New approach to IR study of monomer–dimer
self-association: 2,2-dimethyl-3-ethyl-3-pentanol
in tetrachloroethylene as an example
Jenn-Shing Chen
∗, Cheng-Chang Wu, Dah-Yu Kao
Department of Applied Chemistry, National Chiao-Tung University, Hsin-Chu 30050, Taiwan
Received 10 September 2003; accepted 1 December 2003
Abstract
The dimerization of 2,2-dimethyl-3-ethyl-3-pentanol in tetrachloroethylene in the diluted region has been studied at four temperatures by IR spectroscopy. The aforementioned solute compound is chosen because self-association beyond dimerization is hampered by the steric hindrance generated by the bulky sidechains. The integrated absorbances of the monomer bands were treated based on Eq. (9) to obtain its molar absorptivity and dimerization constant. The same dimerization constant as well as the molar absorptivity of dimer band can be obtained based on Eq. (13) from the data treatment of the integrated absorbances of the dimer band. The disparity between two values of dimerization constant determined by two independent sources offers an opportunity to check the consistency of the determination. The standard enthalpy and entropy of dimerization have also been calculated by means of van’t Hoff plot, respectively, from the data of temperature-dependent dimerization constants obtained from the monomer bands and dimer bands.
© 2003 Elsevier B.V. All rights reserved.
Keywords: Monomer–dimer equilibrium; IR integrated absorbance; Dimerization constant; Enthalpy and entropy of dimerization; Molar monomer and dimer
absorptivities
1. Introduction
The concept of hydrogen bonding pioneered by Pauling and Huggins[1]plays an important role in interpreting the structure and function of the biological molecules, such as the helical or sheet of proteins [2,3], base-pairing of DNA [4], and enzyme kinetics [1]. Hydrogen bonding is also considered to be one of the major factors to render solutions deviant from normal behavior [5]. Since the ad-vent of the establishment by Errera and Mollet[6] for the infrared (IR) spectral characteristics of hydrogen bonding in alcohols, IR spectroscopy has emerged as a major tool to investigate the problems of this sort. Other experimen-tal methods for hydrogen bonding study include nuclear magnetic resonance (NMR), X-ray diffraction, neutron scattering, dielectric polarization, and ultrasonic absorp-tion [1,7,8]. In IR spectra, it is common that several bands related to hydrogen bonding appear concomitantly. Each
∗Corresponding author. Tel.:+886-3-5731636; fax: +886-3-5723764.
E-mail address: jschen@cc.nctu.edu.tw (J.-S. Chen).
band corresponds to the aggregate of a certain size and configuration. Therefore, the data treatment of the spectra should be based on the assumption as to what sizes and what configurations existing in the system [9–13]. The assumption is difficult to justify and, as a rule, leads to un-reliable determination of the spectral and thermodynamic parameters. In order to circumvent this difficulty, associa-tion limited to dimerizaassocia-tion is adopted in this study. It is understood that this limitation will not embrace most of the situations of self-association. However, it is still considered to be crucial in that dimerization is the starting point of any study of self-association. The mere Monomer–dimer self-association can be realized by alcohols with bulky sidechains in the vicinity of hydroxyl group [14–17]. The steric hindrance due to neighboring bulky sidechains would prevent molecules from further association. In this report, we choose dilute solution of 2,2-dimethyl-3-ethyl-3-pentanol in tetrachloroethylene as a model system to investigate monomer–dimer self-association, in view of the fact that the solute exhibits only monomer–dimer association in diluted solution [14,15,17], and that the solvent exhibits almost no absorption within the range 3300–3750 cm−1 1386-1425/$ – see front matter © 2003 Elsevier B.V. All rights reserved.
under study so as not to interfere the absorption of the solute in this range. As can be seen below,Eq. (9) allows us to treat the integrated absorbances of the monomer band to obtain its molar absorptivity and dimerization constant. The dimerization constant, in addition to the molar absorp-tivity of the dimer band, can be also obtained from the data treatment of the integrated absorbances of the dimer band usingEq. (13). Hence, the same dimerization constant can be determined independently either from the data of monomer band or from those of dimer band. This offers an opportunity to check the consistency of the whole deter-mination.
2. Equations for data treatment of the monomer and dimer bands
The dynamic equilibrium in a self-association system,
A + A A2 (1)
is restricted by a relation,
[A] + 2[A2]= [A]0 (2)
and characterized by a temperature-dependent dimerization constant,
K = [A2]
[A]2 (3)
where [A] and [A2] are the equilibrium concentrations of monomer, A, and that of dimer, A2, respectively; [A]0 is the initial concentration of the self-associating species. A combination ofEqs. (2) and (3) allows us to solve for [A] and [A2] in terms of [A]0and K to be
[A] = 2[A]0 (1 + 8K[A]0)1/2+ 1 (4) [A2]= 1 2 (1 + 8K[A]0)1/2− 1 (1 + 8K[A]0)1/2+ 1 [A]0 (5)
Eqs. (4) and (5)have already been derived in the NMR study of monomer–dimer self-association[17–20].
According to Beer–Lambert’s law, the absorbance at a particular wavenumber ˜ν for a vibration taking place in monomer species is given by
Am(˜ν) = εm(˜ν)b[A] (6)
whereεm(˜ν) is the molar absorptivity of the monomer band at ˜ν, b the optical path length of the cell. The integrated absorbance of the whole monomer band, Am, then is obtained by an integration over the whole range of the wavenumber covered by this band. That is
Am= εm(˜ν)b[A]d˜ν = εm(˜ν) d˜νb[A] = 2[A]0 (1 + 8K[A]0)1/2+ 1ε mb (7)
In writing the last equation, Eq. (4) has been used, and
εm
=εm(˜ν) d˜ν
is the molar absorptivity of the monomer band. If we invert the fractions of the first and last terms of the above equation, then multiply them by 2εmb[A]0, we obtain
2εmb[A]0
Am
= (1 + 8K[A]0)1/2+ 1 (8) If we subtract unity from both sides of Eq. (8), then take squares, we finally arrive at
[A]0 (Am)2 = 1 (εmb) 1 Am +(ε2K mb)2 (9) upon dividing by 4ε2mb2[A]0. Eq. (9) allows us to fit the experimental data of y = ([A]0/A2m) vs. x = A−1m to a straight line.εmand K then can be obtained from the slope,
p, and intercept, q, of this regressed line by εm = (1/pb) andK = (q/2p2), respectively.
Along the line of deriving Eq. (7), the integrated ab-sorbance of the dimer band can be derived to be
Ad= εdb[A2]= εdb[A]0 2 (1 + 8K[A]0)1/2− 1 (1 + 8K[A]0)1/2+ 1 (10) where εd(=
εd(˜ν) d˜ν) is the molar absorptivity of the dimer band. If we divide the first and last terms inEq. (10) by εdb[A]0/2 and use the fact that if (a/b) = (c/d), then
(a + b/a − b) = (c + d/c − d), we obtain εdb[A]0+ 2Ad
εdb[A]0− 2Ad = (1 + 8K[A]0)
1/2 (11)
The above equation can be further manipulated by taking squares followed by using the fact if (a/b) = (c/d) then
(a − b/b) = (c − d/d) to yield εdbAd
K = (εdb[A]0− 2Ad)2 (12)
We then take square roots and divide by [A]0on both sides ofEq. (12)to transform it into a linear equation
2Ad [A]0 = εdb − εdb K 1/2(A d)1/2 [A]0 (13) This equation can be used to fit the experimental data of
y = (2Ad/[A]0) versus x = ((Ad)1/2/[A]0) to a straight line. From the slope, p, and intercept, q, of the regressed line,
εdand K can be obtained byεd = (q/b) and K = (q/p2), respectively.
3. Experimental section
The chemical 2,2-dimethyl-3-ethyl-3-pentanol (98%) and tetrachloroethylene (99.98%) were purchased from Aldrich, and Tedia, respectively, and used as received. The sample concentrations on molality scale were prepared with the help of microsyringes, vials, and an analytical balance. The concentrations on molality scale were then converted
into molarity scale based on the density of the solvent at various temperatures obtained from the published data (283 K: 1.63120, 293 K: 1.62260, 303 K: 1.60640, 393 K: 1.44865 g ml−1) [21]. All subsequent calculation, plotting and tabulation were then carried out in terms of molarity scale when concentration is concerned. The spectra were recorded by an FTIR spectrometer (Bio-Rad Spc. 3200) with a CaF2cell window of 0.5 mm optical path length. The cell temperature was controlled by the circulating water, which was from a thermostat, flowing through the cell jacket. A thermal couple was inserted into the jacket to measure the temperature. The error of the temperature was estimated to be ±1 K. A personal computer implanted with a commer-cial software package PeakSolve (Galatic Industries Corp.) was hooked to the spectrometer to perform the task of curve-fitting the overlapped spectra and of calculating the integrated absorbance of the resolved spectra. Other tasks of regression, plotting graphs were performed with the help of Mathematica software on another personal computer.
4. Results and discussion
In this system, the OH fundamental stretching vibration exhibits two absorption bands in the range 3300–3750 cm−1. The sharp one at higher wavenumber corresponds to the OH stretching in the monomer. The broad one at lower wavenumber corresponds to that in the hydrogen-bonded dimer. A collection of the spectra within this region for different concentrations of 2,2-dimethyl-3-ethyl-3-pentanol in tetrachloroethylene at 298 K has been presented in Fig. 1. These two bands are not well separated. They were resolved and their integrated absorbance were cal-culated with the help of a software package PeakSolve. A typical result is presented in Fig. 2 for 0.1958 mol l−1 2,2-dimethyl-3-ethyl-3-pentanol in tetrachloroethylene at 298 K. The integrated absorbances of the resolved monomer
Fig. 2. Curve fit for 0.1958 mol/l 2,2-methyl-3-ethyl-3-pentanol in tetrachloro-ethylene at 298 K.
Fig. 1. A collection of the spectra of the fundamental OH stretching bands for different concentrations of 2,2-methyl-3-ethyl-3-pentanol in tetrachloro-ethylene at 298 K: from bottom to top 0.1621, 0.2100, 0.2610, 0.2917, and 0.3243 mol/l.
band, Am, and that of dimer bands, Ad, for various concen-trations at four different temperatures, 278, 288, 298, 308 K are listed in Table 1. The errors of Am and Ad calculated from several times of fitting are estimated to be within
±0.8% and ±0.5%, respectively.
From this table it is observed that at a given tempera-ture both Am and Ad increase with initial concentration, [A]0, while the ratio Ad/Am decreases. Furthermore, in Fig. 3, Ad/A2m for each temperature keeps constant as [A]0 varies. This is a clear implication of the occurrence of Monomer–dimer self-association. Once Am and Ad for different concentrations of solute at a given temperature have been available. We are in a position to deduce εm and K from the slope and intercept of the linear plot of
Table 1
The integrated absorbances of OH stretching bands from the monomer,
Am, and those of dimer, Ad, for different initial concentrations of
2,2-dimethyl-3-ethyl-3-pentanol in tetrachloroethylene at different temper-atures [A]0/mol l−1 Am Ad 278 K 0.1649 12.65 5.52 0.1821 13.83 6.58 0.1980 14.82 7.61 0.2132 15.81 8.65 0.2293 16.84 9.81 0.2489 18.09 11.25 0.2624 18.87 12.31 0.2796 19.91 13.68 0.2954 20.84 15.02 0.3138 21.95 16.64 0.3300 22.84 18.04 288 K 0.1638 12.31 3.89 0.1805 13.52 4.65 0.1973 14.64 5.45 0.2124 15.68 6.22 0.2285 16.69 7.09 0.2458 17.73 8.06 0.2604 18.71 8.92 0.2767 19.71 9.92 0.2941 20.75 11.04 0.3097 21.73 12.09 0.3333 23.22 13.71 298 K 0.1621 11.96 2.67 0.1813 13.28 3.29 0.1958 14.26 3.79 0.2100 15.15 4.32 0.2265 16.30 4.96 0.2421 17.37 5.61 0.2610 18.68 6.42 0.2747 19.44 7.06 0.2917 20.44 7.85 0.3091 21.63 8.72 0.3243 22.60 9.50 308 K 0.1615 11.86 1.54 0.1772 13.04 1.84 0.1937 14.10 2.17 0.2081 15.11 2.49 0.2244 16.28 2.88 0.2408 17.30 3.28 0.2568 18.47 3.71 0.2728 19.63 4.15 0.2888 20.44 4.62 0.3043 21.70 5.10 0.3203 22.73 5.61
y = ([A]0/(Aobsm )2) versus x = (1/Aobsm ) based onEq. (9), and to deduceεdand K from the slope and intercept of the linear plot ofy = (2Aobsd /[A]0)versus x = ((Aobsd )1/2/[A]0) based onEq. (13). Such plots at four temperatures for the monomer bands, and for the dimer bands are presented in Figs. 4 and 5, respectively. K from the monomer bands and dimer bands, εm, and εd at each temperature thus
deter-0.15 0.20 0.25 0.30 0.35 0.01 0.02 0.03 0.04 A d / A m 2 M / L mol-1
Fig. 3. Plot of Ad/A2m vs. [A]0 to demonstrate the validity of
monomer–dimer self-association and the consistency of the parameter de-termination: (䊉) 278 K, (䊊) 288 K, (䊏) 298 K, (䊐) 308 K.
mined were collected inTable 2. The errors associated with
εm,εdand K were calculated based on a standard method [22]with the assumption of equal variances for Am or Ad.
It is seen that εm and εd decrease with increasing tem-perature. This phenomenon can not be simply explained by Boltzmann distribution. The molar absorptivity,ε (εmorεd), is proportional to the absorption coefficient, α(ω), with a proportionality constant independent of temperature, T,[23]
α(ω) = 2π 3c¯hnω(1 − e −(¯hω/kT)) × ∞ −∞dte −iωt M (0) M (t) (14)
where c is the speed of light, ¯h = 2π/h with h being the Planck constant, n the refractive index of the medium,ω the angular velocity of the radiation absorbed, k the Boltzmann
0.04 0.05 0.06 0.07 0.08 0.09 12 10 8 6 10 4 ¡Ñ[A] 0 / A m 2 1/A m
Fig. 4. Linear plot based on Eq. (9) for different concentrations of 2,2-methyl-3-ethyl-3-pentanol in tetrachloroethylene at different tempera-tures: (䊉) 278 K, (䊊) 288 K, (䊏) 298 K, (䊐) 308 K.
6 8 10 12 14 30 60 90 120 2A d / [A] 0 Ad1/2 / [A]0
Fig. 5. Linear plot based on Eq. (13) for different concentrations of 2,2-methyl-3-ethyl-3-pentanol in tetrachloroethylene at different tempera-tures: (䊉) 278 K, (䊊) 288 K, (䊏) 298 K, (䊐) 308 K.
constant, and the integral represents the Fourier transform of the time correlation function of the dipole moment of the sample, M (0) M (t). The values of the factor
(1 − e−(¯hω/kT)) calculated for ω(= 2πc˜ν) and temperature
involved are fairly close to unity. Thus, it is understood that
M (0) M(t) are mainly responsible for the variation
ofε (εm orεd) with temperature. This may attribute to the temperature variation of the electrostatic field strength in the cavity where solute molecule (monomer or dimer) is seated, and to the strong, direct interaction between the solute and solvent molecules.
The dimerization constants at different temperatures al-low us to obtain the standard enthalpy, H◦, and entropy,
S◦, of dimerization through van’t Hoff plot. For this
sys-tem, the plots can be done either by the dimerization con-stants from Am or by those from Ad. Both plots are jux-taposed in Fig. 6 for better visual comparison. The errors associated with H◦ andS◦ were calculated based on a standard method[22]with unequal variance for K listed in Table 2. In order to assess the goodness of our determina-tion, we plot inFig. 7the original data of Amagainst [A]0at each temperature juxtaposed with the respective theoretical
Table 2
Molar monomer absorptivities (εm) molar dimer absorptivities (εd), dimerization constants (K) for 2,2-dimethyl-3-ethyl-3-pentanol in tetrachloroethylene
at different temperatures, and the standard enthalpy (H◦) and entropy (S◦) of dimerization obtained from monomer bands and dimer bands
From monomer bands From dimer bands
K (l mol−1) εm(l cm−1mol−1) K (l mol−1) εd (l cm−1mol−1)
Temperature (K) 278 0.544± 0.007 1773± 5 0.539± 0.007 10040± 60 288 0.361± 0.013 1670± 9 0.362± 0.005 9820± 60 298 0.235± 0.012 1580± 8 0.238± 0.003 9810± 60 308 0.141± 0.015 1535± 10 0.140± 0.003 9120± 120 H◦(kJ mol−1) −31.07 ± 4.06 −30.71 ± 4.07 S◦ (J mol−1K−1) −116.6 ± 14.1 −115.4 ± 14.1 0.00320 0.00336 0.00352 -2.0 -1.6 -1.2 -0.8 -0.4 0.0 0.4 ln K T-1/ K-1
Fig. 6. van’t Hoff plot to obtain H◦ and S◦ for the dimerization of 2,2-methyl-3-ethyl-3-pentanol in tetrachloroethylene from the data of dimerization constants obtained at different temperatures from the monomer band (䊉), and from the dimer band ( ).
curves calculated by inserting the determined values ofεm and K intoEq. (9). Similar plots for Adagainst [A]0 juxta-posed with the theoretical curves calculated byEq. (13)were also shown inFig. 8. A further assessment can be done by the comparison between the experimental values(Ad/A2m) and the calculated values of(Ad/A2m) = (Kεd/ε2m) for each temperature as shown inFig. 3.
It is observed that the isotherms, i.e. the data points or the theoretical curve at a given temperature, inFigs. 5 and 8, are well separated for the dimer bands. However, this is not the case for monomer bands as shown inFigs. 4 and 7. Such a difference can be explained as follows. Since K de-creases with temperature, [A] would increase with tempera-ture at a given initial concentration [A]0. On the contrary,εm decreases with temperature as seen fromTable 2. Thus two factors counteract against each other, leading to a clustering of the data points for different temperatures at a given ini-tial concentration. But, in case of dimer band, the decrease of bothεd and [A2] with temperature at a given [A]0effect
Adin a parallel way, resulting in a well-separated isotherms as shown inFigs. 5 and 8.
0.16 0.20 0.24 0.28 0.32 12 16 20 24 A m [A]0 / mol L-1
Fig. 7. Plot for comparing the theoretical curves calculated based onEq. (9) with experimental data for Amvs. [A]0 of 2,2-methyl-3-ethyl-3-pentanol
in tetrachloroethylene at different temperatures: (䊉) 278 K, (䊊) 288 K, (䊏) 298 K, (䊐) 308 K.
In IR study of the monomer–dimer self-association, to the best of our knowledge, only monomer band is employed to determine the spectral parameter εm and dimerization constant, K. For example, Liddle and Becker[24]obtained
K from the limiting slope of a plot of apparent absorptivity, Am/[A]0, against [A]0, via the equation
lim [A]0→0 d(Am/[A]0) d[A]0 = −2Kεm (15)
Hereεmis obtained from another limiting slope lim
[A]0→0
dAm d[A]0 = εm
(16) The disadvantage of this determination is that, if either one of the limiting slopes is steep, considerable errors may ensue.
0 5 10 15 20 25 (4) (3) (2) (1) A d [A]0/ mol L-1
Fig. 8. Plot for comparing the theoretical curves calculated based on Eq. (13) with experimental data of Ad vs. [A]0 of
2,2-methyl-3-ethyl-3-pentanol in tetrachloroethylene at different tempera-tures: (1) 278 K, (2) 288 K, (3) 298 K, (4) 308 K.
Prokopenko and Bethea in studying the effect of ring size on the dimerization of lactams[25]adopted to fit the data ofy = Amagainstx = [A]0to the equation
Am= εmb
√
1+ 8K[A]0− 1
4K (17)
to obtain K and εm using the Levenberg-Marquardt non-linear method. Luck [26] in his paper of studying the monomer–dimer self-association of lactams fitted the data ofy = Am/[A]0versusx = A2m/[A]0to the linear equation
Am [A]0 = εm− 2K εm A2 m [A]0 (18) to obtain εm and K. All the above three methods leave the data of the dimer band unattended. Our approach, on the contrary, seems more comprehensive and thoughtful, in the sense that two independent sources, i.e. the data from monomer and dimer bands, are employed to determine K, with the obtainment ofεmandεdas a bonus. The values of
K determined either from the data of monomer or from the
dimer are expected to be identical, since the same entity is referred to. Hence, the disparity between two K values is a good criterion for the merit of determinations. In the same token,H◦ andS◦ of dimerization determined from the data of monomer bands and dimer bands serve the same purpose.
5. Conclusion
In addition to Eq. (9), which treats the integrated ab-sorbances of monomer band to obtain εm and K, we em-ployedEq. (13)to treat those of the dimer band to obtain
εd and K. Thus K can be obtained from two independent sources and the disparity between the respective determined values provides a vehicle to assess the goodness of deter-mination. The standard enthalpy (H◦) and entropy (S0) determined from the temperature-dependent K of two inde-pendent sources also have the same function. Dilute solu-tion of 2,2-dimethyl-3-ethyl-3-pentanol in tetrachloroethene was used as an example to illustrate the usage of Eqs. (9) and (13). It is hoped that this new approach would facilitate the IR study of monomer–dimer self-association.
Acknowledgements
We would like to take this opportunity to express our grat-itude to National Science Council, Taiwan, for the financial support under the project number NSC 92-2133-M-009-023.
References
[1] G.A. Jeffery, An Introduction to Hydrogen Bonding, Oxford, New York, 1997.
[2] L. Pauling, R.B. Corey, Proc. Natl. Acad. Sci. U.S.A. 37 (1951) 729. [3] L. Pauling, R.B. Corey, H.R. Branson, Proc. Natl. Acad. Sci. U.S.A.
37 (1951) 205.
[4] J.D. Waston, F.H.C. Crick, Nature 171 (1953) 737.
[5] J.M. Prausnitz, R.N. Lichtenthaler, E.G. de Azeredo, Molecular Ther-modynamics of Fluid-Phase Equilibra, 3rd ed., Prentice-Hall, Upper Saddle River, New Jersey, 1999.
[6] J. Errera, P. Mollet, Nature 138 (1936) 882.
[7] G.C. Pimental, A.L. McClellan, The Hydrogen Bond, Freeman, San Francisco, 1960.
[8] D. Hadži, H.W. Thompson, Hydrogen Bonding, Pergamon Press, New York, 1959.
[9] W.C. Coburn Jr., E. Grunwald, J. Am. Chem. Soc. 80 (1958) 1318. [10] H. Shekaari, H. Modarress, N. Hadipour, J. Phys. Chem. A 109
(2003) 1981.
[11] N. Asprion, H. Hasse, G. Maurer, Fluid Phase Equilib. 186 (2001) 1. [12] P.L. Huyskens, W.A.P. Luck, T.Z. Huyskens (Eds.), Intermolecular
Force, Springer, Berlin, 1991.
[13] G.M. Frland, F.O. Libnau, O.M. Kvalheim, H. Hiland, Appl. Spectrosc. 50 (1996) 1264.
[14] R. Laenen, K. Simeonidis, R. Ludwig, J. Chem. Phys. 110 (1999) 5897.
[15] R. Laenen, K. Simeonidis, Chem. Phys. Lett. 290 (1998) 94. [16] G.P. Johari, W. Dannhauser, J. Phys. Chem. 72 (1968) 3273. [17] W.C. Luo, J.L. Lay, J.S. Chen, Zeit. Phys. Chem. 216 (2002) 829. [18] J.S. Chen, R.B. Shirts, J. Phys. Chem. 89 (1985) 1643.
[19] J.S. Chen, F. Rosenberger, Tetrahedron Lett. 31 (1990) 3975. [20] J.S. Chen, J. Chem. Soc. Faraday Trans. 90 (1994) 717.
[21] H.F. Mark, D.F. Othmer, C.G. Overberger, G.T. Seaborg (Eds.), Kirk-Othmer, Encyclopedia of Chemical Technology, vol. 5, 3rd ed., Wiley, New York, 1979, p. 755.
[22] P.R. Bevington, D.K. Robinson, Data Reduction and Error Analysis, 3rd ed., McGraw-Hill, Boston, 2003.
[23] R.G. Gordon, Adv. Mag. Reson. 3 (1968) 1.
[24] U. Liddel, E.D. Becker, Spectrochim. Acta A 10 (1957) 70. [25] N.A. Prokopenko, I.A. Bethea, et al., Phys. Chem. Chem. Phys. 4
(2002) 490.
[26] P.L. Huyskens, W.A.P. Luck, T.Z. Huyskens (Eds.), Intermolecular Force, Speringer, Berlin, 1991, p. 163.
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