Extraction of the contribution of kinetic profile for reaction of a chemical species in unknown matrices by standard addition method

1 Introduction

It is not surprising that kinetic methods were introduced as powerful analyzing approaches of very small amounts of substance and to gain better knowledge of reaction mechanisms or order of the reactions in analytical chemistry (Panadero et al., 1996). Kinetic data coupled by chemometric approaches were successfully applied for simultaneous determination in multi-component mixtures (Afkhami and Abbasi-Tarighat, 2009; Bahram and Afkhami, 2008; Madrakian et al., 2007; Afkhami et al., 2005; afkhami and Zarei, 2003; Blanco et al., 1993). In a previously reported work, we introduced first application of continuous wavelet transformation (CWT) for simultaneous kinetic determination of binary mixtures (Afkhami et al. (2007)).

Kinetic information such as the reaction model and rate constants can be evaluated through analysis of spectroscopic data by means of proper chemometric approaches. Partial least squares (PLS) (Sorouraddin et al., 2008) and principal component regression (PCR) (López-Cueto et al., 2000) are well-known multivariate calibration procedures widely used in recent years by means of kinetic spectrophotometric procedures. A large number of papers have been published to solve kinetic process, including iterative target transformation factor analysis (Zhu et al., 2002a), weighted target testing (Imlinger et al., 2009), target testing (Furusjö et al., 2003), MILCA and SIMPLISMA algorithms (Monakhova et al., 2011) rank annihilation factor analysis (RAFA) (Zhu et al., 2002a,b; 2004), curve resolution (Bijlsma and Smilde, 1999a), three-linear methods (Bijlsma et al., 1998; 1999b) mean centering ratio spectra (Bahram, 2007) and genetic algorithms (Maeder et al., 2004).

Windig et al. (1999) developed a new multivariate analysis tool, namely DECRA (Direct exponential curve resolution algorithm) that was based on the generalized rank annihilation method (GRAM) for resolving of consecutive first order reactions. In spite of advances of these methods, determination of the analyte concentration and rate constant determination when the interferent is not known remains an unresolved problem in the chemical analysis field. There are always some restrictions in proposed solutions for example such as PCR, PLS or RAFA.

PCR and PLS require a set of calibration samples with known concentrations of analytes to be determined and the concentration of each interferent in different samples to be varied to allow the calibration algorithm to model its effect. The accuracy of results obtained by RAFA depends on the noise, the number of components with overlapping analytical signals and relative concentration of the overlapping species. Vega-Montoto and Wentzell (2006) recently applied the implementation of maximum likelihood parallel factor analysis (MLPARAFAC) in conjunction with the direct exponential curve resolution algorithm (DECRA) to resolution of simulated and real experimental data of consecutive-first order reactions.

In some situations kinetic processes undergo complex mechanisms or the order of the reaction may be changed during the reaction, therefore such kinetic process are hard to be expressed by simple mathematical models. Several methods have been introduced to solve such problems. These include self modeling curve resolution (Quinn et al., 1999), hard-soft multivariate curve resolution (Juan et al., 2005), iterative target transformation factor analysis (Zhu et al., 2002a), and non-linear least squares regression (Zhao et al., 2006).

Although, these techniques can give much information about interaction between examined parameters but several factors can be responsible for the breakdown of accuracy of this information. Strongly, matrix effect can alter the m sample analytical signal relative to that of a standard of the same analyte concentration. It means that in the presence of matrix effect the applicability of these methods is hindered.

In the other words, for quantitative analysis the advantages of the proposed method are analysis of unknown sample contain unknown components and also analysis do not necessary calibration steps. Only the analyte should be known.

If impurities present in the unknown mixture produce an instrumental response or interact with the analyte to change the instrumental response, then a calibration curve based on pure analyte samples gives an incorrect determination. Hence, standard addition can be applied for most analytical techniques in order to solve the matrix effect problem.

Diphenylcarbazide (DPCI) and diphenylcarbazone (DPCO) belong to the group of chemicals derived from hydrazine and hydrazone and was used as ligands to form colored complexes with some metal ions at acidic pHs (Cheng et al., 1990; Clarke, 1950; Gerlach and Frazier, 1958; Salinas-Hernandez et al., 2003).

For example, DPCI which can form a complex with chromium(VI) in acidic medium. Cr-complex with DPC is very sensitive and inexpensive procedure and also permits the speciation of chromium. Cr-complex with DPCI will form red to violet colored chromophore and it is a chelate of Cr(III) and diphenylcarbazone. The latter is produced and simultaneously combines with chromium when diphenylcarbazide is oxidized by Cr(VI). The reaction can be summarized as follow: $$2{\text{CrO}}_4^{2-}+3{\text{H}}_4\text{L}+8{\text{H}}^{+}\iff {[\text{Cr}]}^{\text{III}}{(\text{HL})}_2+{\text{Cr}}^{3+}+{\text{H}}_2\text{L}+8{\text{H}}_2\text{O}$$ where H₄L is diphenylcarbazide (C₆H₅—NH₂—C⚌O—(NH)₂—C₆H₅), H₂L is diphenylcarbazone (C₆H₅—N≡N—C⚌O—(NH)₂—C₆H₅). Direct reaction of Cr(III) with diphenylcarbazone does not occur to any appreciable extent on account of the well known inertness of the Cr(III) aquo-complex (Gardner and Comber, 2002).

Salinas-Hernandez et al., studied thermodynamic and kinetics of the DPCI at various pH values (Salinas-Hernandez et al., 2003). In a previously reported work we used successfully the standard addition method (SAM) to obtain analyte spectrum at unknown matrices (Afkhami et al., 2008). Compensation of matrix effect was performed using successive standard addition of analyte to sample solution. The spectrum of the analyte was obtained by averaging the resulting spectra, obtained by subtraction of the solutions spectra after and before standard addition. The result at simulated and real data sets (pharmaceutical, biological and alloy samples) showed that proposed strategy for considering matrix effect was quite efficient.

The aim of this study was introducing a novel, simple and precise method (which can conveniently and satisfactorily solves the above difficulty) for extraction of the kinetic profile for a reaction in an unknown matrix and determination of the rate constant using the obtained kinetic profile. In other words, we decide to examine the applicability of SAM in the compensation of matrix effects on kinetic data. In order to evaluate the applicability of the proposed method, several model data sets as well as an experimental data set were examined. The hydroxylation reaction of DPCI in the presence of DPCO was analyzed in detail at various pHs as a real kinetic reaction model. It was proved that this approach can be successfully applied to the analysis of practical systems as well.

2 Theory

3 Application of method with simulate data

In order to evaluate the performance of the method, several sets of simulated spectrophotometric kinetic profiles was created. Pseudo first order reaction of two species C and B with the common reagent R with different products was considered (Fig. 1a).

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Figure 1

First order kinetic profiles of (a) (−) species A product and (Δ) B and its products with $\frac{k_A}{k_B}=2$ in presence of constant interferent (×) and (b) kinetic profile of sample (×) and (□) for different additions of analyte A.

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It was decided to obtain kinetic profile of product of C (P_C) and data monitoring was performed at λ_max of P_C. Fig. 1a shows, the kinetic profile of two species C and B which k_C = 2k_B where there is presented a constant interferent, too. Random noise was added to the generated set of artificial data in order to test the method more rigorously. In this step, after creation of the kinetic profiles of the mixtures, random noise of ±0.002 absorbance unit was added to each profile and then the procedure was followed as explained above and the rate constants of the analyte in the reaction in the noisy signal was predicted.

Standard addition of component A was performed. The kinetic profiles of the solution were recorded after and before standard addition steps (Eqs. (1)–(4)). The created kinetic profiles were subtracted from the initial kinetic profile of the solution. Average of several runs was calculated according to Eq. (5) and used as kinetic profile for P_C in unknown matrix. Created kinetic profile can be used for determination of rate constant of reaction C. Rate constant was calculated by solver-fitting in Excel.

The same procedure was performed using kinetic profiles with k_B = 2k_C. In this case, similar procedure was repeated and kinetic profiles in the presence of constant interferent and species B were resolved and rate constant was calculated. At two simulated data sets, k_C was obtained as 0.05 ± 0.0. Plotting predicted kinetic profile vs. the simulated one showed a good agreement between simulated and predicted kinetic profiles. Regression coefficients and regression equations were obtained as 1.0 and regression equation Y = 1k_i + 0.001, was achieved, respectively.

The results showed that the proposed method can resolve kinetic profile of known analyte in the presence of unknown mixture. So, it is not necessary to know the kind or number of interferences in the mixture. The proposed method will work in case when there is more than one compound to be determined. Standard addition of each component can be performed on samples and rate constant and order on this component can be obtained. For example, in order to determine B species, the standard addition of B species should be performed.

4 Estimation of reaction order

A the absorbance, according to Beer–Lambert law, is directly proportional to the concentration, the order of the reaction can be obtained for zero order, first order and second order reactions from the absorbance of the solution with time by the following equations:

For a zero- order reaction plotting A_∞ − A_t as a function of t, fits to a straight line with the slope −k. On the other hand, when order is one, the plot $\ln \left[\frac{\left(A_{\infty }-A_t\right)}{\left(A_{\infty }-A_0\right)}\right]$ as a function of t, will produce straight line with slope of −k. And plotting $\left[\left(\frac{1}{A_{\infty }-A_t}\right)\right]$ as a function of t, fits to straight line with slope k for second order reactions. In other words, these equations should be examined and the equation which gave the best fit for the experimental data will produce the order of the reaction.

As Fig. 2a shows, for simulated data sets, plotted $A_{\infty }-A_t$ values vs. time do not produce straight line, because the order of species is not zero. At Fig. 2b, $\ln \left[\frac{\left(A_{\infty }-A_t\right)}{\left(A_{\infty }-A_0\right)}\right]$ and $\ln [(A_{\infty }-A_0)]$ was plotted as a function of t, in this case the straight line was obtained which indicates the order of the reaction with respect to A is one. Therefore, order of reaction with respect to desired species can be simply calculated using obtained kinetic profiles by proposed procedure.

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Figure 2

Estimation of reaction order by plotting (a) $A_{\infty }-A_t$ vs time for zero order reaction and (b); $\ln \left[\frac{(A_{\infty }-A_0)}{(A_{\infty }-A_0)}\right]$ (□) or $\ln [A_{\infty }-A_0]$ (Δ) for first or pseudo first order reactions.

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5 Experimental

6 Results and discussion

6.2

6.2 Kinetic profile extraction and rate constant determination for DPCI hydroxylation reaction

The rate constants of DPCI were determined at different pHs using solution of DPCI and DCPO. Good fit was obtained by plotting $\ln \left[\frac{A_{\infty }-A_t}{A_{\infty }-A_0}\right]$ as a function of time for the first 1500 s of the reaction. The created straight line with negative slope at pH 9–11.25, showed the reaction is first order with respect to DPCI. But at pH 12.15 kinetic results did not fit with first order reaction. The results showed that the order of reaction is 2.

These conclusions are correct because Sivakumar et al. showed that solvents can interact with carbonyl or $\succ {\text{NH}}_2$ moiety of diphenylcarbazide (Sivakumar et al., 2005). Water can act as a proton donor solvent and interaction of $\succ {\text{NH}}_2$ moiety is larger than that of the carbonyl group. Also, Salinas-Hernandez et al. showed that at low OH^- concentrations the attack to the DPCI is done by water molecules (2003). Hence mechanism will be independent on OH^- concentration and the reaction will be pseudo firs order. But at higher pH values the attack is directly by hydroxide ion, therefore OH⁻ concentration affects the reaction mechanism and the order of the reaction is 2.

Fig. 3 shows the kinetic profiles of DPCO, DPCI and their mixtures at pH 11.25. Kinetic profiles of three separate mixtures of DPCO and DPCI with the same compositions were recorded and average of them was used as final kinetic profile. The kinetic profiles of the solution at this pH were recorded with the addition of known amounts of DPCI to the mixtures. The procedure was run three times by the same time duration. By subtraction of these kinetic profiles from the kinetic profile before standard addition of DPCI, three new kinetic profiles were created. The average of these kinetic profiles was extracted as product kinetic profile and used for rate constant determination of DPCI at pH 11.25. Rate constant was calculated by solver-fitting in Excel.

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Figure 3

The absorption kinetic profiles of (a) mixtures contain 1 × 10⁻⁵ M and 5 × 10⁻⁵ M from DPCO and DPCI, respectively; (b) mixtures (a) and addition of 5 × 10⁻⁵ M DPCI; (c) DPCI by proposed method (see text).

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The same procedure was repeated for rate constant determination of DPCI at pHs 9.4, 10.0, 10.6 and 11.0 the presence of DPCO. The kinetic profiles for the reaction of pure DCPI solutions at different pHs were also obtained and the corresponding rate constants were calculated by solver-fitting in Excel. The results of calculated rate constants are given in Table 1.

Table 1 Apparent rate constant and standard deviation (SD) for the DPCI hydroxylation reaction in the presence and absence of DPCO at various pHs (ratios of DPCI:DPCO is 1:3).

pH	Rate constant ± SD a	Rate constant ± SD b	tcalcultedc
9	1.82 × 10−4 ± 1.08 × 10−5	1.91 × 10−4	1.55
9.4	1.56 × 10−4 ± 1.08 × 10−5	1.68 × 10−4	1.49
10	1.64 × 10−4 ± 2.6 × 10−5	10−4 × 1.74	1.38
10.6	1.55 × 10−4 ± 0.24 × 10−5	1.45 × 10−4	1.82
11.0	3.41 × 10−4 ± 1.2 × 10−5	3.25 × 10−4	1.75

a DPCI hydroxylation reaction in the presence of DPCO.

b DPCI hydroxylation reaction in the absence of DPCO.

c t_critical = 4.3 (α = 0.05).

We found at pH 12.15 by increasing amount of DPCO at solution a decrease in kinetic profiles of DPCI was appeared. Matrix effect was studied at more detail at pHs 12.15 and 11.25. Different concentrations of standard DPCI were added to mixture solutions and proposed procedure was used to obtain kinetic profiles for the reaction of DPCI. At DPCO:DPCI concentration ratios of 1:4, 1:10 and 1:20 at pH 12.15 and 1:5, 2:5 and 3:1 at pH 11.25 kinetic profiles was recorded and rate constant of DPCI hydroxylation process in the presence of DPCO was calculated. Calculated rate constants are given in Table 2.

Table 2 Calculated rates constant at pHs 11.25 and 12.15 with different concentration ratios of DPCI:DPCO (for three determinations).

pH	Ratios of DPCI:DPCO	Concentration		Rate constant ± standard deviation
		DPCO	DPCI
11.25	5:1	2.0 × 10−5	1.0 × 10−4	2.40 × 10−4 ± 1.20 × 10−5
11.25	2:1	5.0 × 10−5	1.0 × 10−4	1.82 × 10−4 ± 1.02 × 10−5
11.25	1:3	3.0 × 10−6	1.0 × 10−6	1.60 × 10−6 ± 1.06 × 10−5
12.15	1:10	1.0 × 10−4	1.0 × 10−5	1.60 × 10−6 ± 1.00 × 10−5
12.15	1:4	2.0 × 10−4	5.0 × 10−5	1.90 × 10−4 ± 6.0 × 10−5
12.15	1:20	2.0 × 10−4	1.0 × 10−5	4.90 × 10−4 ± 1.20 × 10−5

The results showed that at lower concentrations ratio of DPCI: DPCO the hydroxylation process of DPCI became slower. Decreases at rate constants values may be due to decrease in OH⁻ concentration by rapid reaction rate of DPCO. It was concluded molecular decomposition of DPCI is not taking place at these situations. In the other words these results prove matrix effect is strongly presented at higher concentrations ratios by DPCO.

7 Conclusion

A simple, sensitive, precise, and easy to understand method for resolving kinetic profile of known analyte in unknown mixtures was introduced. The results are all satisfactory. The method does not decrease S/N ratio and do not need to use any separation steps. The main advantage of the proposed method is its applicability to extraction of kinetic profile at matrices contains matrix effects (additive or multiplicative interferences). Another important advantage of the method is that it can be used to the resolving of the kinetic profiles of unknown reaction models. In the other words the method is suitable for not only the first order reactions, but also for the reactions with second orders or fractional orders. The proposed method was applied for apparent rate constant determination of DPCI at various pHs, which at pHs 9.0–11.25 apparent rate constant order was first order but apparent rate constant order was second order at pH 12.