Associative Behavior of Benzyl Alcohol in Carbon Tetrachloride

Self-association of benzyl alcohol in carbon tetrachloride has been studied by infrared absorption ... to use a multivariate factor analysis method on...
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6960

J. Phys. Chem. B 1997, 101, 6960-6969

Associative Behavior of Benzyl Alcohol in Carbon Tetrachloride Solutions Geir M. Førland,* Yizeng Liang,† Olav M. Kvalheim, Harald Høiland, and Annick Chazy Department of Chemistry, UniVersity of Bergen, Allegt. 41, N-5007 Bergen, Norway ReceiVed: February 5, 1997; In Final Form: May 29, 1997X

Self-association of benzyl alcohol in carbon tetrachloride has been studied by infrared absorption spectroscopic measurements in the fundamental OH-stretching vibration region. Infrared spectra were acquired at 30, 40, and 50 °C for varying alcohol molalities, the highest concentration being 0.2 mol/kg. The spectra were collected in a data matrix and analyzed by multivariate resolution methods in order to determine the numbers of different components present in the solution and to find the spectra and concentration profiles of each component. The result indicates that the spectral variance can be described by three components, free alcohol monomers, open chain oligomers, and cyclic oligomers. The resolved spectra show one sharp band at 3620 cm-1 for the component representing the hydroxyl group of free alcohol molecules and one broad band at 3300 cm-1 for the component representing the hydroxyl groups of cyclic alcohol aggregates. The component representing the hydroxyl groups of open chain aggregates appears with two absorption bands, one broad and asymmetric band at 3500 cm-1 and a smaller band at 3600 cm-1. The small band at 3600 cm-1 is connected to the hydroxyl group situated at the “free-end” of the chain, while a broad and asymmetric band at 3500 cm-1 is connected to the hydroxyl groups situated inside and at the “bound-end” of the chain. The average number of alcohol monomers in the H-bonded aggregates decreases with increasing temperature in the solution. It was equal to 4 for open chain aggregates at 30 °C and equal to 3 at 40 and 50 °C. The corresponding numbers for the cyclic aggregates were 7 at 30 °C and 6 at 40 and 50 °C. The calculated concentration profiles show that the alcohol monomers are the dominating component throughout the concentration range investigated and that there are more open chain oligomers than cyclic aggregates present.

Introduction The general spectral characteristics of hydrogen-bonded alcohol molecules were first established by Errera et al. in 1936.1 Since then, the formation of self-associated alcohol species has been a subject for frequent investigation. In spite of many attempts to determine the structure of the species, the association number, and some thermodynamic properties like the hydrogenbond energies and the equilibrium constants between the monomers and various proposed self-assembled oligomers, no uniform physical picture of the association processes has been established. This problem is due to a lack of capable and simple analytical methods for distinguishing the various aggregated species of the liquid phase. In spite of their low binding energy, the H-bonds between alcohol molecules are responsible for many interesting physical phenomena and are of great importance in chemistry, biology, and biochemistry.2-6 In pure alcohols, many possible modes of self-association can occur simultaneously. Therefore, previous studies have often been directed toward the self-association in simple dilute binary liquids containing alcohols and apolar and inert solvents. A number of analytical methods have been utilized in such studies, including spectrophotometric, dielectric, vapor pressure, NMR, and thermodynamic techniques.7-11 However, the results are often conflicting. This seems to be due mainly to the varying methods used. Most of the analytical techniques yield only indirect information, and a reliable association model is needed in order to determine acceptable physical and thermodynamical parameters. Infrared (IR) spectroscopy appears as a promising analytical tool for evaluating the problem since the spectral data in the * Corresponding author. Fax: 47 55 58 94 90. Phone: 47 55 58 33 73. E-mail: [email protected]. † On leave from Department of Chemistry, Hunan University, Changsha 410082, PRC. X Abstract published in AdVance ACS Abstracts, August 1, 1997.

S1089-5647(97)00452-5 CCC: $14.00

frequency range of the hydroxyl stretching mode contains detailed information about the position of the alcohol molecule in a self-assembled aggregate. One of the major difficulties in using this method is interpretation of the complex spectra. This is especially the case in the near-infrared region. Another difficulty is caused by the large absorption coefficients deriving from the OH-stretching frequencies, giving a marked interference between the absorption band originating from free species and from H-bonded species. In order to resolve the problems concerning spectral interference, some curve resolution methods using multivariate data analysis have been proposed.12-14 The aim of these methods is to determine the number of components that give a unique contribution to the spectral variance and to resolve the complex spectra into pure spectra of each component. The first attempt to use a multivariate factor analysis method on IR data of associated alcohols can be traced back to 1976 when Shurvell15 and co-workers studied the self-association of phenol in CCl4. Later, Malinowski and his colleagues14 tried to use an evolutionary factor analysis method to solve the problem for stearyl alcohol in CCl4. More recently, several multivariate resolution methods have been developed such as iterative target transformation factor analysis,16,17 alternating least squares,18,19 evolving factor analysis,20-23 fixed size moving window evolving factor analysis,24 heuristic evolving latent projections,25-27 and window factor analysis.28 In the present work, we study the self-association of benzyl alcohol (BA) in carbon tetrachloride (CCl4) solutions by using Fourier transformation infrared (FT-IR) spectroscopy. The spectra originating from the fundamental OH-stretching vibration region were acquired for solutions containing alcohol concentrations up to 0.2 mol/kg and at the temperature 30, 40, and 50 °C. The spectral data was then analyzed by advanced multivariate methods in order to determine the number of species © 1997 American Chemical Society

Self-Association of Benzyl Alcohol in CCl4

J. Phys. Chem. B, Vol. 101, No. 35, 1997 6961

present in the solution and to find the spectral profile for each species. The results were interpreted in accordance with a simple association model in order to determine the equilibrium constants and the enthalpy of hydrogen-bond formation. Experimental Sections Materials. The CCl4 was delivered by Riedel de Haen (min 99.8% pure), and the benzyl alcohol was a p.a. quality product delivered by Merck. The products were used without further purification. Infrared Absorption Measurements. The measurements were made with a Nicolet 800 FT-IR spectrophotometer and a medium-band mercury cadmium telluride detector. A transmission cell with NaCl windows and a 0.2 mm spacer was used. The scans were acquired at a nominal resolution of 2 cm-1 and a data point resolution of approximately 1 cm-1. The cell was modified in order to be connected to a thermostated water bath during the measurements and to be filled from outside the cell chamber. Absorption of light from water vapor was avoided by flushing the instrument with dry air for at least 20 h prior to the measurements. The temperature stability in the cell during the measurements was determined to be (0.1 °C as measured by a Hewlett-Packard quartz thermometer.

A

X ) UGPt + E ) TPt + E )

tipit + E ∑ i)1

(2)

Comparing eq 1 with eq 2, one can easily see that the matrix C and T and the matrix St and Pt span the same linear space. By using X+X ) PG-1UtUGPt ) PPt to construct a projection matrix, one can project a series of so-called needle targets, which are described as follows:

g1t ) (1, 0, 0, ..., 0) g2t ) (0, 1, 0, ..., 0) ...

Theory and Methodology The task of using the multivariate self-modeling curve resolution methods is 2-fold: firstly to determine the number of significant components with a unique contribution to the spectral variation and secondly, to resolve the spectral bands corresponding to the significant components without the need for any assumption regarding peak shape, location, or identity. If spectra of N samples with different ratios of solvent and alcohol are acquired at M wavenumbers, they define a twoway data matrix X of size N × M. According to the LambertBeer law, the matrix X can be decomposed as shown by A

X ) CSt + E )

evolving factor analysis (EFA)21-23 and eigenstructure tracking analysis (ETA),27 have been described previously. In addition, a quite new technique, the needle plot,29 can be used both for determining the number of components in the system and for locating the peak maxima of the component spectra. This has been used in the present work to confirm the results obtained by LPCA. The principle of the method is quite simple. The data matrix X is first decomposed into scores T and loadings Pt:

cisit + E ∑ i)1

(1)

Here C ) [c1, c2, ..., cA] is the concentration profile matrix of size N × A, while St is the spectral matrix of component spectra of size A × M. A denotes the number of significant components of the data matrix X. E is the error matrix of size N × M representing experimental and instrumental noise. By using the notation in eq 1, the task of resolution methods is first to find the number A and then the concentration profiles C and the spectral profiles St of the components. The following provides a brief description of the resolution procedure used in this work. Global and Local Principal Component Analysis, Needle Plot. The determination of the number of components in the studied systems presents a rather difficult problem. Global principal component analysis can give ambiguous results as described by Shurvell et al.15 Fortunately, the self-association of alcohol infrared spectral data possesses an evolving feature, that is, the absorption peaks appear sequentially as the total alcohol concentration increases. This makes the use of the local principal component analysis (LPCA) meaningful. Moreover, we can also reveal the presence of selective regions, i.e. onecomponent regions, in spectral direction. The major advantage of using the LPCA lies in its ability to focus on small informative parts of the data matrix, which can significantly reduce the accumulation effect of measurement noise and can also give a rank map in both concentration and spectral directions. The LPCA techniques used in this work, including

gMt ) (0, 0, 0, ..., 1) Here, X+ is the generalized (Moore-Penrose) inverse matrix. When the distances, e.g. ||git - gitPPt|| (i ) 1, 2, ..., M), are plotted vs their respective wavenumbers, the local minima in the plot will coincide with the position of the real spectral maxima and the number of the local minima will equal the real number of the components in the system. If one includes a number of principal components which is larger than the real number of the components into the project matrix, the needle plot will show a noisy curve. For more details, see ref 29. Resolution by Iterative Target Transformation and Selective Information. It is well known that if the target spectrum is a true component spectrum, the spectrum obtained after such a projection described above should be the same as itself. Gemperline16 and Vanteginste et al.17 used this principle to construct an iterative procedure to approach the true component spectra. The iterative procedure goes as follows: Choose a target spectrum which is supposed to be similar to the component spectrum vt(0). For i ) 1 ...

vt,(i+1) ) vit,(i)PPt

(3)

until |vit,(i+1) - vit,(i)| e , where  is a predefined convergence criterion for stopping the iteration. It should be pointed out that since the IR absorbances cannot be less than zero, negative values in the target vector vit,(i) should be changed to zero before the next iteration. The key point of this iterative resolution procedure is to find a good initial target spectrum. One can obtain the nearly true component spectra from such an iterative procedure if either the initial target spectrum is good enough or there is no significant overlap between the component spectra. In order to improve the results obtained by the iterative procedure described above, the alternating least squares and selective information detected by local principal component analysis were used. The resolved spectra are used to estimate the concentration profiles, that is,

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Førland et al. TABLE 1: The First 10 Eigenvalues and Their Corresponding Accumulated Percentages temp, °C

i

λi × 104

percentage, %

30

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

4.1769 0.5821 0.0528 0.0086 0.0047 0.0019 0.0014 0.0009 0.0008 0.0007 2.6381 0.5883 0.0547 0.0060 0.0032 0.0022 0.0011 0.0009 0.0008 0.0006 2.2108 0.5083 0.0530 0.0052 0.0033 0.0022 0.0017 0.0013 0.0012 0.0009

86.43 98.47 99.56 99.74 99.84 99.87 99.90 99.92 99.94 99.95 79.97 97.81 99.47 99.65 99.75 99.81 99.85 99.87 99.90 99.92 79.21 97.42 99.32 99.51 99.62 99.70 99.76 99.81 99.86 99.89

40

50

Figure 1. Graphic presentation of the OH-stretching absorption bands in the fundamental wavenumber region from 3100 to 3700 cm-1 and at various alcohol molalities up to 0.2 mol/kg at (a) 30, (b) 40, and (c) 50 °C.

C ) XS(StS)-1

(4)

Here S is the resolved component spectral matrix obtained by iterative target transformation. Some columns of the obtained concentration profiles matrix C will be replaced by the selective information found in wavenumber direction by ETA. The spectral profiles will be reobtained by least squares using the new concentration profiles Cnew,

Snewt ) (CnewtCnew)-1CnewtX

(5)

This procedure can reduce the residual noise significantly. Results and Discussion The Number of Different Components and the Corresponding Spectral Resolution. A graphic presentation of the fundamental OH-stretching absorption bands of benzyl alcohol at various concentrations of the alcohol is shown in Figure 1.

The labels a, b, and c represent the spectra taken at 30, 40, and 50 °C, respectively. Three major peaks can be seen in the spectra. The first one gives rise to absorption in the 36003650 cm-1 region. As the alcohol concentration increases, a second broad peak appears around 3500 cm-1. The third large and broad peak appears near 3300 cm-1 at even higher alcohol concentrations. The second and third peaks are normally explained by OH-stretching vibrations in various hydrogenbonded alcohol aggregates.6,8,11,30,31 In order to determine the number of different components (the different noncorrelated OH-stretching frequencies) from global and local principal component analysis (PCA), it is crucial to separate the variation due to the chemical components from that due to the background noise. Table 1 and Figure 2 present the result calculated from global PCA. Table 1 shows the first 10 eigenvalues (λi) and their corresponding accumulated percentages. For all three systems, the fourth eigenvalue contribute by less than 0.2% variation, which is close to the noise level of the IR spectra. The small variations originating from the fourth, fifth, and sixth eigenvalues support a threecomponent system. Figure 2 shows the first six loadings for the system at 40 °C. Quite similar plots were obtained for the systems at 30 and 50 °C. The first three loadings are quite similar and structurally equivalent. The fourth loading seems to contain some structure which decreases with increasing temperature. The fifth and sixth loadings are simply representing noise. In conclusion, the results from global PCA analysis cannot tell us whether we have a three- or a four-component system. Since the systems possess evolving features (i.e. the chemical species appear sequentially with increasing concentration of alcohol), we investigate them by using local PCA techniques. These techniques can be looked upon as datascope techniques which successively investigate various parts of the data matrix. Thus, the local PCA is normally better suited than the global PCA for analyzing this kind of dynamic system. The result

Self-Association of Benzyl Alcohol in CCl4

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Figure 4. Eigenstructure tracking analysis in the wavenumber direction. Top shows the mean spectra of the system (40 °C). Bottom shows the log eigenvalues.

Figure 2. First six spectra loadings for the system at 40 °C.

Figure 5. Needle plot representing the system at 40 °C.

Figure 3. Graphic presentation of the evolving factor analysis in the concentration direction (40 °C).

obtained from evolving factor analysis of the system at 40 °C is shown in Figure 3. The plots show that the first three eigenvalues increase steadily, while the fourth flattens at the noise level. This was also observed for the system at 30 and 50 °C. Thus, it appears that the system can be described as a three-component system at all temperatures measured. Furthermore, a three-component system seems reasonable since the factor analysis indicates that the fourth component is responsible for spectral variations which are comparable to the variations originating from the background noise. Figure 4 shows the results from the ETA method in the wavenumber direction. The data shows a similar structure for all three systems. This indicates that they contain the same number of components independent of the temperature. The plots show a onecomponent section (a section where the spectral variation

originates from only one type of OH bonds) in the lowwavenumber region at about 3100-3300 cm-1 and in the wavenumber region at 3500-3550 cm-1. A two-component section is observed in the wavenumber regions between 3300 and 3500 cm-1 and between 3550 and 3650 cm-1. This supports the result obtained from the EFA analysis that indicates that we have a three-component system. The absorption band at 3500 cm-1 interferes with the other bands, and one-component areas can only be found for the bands representing the H-bonded species. In order to further confirm the results obtained in the local factor analysis, a technique called needle plot was used. Figure 5 shows the needle plot for the system at 40 °C. The plot displays a relatively clear structure with three minima corresponding to the above-mentioned bands when three principal components are included in the calculation. However, it gives a noisy curve from which it is impossible to conclude how many components are in it when four principal components are selected. This supports the result obtained from all the local rank analyses, favoring a three-component system. Similar plots were obtained for the system at 30 and 50 °C. The data in Table 1 shows that nearly 99.7% of the structural information of the spectra is obtained by the first three components. Thus, matrices with a chemical rank equal to 3 form the basis for further quantitative analysis of the spectral data. Figure 6 shows the resolved spectra for all three-component systems. The first component displays a sharp band at about

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Figure 7. Schematic presentation of various alcohol aggregates.

Figure 6. Resolved spectra for all three component systems at (a) 30, (b) 40, and (c) 50 °C.

3620 cm-1. For the second component, a broad and asymmetric band appears at about 3500 cm-1, while a smaller band appears at 3600 cm-1, near the band of the first component. The third component shows a single, broad and symmetric band at about 3300 cm-1. The resolved spectra correspond well with the spectra presented in Figure 1. The residuals between the reconstructed spectra and the spectra obtained from the raw data show only 0.3% spectral variation which is close to the noise level in the spectra. Interpretation of the Spectral Observations. A solution containing hydrogen-bonded alcohol species is a “black system”14 for which it is necessary to first determine the number of different alcohol species present and then to determine and connect the structures of the species to the observed absorption frequencies. Since this connection has not been firmly estab-

lished, we will carefully describe our interpretation of the spectra in terms of the various types of hydroxyl bonds using the structures of self-assembled alcohol aggregates presented in Figure 7. According to this figure, there exist eight types of OH bonds (termed a-h) that can contribute to the spectra. Since only three clear bands can be seen from the spectral curves in Figure 1, some of the suggested structures may not exist to a significant extent or some of the bonds may be spectrally equivalent. The association process can be considered as a dynamic process where the alcohol molecules successively associate into larger alcohol aggregates.9 Brink and Glasser10 have demonstrated that dielectric measurements of ethanol in cyclohexane indicate a successive association which involves the presence of different alcohol association modes. Besides the free alcohol monomers, some highly polar species were determined at low alcohol concentrations. At higher alcohol concentrations, near 0.2 mol/kg, aggregates with low dipole moment were determined. These results support an association model where the alcohol molecules first associate into open chain aggregates and then, at higher alcohol concentration, form cyclic aggregates. Small open chain aggregates, i.e. dimers and trimers, are supposed to be the dominating H-bonded species at low alcohol concentrations.32,33 These species associate into large open chain aggregates with increasing alcohol content.8,34 As the number of alcohol molecules in the aggregates becomes sufficiently high, they apparently transform into cyclic structures.8,31 This sequential association behavior is also supported by the evolving feature seen in the spectra. If the alcohol is sufficiently diluted, one can observe an OHstretching band free from intermolecular bridges. This band appears as a sharp peak at 3650 cm-1, and it is the first one to appear in the low alcohol molality region. This band is therefore connected to the bond labeled a in Figure 7.

Self-Association of Benzyl Alcohol in CCl4 The absorption band at 3500 cm-1 is the first band from H-bonded alcohols that can be seen in the spectra as the total alcohol concentration increases. This band represents the second component and is connected to the OH-stretching frequencies in open chain structures. The resolved spectra in Figure 6 show that the second component gives rise to two absorption bands. In an open chain aggregate with more than two molecules, the OH-stretching frequencies can at least be divided into three groups, depending on the position of the molecules in the aggregate. The “free-end” molecule (the molecule which only has a hydrogen-bonded oxygen atom) contributes to the absorption in a relatively high-frequency region, near the wavenumber region where absorption originates from free monomeric molecules.8 On the other side of the oligomeric chain, a “boundend” molecule (the molecule which is bonded with the hydrogen atom) will possess an absorption at a lower wavenumber because the H-bonding is directly connected to the small hydrogen atom in the molecule. Moreover, the internal molecules have the hydrogen atom as well as the oxygen atom H-bonded and contribute to absorption at even lower wavenumbers. Thus, we assume that the hydroxyl bonds labeled c and f in the free-end molecules can be connected to the small absorption peak at 3600 cm-1 as seen in the resolved spectra in Figure 6. Furthermore, the internal molecules and the bound-end molecule are connected to the highly asymmetric and broad band which absorbs in the region 3300-3550 cm-1. The asymmetry and the broadening of the band can be explained by the various stretching frequencies of the hydroxyl groups originating from molecules which are situated either inside or at the end of the chain. The highest intensity can be seen in the high-wavenumber range of the band, indicating a relatively high content of small aggregates, i.e. dimers and trimers. Moreover, a contribution from the second component to the absorption at low wavenumbers near 3300 cm-1 takes place due to the presence of molecules situated inside larger open chain aggregates. On this basis, we suggest that the hydroxyl groups labeled d and e will belong to the broad absorption band represented by the second component in the resolved spectra (Figure 6). In a recently published work,35 Brot shows that cyclic dimers would imply high strain within the structure and are not energetically favorable. This is also concluded by Peeters and Leroy33 who suggest that the smallest ring one can imagine is the cyclic trimer. They showed that cyclic trimers contain some strain which is partly released in the tetramer. In order to satisfy the requirement of cyclic structures with a nearly linear O‚‚‚H-O hdyrogen bond, we suggest that cyclic dimers are absent or not present in any considerable amount. We are now left with the hydroxyl groups in cyclic polymers (bond h) and a broad and apparently symmetric absorption band at 3300 cm-1. It is reasonable to suggest that the frequency shift between the hydroxyl groups situated inside a large acyclic aggregate and the hydroxyl groups in a corresponding cyclic aggregate are not large. This is also the case for the absorption bands in the resolved spectra in Figure 6. A relatively small wavenumber shift (50-100 cm-1) can be seen between the absorption caused by the hydroxyl groups in the acyclic species (the low-frequency side of the asymmetric band of component 2) and the hydroxyl groups in a cyclic structure (the band of component 3). Most data indicates that the cyclic structures consist of only a few alcohol monomers, i.e. a cyclic oligomer.10,36 Therefore, we suggest that the third absorption band observed at 3300 cm-1 originates from hydroxyl groups in cyclic oligomeric structures. In this way, the overall spectral variance can be interpreted in accordance with an association model which contains three different chemical components: monomers, open chain oligomers, and cyclic oligomers.

J. Phys. Chem. B, Vol. 101, No. 35, 1997 6965 TABLE 2: The Wavenumbers for the Band Maxima temp, °C

component 1, cm-1

component 2, cm-1

component 3, cm-1

30 40 50

3620 3620 3620

3600, 3470 3600, 3485 3600, 3490

3310 3330 3340

In solutions where the self-association of alcohol is a dynamic and evolving process, one normally suggests that the alcohol is present as a range of associated species like dimers, trimers, tetramers, and even higher associated species. This leads to a wide envelope of absorptions and, hence, to broadening of the absorption bands. The stability of all the H-bonded aggregates and thus the shape of the corresponding absorption bands will be modified when the temperature in the solution changes. A higher content of free alcohol molecules and small alcohol aggregates will be present in the solutions measured at 50 °C as compared to that in the corresponding solutions at lower temperatures. The spectra presented in Figure 1 support this behavior. They show that the intensity of the band at 3300 cm-1, representing cyclic aggregates, is significantly reduced while the intensity of the absorption band at 3620 cm-1, representing free alcohol monomers, increases with increasing temperature. The difference in the shape of the absorption band with increasing temperature in the solution indicates a reduction of the alcohol association degree. Figure 8 shows the resolved absorption bands representing the hydroxyl groups in open chain aggregates (a) and the hydroxyl groups in cyclic aggregates (b) at all three temperatures. Figure 8a shows that the intensity at the low-wavenumber side of the asymmetric band decreases while the intensity at the high-wavenumber side increases with increasing temperature. The observed difference in the shape indicates a reduction in the average size of the open chain alcohol oligomers when the temperature increases. The most striking difference is observed between 40 and 50 °C. Table 2 lists the wavenumbers for the maximum absorbance for all three components at the three temperatures measured. The wavenumber for the band representing the hydroxyl groups in cyclic aggregates increases with increasing temperature. A similar but not so large increase is observed for the asymmetric band representing the bonded hydroxyl groups in open chain aggregates. These shifts may be attributed to smaller intermolecular distances as a result of volume contraction, but more likely, they may be due to the formation of smaller hydrogen-bonded aggregates at higher temperatures. Calculation of Association Degree. Several association models have been described in the literature.9,37,38 The more rigorous models describing a successive association process require information about the size (the number of monomers in the aggregate) and size distribution of the alcohol species to be known in order to be used in a reliable way. Due to lack of this type of information and in order to make further progress, we adopt the simple but commonly used single-parameter model which describes the association between the monomers and one type of n-mer. This type of association can be supported from the reported existence of an isosbestic point.7,40 In this work, we use the variation of the absorption intensities against the concentration of alcohol to determine the average number of molecules (n) that are linked by hydrogen-bonds to form acyclic (na), and cyclic (nc) aggregates. The equilibrium constants can then be written

naR1 a Rna; K1-na )

[Rna] [R1]na

(6)

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Førland et al.

K1-n )

ARnRn 1ln-1 ARn 1Rn

(9)

By assembling the terms which are constant with the alcohol concentration and reorganizing the expression, the following equation is derived

AR1/nn ) R1/n n AR1

(10)

where Rn is a constant. The above relationship can be used to obtain the average number of monomers (n) in the associated oligomer. From a plot of the determined absorbance values, using n-values 2, 3, 4, ... a straight line through the origin is obtained when the right association degree is used. Figures 9 and 10 show the results achieved for the acyclic and cyclic species, respectively. The data indicate that the average number of monomers in the aggregates increases as the temperature decreases. It is approximately equal to 3 at 50 and 40 °C and equal to 4 at 30 °C for the open chain aggregates. The results obtained for the cyclic oligomers are 6 monomers at 50 and 40 °C, and 7 monomers at 30 °C. Calculation of the Concentration Profiles and Absorption Coefficients. If the spectra of N mixtures with different alcohol molalities are acquired at M wavenumbers, they define a twoway data matrix X of size N × M. According to the LambertBeer law, the matrix X can be decomposed as shown by eq 1. Here, X is the absorption matrix, normalized with respect to the path length of the measurement cell. C ) [c1, c2, ..., cA] is the alcohol concentration profile matrix of size N × A. St is a spectral matrix of size A × M. Note that if the alcohol concentration profile matrix contains the real molalities of the alcohol components in the solution, then the spectral profile matrix obtained by

St ) (CtC)-1CtX

Figure 8. Resolved spectra for the system at 30, 40, and 50 °C: (a) resolved spectra for the second component (open chain aggregates) and (b) resolved spectra for the third component (cyclic aggregates).

ncR1 a Rnc; K1-nc )

[Rnc] [R1]nc

(7)

where R1 represents the monomer, Rna represents the open chain or acyclic oligomer, Rnc represents the cyclic oligomer, and the brackets indicate that we operate with concentrations. K is the equilibrium constant. The resolved spectral profiles give us the absorbance of the species in the solutions. The connection between concentration and absorbance for a component R is given by the LambertBeer law, and can be written as

[R] )

AR Rl

(8)

The equilibrium constant K1-n for the formation of species n from the monomeric species can now be written as

(11)

should contain the corresponding molal absorption coefficients of each chemical species at every wavenumber in the investigated spectra region. However, the concentration profiles obtained from the spectral resolution are not the molality profiles of the alcohol species in the solution but rather are some profiles directly proportional to the molality profiles. This means that we can find the real concentration profiles cRi (molality profiles of the alcohol species) from the following linear equations:

cg ) n1cR1 + n2cR2 + n3cR3 ) n1k1c1 + n2k2c2 + n3k3c3 (12) where n1, n2, and n3 are the numbers of molecules in the alcohol species, cg is the gross molality profile of the alcohol, and c1, c2, and c3 are the concentration profiles obtained from the spectral resolution. Since c1, c2, and c3 are known from the self-modeling spectral resolution methods for all the different gross alcohol molalities, one can easily get the k-values by solving the following linear equations:

cg1 ) n1k1c11 + n2k2c21 + n3k3c31 cg2 ) n1k1c12 + n2k2c22 + n3k3c32 ... cgN ) n1k1c1N + n2k2c2N + n3k3c3N

(13)

Self-Association of Benzyl Alcohol in CCl4

Figure 9. Plots of eq 10 with least-square fit. The degree (n) of open chain association is determined from the line which intercepts the ordinate nearest to zero: (a) 30, (b) 40, and (c) 50 °C.

Here, {cg1, cg2, ..., cgN} are the different gross stoichiometric alcohol molalities, and {ci1, ci2, ..., ciN} where i ) 1, 2, 3 are the concentration profiles obtained from resolution procedure. Using the k-values obtained above, we can obtain the real concentration profiles, cR1, cR2, and cR3, respectively. Note that the concentration profile obtained in this way is based on the monomeric alcohol units. So, the molality profiles for the open chain and cyclic aggregates are based on the alcohol monomers

J. Phys. Chem. B, Vol. 101, No. 35, 1997 6967

Figure 10. Plots of eq 10 with least-square fit. The degree (n) of cyclic association is determined from the line which intercepts the ordinate nearest to zero: (a) 30, (b) 40, and (c) 50 °C.

present in the associated form and not on the aggregated units. Figure 11 shows the obtained concentration profiles for the alcohol species. One can see that most of the alcohol molecules are present as free monomeric units, even in the solution containing 0.2 mol/kg BA. The concentration of free alcohol molecules increases while the concentration of H-bonded alcohol molecules decreases as the temperature increases from 30 to 50 °C. The concentration lowering of H-bonded alcohol molecules is especially remarkable for the cyclic oligomers

6968 J. Phys. Chem. B, Vol. 101, No. 35, 1997

Førland et al. TABLE 3: The Molal Absorption Coefficients for the Three Alcohol Components, i.e. the Monomers, the Open Chain Oligomers, and the Cyclic Oligomersa , kg mol-1 cm-1 temp, °C

monomers

open chain aggregates

cyclic aggregates

30 40 50

112 (70) 107 (67) 98 (61)

94 (59) 84 (52) 83 (52)

182 (114) 182 (114) 181 (113)

a The values in the parentheses are the corresponding molar absorption coefficients.

TABLE 4: The Equilibrium Constants Based on Molality Units for the Formation of Acyclic Oligomers, K1-na, and Cyclic Oligomers, K1-nc, from the Monomersa temp, °C

K1-na

K1-nc

30

573 ( 26 (2347 ( 107) 18.1 ( 0.6 (46.3 ( 1.6) 10.2 ( 0.2 (26.0 ( 0.6)

7.20 × 105 ( 4.5 × 104 (1.21 × 107 ( 7.5 × 105) 3149 ( 102 (3.30 × 104 ( 1074) 675 ( 35 (7085 ( 367)

40 50

a The values in the parentheses are the corresponding equilibrium constants based on the molarity units.

coefficients (-coefficients) taken at the maximum absorbance in the three absorption bands corresponding to the three alcohol species. Determination of Equilibrium Constants. Once the aggregation numbers and the absorption coefficients are determined, one can readily calculate some thermodynamic parameters like the equilibrium constants and the hydrogen-bond energies. By using eqs 6 and 7, one can obtain the equilibrium constants K1-na and K1-nc for the formation of acyclic and cyclic oligomers, respectively. The hydrogen-bond energies can be determined from the temperature dependence of the equilibrium constants for the formation of acyclic trimers and cyclic hexamers found at 40 and 50 °C by using the van’t Hoff equation

d(ln K) ∆H ) 2 dT RT

Figure 11. Concentration profiles in molality units (mol/kg) for the monomer, open chain oligomer, and cyclic oligomer. All alcohol molalities are based on monomeric alcohols: (a) 30, (b) 40, and (c) 50 °C.

while it is almost negligible for the open chain aggregates. This is quite reasonable as the H-bonds are known to be thermolabile and will break upon increasing temperature. This instability makes the aggregates dissociate gradually into smaller H-bonded aggregates and finally into free monomers at the higher temperatures. Thus, Figure 9 shows that the average number of monomers in the open chain oligomers decreases as the temperature increases. The absorption coefficients of each chemical species at any wavenumber in the investigated spectral region can be obtained by using eq 11. Table 3 contains the calculated molal absorption

(14)

where ∆H is the bond enthalpy for the associated alcohol. The enthalpy can be determined from the slope of the line when ln K is plotted against the temperature. Since the aggregation numbers seem to vary with temperature, this calculation will at best be approximate, and the result will only be presented in order to show that it gives a reasonable value as compared with previous data. Table 4 contains the determined equilibrium constants based on molality units. The values presented in the parentheses are the more frequently used constants based on molarity units. Each of the constants was determined as the average value obtained from the five solutions containing the highest amount of alcohol, and the corresponding error is the calculated standard deviation. The calculated molar enthalpies for the formation of hydrogen bonds (∆H°) were -24.5 kJ mol-1 bond-1 in open chain aggregation and -21.8 kJ mol-1 bond-1 in cyclic aggregation. The magnitude of the molar enthalpy is close to the values obtained by others on association of various alcohols3,9,37,39,40 and represents a reasonable and medium strength H-bond energy. Previous data show that the degree of self-association, and thus the enthalpy changes for the formation of H-bonds is smaller for alcohol molecules in carbon tetrachloride than those for alcohol molecules in alkane solutions.42 However, most enthalpy data have been reported for alcohols in alkane systems while only few such data have been reported on alcohols in

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