Anal. Chem. 1999, 71, 602-608
Detection and Identification of a Methanol-Water Complex by Factor Analysis of Infrared Spectra Zhouming Zhao and Edmund R. Malinowski*
Department of Chemistry and Chemical Biology, Stevens Institute of Technology, Hoboken, New Jersey 07030
Fourier transform infrared absorption spectra, between 1850 and 2700 cm-1, were used to study the intermolecular interactions between methanol and water. Digitized spectra obtained from methanol-water mixtures, with methanol mole fraction varying from 0 to 1.00, were subjected to factor analysis. Principal factor analysis indicated that three chemical factors were responsible for the data. Window factor analysis, a model-free method, was used to extract the concentration profiles of three species, which were identified as water, methanol, and a methanol-water complex. The profiles were used to deduce the stoichiometry of the complex, which was found to consist of one molecule of methanol and two molecules of water. The dissociation constant of the complex was determined to be 306 ( 33 M2. These results differ from those reported by other investigators. Methanol-water mixtures have excellent solvent properties and are widely used as mobile phases in reversed-phase highperformance liquid chromatography. This solvent system is usually considered to be a binary system.1,2 However, the binary concept of methanol-water mixtures does not conform to solute distribution theory.3 Similar failures has been reported for acidbase studies in methanol-water mixtures,4-7 which is important for pH standardization of the mobile phase. Since the retention time of a solute is dependent upon its distribution between stationary and mobile phases, it is important to know the exact composition of the mobile phase. The wide use of methanolwater mixtures in analytical chemistry has spurred the need for further explorations of this important solvent system. Based on measurements of volume change upon mixing, refractive index, and density, Katz et al.3,8 concluded that methanolwater mixtures act as a ternary system. The system was assumed to consist of clusters of methanol, water, and a 1:1 methanolwater complex, with all three species coexisting in equilibrium. * Address correspondence to E. R. Malinowski, 5042 SE Devenwood Way, Stuart, FL 34997. (1) Valko, I. E.; Siren, H.; Riekkola, M. L. LC-GC, 1997, 15, 560-567. (2) Gmehling, J.; Onken, U.; Arlt, W. Vapor-Liquid Equilibrium Data Collection; Schon and Wetzel GmbH: Frankfurt, 1981. (3) Katz, E. D.; Ogan, K.; Scott, R. P. W. J. Chromatogr. 1986, 352, 67-90. (4) Bosch, E.; Rafols, C.; Roses, M. Anal. Chim. Acta 1995, 302, 109-119. (5) Bosch, E.; Bou, P.; Allemann, H.; Roses, M. Anal. Chem. 1996, 68, 36513657. (6) Mussini, T.; Longhi, P.; Rondinini, S.; Tettamanti, M. Anal. Chim. Acta 1985, 174, 331-337. (7) Niazi, M. S. K.; Khan, M. Z. I. J. Solution Chem. 1993, 22, 437-456. (8) Katz, E. D.; Lochmuller, C. H.; Scott, R. P. Anal. Chem. 1989, 61, 349355.
602 Analytical Chemistry, Vol. 71, No. 3, February 1, 1999
The complex was believed to be the result of hydrogen bonding between molecules of methanol and water. Using an assumed value for the molar volume of the complex, Katz et al. estimated the disassociation constant of the complex. Bosch et al.5 expressed the equilibrium between S1 (water), S2 (methanol), and S12 (mixed structure of 1:1 methanol-water) as S1 + S2 ) 2(S12). Two molecules of S12 were used in the equilibrium expression in order to keep the number of solvent molecules constant in the system. Using the same data that Katz and co-workers used, these investigators found the logarithm of the formation constant of S12 to be 0.8 on a mole fraction scale. Alam and Callis9 studied methanol-water mixtures by multivariate analysis of the second derivatives of the near-infrared spectra. Two chemical models were examined, a single ideal association model and a double ideal association model. Evolving factor analysis (EFA)10 and principal component analysis (PCA)11 techniques were employed to determine the rank (the number of controlling factors) of methanol-water mixtures. The double association model was found to fit the experimental data better than the single association model. The number of species involved in the mixtures was estimated to be 4, i.e., methanol, water, the 1:1 methanol-water complex, and a 1:5 methanol-water complex which occurs at low methanol concentrations. No meaningful association constants could be determined because the association constants and the extinction coefficients of the complexes were not independent. All of the above investigations rely on models and assumptions. Yet, to date, the superiority of any one model remains to be demonstrated. A new chemometric methodology, called window factor analysis (WFA),12 is ideally suited for this purpose. WFA is a model-free technique for obtaining the composition and concentration of the species in solutions without requiring any a priori information. Malinowski and his students12-17 have successfully applied WFA for the study of complexation in solutions. The present work concerns the investigation of methanolwater mixtures by WFA of digitized FT-IR absorption spectra. We describe, herein, how WFA can be used to determine the solution (9) Alam, M. K.; Callis, J. B. Anal. Chem. 1994, 66, 2293-2301. (10) Maeder, M. Anal. Chem. 1987, 59, 527-530. (11) Malinowski, E. R. Factor Analysis in Chemistry, 2nd ed.; Wiley: New York, 1991. (12) Malinowski, E. R. J. Chemom. 1992, 6, 29-40. (13) Den, W.; Malinowski, E. R. J. Chemom. 1993, 7, 89-98. (14) Schostack, K. J.; Malinowski, E. R. Chemom. Intell. Lab. Syst. 1993, 20, 173-182. (15) Lee, I. Master’s Thesis, Stevens Institute of Technology, 1993. (16) Wang, S. Master’s Thesis, Stevens Institute of Technology, 1994. (17) Darj, M.; Malinowski, E. R. Anal. Chem. 1996, 68, 1593-1598. 10.1021/ac9809480 CCC: $18.00
© 1999 American Chemical Society Published on Web 12/29/1998
composition, the chemical formula of the complex, and its dissociation constant without any a priori speculation about the nature of the species. THEORY WFA12,13 takes advantage of the fact that each component lies in a specific region along the evolutionary axis, called the “window”. By specifying the window of existence for a particular component of an evolutionary process and performing principal factor analysis (PFA) on the data after deleting the window data, the concentration profile of the component can be determined. The mathematics involved has been derived previously17 and is summarized in this section. The full absorbance data matrix A may be written as
n-1
A)
n
∑s (∑β c 0
j
T
ij i
j)1
) + βnnsn0cnT
(5)
i)1
Multiplying eq 7 by sj0T, which is orthonormal, gives n
sj0TA )
∑β c
T
ij i
(6)
i)1
Inserting eq 6 into eq 5 gives n-1
A)
∑s
0 j
(sj0TA) + βnnsn0cnT
(7)
j)1
Equation 7 can be rearranged to n
A)
n
∑A ) ∑s c j
T
j j
j)1
) SC
(1)
n-1
∑s
0 j
cj0T ) S0C0
(2)
j)1
where S0 is an abstract matrix containing (n - 1) orthogonal spectral vectors sj0, and C0 is an abstract matrix containing (n 1) orthogonal concentration vectors cj0T. These abstract vectors can be readily obtained by singular value decomposition (SVD). Because the (n - 1)-dimensional subspace of A0 lies inside the n-dimensional space of A, the true spectral vectors of the (n - 1) components can be expressed as linear combinations of the abstract vectors, with linear coefficients bij, as shown in eq 3. n-1
∑β s
si )
0
ij j
(3)
j)1
The spectrum of the nth component, which lies inside the n-dimensional space, has a projection into the (n - 1)-dimensional subspace. To express its spectrum as a linear combination requires the addition of vector, sn0, which is orthogonal to the other vectors. n-1
sn )
(8)
j)1
where sj is a vector representing the spectrum of component j, cj is a vector representing the concentration profile of component j, and the superscript T is used to indicate the transpose of the vector. Aj ) sjcjT is the contribution of component j to the matrix of absorption spectra. The columns of S contain the digitized spectra of the components, whereas the rows of C contain the respective concentration profiles of the components. Let A0 represent a submatrix of A constructed by deleting the spectral data columns indigenous to component n. The deleted set of data columns is called the “window” of n. Submatrix A0 can be mathematically expressed as shown in eq 2,
A0 )
(I - S0S0T)A ) Mn ) βnnsn0cnT
∑β
0
njsj
+ βnnsn0
j)1
Inserting eqs 3 and 4 into eq 1 and rearranging gives
In eq 8, I is the identity matrix and Mn is a matrix of rank 1, βnn is an unknown constant, cn is the concentration profile of component n, and sn0 is the part of the spectral vector of component n which is orthogonal to the (n - 1) factor space defined by S0. Mn is readily calculable because I and A are known, and S0 is obtainable from SVD. Because Mn has unit rank, each row is directly proportional to the concentration profile. Hence, by using eq 8, the uncalibrated concentration profile of component n can be extracted from the absorbance data. By performing WFA, the concentration profiles of each component in a mixture can be determined by systematically considering all possible windows, retaining only those profiles that are well-behaved, i.e., those having a single maximum with essentially no negative concentration points. The WFA method is particularly exciting because it permits extraction of a single profile without recourse to any information concerning the other species. Each concentration profile is extracted independently without any knowledge of the profiles of the other components and, most importantly, without requiring any preconceived model of the system. The method, however, requires that each component has a distinguishable spectrum and that the profile of the component does not encase the complete profile of any other component, although the other components may have signals inside the designated window. This requirement is called the “inclusion-exclusion” rule. If this rule is violated, then the resulting concentration profile will be a mixed profile containing a contribution from the profile of the offending component. The degree of mixing depends on how badly the rule has been violated. In many cases, because of the chemical nature of the controlling equilibria, strict adherence to this rule should not be expected. Assessment of the error produced by the violation requires testing the results using chemical concepts, chemical constraints, chemical balance, simulation studies, etc. Finally, WFA is based on the conditions that the spectral measurement is a linear sum of component spectra and that the correct number of factors is used in the computations.
(4) EXPERIMENTAL SECTION Chemicals. Methanol (99.9+%, ACS HPLC grade) was purchased from Sigma Chemical Co. (St. Louis, MO). Ultrahigh-purity distilled water was obtained from Electrified (East Orange, NJ). Analytical Chemistry, Vol. 71, No. 3, February 1, 1999
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Table 1. Mole Fractions, Initial Molarities, and Equilibrium Molarities from WFA and Ratios of Water to Methanol in the Complex methanol mole fraction, Xa
initial methanol, CM0 (M)b
initial water, CW0 (M)b
initial total, C0 (M)c
WFA methanol, CM (M)d
WFA water CW (M)d
WFA complex (m + n)CX (M)e
CW0 - CW/CM0 - CM (n/m)f
0 0.0090 0.0182 0.0276 0.0.372 0.0470 0.0570 0.0672 0.0777 0.0883 0.0988 0.1099 0.1213 0.1328 0.1445 0.1565 0.1689 0.1816 0.1944 0.2077 0.2214 0.2350 0.2495 0.2641 0.2793 0.2947 0.3102 0.3270 0.3428 0.3617 0.3805 0.3994 0.4191 0.4392 0.4610 0.4834 0.5057 0.5300 0.5550 0.5822 0.6103 0.6398 0.6709 0.7032 0.7373 0.7729 0.8137 0.8536 0.8955 0.9439 1.0000
0 0.49 0.99 1.48 1.98 2.47 2.96 3.46 3.95 4.45 4.94 5.43 5.93 6.42 6.92 7.41 7.90 8.40 8.89 9.39 9.88 10.37 10.87 11.36 11.86 12.35 12.84 13.34 13.83 14.33 14.82 15.31 15.81 16.30 16.80 17.29 17.78 18.28 18.77 19.27 19.76 20.25 20.75 21.24 21.74 22.23 22.72 23.22 23.71 24.21 24.70
55.34 54.34 53.24 52.24 51.13 50.08 49.03 47.98 46.93 45.93 45.05 43.99 42.94 41.95 40.95 39.95 38.90 37.85 36.86 35.80 34.75 33.76 32.70 31.65 30.60 29.55 28.55 27.45 26.40 25.29 24.13 23.02 21.91 20.81 19.64 18.48 17.38 16.21 15.05 13.83 12.62 11.40 10.18 8.97 7.75 6.53 5.20 3.98 2.77 1.44 0
55.34 54.83 54.23 53.72 53.11 52.55 51.99 51.44 50.88 50.38 49.99 49.42 48.87 48.37 47.87 47.36 46.80 46.25 45.75 45.19 44.63 44.13 43.57 43.03 42.46 41.90 41.39 40.79 40.23 39.62 38.95 38.33 37.72 37.11 36.44 35.77 35.16 34.49 33.82 33.10 32.38 31.65 30.94 30.21 29.48 28.76 27.93 27.20 26.48 25.65 24.70
-0.29 0.10 -0.12 0.07 -0.13 -0.02 0.07 0.09 0.17 0.39 0.57 0.83 1.02 1.18 1.54 2.03 2.49 2.73 2.81 3.36 3.72 4.18 4.33 4.90 5.39 6.01 6.45 7.03 7.50 8.11 8.66 9.29 9.92 10.55 11.17 11.88 12.50 12.85 13.21 14.74 15.49 16.22 17.18 18.09 19.00 20.03 21.07 21.72 23.22 24.23 25.55
54.46 52.78 50.75 49.17 46.77 45.34 43.50 41.69 39.89 38.37 37.37 35.37 33.84 32.39 31.59 29.81 28.57 26.98 26.51 24.56 23.04 22.23 21.71 20.12 18.75 17.33 16.46 15.36 14.25 13.44 12.59 11.59 10.58 9.62 8.91 8.06 7.03 6.63 6.23 4.65 3.78 2.95 2.29 1.75 1.20 0.85 0.50 0.34 0.04 -0.16 0.04
0.08 1.29 3.22 4.27 6.20 7.11 8.43 9.82 11.16 11.96 12.37 13.50 14.38 15.24 15.17 15.70 17.00 17.08 17.44 17.87 18.22 17.99 18.17 18.12 18.19 18.07 17.98 17.84 17.94 17.55 17.30 17.02 16.73 16.39 15.88 15.32 14.90 14.73 14.57 13.41 12.72 12.08 11.19 10.23 9.27 8.03 6.78 5.36 3.80 2.08 -0.02
3.02 3.95 2.25 2.17 2.07 1.90 1.91 1.87 1.86 1.86 1.76 1.87 1.86 1.82 1.74 1.88 1.91 1.92 1.70 1.87 1.90 1.86 1.68 1.78 1.83 1.93 1.89 1.92 1.92 1.91 1.87 1.90 1.92 1.95 1.91 1.93 1.96 1.77 1.59 2.02 2.07 2.10 2.21 2.29 2.39 2.58 2.84 2.44 5.55 -68.41 0.05
a Before mixing. X ) C /C . b Molar concentration after mixing. c Total molar concentration after mixing. C ) C d M0 0 0 M0 + CW0. Molar concentration obtained from WFA after applying eq 11. e Because m ) 1 and n ) 2, these values are 3 times the concentration of the complex. f This ratio represents the number of water molecules per methanol molecule in the complex.
Sample Preparation. Fifty-one methanol-water mixtures were prepared by adding 0-50 mL of methanol to separate 50mL volumetric flasks in 1.00-mL increments. Distilled water was added to each flask to the 50-mL mark. The amounts of methanol and water were accurately measured using burets. The mixtures contained methanol ranging from 0 to 100%. The methanol mole fractions (before mixing) and the initial (stoichiometric) molarities of methanol and water in each mixture are shown in the first three columns in Table 1. Measurements. The infrared spectra of the sample solutions were recorded at 25 ( 1 °C with a Perkin-Elmer PARAGON 1000 604 Analytical Chemistry, Vol. 71, No. 3, February 1, 1999
PC FT-IR spectrometer manufactured by Perkin-Elmer Instruments Corp. (Norwalk, CT). The spectrometer, interfaced with a DX2-66 PC486 computer, was controlled by the GRAMS Analyst 1000 software. A CaF2 permanent-sealed cell with path length of 0.104 mm was employed. All spectra were ratioed against an empty cell. The spectra were then converted to absorbance scale. Zero baseline was set at the minimum absorbance point in each spectrum by running the “Y-Offset” program. Digitized spectra between wavenumbers of 1850 and 2700 cm-1 were selected for the analysis because of the reasonable magnitude of absorbance (0.0200-1.0000) and
Figure 1. Fourier transform infrared spectra of methanol-water mixtures with methanol mole fractions varying from 0 to 1.00.
sensitivity. The selected spectra were transformed into ASCII format using the GRAMS Analyst 1000 software for further chemometric analysis. Computer Programs. The programs used in this study were written by Professor Edmund R. Malinowski18 in MATLAB.19 RESULTS Figure 1 displays the Fourier transform infrared spectra of 51 methanol-water mixtures. The spectra exhibit an interference fringe with a 96-cm-1 period that is constant for each mixture and each wavelength. These fringes result from reflection off the highly polished surfaces of the 0.1-mm sample cell (1/96 cm-1 ) 0.104 mm). Although the fringes are not caused by the chemical species themselves, their effects are superimposed upon the spectra of the components but will not affect the concentration profiles extracted by WFA. Figure 2 is a three-dimensional plot of the spectra. The lack of smoothness in this plot is the result of experimental error. FACTOR ANALYSIS Principal Factor Analysis. To determine the experimental uncertainty in the absorbance measurements, 10 replicate spectra were recorded for both pure water and pure methanol, within the same wavenumber region as in Figure 1. The spectra were assembled into two 426 × 10 data matrixes, one for each pure solvent. These matrixes were subjected to principal factor analysis (PFA).11 Because each matrix was produced by only one pure chemical component, the real error (RE)11 associated with the first abstract factor represents the uncertainty in the absorbance measurement. In this way, the uncertainties were determined to be 1.4 × 10-3 and 1.6 × 10-3 AU for pure water and pure methanol, respectively. These values are a composite of undesirable artifacts such as instrument drift, filter switching, thermal noise, etc. They (18) Malinowski, E. R. Tool Kit for Chemical Factor Analysis; 5042 SE Devenwood Way, Stuart, FL 34997. (19) MATLAB is a matrix laboratory, copyrighted by The Mathworks, Inc., (Cochituate Place, 24 Prime Park Way, Natick, MA 01760).
Figure 2. Three-dimension view of the FT-IR spectra of methanolwater mixtures.
represent a lower limit of uncertainty because only pure reagents were involved in their determinations. When dealing with a set of mixtures, the uncertainties can be expected to be greater than these values because of the extra handling involved in their preparation and spectral measurements. A 426 × 51 data matrix suitable for factor analysis was constructed by digitizing the FT-IR spectra in Figure 1 at 2.0cm-1 intervals. This matrix was subjected to PFA. The first 20 factors resulting from the PFA are listed in Table 2. The number of factors, N, responsible for the data was determined by comparing the real error (RE) obtained from PFA with the known experimental uncertainty.11 Table 2 shows that the RE is less than the expected uncertainty, 1.5 × 10-3 , when N > 3; hence, we conclude that there are three chemical factors involved in the process. This was further substantiated by the fact that, when four chemical factors were considered, the concentration profiles obtained by WFA were extremely noisy and/or exhibited large negative values. It is academically interesting to examine the results depicted in the last three columns of Table 2, concerning the indicator function (IND) and the percent significance levels (%SL) determined by the methods of Malinowski11 and by Faber and Analytical Chemistry, Vol. 71, No. 3, February 1, 1999
605
Table 2. Results of Principal Factor Analysis of Digitized FT-IR Spectra from Methanol-Water Mixtures (Concerning Only the First 20 Factors)
factor number, N 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
eigenvalue, EV
significance level, %SL indicator real error, function, FaberRE × 104 IND × 107 Malinowski Kowalski
3.5105 × 103 824.27 329.71 1.4278 × 102 96.281 40.100 1.8655 18.306 7.9454 -2 3.8071 × 10 12.333 5.5830 -2 2.1995 × 10 6.5699 3.1049 -3 6.5616 × 10 3.1455 1.5533 1.2574 × 10-3 1.8469 0.95398 4.0043 × 10-4 1.1421 0.61768 5.9700 × 10-5 1.0009 0.56739 2.6083 × 10-5 0.93641 0.55705 1.3833 × 10-5 0.90421 0.56513 -5 1.3148 × 10 0.87145 0.57295 -5 1.2255 × 10 0.83887 0.58094 9.4327 × 10-6 0.81417 0.59472 8.9006 × 10-6 0.78947 0.60916 8.2423 × 10-6 0.76536 0.62479 7.5800 × 10-6 0.74208 0.64193 6.9529 × 10-6 0.71966 0.66084 -6 6.1516 × 10 0.69926 0.68287 -6 5.8858 × 10 0.67836 0.70589
0 0 0 0 0 0 0 0 1.2878 7.3597 17.364 16.888 16.770 21.136 21.062 21.399 21.900 22.473 23.965 23.596
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Kowalski.20
These three functions are useful when there is no knowledge of the uncertainty in the data. The results in Table 2 show that the number of factors is 9 at the 5% SL (Malinowski) and 10 at the 10% SL (Malinowski). The indicator function (IND) reaches a minimum at the 10 factor level. This is consistent with the observation that the minimum in the IND lies between the 5% and 10% SL. However, the real error (RE) at N ) 10, as can be seen in Table 2, is 0.94 × 10-4, an order of magnitude below the expectation. IND and %SL are sensitive to instrumental artifacts such as baseline drift, aberrations caused by filter switching, thermal noise, etc. Such artifacts produce “ghost” factors, leading to a larger number of factors than chemical components. The fact that the %SL (Faber) is zero for the first 20 eigenvalues gives strong evidence that the errors in the spectroscopic absorbances do not have a Gaussian distribution. How these functions can be used to determine the number of chemical factors in the presence of instrumental and experimental artifacts is the subject of a separate study21 involving factor analysis of data with multiple sources of error. This new technique indicates that only three factors are responsible for the methanol-water spectra reported in the current paper. Window Factor Analysis. The 426 × 51 spectral data matrix was subjected to window factor analysis based on three factors resulting from three chemical species. The concentration profiles, labeled A, B, and C in Figure 3, were obtained using the windows designated in Table 3. For each of these profiles, WFA arbitrarily sets the maximum concentration to unity. To convert these profiles into molar concentrations, we make use of the known stoichiometry. Namely, the sum of the concentrations of the three species in any particular solution must correspond to the sum of the initial concentrations of methanol and water in the solution. Mathematically, this
(20) Faber, K.; Kowalski, B. Anal. Chim. Acta 1997, 337, 57-71. (21) Malinowski, E. R. Abstract Factor Analysis of Data with Multiple Sources of Error and a Modified Faber-Kowalski F-test. J. Chemom. In press.
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Analytical Chemistry, Vol. 71, No. 3, February 1, 1999
Figure 3. Concentration profiles of three species extracted by WFA from digitized FT-IR spectra of methanol-water mixtures (A ) water, B ) complex, and C ) methanol). Table 3. Windows Used in the Analysis of the FT-IR Spectra of Methanol-Water Mixtures and the Calculated b Coefficients
species
window spectral numbers
methanol mole fraction
calibration coefficient, b
chemical species identified
A B C
1-48 2-50 4-51
0.0000-0.8536 0.0090-0.9439 0.0276-1.0000
54.45 18.22 25.55
water complex methanol
correspondence can be educed as follows. Let CWFA represent a matrix constructed from the uncalibrated concentration profiles obtained from WFA, where each row of CWFA represents one of the profiles. For this study, CWFA is a 3 × 51 matrix. Premultiplying CWFA by b, a 1 × 3 row vector,
bCWFA ) C0
(9)
should yield C0, a 1 × 51 matrix composed of the sum of the initial concentrations of methanol, CM0, and water, CW0. The fourth column in Table 1 represents C0. Vector b was calculated from eq 10, which results from rearrangement of eq 9.
b ) C0CWFA+
(10)
In eq 10, CWFA+ is the pseudoinverse of CWFA. The molar concentration profiles, shown in Figure 4, were obtained by multiplying each row of CWFA by the respective element of b. In other words, the concentration profile matrix, CCPM, was obtained by premultiplying CWFA by a matrix B whose diagonal elements are the elements of b and the off-diagonal elements are zero, as expressed by eq 11.
CCPM ) BCWFA
(11)
Each row of CCPM represents, quantitatively, the concentration profile of one of the chemical species. It is important to recognize at this point in the analysis that these profiles were obtained without requiring any chemical model nor any preconceived notions concerning their origin. DISCUSSION Deducing the Formula of the Complex. In Figure 4, the profiles labeled A, B, and C, obtained by means of eq 11, represent
Figure 4. Molar concentration profiles converted from the concentration profiles extracted by WFA from digitized FT-IR spectra of methanol-water mixtures (A ) [H2O], B ) (m + n)[(CH3OH)m(H2O)n], and C ) [CH3OH)]).
Figure 5. Corrected molar concentration profiles (solid lines) obtained from WFA and theoretical concentrations (+) based on kd ) 306 M2 for each of the 51 mixtures.
unassociated water, a methanol-water complex, and unassociated methanol, respectively. It is obvious that A represents water because its concentration is 54.46 M, the concentration of pure water, when the methanol fraction is zero. It is also obvious that C represents methanol because its concentration is 25.55 M, the concentration of pure methanol, when the methanol volume fraction reaches 1. These values are not exactly equal to the true values, 55.34 and 24.70, due to experimental error in the spectra as well as round-off error produced by finite window designations required by WFA. Species B is an intermediate complex formed between water and methanol. This is evident from the fact that its concentration increases from zero to a maximum and then decreases to zero as the methanol fraction approaches 1. Because the chemical composition of the complex is unknown at this stage of the analysis, we will designate its formula as (CH3OH)m(H2O)n, where m and n represent the number of methanol and water molecules bound to the complex. The exact formula of the complex can be deduced from the molar concentration profiles of water and methanol. The differences between the initial molar concentrations and those obtained from WFA (CW0 - CW and CM0 - CM) represent the amounts of the pure solvent components bound to the complex. The ratio between these two differences should equal the number of water molecules per methanol molecule (n/m) in the complex. These ratios, calculated for each of the 51 compositions, are shown in Table 1. The ratios are unreliable at the extreme end of the composition because as the denominator approaches zero the relative error dramatically increases. The ratios in Table 1 give evidence that the complex consists of one molecule of methanol and two molecules of water. Determining the Dissociation Constant of the Complex. The molar concentration profile of the 1:2 methanol-water complex, (CH3OH)m(H2O)n portrayed in Figure 4, is actually 3 times the true concentration, CX. This is true because the sum of initial concentrations of pure methanol and pure water, C0, is related to the equilibrium concentrations of the three species as given by eq 12.
4 correspond to CW and CM. However, the concentration profile of B does not correspond to CX but actually corresponds to (m + n)CX, which equals 3CX since m ) 1 and n ) 2. The corrected concentration profiles are illustrated as solid lines in Figure 5. The three species exist in dynamic equilibrium as expressed by eq 13.
C0 ) CW + (m + n)CX + CM
(12)
According to eq 12, the concentration profiles of A and C in Figure
kd
(CH3OH)(H2O)2 y\z CH3OH + 2H2O
(13)
Assuming that concentrations equal the activities, the dissociation constant of the complex can be expressed by eq 14.
kd ) CMCW2/CX
(14)
From the results given in Table 1, kd is estimated to be 306 ( 33 M2. Because the relative error increases dramatically as CX approaches zero, this value represents the average of the values between 0.2077 and 0.3617 methanol mole fraction. Comparison with Theory. The (+) points shown in Figure 5 were calculated theoretically using the determined dissociation value (306 M2) and the initial concentrations of methanol and water for each of the 51 mixtures. The theoretical points fit reasonably well at the low methanol volume fraction but do not fit well at the high fraction region. Close examination of this figure shows that the inclusion-exclusion rule required by WFA is obeyed at the low region but not at the high region. All of the theoretical points of the complex lie under the theoretical profile of water. In the high fraction region, where both the water and the complex concentrations approach zero, this disobeyance becomes acute forcing the WFA to find a solution that does not violate the inclusion-exclusion rule. Notice that the WFA profiles of these two components (solids lines) cross near 0.6 mole fraction, forcing their profiles to obey the rule. In this case, the theoretically calculated profiles are actually a better representation of the true situation than the WFA profiles. Molar Absorptivities. The molar absorptivities, , of the three species (see Figure 6) were obtained using the concentrations, CCPM, gleaned from WFA. These spectra were determined by Analytical Chemistry, Vol. 71, No. 3, February 1, 1999
607
Figure 6. Molar absorptivities obtained from WFA based on eq 15.
applying eq 15, which is simply the pseudoinverse of eq 1.
)
S ) b
ACCPM+ b
(15)
In eq 15, b is the cell path length, which, in this case, is 0.0104 cm. The pseudoinverse is represented by the superscript, +. It is interesting to note that when the theoretical concentrations were inserted in eq 15 in place of CCPM, the same absorptivity spectra were obtained. This shows that the violation of the inclusion-exclusion rule is not serious in this case. It is also interesting to notice that the interference fringes, produced by reflection off the cell surfaces, are superimposed upon each of the spectra in Figure 6. Because the fringes act as a constant perturbation throughout the spectral measurements, their appearance does not affect the concentration profiles extracted by WFA. Validation of the WFA Results. To provide evidence that the results are reliable, although the inclusion-exclusion rule has been violated in this study, the following computations were carried out. A model data matrix was constructed by premultiplying the theoretical concentration matrix portrayed in Figure 5 by the digitized spectra of the components portrayed in Figure 6, in accord with eq 1. To simulate a more realistic matrix, random numbers with a Gaussian distribution and a standard deviation of 1.5 × 10-3 were added to the matrix. This standard deviation is the same order of magnitude as the uncertainty in the measured absorbances. The resulting matrix was then subjected to WFA using the same window designations as used in the original study (see Table 3). The concentration profiles obtained by WFA of the model data are shown as solid lines in Figure 7. This figure also shows the theoretical concentrations (+ points) after standardization to unit height, so that a direct comparison can be made. Exact recovery of the original concentrations has not been achieved for two reasons: (1) the Gaussian errors perturb the profiles and (2) the inclusion-exclusion rule has not been strictly obeyed. Despite (22) Zhao, Z.; Malinowski, E. R. Determination of the Hydration of Methylene Blue Aggregates and Their Dissociation Constants Using Visible Spectroscopy. J. Chemom. In press.
608 Analytical Chemistry, Vol. 71, No. 3, February 1, 1999
Figure 7. Concentration profiles (solid lines) obtained from WFA of model data and theoretical concentrations, (+) based on kd ) 306 M2, after standardization to unit height.
these disturbances, the agreement between the true profiles and those extracted by WFA is well within 95%. This lends support to the results obtained by WFA in this investigation. CONCLUSIONS This investigation has shown that factor analysis of FT-IR spectra can be used to determine (1) the number of chemical species in a mixed solvent system, (2) the chemical formula of the intermediate complex, and (3) the dissociation constant of the complex. Unlike previous studies, this work does not require any a priori speculation concerning the nature of the complex. The method, however, requires (1) data that can be expressed as a linear sum of product terms, e.g., obeyance to Beer’s law, (2) data obtained from an evolutionary process where each component appears and disappears, exhibiting a single maximum in its concentration profile, (3) components that have unique spectra, and (4) components that have unique concentration windows, where all other components have signals outside the component window. It is evident from this study that methanol-water mixtures meet all of these requirements except the last one. Despite this, reasonable approximations to the true situation were obtained. It will be interesting to see if other solvent systems can be similarly deciphered by window factor analysis. In our studies of the hydration of methylene blue and its aggregates in aqueous solutions,22 we make important use of the results gleaned from this study. These results can also provide the basis for interpretation of the elution behavior of methanolwater gradients used in reversed-phase liquid chromatography. ACKNOWLEDGMENT This work was presented at the 1998 Eastern Analytical Symposium in Somerset, NJ, on Nov 17, 1998. Material in this paper is based on part of the dissertation written by Zhouming Zhao in partial fulfillment of the requirements for the degree of Doctor of Philosophy at Stevens Institute of Technology. Received for review August 24, 1998. Accepted November 9, 1998. AC9809480
