.20	-0.24	-0.24	-0.14	0.13	1.00
X7	-0.75	-0.77	-0.77	-0.71	-0.49	0.80	1.00
X8	-0.86	-0.88	-0.88	-0.82	-0.64	0.68	0.98	1.00

Table 1C. Percentage explainability of Methyl red

Eigen values	Percentage explainability (PE)	Cumulative percentage explainability (CPE)	Landa
1	74.4075	74.4075	11.5782
2	23.8232	98.0307	1.1671
3	1.2516	99.2823	0.003276
4	0.39288	99.6751	0.00032283

Example 2: Acido basic equilibria of chloranilic acid

The spectra of chloranilic¹³ acid (21 ´ 12 matrix) at different acid concentrations are taken for analysis.The absorbance values of chloranilic acid¹³ for solutions with 12 concentrations of the acid are recorded as a function of wavelength (Table 2A). Contour plots are drawn and 2D representation of 3D surface in variable axes as iso-response values. Each iso-contour represents the lines of equal responses. It gives quantitative information of the variation of the response with the simultaneous variation of the two variables.

The correlation between column vectors is 1.0 and angle is < 3°. All the variance (99.96 %) in the absorbance matrix is explained by the first eigen vector (Table 2B). The chemical validity of one species for chloranilic acid in 3M perchloric acid comes from the fact that no ionization takes place at this acid concentration and clearly demonstrates the presence of one species for chloranilic acid.

Table 2A . Spectra of chloranilic acid of different concentrations

	Absorbance values of [H2Ch] X 10^4, mol dm^-3
Wave length (nm)	0.08	0.16	0.24	0.32	0.40	0.48	0.56	0.64	0.72	0.80	0.88	0.96
360	0.007	0.007	0.008	0.009	0.012	0.012	0.012	0.012	0.031	0.023	0.031	0.027
355	0.007	0.007	0.008	0.010	0.012	0.012	0.012	0.014	0.032	0.024	0.037	0.029
350	0.007	0.007	0.009	0.011	0.012	0.012	0.012	0.016	0.033	0.025	0.038	0.031
345	0.007	0.007	0.010	0.012	0.013	0.013	0.013	0.018	0.037	0.027	0.038	0.032
340	0.007	0.007	0.011	0.012	0.014	0.017	0.017	0.021	0.042	0.032	0.046	0.037
335	0.007	0.008	0.013	0.017	0.018	0.017	0.020	0.023	0.048	0.037	0.054	0.045
330	0.008	0.010	0.018	0.022	0.026	0.028	0.030	0.037	0.064	0.053	0.073	0.061
325	0.013	0.016	0.032	0.038	0.049	0.056	0.062	0.075	0.107	0.101	0.123	0.122
320	0.027	0.048	0.073	0.096	0.117	0.136	0.158	0.182	0.230	0.237	0.275	0284
315	0.062	0.114	0.173	0.232	0.288	0.342	0.398	0.452	0.544	0.574	0.650	0.688
310	0.114	0.231	0.343	0.456	0.571	0.678	0.792	0.895	1.047	1.133	1.272	1.348
305	0.158	0.317	0.473	0.627	0.782	0.935	1.098	1.237	1.430	1.536	1.734	1.808
300	0.163	0.324	0.487	0.642	0.803	0.960	1.118	1.273	1.464	1.573	1.757	1.893
295	0.146	0.288	0.438	0.572	0.725	0.860	1.005	1.132	1.332	1.414	1.598	1.723
290	0.122	0.239	0.361	0.482	0.603	0.720	0.835	0.962	1.127	1.218	1.351	1.436
285	0.091	0.177	0.270	0.358	0.448	0.528	0.614	0.700	0.845	0.900	1.018	1.072
280	0.062	0.118	0.183	0.222	0.308	0.359	0.422	0.478	0.593	0.618	0.712	0.740
275	0.039	0.078	0.116	0.158	0.198	0.222	0.264	0.306	0.404	0.405	0.474	0.465
270	0.026	0.043	0.071	0.097	0.123	0.137	0.158	0.185	0.269	0.254	0.315	0.304
265	0.014	0.024	0.043	0.058	0.075	0.077	0.088	0.107	0.106	0.156	0.208	0.183
260	0.011	0.012	0.026	0.034	0.040	0.044	0.048	0.063	0.138	0.098	0.148	0.117

The number of significant eigen values are arrived at based on statistical tests like IE, XE, FIND etc. The further information related to statistical tests are given below.

RSD = SUM (LANDA) / RX / (CX-NPE)

IE = * RSD

XE = * RSD

RE =

FIND =

F₁, c-p = (CX-NPE)*LANDA (NPE) /

SUM (LANDA (NPE + 1: CX))

RX, CX: Rows and columns; LANDA: eigen values

NPE: No. of primary eigen values

IE: Imbedded error

FIND: Factor indicator function

RE: Residual error

XE: Extracted error

Example 3: Hafnium-chloranilic acid

Chloranilic acid HL, forms ML and ML₂ type of complexes with metal ions like Hf and Zr. Thus, one expects three absorbing species viz., ML, ML₂ and free HL in 3M perchloric acid.

Table 3A represents the spectra of Hf-chloranilic acid¹³ at different wavelengths. This is a data matrix of the size 20´12 and use this to find the eigen values. The three eigen values explain to an extent of 99.96% (Table 3B) which establishes clearly the presence of three species. The absorption spectra of chloranilic acid and ML overlap largely indicating two maxima in the 3D surface and contour diagram .The scree plot and modified scree plot as well as the different tests implemented by Malinowski¹⁴ parameters and as clearly demonstrate the presence of three species for Hf - Chloranilic acid.

Table 2B. Percentage explainability of Chloranilic acid

Eigen values	Percentage explainability (PE)	Cumulative percentage explainability (CPE)	Landa
1	99.96	99.96	75.8
2	0.02089	99.99	0.01584
3	0.01111	100.00	0.008424

EXPERIMENTAL DATA SETS:

Example 1: Hydrolysis of hexaaquochromium(III)

The spectra (370-700 nm) has been recorded of hexaaquo- chromium (III) (5.0´10^{-2 mol} dm^-3) in the pH range (3.00-5.00) with Milton Roy Spectronic 1201 spectrophotometer. The data matrix (70´5) of the spectra is represented in Table 4A. The absorption spectra are plotted in 2D (Figure 1a). The spectra show two maxima, one around 420 nm and the other at around 578 nm. With increase in pH, there is an increase in absorbance value as well as a slight shift in the wavelength of maximum absorption. Eigen vector analysis indicates the presence of two species [(Cr(H₂O)₆³⁺ and Cr(H₂O)₅(OH)²⁺]. The first two eigen values explain 93.99% and 3.38% variance in data matrix as seen in Table 4B.

Table 3A. Spectra of Hf-Chloranilic acid complex

	Absorbance values at different mole ratios of chloranic acid to Hafnium [Chloranic acid] = 0.08 to 0.96 X 10^-4 M [Hf] = 0.92 to 0.04 X 10^-4 M
Wave length (nm)	0.087	0.190	0.316	0.471	0.667	0.923	1.28	1.79	2.59	4.00	7.33	24.0
360	0.020	0.024	0.034	0.038	0.042	0.042	0.043	0.038	0.037	0.032	0.022	0.018
355	0.028	0.041	0.051	0.062	0.067	0.068	0.069	0.066	0.062	0.053	0.037	0.027
350	0.038	0.062	0.081	0.092	0.103	0.109	0.107	0.098	0.093	0.076	0.055	0.032
345	0.053	0.088	0.114	0.136	0.145	0.154	0.150	0.143	0.128	0.105	0.075	0.040
340	0.066	0.116	0.152	0.177	0.196	0.203	0.201	0.188	0.169	0.138	0.100	0.053
335	0.080	0.138	0.182	0.214	0.225	0.248	0.242	0.228	0.203	0.167	0.118	0.066
330	0.087	0.152	0.200	0.241	0.266	0.279	0.275	0.258	0.234	0.194	0.143	0.087
320	0.090	0.162	0.207	0.272	0.317	0.338	0.353	0.362	0.357	0.345	0.317	0.286
315	0.096	0.178	0.252	0.321	0.388	0.443	0.494	0.541	0.582	0.613	0.625	0.670
310	0.108	0.211	0.313	0.417	0.516	0.620	0.737	0.845	0.960	1.056	1.166	1.287
305	0.118	0.232	0.353	0.476	0.616	0.765	0.915	1.067	1.242	1.409	1.580	1.726
300	0.114	0.223	0.338	0.464	0.607	0.747	0.900	1.064	1.268	1.452	1.695	1.775
295	0.109	0.202	0.302	0.382	0.534	0.663	0.803	0.944	1.161	1.317	1.478	1.630
290	0.112	0.186	0.268	0.359	0.459	0.573	0.686	0.805	1.026	1.163	1.293	1.425
285	0.113	0.169	0.231	0.296	0.373	0.454	0.538	0.623	0.862	0.967	1.038	1.147
280	0.118	0.158	0.202	0.249	0.300	0.352	0.406	0.466	0.732	0.792	0.820	0.910
275	0.127	0.156	0.184	0.213	0.249	0.282	0.314	0.350	0.633	0.675	0.673	0.720
270	0.137	0.152	0.168	0.192	0.206	0.227	0.248	0.267	0.575	0.594	0.564	0.595
265	0.137	0.150	0.162	0.173	0.191	0.197	0.206	0.213	0.533	0.537	0.492	0.505
260	0.134	0.144	0.154	0.163	0.176	0.177	0.182	0.183	0.492	0.488	0.441	0.445

Table 3B . Percentage explainability of Hf- Chloranilic acid

Eigen values	Percentage explainability (PE)	Cumulative percentage explainability (CPE)	Landa
1	90.4019	90.4019	206.1163
2	6.2658	96.6677	14.2860
3	3.2905	99.9582	7.5024
4	0.017708	99.9759	0.04037
5	0.010497	99.9864	0.02393
6	0.0080483	99.9944	0.01835

Table 4A. Hydrolysis of Hexaaquochromium(III) at different pH values

Wave Length (nm)	Absorbance values at different pH values [Cr(H2O)6^3+] = 5.0 ´ 10^-2mol dm^-3 , t / ^oC = 30.0 ± 0.1
	3.00	3.50	4.00	4.50	5.00
370	0.300	0.321	0.331	0.330	0.303
380	0.445	0.471	0.487	0.494	0.476
390	0.606	0.637	0.664	0.687	0.694
400	0.764	0.812	0.858	0.894	0.920
402	0.783	0.838	0.885	0.922	0.955
404	0.797	0.857	0.911	0.952	0.985
406	0.804	0.870	0.930	0.970	1.007
408	0.810	0.887	0.949	0.994	1.031
410	0.813	0.894	0.966	1.011	1.050
412	0.814	0.904	0.986	1.027	1.066
414	0.813	0.912	0.994	1.040	1.079
416	0.811	0.917	1.007	1.052	1.091
418	0.805	0.921	1.018	1.063	1.101
420	0.797	0.923	1.023	1.070	1.108
422	0.790	0.926	1.034	1.079	1.115
424	0.770	0.913	1.027	1.072	1.106
426	0.752	0.906	1.024	1.068	1.101
428	0.745	0.904	1.030	1.072	1.103
430	0.754	0.904	1.047	1.088	1.117
432	0.734	0.904	1.039	1.078	1.107
434	0.713	0.889	1.025	1.063	1.091
436	0.691	0.871	1.010	1.046	1.073
438	0.666	0.851	0.990	1.025	1.050
440	0.643	0.829	0.970	1.002	1.026
450	0.532	0.805	0.850	0.872	0.880
460	0.426	0.591	0.712	0.725	0.722
470	0.345	0.487	0.588	0.593	0.578
480	0.292	0.399	0.473	0.471	0.448
490	0.274	0.347	0.397	0.394	0.368
500 510	0.279 0.311	0.328 0.343	0.361 0.367	0.362 0.374	0.341 0.362
520	0.362	0.384	0.405	0.422	0.422
530	0.430	0.448	0.469	0.497	0.497
532	0.445	0.462	0.483	0.513	0.529
534	0.459	0.475	0.497	0.528	0.548
536	0.473	0.489	0.512	0.545	0.568
538	0.487	0.504	0.528	0.564	0.589
540	0.503	0.520	0.544	0.583	0.611
542	0.518	0.534	0.560	0.600	0.632
544	0.531	0.548	0.574	0.616	0.651
546	0.545	0.562	0.590	0.636	0.671
548	0.559	0.577	0.605	0.651	0.692
550	0.572	0.590	0.619	0.667	0.712
552	0.582	0.601	0.631	0.682	0.729
554	0.596	0.615	0.647	0.699	0.750
556	0.607	0.626	0.660	0.714	0.766
558	0.616	0.637	0.671	0.727	0.781
560	0.627	0.648	0.683	0.741	0.799
562	0.638	0.661	0.696	0.756	0.817
564	0.646	0.669	0.707	0.768	0.831
566	0.652	0.677	0.714	0.777	0.844
568	0.657	0.682	0.721	0.785	0.853
570	0.661	0.687	0.728	0.793	0.863
572	0.665	0.692	0.733	0.800	0.872
574	0.667	0.695	0.738	0.805	0.881
576	0.668	0.697	0.741	0.809	0.886
578	0.669	0.699	0.743	0.812	0.890
580	0.667	0.699	0.744	0.815	0.893
590	0.647	0.685	0.736	0.807	0.889
600	0.605	0.649	0.704	0.772	0.857
610	0.542	0.592	0.651	0.717	0.798
620	0.471	0.527	0.585	0.645	0.716
630	0.395	0.451	0.510	0.557	0.615
640	0.332	0.390	0.447	0.485	0.531
650	0.268	0.327	0.382	0.413	0.453
660	0.217	0.273	0.321	0.345	0.373
670	0.192	0.240	0.278	0.294	0.310
680	0.155	0.201	0.238	0.252	0.266
690	0.122	0.165	0.200	0.211	0.224
700	0.099	0.136	0.163	0.169	0.176

Table 4B. Singular values and percentage explainability

Eigen values	Singular value	Percentage explainability	Cumulative percentage explainability	Landa
1	13.3958	93.9968	93.9968	179.4484
2	0.48266	3.3868	97.3836	0.23296
3	0.24918	1.7484	99.1321	0.062089
4	0.072726	0.51031	99.6434	0.0052891
5	0.050966	0.35762	100.00	0.0025976

Table 4C. Malonowski Parameters

Landa	I.E	RE	XE	FIND	RMS	ER
179.4484	0.01471	0.032893	0.02942	0.016446	0.30294	770.2831
0.23296	0.011545	0.018254	0.01414	0.010539	0.017494	3.7521
0.062089	0.0058138	0.0075055	0.0047469	0.0053072	0.00087629	11.7391
0.0052891	0.0054485	0.0060916	0.0027243	0.0060916	0.00016235	2.0362

Table 5A. Kinetic spectra of Hexaaquochromium(III) with EDTA

	Absorbance values at different Wavelengths(nm)
	[Cr(H2O)6^3+] = 4.0 ´ 10^- 3mol dm^-3 ; [EDTA] = 8.0 ´ 10^-2mol dm^-3 m = 1.0 M, pH = 4.00 , t / ^oC = 30.0 ± 0.1
Time minutes	520	530	540	545	550	560	570
0.30	0.040	0.047	0.054	0.057	0.060	0.066	0.068
2	0.051	0.059	0.066	0.070	0.074	0.079	0.083
4	0.068	0.079	0.089	0.094	0.098	0.103	0.105
6	0.097	0.113	0.124	0.130	0.133	0.136	0.137
8	0.132	0.150	0.164	0.169	0.172	0.174	0.171
10	0.170	0.192	0.206	0.210	0.214	0.212	0.207
12	0.208	0.231	0.248	0.252	0.253	0.249	0.239
14	0.244	0.270	0.285	0.289	0.291	0.284	0.270
16	0.274	0.305	0.322	0.326	0.326	0.317	0.300
18	0.307	0.338	0.356	0.357	0.358	0.348	0.326
20	0.334	0.367	0.384	0.387	0.385	0.373	0.350
22	0.360	0.394	0.410	0.413	0.412	0.397	0.371
24	0.382	0.417	0.435	0.437	0.435	0.418	0.391
26	0.403	0.440	0.457	0.458	0.457	0.439	0.408
28	0.421	0.459	0.476	0.478	0.476	0.454	0.425
30	0.437	0.477	0.494	0.494	0.492	0.472	0.440
32	0.452	0.492	0.511	0.509	0.508	0.484	0.452
34	0.466	0.506	0.526	0.525	0.521	0.499	0.465
36	0.479	0.520	0.538	0.539	0.534	0.510	0.474
38	0.490	0.531	0.548	0.552	0.549	0.522	0.483
40	0.500	0.543	0.561	0.561	0.557	0.532	0.495
45	0.521	0.567	0.585	0.587	0.582	0.555	0.515
50	0.541	0.587	0.605	0.606	0.601	0.572	0.530
55	0.556	0.603	0.622	0.623	0.614	0.588	0.543
60	0.569	0.617	0.635	0.636	0.631	0.602	0.557
65	0.580	0.629	0.647	0.649	0.642	0.612	0.566
70	0.591	0.639	0.659	0.658	0.652	0.621	0.576
75	0.599	0.649	0.669	0.668	0.662	0.630	0.582
80	0.606	0.658	0.677	0.676	0.669	0.637	0.590
85	0.613	0.665	0.683	0.683	0.674	0.643	0.595
90	0.620	0.672	0.690	0.690	0.679	0.649	0.600

Table 5B. Singular values and percentage explainability of Hexaaquochromim (III) with EDTA

Eigen values	Singular value	Percentage explainability	Cumulative percentage explainability	Landa
1	6.838	98.2394	98.2394	46.7586
2	0.098612	1.4167	99.6562	0.0097243
3	0.0073557	0.10598	99.7618	5.4106e-005
4	0.0052137	0.074904	99.8367	2.7183e-005
5	0.0048736	0.070017	99.9068	2.3752e-005
6	0.003607	0.051821	99.9586	1.3011e-005
7	0.0028833	0.041423	100.00	8.3133e-006

Figure 1a. Absorbance spectra of Chromium (III) at different pH Values

Residuals in the absorbance values considering eigen values up to 1,2,3,4 and 5 are calculated. The results clearly demonstrate the residuals are less than ± 0.01 absorbance units when first two eigen values are considered. Further, the presence of two species is confirmed by scree plot, modified scree plot and from the plots of the different errors vs eigen values [Figure 1b, Table 4C].

Example 2: Kinetics of Hexaaquochromium(III) with EDTA

12.5 ml of disodium salt of EDTA (0.16M) solution is taken in 50 ml beaker and the ionic strength (m=1.0) is maintained with sodium perchlorate. This solution is adjusted to the pH 4.00 using dilute solution of sodium hydroxide and quantitatively transferred into 25ml volumetric flask. A 2.5 ml portion of hexaaquochromium(III) solution is added to it and made up to the mark with distilled water. Immediately the solution is transferred into the optical cell and the absorbance values are taken at multiple wavelengths (520 nm-570 nm) at 30.0 °C as a function of time (Table 5A). The data matrix is of the size of 31´7. The absorbance time profiles at different wavelengths and 3D surface plot of absorbance vs time and wavelength are seen in figure 2a respectively. Figure 2b shows the isoabsorbance surface of time vs wavelength. The eigen vector analysis indicates the presence of two species [Cr(H₂O)₆³⁺ and Cr(EDTA)(H₂O)^-1 ].Basing on the singular values and percentage explainability (Table 5B) the first two eigen values explain 98.23% and 1.41%. The residuals in absorbance after considering eigen values 1, 2 and 3 are pictorially represented in figure 2c. The residuals in the range of ± 3´10^-3 (absorbance) units when two eigen values are used indicates the presence of two species.

Figure 1b. The number of eigen values vs errors (Based on different statistical tests)

Figure 2a. 3D surface plot Absorbance vs time vs wavelength X is attribute

Figure 2b. Isoresponse surface of wavelength vs time; X is attribute

CONCLUSIONS:

SVD technique is used for identifying the number of species involved in a kinetic investigation. We examined the simulated data sets, literature reported data sets and experimental data sets by using this chemo metric tool. In this process, the identified number of species in a particular reaction is well in agreement with experimental. Therefore, we are suggesting this method for identifying the number of species in a chemical reaction by using the S V D technique.

Figure 2c.Eigen values.

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Received on 16.02.2013 Accepted on 26.08.2013

Modified on 20.09.2013 ©A&V Publications all right reserved

Research J. Science and Tech 5(4): Oct.- Dec.., 2013 page 412-420