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Study of Complexation in Acetone-Chloroform Mixtures by Infrared Spectroscopy

Study of Complexation in Acetone-Chloroform Mixtures by Infrared Spectroscopy Hindawi Publishing Corporation Journal of Atomic and Molecular Physics Volume 2014, Article ID 106178, 6 pages http://dx.doi.org/10.1155/2014/106178 Research Article Study of Complexation in Acetone-Chloroform Mixtures by Infrared Spectroscopy Oleksii O. Ilchenko, Andrii M. Kutsyk, and Vyacheslav V. Obukhovsky Radiophysical Department, National Taras Shevchenko University of Kyiv, Volodymyrska Street 64, Kyiv, Ukraine Correspondence should be addressed to Oleksii O. Ilchenko; radastasi@gmail.com Received 29 April 2013; Revised 28 December 2013; Accepted 29 December 2013; Published 24 February 2014 Academic Editor: Keli Han Copyright © 2014 Oleksii O. Ilchenko et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FTIR spectra of acetone-chloroform system with various component ratios were investigated within the spectral range 3950– −1 4550 cm . Methods of multivariate curve resolution were applied to decompose the FTIR spectra into specific components of different composition. A method of decomposition based on structural model of solution which contains acetone, chloroform, and complex acetone/chloroform (1 : 1) was proposed. Results of both approaches are in good agreement within the range of measuring error. 1. Introduction excitation frequency. The temperature of liquid samples was 25 ± 0.2 C. The chloroform and acetone with purity 99.9% Acetone-chloroform mixture is a prominent example of a were used in this research. eTh concentration of components system with a pronounced negative deviation from ideal was changed from 0% to 100% (in volume %) with step 10%. solution behavior. So it has been the subject of many theo- Spectra recording was repeated 5 times for every sample. retical and experimental researches [1–6]. The main problem Thereaer ft the average spectra were calculated for every is the structure and composition of associated species of these concentration and used in further analysis. mixtures. Infrared spectroscopy is powerful tool of investigation of liquid systems [7]. The IR spectra are very sensitive 3. Theory to structural changes caused by intermolecular interactions 3.1. Multivariate Curve Resolution. Resolution methods between components of solution. Multivariate regression can be used for the determination of quantitative information decompose mathematically a global mixed instrumental response into the pure contributions due to each component from IR spectra, such as number and concentration of species in the system [9]. This mixed signal is organized in the matrix in the mixture [8]. eTh purposeofthisworkisquantitativeanalysisof D containing raw information about all the components present in the data set. Resolution methods allow for the acetone-chloroform mixture using model-free and model- based multivariate regression approaches. decomposition of the initial mixture data matrix D into the product of two data matrices C and S , each of them including the pure response profiles of the 𝑛 mixture or 2. Experiment process components associated with the row and the column direction of the initial data matrix, respectively. In matrix FTIR transmission spectra were measured with er Th mo Sci- notation, the expression valid for all resolution methods is enticfi spectrometer Nicolet 6700 with a spectral resolution −1 of 4 cm . Optical path length of the quartz cell was 1 mm. The heating of the sample almost did not occur during the D = CS + E, (1) measurements due to small value of absorption coefficient at 2 Journal of Atomic and Molecular Physics where D(𝑟×)𝑐 is the original data matrix, C(𝑟×𝑛 )and S (𝑛×)𝑐 3.2. Estimation of Band Boundaries of Feasible MCR Solutions. arethe matrices containing thepureresponseprofiles related Complete resolution of a two-way data set without ambigui- to the data variation in the row direction and in the column ties is only possible in some favorable cases where selectivity direction, respectively, and E(𝑟 × )𝑐 is the error matrix, that is, [11] or local rank conditions [13] are present. When these the residual variation of the data set that is not related to any resolution conditions are not present in the system, resolution chemical contribution. Parameters 𝑟 and 𝑐 are the number of without ambiguities is not possible even if constraints such rows and the number of columns of the original data matrix, as nonnegativity, unimodality, or closure are applied [14]. In respectively, and 𝑛 is the number of chemical components in these cases, instead of unique profiles, a range or band of feasible profiles tfi ting equally well the experimental data and the mixture or process. C and S often refer to concentration fulfilling the physical and chemical constraints of the system profiles and spectra (hence their names). hastobeconsidered. With the measurement matrix D,the probleminmul- Boundaries of feasible solutions are related to rotational tivariate curve resolution is to estimate the pure variables, matrices T and T for each species 𝑘 . Maximum and C and S,interms of thebilinearmodel andsomegeneric min,𝑘 max,𝑘 minimum band boundaries for concentration and spectral knowledgeabout thepurevariables.Accordingly, thebasic profiles may be defined by the following equation: principle of curve resolution is to seek for a bilinear model that gives best t, fi in the sense of least squares or weighted least T −1 T T D = CS = CT T S = C S squares, to the two-way data D.That is to say, oneisseeking min min min min (4) forthe estimatesofpurevariables, C and S, which minimize −1 T T = CT T S = C S . the norm max max max max eTh goal is calculation of matrices T and T 󵄩 󵄩 min,𝑘 max,𝑘 󵄩 󵄩 󵄩 󵄩 ‖E‖ = D − CS 󳨀→ min . (2) 󵄩 󵄩 for nding fi band boundaries. The problem is considered in 󵄩 󵄩 frame of a constrained nonlinear optimization problem. It is described mathematically using the equation [15] The mathematical decomposition of a single data matrix D or, in other words, the norm minimization problem (2) 𝑓 (T) 󳨀→ opt, is known to be subject to ambiguities [10]. This means that many sets of paired C-and S -type matrices can reproduce (5) theoriginaldataset with thesamefitquality.But some of the g (T) =0, obtained solutions do not have any physical meaning. In plain g (T) ≤0, ineq words, the correct reproduction of the original data matrix can be achieved by using response profiles differing in shape where 𝑓( T) is an objective function; g is vector of ineq (rotational ambiguity) or in magnitude (intensity ambiguity) inequality constraints (such as nonnegativity, selectivity, and from the sought (true) ones [11]. Mathematically it can be unimodality); g is vector of equality constraints (mass written as follows: balance, known values of concentration, etc.). Gemperline [16]and Tauler [15] have shown that the calculation of the T −1 T T ̃ ̃ (3) D = CS = CTT S = CS , band boundaries of feasible solutions for every species is possible when an objective optimization function is defined in terms of the ratio of the signal contribution of that species where T(𝑛 × 𝑛) is invertible matrix. If T is orthogonal matrix ambiguity is rotational. The intensity ambiguity is not a to the whole signal contribution for the mixture of all species. For example, Tauler proposed to define objective function as serious problem in qualitative analysis (spectral identifica- tion), but it is a serious problem in quantitative analysis. 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 The most important is the rotational ambiguity which always c s 󵄩 󵄩 𝑘 𝑘 󵄩 󵄩 (6) 𝑓 (T) = , 𝑘 󵄩 󵄩 occurs when there are several overlapped profiles of compo- 󵄩 󵄩 CS 󵄩 󵄩 󵄩 󵄩 nents. The presence of this ambiguity means that estimated spectrum for any of component will be unknown linear where c and s are concentration and spectral profiles 𝑘 𝑘 combination of true components spectra. (column-vectors) of species 𝑘 in mixture. There are many methods of decomposition experimental eTh method needs rfi stly the estimation of one of the matrix D [12]. The one of most popular techniques is Mul- feasible solutions within the range of all possible solutions, for tivariate Curve Resolution-Alternating Least Squares (MCR- instance, using alternating least squares with constraints [9]. ALS) [9]. MCR-ALS solves iteratively (2)byanalternating Secondly, once this feasible solution is available, a nonlinear least squares algorithm which calculates concentration C and constrained optimization is initiated looking for the bound- pure spectra S matrices optimally tfi ting the experimental aries of the whole set of feasible solutions. For each compo- data matrix D.Thealgorithmcomprisesaniterativesolvingof nent 𝑘 we must solve problem (5). When the optimization is two alternating least squares problems, that is, minimization implemented as a minimization of the objective function as of (2)over C for xe fi d S as well as minimization of (2)over in (4), we will nfi dthe minimumbandboundaries, whereas S for xe fi d C. This optimization is carried out for a proposed if the optimization is implemented as a maximization of number of components and using initial estimates of either C the objective function with changed sign, we will n fi d the or S . maximum band boundaries functions. Implementation prior Journal of Atomic and Molecular Physics 3 information about investigated system in form of equality and 3.5 inequality constraints decreases the range of feasible solutions 3.0 [15]. 2.5 Pure acetone 2.0 3.3. Model-Based Approach to Decomposition of Spectra. As it was mentioned above, multivariate curve resolution is 1.5 model-free analysis and its solutions are not unique. Unlike Pure chloroform 1.0 this, model-based analysis gives unique solution and basic parameter of the process, that is, the rate constant and 0.5 the equilibrium constant in equilibrium investigations [17]. 0.0 Model-based approaches of tfi ting multivariate spectral data 4000 4100 4200 4300 4400 4500 are based on mathematical relationships, which describe −1 (cm ) the measurements quantitatively. In chemical kinetics, the analysis is based on the kinetic model or reaction mechanism, Figure 1: IR absorbance spectra of acetone-chloroform mixtures which quantitatively describes the reactions and all concen- with different component ratios. trations in the solution under investigation. For equilibrium studies the analysis is based on the law of mass action [18]. We can use the structural model of mixture for equi- 4. Results and Discussion librium studies. It is well known that the dissolution of one substance in another is accompanied by the formation 4.1. IR Measurements. FTIR transmission spectra (Figure 1) of molecular complexes arising due to the intermolecular −1 were detected in spectral range 3950–4550 cm .Thisspectral interaction [19]. In the frame of the posed problem, we now rangewas chosen duetothe weak absorption of overtones consider a mixture of two molecular substances A and B and composite frequencies in Near-IR region. Weak absorp- in the liquid state. es Th e components are not necessarily tion valueallowsusing theliquidcellwitharelatively large monomolecular. In many cases the structural forms of liquid value of the optical beam path (1 mm). Measurements in stateare dimers,trimmers, andsoforth.Theinteraction Mid-IR region require to use the optical beam path around between molecules leads to formation of complex AB (it is 10–25 𝜇 m. eTh refore, this range is not preferred for the considered as an effective type of complex). In what follows, quantitative analysis. However, the Near-IR range allows we will consider only the “formation-decay” reactions: carrying out reproducible concentration analysis with relative accuracy of at least 1%. 𝑛𝐴 + 𝐵𝑚 󴀕󴀬 𝐴𝐵.𝑝 (7) 4.2. MCR-ALS Analysis. Our approach is based on a three- Forthe formationofthe species AB, according to (7), we can component MCR-ALS analysis of the FTIR spectra of water- write equilibrium constant 𝐾 as follows: methanol solutions. During ALS optimization nonnegativity, unimodality, and closure constraints were applied. We found [ ] 𝐾 = , (8) 𝑐 that three components are required to obtain a good tfi to the 𝑛 𝑚 [𝐴 ] [𝐵 ] data (accuracy better than 1%). The “pure” components were identified as “free acetone (C H O)” and “free chloroform 3 6 where square brackets denote the molar concentration of (CHCl ),” and the third component as “acetone-chloroform species (mol/L). complex.” eTh graphical user interface (GUI) in the MATLAB If we know structural model of mixture and values of environment developed by Jaumot et al. [20]was used for components concentrations before mixing, we can estimate determination of concentration and spectral matrixes. the matrix of concentrations. We could get elements of matrix We used MCR-BANDS GUI [16]todecreaserotational C by solving (8)atfixedvalue of 𝐾 . ambiguity of obtained solutions; in other words it was used Hence, we can nd fi the matrix of spectral profiles S: to solve nonlinear constrained optimization problem (5). MCR-ALS solutions were used as initial estimation and −1 T T + (9) S =(C C) C D = C D. nonnegativity, unimodality, and closure constraints were also applied. eTh results on the resolved FTIR spectra ( S )and the Using (1)and (9), matrix of residuals can be transformed: concentrations (C) (band boundaries of feasible solutions) are shown in Figure 2. 𝜙 is volume fraction of acetone before E = D − CC D. (10) mixing; 𝜙 ,𝑖 = ,𝐴 ,𝐵 and , is volume fraction of species in the mixture. Firstly, we need to n fi d 𝐾 to calculate concentration profiles. eTh refore, the next optimization problem should be 4.3. Model-Based Analysis. It is possible to consider acetone- solved: chloroform mixture as ternary. er Th e is no doubt as to formation of equimolar complex AB [1]. The formation of ‖E‖ 󳨀→ min,𝐾 ≥0. (11) AB complex explains qualitatively the negative values of the IR absorption (a.u.) 𝐴𝐵 𝐴𝐵 4 Journal of Atomic and Molecular Physics 1.0 3.6 Chloroform 3.0 0.8 Complex 2.4 0.6 1.8 Acetone 0.4 1.2 0.2 0.6 0.0 0.0 0.0 1.0 0.2 0.4 0.6 0.8 4050 4200 4350 4500 −1 1 (cm ) Chloroform Acetone Complex (a) (b) Figure 2: Boundaries of concentration (a) and spectral profiles (b) of “pure” components obtained by MCR-BANDS analysis ( 𝜙 , 𝜙 ,𝑖 = 1 𝑖 𝐴, 𝐵, : volume fraction of acetone before mixing and volume fraction of species in mixture). excess thermodynamic functions [2]. But there are evidences Substituting 𝜙 in (15)by (16), we obtain 𝐴,𝐵 of existence of AB complex [3–6]. In our investigation we neglected the presence of complex due its small amount. 𝜙 =𝐾 (𝜙 −𝜂 𝜙 )(1 − 𝜙 −𝜂 𝜙 ). (17) 1 𝐴 1 𝐵 eTh self-association into dimers of pure acetone and pure chloroform was taken into account. u Th s, the “formation- We can obtain volume fraction of AB complex by solving decay” reactions become: (17)withknown values of 𝐾 and 𝜙 ,and thus we canobtain volume fraction of species A and B using (16). Matrix C can 𝐴 +𝐵 2𝐴𝐵. (12) 2 2 be obtained by changing values of 𝜙 . The equilibrium constant 𝐾 is defined by the equation We solved (11) numerically with optimization parameter 2 𝐾 .Theoptimal valueis 𝐾 = 6.76 .In Figure 3 concentration [ ] 𝐾 = . (13) and spectral profiles at this value of equilibrium constant are [𝐴 ][𝐵 ] 2 2 shown. We hold total volume of components before mixing constant: 4.4. Comparison. We compared theresolvedresults by 0 0 model-based analysis concentration profiles with those (14) ] 𝑉 + ] 𝑉 =𝑉 = const, 𝐴 𝐵 𝑚 1 2 obtained by MCR-ALS. The result of comparison is shown in 0 0 where ] , ] are amounts of acetone and chloroform before Figure 4. eTh results of both approaches are in good agree- 1 2 ment within the range of measuring error. eTh concentration mixing (in mol); 𝑉 , 𝑉 are molar volumes of acetone and 𝐴 𝐵 profile of complex obtained by MCR-ALS is skew. It may be chloroform, respectively. Change in volume aer ft mixing is explained by existence of small amount of complex at small; itsmaximum valueisabout 0.25%ofmolar volume of low concentration of acetone. Because MCR-ALS analysis is mixture [2]. So we neglected this change and considered vol- model-free we do not know what type (or types) of complex umeofmixture as constant.Thus we canexpress equilibrium forms this concentration profile. u Th s complex profile in constant in terms of volume fractions of component: model-free analysis may be considered as integral profile of 𝜙 𝑉 all complexes which are existent in mixture. (15) 𝐾= =𝐾 , 𝜙 𝜙 𝑉 𝑉 𝐴 𝐵 𝐴 𝐵 5. Conclusions where 𝜙 =(] 𝑉 )/𝑉 ,𝑖 = ,𝐴 ,𝐵 𝐵𝐴 , is volume fraction of 𝑖 𝑖 𝑖 𝑚 species in mixture; ] , 𝑉 are amount of species and its molar 𝑖 𝑖 The analysis of FTIR spectra at different acetone concentra- volume, respectively. So in (1)matrix C consists of volume tions in acetone-chloroform system using MCR-ALS method fractions of component. eTh relationship between fractions was carried out. It can give very necessary information about before and aeft r mixing is as follows: complex formation in this mixture. Three-component model of the mixture was chosen for the analysis. MCR-BANDS 𝐴,𝐵 𝜙 =𝜙 −𝜂 𝜙 ,𝜂 = . (16) technique was used to obtain the band boundaries of MCR- 𝐴,𝐵 1,2 𝐴,𝐵 𝐴,𝐵 𝑉 + 𝑉 𝐴 𝐵 ALS solutions. IR absorption (a.u.) 𝐴𝐵 𝐴𝐵 𝐴𝐵 𝐴𝐵 𝐴𝐵 󴀕󴀬 𝐴𝐵 𝐴𝐵 𝐴𝐵 𝐴𝐵 𝐴𝐵 Journal of Atomic and Molecular Physics 5 3.6 1.0 3.0 0.8 2.4 0.6 1.8 0.4 1.2 0.2 0.6 0.0 0.0 4000 4100 4200 4300 4400 4500 0.0 0.2 0.4 0.6 0.8 1.0 −1 (cm ) Acetone Complex Complex Acetone Chloroform Chloroform (a) (b) Figure 3: Concentration (a) and spectral profiles (b) of “pure” components obtained by model-based analysis ( 𝜙 , 𝜙 ,𝑖 = ,𝐴 𝐵,𝐵𝐴 : volume 1 𝑖 fraction of acetone before mixing and volume fraction of species in mixture). 1.0 Conflict of Interests eTh authors declare that there is no conflict of interests. 0.8 References 0.6 [1] A. N. Campbell and E. M. Kartzmark, “The energy of hydrogen bonding in the system acetone-chloroform,” Canadian Journal 0.4 of Chemistry,vol.38, no.5,pp. 652–655, 1960. [2] A. Apelblat, A. Tamir, and M. Wagner, “Thermodynamics of 0.2 acetone-chloroform mixtures,” Fluid Phase Equilibria,vol.4,no. 3-4, pp. 229–255, 1980. 0.0 [3] V. A. Durov and I. Y. Shilov, “Molecular structure and physic- 0.0 0.2 0.4 0.6 0.8 1.0 ochemical properties of acetone-chloroform mixtures,” Journal of the Chemical Society—Faraday Transactions,vol.92, no.19, pp. 3559–3563, 1996. Boundaries of MCR-ALS solutions [4] A. L. Capparelli, H. G. Hertz, B. Kvvatra, and R. Tutsih, “On Model-based analysis the nature of association in the system chloroform-acetone. A Figure 4: eTh result of comparison of resolved concentration nuclear magnetic relaxation study,” Zeitschrift f ur ¨ Physikalische profiles using model of mixture (solid curves) and boundaries of Chemie,vol.103,no. 5-6, pp.279–294,1976. MCR solutions (dotted curves). [5] G.A.Alper,M.Y.Nikiforov,A.N.Petrov, andG.A.Krestov, “Investigations of complex generation in acetone-chloroform by proton-magnetic-resonance spectroscopy,” Zhurnal Obshchei Khimii,vol.61, no.4,pp. 837–839, 1991. [6] L. Domonkos and F. Ratkovics, “Determination of the equilib- The most difficult aspect of model-based approach is the rium constants of association from IR-studies in the acetone- determination of correct model. eTh process of fitting several chloroform system,” Monatsheeft f ur Chemie,vol.116,no. 4, pp. models and comparing the results can be tedious. Three- 437–442, 1985. component model of the mixture was used successfully for [7] D. Steele, “Infrared spectroscopy: theory,” in Handbook of model-based approach to decomposition of spectra. Despite Vibrational Spectroscopy,J.M.Chalmersand P. R. Gritffi s,Eds., simplifying model usage, the concentration profiles obtained vol. 1, John Wiley&Sons,New York,NY, USA, 2001. by model-free and model-based approaches are in good [8] R. Wehrens, Chemometrics with R, Springer, Berlin, Germany, agreement within the range of measuring error. complex forming will be subject of our further investigation. [9] R. Tauler, B. Kowalski, and S. Fleming, “Multivariate curve Both approaches should be used in decomposition liquid resolution applied to spectral data from multiple runs of an mixture vibrational spectra. eTh model-free analysis can be industrial process,” Analytical Chemistry, vol. 65, no. 15, pp. invaluable in supporting the model choice. 2040–2047, 1993. IR absorption (a.u.) 𝐴𝐵 6 Journal of Atomic and Molecular Physics [10] W. H. Lawton and E. A. Sylvestre, “Self modeling curve resolution,” Technometrics,vol.13, pp.617–633,1971. [11] R. Tauler, A. K. Smilde, and B. R. Kowalski, “Selectivity, local rank, three-way data analysis and ambiguity in multivariate curve resolution,” Journal of Chemometrics,vol.9,no. 1, pp.31– 58, 1995. [12] J.-H. Jiang and Y. Ozaki, “Self-modeling curve resolution (SMCR): principles, techniques, and applications,” Applied Spectroscopy Reviews,vol.37, no.3,pp. 321–345, 2002. [13] R. Manne, “On the resolution problem in hyphenated chro- matography,” Chemometrics and Intelligent Laboratory Systems, vol. 27,no. 1, pp.89–94,1995. [14] A. de Juan, Y. Vander Heyden, R. Tauler, and D. L. Massart, “Assessment of new constraints applied to the alternating least squares method,” Analytica Chimica Acta,vol.346,no. 3, pp. 307–318, 1997. [15] R. Tauler, “Calculation of maximum and minimum band boundaries of feasible solutions for species profiles obtained by multivariate curve resolution,” Journal of Chemometrics,vol.15, no. 8, pp. 627–646, 2001. [16] P. J. Gemperline, “Computation of the range of feasible solutions in self-modeling curve resolution algorithms,” Analytical Chem- istry,vol.71, no.23, pp.5398–5404,1999. [17] S. Norman and M. Maeder, “Model-based analysis for kinetic and equilibrium investigations,” Critical Reviews in Analytical Chemistry,vol.36, no.3-4,pp. 199–209, 2006. [18] A. E. Martell and R. J. Motekaitis, The Determination and Use of Stability Constants, Wiley-VCH, New York, NY, USA, 1992. [19] A. Apelblat, “The concept of associated solutions in historical development—part 1: the 1884–1984 period,” Journal of Molec- ular Liquids,vol.128,no. 1–3, pp.1–31, 2006. [20] J. Jaumot, R. Gargallo, A. de Juan, and R. Tauler, “A graphical user-friendly interface for MCR-ALS: a new tool for multivari- ate curve resolution in MATLAB,” Chemometrics and Intelligent Laboratory Systems,vol.76, no.1,pp. 101–110, 2005. 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Study of Complexation in Acetone-Chloroform Mixtures by Infrared Spectroscopy

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Hindawi Publishing Corporation Journal of Atomic and Molecular Physics Volume 2014, Article ID 106178, 6 pages http://dx.doi.org/10.1155/2014/106178 Research Article Study of Complexation in Acetone-Chloroform Mixtures by Infrared Spectroscopy Oleksii O. Ilchenko, Andrii M. Kutsyk, and Vyacheslav V. Obukhovsky Radiophysical Department, National Taras Shevchenko University of Kyiv, Volodymyrska Street 64, Kyiv, Ukraine Correspondence should be addressed to Oleksii O. Ilchenko; radastasi@gmail.com Received 29 April 2013; Revised 28 December 2013; Accepted 29 December 2013; Published 24 February 2014 Academic Editor: Keli Han Copyright © 2014 Oleksii O. Ilchenko et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FTIR spectra of acetone-chloroform system with various component ratios were investigated within the spectral range 3950– −1 4550 cm . Methods of multivariate curve resolution were applied to decompose the FTIR spectra into specific components of different composition. A method of decomposition based on structural model of solution which contains acetone, chloroform, and complex acetone/chloroform (1 : 1) was proposed. Results of both approaches are in good agreement within the range of measuring error. 1. Introduction excitation frequency. The temperature of liquid samples was 25 ± 0.2 C. The chloroform and acetone with purity 99.9% Acetone-chloroform mixture is a prominent example of a were used in this research. eTh concentration of components system with a pronounced negative deviation from ideal was changed from 0% to 100% (in volume %) with step 10%. solution behavior. So it has been the subject of many theo- Spectra recording was repeated 5 times for every sample. retical and experimental researches [1–6]. The main problem Thereaer ft the average spectra were calculated for every is the structure and composition of associated species of these concentration and used in further analysis. mixtures. Infrared spectroscopy is powerful tool of investigation of liquid systems [7]. The IR spectra are very sensitive 3. Theory to structural changes caused by intermolecular interactions 3.1. Multivariate Curve Resolution. Resolution methods between components of solution. Multivariate regression can be used for the determination of quantitative information decompose mathematically a global mixed instrumental response into the pure contributions due to each component from IR spectra, such as number and concentration of species in the system [9]. This mixed signal is organized in the matrix in the mixture [8]. eTh purposeofthisworkisquantitativeanalysisof D containing raw information about all the components present in the data set. Resolution methods allow for the acetone-chloroform mixture using model-free and model- based multivariate regression approaches. decomposition of the initial mixture data matrix D into the product of two data matrices C and S , each of them including the pure response profiles of the 𝑛 mixture or 2. Experiment process components associated with the row and the column direction of the initial data matrix, respectively. In matrix FTIR transmission spectra were measured with er Th mo Sci- notation, the expression valid for all resolution methods is enticfi spectrometer Nicolet 6700 with a spectral resolution −1 of 4 cm . Optical path length of the quartz cell was 1 mm. The heating of the sample almost did not occur during the D = CS + E, (1) measurements due to small value of absorption coefficient at 2 Journal of Atomic and Molecular Physics where D(𝑟×)𝑐 is the original data matrix, C(𝑟×𝑛 )and S (𝑛×)𝑐 3.2. Estimation of Band Boundaries of Feasible MCR Solutions. arethe matrices containing thepureresponseprofiles related Complete resolution of a two-way data set without ambigui- to the data variation in the row direction and in the column ties is only possible in some favorable cases where selectivity direction, respectively, and E(𝑟 × )𝑐 is the error matrix, that is, [11] or local rank conditions [13] are present. When these the residual variation of the data set that is not related to any resolution conditions are not present in the system, resolution chemical contribution. Parameters 𝑟 and 𝑐 are the number of without ambiguities is not possible even if constraints such rows and the number of columns of the original data matrix, as nonnegativity, unimodality, or closure are applied [14]. In respectively, and 𝑛 is the number of chemical components in these cases, instead of unique profiles, a range or band of feasible profiles tfi ting equally well the experimental data and the mixture or process. C and S often refer to concentration fulfilling the physical and chemical constraints of the system profiles and spectra (hence their names). hastobeconsidered. With the measurement matrix D,the probleminmul- Boundaries of feasible solutions are related to rotational tivariate curve resolution is to estimate the pure variables, matrices T and T for each species 𝑘 . Maximum and C and S,interms of thebilinearmodel andsomegeneric min,𝑘 max,𝑘 minimum band boundaries for concentration and spectral knowledgeabout thepurevariables.Accordingly, thebasic profiles may be defined by the following equation: principle of curve resolution is to seek for a bilinear model that gives best t, fi in the sense of least squares or weighted least T −1 T T D = CS = CT T S = C S squares, to the two-way data D.That is to say, oneisseeking min min min min (4) forthe estimatesofpurevariables, C and S, which minimize −1 T T = CT T S = C S . the norm max max max max eTh goal is calculation of matrices T and T 󵄩 󵄩 min,𝑘 max,𝑘 󵄩 󵄩 󵄩 󵄩 ‖E‖ = D − CS 󳨀→ min . (2) 󵄩 󵄩 for nding fi band boundaries. The problem is considered in 󵄩 󵄩 frame of a constrained nonlinear optimization problem. It is described mathematically using the equation [15] The mathematical decomposition of a single data matrix D or, in other words, the norm minimization problem (2) 𝑓 (T) 󳨀→ opt, is known to be subject to ambiguities [10]. This means that many sets of paired C-and S -type matrices can reproduce (5) theoriginaldataset with thesamefitquality.But some of the g (T) =0, obtained solutions do not have any physical meaning. In plain g (T) ≤0, ineq words, the correct reproduction of the original data matrix can be achieved by using response profiles differing in shape where 𝑓( T) is an objective function; g is vector of ineq (rotational ambiguity) or in magnitude (intensity ambiguity) inequality constraints (such as nonnegativity, selectivity, and from the sought (true) ones [11]. Mathematically it can be unimodality); g is vector of equality constraints (mass written as follows: balance, known values of concentration, etc.). Gemperline [16]and Tauler [15] have shown that the calculation of the T −1 T T ̃ ̃ (3) D = CS = CTT S = CS , band boundaries of feasible solutions for every species is possible when an objective optimization function is defined in terms of the ratio of the signal contribution of that species where T(𝑛 × 𝑛) is invertible matrix. If T is orthogonal matrix ambiguity is rotational. The intensity ambiguity is not a to the whole signal contribution for the mixture of all species. For example, Tauler proposed to define objective function as serious problem in qualitative analysis (spectral identifica- tion), but it is a serious problem in quantitative analysis. 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 The most important is the rotational ambiguity which always c s 󵄩 󵄩 𝑘 𝑘 󵄩 󵄩 (6) 𝑓 (T) = , 𝑘 󵄩 󵄩 occurs when there are several overlapped profiles of compo- 󵄩 󵄩 CS 󵄩 󵄩 󵄩 󵄩 nents. The presence of this ambiguity means that estimated spectrum for any of component will be unknown linear where c and s are concentration and spectral profiles 𝑘 𝑘 combination of true components spectra. (column-vectors) of species 𝑘 in mixture. There are many methods of decomposition experimental eTh method needs rfi stly the estimation of one of the matrix D [12]. The one of most popular techniques is Mul- feasible solutions within the range of all possible solutions, for tivariate Curve Resolution-Alternating Least Squares (MCR- instance, using alternating least squares with constraints [9]. ALS) [9]. MCR-ALS solves iteratively (2)byanalternating Secondly, once this feasible solution is available, a nonlinear least squares algorithm which calculates concentration C and constrained optimization is initiated looking for the bound- pure spectra S matrices optimally tfi ting the experimental aries of the whole set of feasible solutions. For each compo- data matrix D.Thealgorithmcomprisesaniterativesolvingof nent 𝑘 we must solve problem (5). When the optimization is two alternating least squares problems, that is, minimization implemented as a minimization of the objective function as of (2)over C for xe fi d S as well as minimization of (2)over in (4), we will nfi dthe minimumbandboundaries, whereas S for xe fi d C. This optimization is carried out for a proposed if the optimization is implemented as a maximization of number of components and using initial estimates of either C the objective function with changed sign, we will n fi d the or S . maximum band boundaries functions. Implementation prior Journal of Atomic and Molecular Physics 3 information about investigated system in form of equality and 3.5 inequality constraints decreases the range of feasible solutions 3.0 [15]. 2.5 Pure acetone 2.0 3.3. Model-Based Approach to Decomposition of Spectra. As it was mentioned above, multivariate curve resolution is 1.5 model-free analysis and its solutions are not unique. Unlike Pure chloroform 1.0 this, model-based analysis gives unique solution and basic parameter of the process, that is, the rate constant and 0.5 the equilibrium constant in equilibrium investigations [17]. 0.0 Model-based approaches of tfi ting multivariate spectral data 4000 4100 4200 4300 4400 4500 are based on mathematical relationships, which describe −1 (cm ) the measurements quantitatively. In chemical kinetics, the analysis is based on the kinetic model or reaction mechanism, Figure 1: IR absorbance spectra of acetone-chloroform mixtures which quantitatively describes the reactions and all concen- with different component ratios. trations in the solution under investigation. For equilibrium studies the analysis is based on the law of mass action [18]. We can use the structural model of mixture for equi- 4. Results and Discussion librium studies. It is well known that the dissolution of one substance in another is accompanied by the formation 4.1. IR Measurements. FTIR transmission spectra (Figure 1) of molecular complexes arising due to the intermolecular −1 were detected in spectral range 3950–4550 cm .Thisspectral interaction [19]. In the frame of the posed problem, we now rangewas chosen duetothe weak absorption of overtones consider a mixture of two molecular substances A and B and composite frequencies in Near-IR region. Weak absorp- in the liquid state. es Th e components are not necessarily tion valueallowsusing theliquidcellwitharelatively large monomolecular. In many cases the structural forms of liquid value of the optical beam path (1 mm). Measurements in stateare dimers,trimmers, andsoforth.Theinteraction Mid-IR region require to use the optical beam path around between molecules leads to formation of complex AB (it is 10–25 𝜇 m. eTh refore, this range is not preferred for the considered as an effective type of complex). In what follows, quantitative analysis. However, the Near-IR range allows we will consider only the “formation-decay” reactions: carrying out reproducible concentration analysis with relative accuracy of at least 1%. 𝑛𝐴 + 𝐵𝑚 󴀕󴀬 𝐴𝐵.𝑝 (7) 4.2. MCR-ALS Analysis. Our approach is based on a three- Forthe formationofthe species AB, according to (7), we can component MCR-ALS analysis of the FTIR spectra of water- write equilibrium constant 𝐾 as follows: methanol solutions. During ALS optimization nonnegativity, unimodality, and closure constraints were applied. We found [ ] 𝐾 = , (8) 𝑐 that three components are required to obtain a good tfi to the 𝑛 𝑚 [𝐴 ] [𝐵 ] data (accuracy better than 1%). The “pure” components were identified as “free acetone (C H O)” and “free chloroform 3 6 where square brackets denote the molar concentration of (CHCl ),” and the third component as “acetone-chloroform species (mol/L). complex.” eTh graphical user interface (GUI) in the MATLAB If we know structural model of mixture and values of environment developed by Jaumot et al. [20]was used for components concentrations before mixing, we can estimate determination of concentration and spectral matrixes. the matrix of concentrations. We could get elements of matrix We used MCR-BANDS GUI [16]todecreaserotational C by solving (8)atfixedvalue of 𝐾 . ambiguity of obtained solutions; in other words it was used Hence, we can nd fi the matrix of spectral profiles S: to solve nonlinear constrained optimization problem (5). MCR-ALS solutions were used as initial estimation and −1 T T + (9) S =(C C) C D = C D. nonnegativity, unimodality, and closure constraints were also applied. eTh results on the resolved FTIR spectra ( S )and the Using (1)and (9), matrix of residuals can be transformed: concentrations (C) (band boundaries of feasible solutions) are shown in Figure 2. 𝜙 is volume fraction of acetone before E = D − CC D. (10) mixing; 𝜙 ,𝑖 = ,𝐴 ,𝐵 and , is volume fraction of species in the mixture. Firstly, we need to n fi d 𝐾 to calculate concentration profiles. eTh refore, the next optimization problem should be 4.3. Model-Based Analysis. It is possible to consider acetone- solved: chloroform mixture as ternary. er Th e is no doubt as to formation of equimolar complex AB [1]. The formation of ‖E‖ 󳨀→ min,𝐾 ≥0. (11) AB complex explains qualitatively the negative values of the IR absorption (a.u.) 𝐴𝐵 𝐴𝐵 4 Journal of Atomic and Molecular Physics 1.0 3.6 Chloroform 3.0 0.8 Complex 2.4 0.6 1.8 Acetone 0.4 1.2 0.2 0.6 0.0 0.0 0.0 1.0 0.2 0.4 0.6 0.8 4050 4200 4350 4500 −1 1 (cm ) Chloroform Acetone Complex (a) (b) Figure 2: Boundaries of concentration (a) and spectral profiles (b) of “pure” components obtained by MCR-BANDS analysis ( 𝜙 , 𝜙 ,𝑖 = 1 𝑖 𝐴, 𝐵, : volume fraction of acetone before mixing and volume fraction of species in mixture). excess thermodynamic functions [2]. But there are evidences Substituting 𝜙 in (15)by (16), we obtain 𝐴,𝐵 of existence of AB complex [3–6]. In our investigation we neglected the presence of complex due its small amount. 𝜙 =𝐾 (𝜙 −𝜂 𝜙 )(1 − 𝜙 −𝜂 𝜙 ). (17) 1 𝐴 1 𝐵 eTh self-association into dimers of pure acetone and pure chloroform was taken into account. u Th s, the “formation- We can obtain volume fraction of AB complex by solving decay” reactions become: (17)withknown values of 𝐾 and 𝜙 ,and thus we canobtain volume fraction of species A and B using (16). Matrix C can 𝐴 +𝐵 2𝐴𝐵. (12) 2 2 be obtained by changing values of 𝜙 . The equilibrium constant 𝐾 is defined by the equation We solved (11) numerically with optimization parameter 2 𝐾 .Theoptimal valueis 𝐾 = 6.76 .In Figure 3 concentration [ ] 𝐾 = . (13) and spectral profiles at this value of equilibrium constant are [𝐴 ][𝐵 ] 2 2 shown. We hold total volume of components before mixing constant: 4.4. Comparison. We compared theresolvedresults by 0 0 model-based analysis concentration profiles with those (14) ] 𝑉 + ] 𝑉 =𝑉 = const, 𝐴 𝐵 𝑚 1 2 obtained by MCR-ALS. The result of comparison is shown in 0 0 where ] , ] are amounts of acetone and chloroform before Figure 4. eTh results of both approaches are in good agree- 1 2 ment within the range of measuring error. eTh concentration mixing (in mol); 𝑉 , 𝑉 are molar volumes of acetone and 𝐴 𝐵 profile of complex obtained by MCR-ALS is skew. It may be chloroform, respectively. Change in volume aer ft mixing is explained by existence of small amount of complex at small; itsmaximum valueisabout 0.25%ofmolar volume of low concentration of acetone. Because MCR-ALS analysis is mixture [2]. So we neglected this change and considered vol- model-free we do not know what type (or types) of complex umeofmixture as constant.Thus we canexpress equilibrium forms this concentration profile. u Th s complex profile in constant in terms of volume fractions of component: model-free analysis may be considered as integral profile of 𝜙 𝑉 all complexes which are existent in mixture. (15) 𝐾= =𝐾 , 𝜙 𝜙 𝑉 𝑉 𝐴 𝐵 𝐴 𝐵 5. Conclusions where 𝜙 =(] 𝑉 )/𝑉 ,𝑖 = ,𝐴 ,𝐵 𝐵𝐴 , is volume fraction of 𝑖 𝑖 𝑖 𝑚 species in mixture; ] , 𝑉 are amount of species and its molar 𝑖 𝑖 The analysis of FTIR spectra at different acetone concentra- volume, respectively. So in (1)matrix C consists of volume tions in acetone-chloroform system using MCR-ALS method fractions of component. eTh relationship between fractions was carried out. It can give very necessary information about before and aeft r mixing is as follows: complex formation in this mixture. Three-component model of the mixture was chosen for the analysis. MCR-BANDS 𝐴,𝐵 𝜙 =𝜙 −𝜂 𝜙 ,𝜂 = . (16) technique was used to obtain the band boundaries of MCR- 𝐴,𝐵 1,2 𝐴,𝐵 𝐴,𝐵 𝑉 + 𝑉 𝐴 𝐵 ALS solutions. IR absorption (a.u.) 𝐴𝐵 𝐴𝐵 𝐴𝐵 𝐴𝐵 𝐴𝐵 󴀕󴀬 𝐴𝐵 𝐴𝐵 𝐴𝐵 𝐴𝐵 𝐴𝐵 Journal of Atomic and Molecular Physics 5 3.6 1.0 3.0 0.8 2.4 0.6 1.8 0.4 1.2 0.2 0.6 0.0 0.0 4000 4100 4200 4300 4400 4500 0.0 0.2 0.4 0.6 0.8 1.0 −1 (cm ) Acetone Complex Complex Acetone Chloroform Chloroform (a) (b) Figure 3: Concentration (a) and spectral profiles (b) of “pure” components obtained by model-based analysis ( 𝜙 , 𝜙 ,𝑖 = ,𝐴 𝐵,𝐵𝐴 : volume 1 𝑖 fraction of acetone before mixing and volume fraction of species in mixture). 1.0 Conflict of Interests eTh authors declare that there is no conflict of interests. 0.8 References 0.6 [1] A. N. Campbell and E. M. Kartzmark, “The energy of hydrogen bonding in the system acetone-chloroform,” Canadian Journal 0.4 of Chemistry,vol.38, no.5,pp. 652–655, 1960. [2] A. Apelblat, A. Tamir, and M. Wagner, “Thermodynamics of 0.2 acetone-chloroform mixtures,” Fluid Phase Equilibria,vol.4,no. 3-4, pp. 229–255, 1980. 0.0 [3] V. A. Durov and I. Y. Shilov, “Molecular structure and physic- 0.0 0.2 0.4 0.6 0.8 1.0 ochemical properties of acetone-chloroform mixtures,” Journal of the Chemical Society—Faraday Transactions,vol.92, no.19, pp. 3559–3563, 1996. Boundaries of MCR-ALS solutions [4] A. L. Capparelli, H. G. Hertz, B. Kvvatra, and R. Tutsih, “On Model-based analysis the nature of association in the system chloroform-acetone. A Figure 4: eTh result of comparison of resolved concentration nuclear magnetic relaxation study,” Zeitschrift f ur ¨ Physikalische profiles using model of mixture (solid curves) and boundaries of Chemie,vol.103,no. 5-6, pp.279–294,1976. MCR solutions (dotted curves). [5] G.A.Alper,M.Y.Nikiforov,A.N.Petrov, andG.A.Krestov, “Investigations of complex generation in acetone-chloroform by proton-magnetic-resonance spectroscopy,” Zhurnal Obshchei Khimii,vol.61, no.4,pp. 837–839, 1991. [6] L. Domonkos and F. Ratkovics, “Determination of the equilib- The most difficult aspect of model-based approach is the rium constants of association from IR-studies in the acetone- determination of correct model. eTh process of fitting several chloroform system,” Monatsheeft f ur Chemie,vol.116,no. 4, pp. models and comparing the results can be tedious. Three- 437–442, 1985. component model of the mixture was used successfully for [7] D. Steele, “Infrared spectroscopy: theory,” in Handbook of model-based approach to decomposition of spectra. Despite Vibrational Spectroscopy,J.M.Chalmersand P. R. Gritffi s,Eds., simplifying model usage, the concentration profiles obtained vol. 1, John Wiley&Sons,New York,NY, USA, 2001. by model-free and model-based approaches are in good [8] R. Wehrens, Chemometrics with R, Springer, Berlin, Germany, agreement within the range of measuring error. complex forming will be subject of our further investigation. [9] R. Tauler, B. Kowalski, and S. Fleming, “Multivariate curve Both approaches should be used in decomposition liquid resolution applied to spectral data from multiple runs of an mixture vibrational spectra. eTh model-free analysis can be industrial process,” Analytical Chemistry, vol. 65, no. 15, pp. invaluable in supporting the model choice. 2040–2047, 1993. IR absorption (a.u.) 𝐴𝐵 6 Journal of Atomic and Molecular Physics [10] W. H. Lawton and E. A. Sylvestre, “Self modeling curve resolution,” Technometrics,vol.13, pp.617–633,1971. [11] R. Tauler, A. K. Smilde, and B. R. Kowalski, “Selectivity, local rank, three-way data analysis and ambiguity in multivariate curve resolution,” Journal of Chemometrics,vol.9,no. 1, pp.31– 58, 1995. [12] J.-H. Jiang and Y. Ozaki, “Self-modeling curve resolution (SMCR): principles, techniques, and applications,” Applied Spectroscopy Reviews,vol.37, no.3,pp. 321–345, 2002. [13] R. Manne, “On the resolution problem in hyphenated chro- matography,” Chemometrics and Intelligent Laboratory Systems, vol. 27,no. 1, pp.89–94,1995. [14] A. de Juan, Y. Vander Heyden, R. Tauler, and D. L. Massart, “Assessment of new constraints applied to the alternating least squares method,” Analytica Chimica Acta,vol.346,no. 3, pp. 307–318, 1997. [15] R. Tauler, “Calculation of maximum and minimum band boundaries of feasible solutions for species profiles obtained by multivariate curve resolution,” Journal of Chemometrics,vol.15, no. 8, pp. 627–646, 2001. [16] P. J. Gemperline, “Computation of the range of feasible solutions in self-modeling curve resolution algorithms,” Analytical Chem- istry,vol.71, no.23, pp.5398–5404,1999. [17] S. Norman and M. Maeder, “Model-based analysis for kinetic and equilibrium investigations,” Critical Reviews in Analytical Chemistry,vol.36, no.3-4,pp. 199–209, 2006. [18] A. E. Martell and R. J. Motekaitis, The Determination and Use of Stability Constants, Wiley-VCH, New York, NY, USA, 1992. [19] A. Apelblat, “The concept of associated solutions in historical development—part 1: the 1884–1984 period,” Journal of Molec- ular Liquids,vol.128,no. 1–3, pp.1–31, 2006. [20] J. Jaumot, R. Gargallo, A. de Juan, and R. Tauler, “A graphical user-friendly interface for MCR-ALS: a new tool for multivari- ate curve resolution in MATLAB,” Chemometrics and Intelligent Laboratory Systems,vol.76, no.1,pp. 101–110, 2005. 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