Application of the Quantum Cluster Equilibrium (QCE) Model for the

ARTICLE pubs.acs.org/JPCB

Application of the Quantum Cluster Equilibrium (QCE) Model for the Liquid Phase of Primary Alcohols Using B3LYP and B3LYP-D DFT Methods Gergely Matisz,†,‡ Anne-Marie Kelterer,§ Walter M. F. Fabian,‡ and Sandor Kunsagi-Mate*,† †

Department of General and Physical Chemistry, University of Pecs, Pecs H-7624, Hungary Institute of Chemistry, Karl-Franzens University Graz, Heinrichstrasse 28, Graz A-8010, Austria § Institute of Physical and Theoretical Chemistry, Graz University of Technology, Stremayrgasse 9/I, A-8010 Graz, Austria ‡

bS Supporting Information ABSTRACT: The Quantum Cluster Equilibrium (QCE) model was applied to the liquid phase for the first few members of the homologous series of unbranched aliphatic primary alcohols, methanol, ethanol, propan-1-ol, and butan-1-ol. Cluster structures and energies were calculated by density functional theory [B3LYP/6-311þþG(2d,2p)]. For butan-1-ol the dispersion interaction was also considered with the B3LYP-D method. In agreement with previous findings, cyclic cluster structures are the most probable ones. In addition, weak CH...O interactions as well as dispersion interactions between the longer alkyl chains were found to be important in the cluster formation. The reliability of the model was assessed by the calculated constant pressure heat capacity (Cp) values. Larger deviations between theory and experiment were found for higher homologes (propan-1-ol, butan1-ol) with the B3LYP method. When the B3LYP-D method was applied for butan-1-ol, adequate agreement was found between experimental and calculated Cp values.

’ INTRODUCTION Hydrogen bonded liquids, like water16 and various alcohols,715 were investigated previously in several publications via the Quantum Cluster Equilibrium (QCE) model16,17 of F. Weinhold. The advantage of this model is that the cluster distribution in liquid phase can be obtained rather easily using ab initio cluster geometries and cluster properties. To the best of our knowledge, no complete systematic study for the homologous series of unbranched aliphatic primary alcohols was presented until now regarding their structure within the QCE model. Although force field based MD calculations are possible, they are not reliable enough to predict the liquid structure adequately.18,19 All the studies in the literature which deal with the structures of hydrogen bonded clusters formed in the liquid phase show that only cyclic clusters, where no sterical hindrance is present,14,15 exist in this phase. In all cases of ring formation the alcohol molecules are bicoordinated. Not surprisingly, the most extensive study in the literature among the alcohols is published for methanol. Borowski et al.7 have found that cyclic tetramer and heptamer structures are the most probable ones at room temperature, followed by the cyclic hexamers and pentamers. In the work of R. Ludwig8 these hexamer and pentamer structures were found as the most probable clusters, 60% and 32%, respectively. The results of R. Ludwig based on the B3LYP density functional are in agreement with our previous work20 regarding methanol structures calculated at the MP2(fc) theoretical level.21 In contrast to methanol, studies for higher r 2011 American Chemical Society

homologes are rather limited. For ethanol, in the work of Borowski et al.,7 only the pentamer structure was found in the liquid, which shows an interchange with the monomer form at getting closer to the boiling point. At the HF level quite a large basis set dependence has been found.1214 For instance, HF/631G(d) calculations by R. Ludwig et al. indicated a nearly equal abundance of the pentamer followed by the tetramer clusters at room temperature in the liquid. Studies for other alcohols, namely for benzyl-alcohol14 and for 2,2-dimethyl-3-ethyl-3pentanol,14,15 were done with the RHF method too, where cyclic tetramers and simple dimers, respectively, were suggested as the most abundant cluster structures. In this current work the temperature-dependence of the distribution of cyclic clusters formed in the most common liquid primary alcohols, methanol, ethanol, propan-1-ol, and butan-1ol, were studied. The Quantum Cluster Equilibrium theory was applied to calculate the distribution of the clusters whose geometries and energies were determined by DFT calculations augmented with dispersion correction in the butan-1-ol case.

’ COMPUTATIONAL DETAILS In the first part of the work only the OH...O hydrogen-bond interaction was considered as the determining factor of the cluster Received: October 17, 2010 Revised: February 28, 2011 Published: March 18, 2011 3936

dx.doi.org/10.1021/jp109950h | J. Phys. Chem. B 2011, 115, 3936–3941

The Journal of Physical Chemistry B

ARTICLE

calculated electronic, vibrational, rotational, translational partition functions, finally leading to the distribution of the clusters at the equilibrium (eq 1). In this equation k(i) is the number of molecules in the i-th cluster, and NA is the Avogadro constant while ni is the mol number of the i-th cluster. !kðiÞ n1 kðiÞ 1 ni ¼ qi NA ð1Þ q1

Figure 1. Temperature dependence of the density of methanol, ethanol, propan-1-ol, and butan-1-ol from 0 °C to the boiling temperature at standard pressure.

structures and energies (i.e., no dispersion). Density functional calculations [B3LYP/6-31þG(d,p)]2227 were used for geometry optimizations. The geometries of cyclic clusters of (ROH)n, where n = 47, and the dimer and monomer structures of methanol, ethanol, propan-1-ol, and butan-1-ol were chosen in the cluster sets. The conformation of the alcohol molecules was selected in agreement with previous work28 of primary alcohols at the B3LYP level up to the hexan-1-ol, where, generally, the gþtt or equivalently the gtt structures were associated with the global minima of the conformational potential energy surface. According to the convention describing the conformation of alcohols28 or normal alkanes,2931 the first letter (gþ or g (∼60°) in the present case) represents the H-O-C-C dihedral angle; the following ones, t (∼180°), correspond to those of the alkyl chain. No cyclic clusters with n = 3 were considered because of their low probability in the liquid according to the quite large steric strain.32 Based on the findings for methanol20 neither larger cyclic nor lasso structures were considered. Counterpoise (CP) corrected energy calculations on the optimized cluster geometries were done using the 6-311þþ G(2d,2p) basis set. The Gaussian 09 program33 was used for the B3LYP structure and energy calculations. van der Waals interactions between butan-1-ol molecules which have the longest alkyl chain were included by the dispersion corrected34 B3LYP method [B3LYP-D]. Dispersion interactions might stabilize other cluster types than those resulting from preferential OH...O hydrogenbond interaction. Hence, in addition to the cyclic clusters described above, especially, those where weak CH...O hydrogen bonds are forming were also considered. Furthermore, clusters with different alkyl chain positions and conformations and the trimer (BuOH)3 where some stabilization effect can come from the dispersion interaction between the alkyl chains were taken into account. The B3LYP-D calculations were performed in the ORCA program.35 The density data for the liquid phase of the alcohols at the corresponding temperatures (Figure 1) were obtained from available experimental data, for methanol,36 ethanol,3742 propan-1-ol,4348 and butan-1-ol38,42,44,4953 by parabolic fitting (Tables S1 and S2 of the Supporting Informatin). The vapor density data at the boiling temperature are collected in Table S3 together with the van der Waals constants (a, b)54 of the corresponding alcohols. The QCE model was applied here the same way as described in our previous paper20 on liquid methanol. Consequently, only a very brief description of this model will be given. It relies on statistical thermodynamics and is based on the chemical potential equality of the molecules in different cluster forms. The cluster partition functions (qi) can be obtained by multiplying the

Two empirical parameters are inherent in the QCE model. The first one, bxv in eq 2, which appears in the translational partition function, is used to take the exact available free volume for translation into account. Vexcl means the volume which is occupied by the clusters, and Vi is the cluster volume calculated by applying a 0.001 e/bohr3 electron density envelope.

∑i ni NA Vi

Vexcl ¼ bxv

ð2Þ

The other parameter (amf) is used to include the intercluster interaction in the electronic partition function, eq 3. It is possible to use an exponential scaling for the intercluster interaction as in the work of Borowski et al.,7 instead of the linear relationship with the cluster size; however, this latter is used conventionally in the QCE model and also in this work. Here ΔEi is the energy of the cluster relative to the monomer units. All the cluster energies are counterpoise corrected. Both the cluster and monomer energies contain the ZPE correction. ΔEi ¼ Ei, clust kðiÞamf =Vm kðiÞEref

ð3Þ

The fitting procedure is carried out for the experimental densities and for the standard pressure boiling temperatures, while the standard pressure condition was requested (eq 4). Three different approaches have been used previously for the parameter fitting. Here we used that one where the Gibbsfunction of the liquid and vapor phase intersect at the boiling temperature at standard pressure, thus obtaining an amf bxv parameter pair; then for other temperatures the amf parameter was kept fixed, while the bxv parameter was optimized for each temperature (model 1 in ref 20). 1 0 c D Ni ½ln qi, elec þ ln qi, trans C B Dln Q C B C p ¼ kT ¼ kT B i ¼ 1 A @ DV DV T

∑

0

c

"

c

"

∑

ΔEi =kT þ ln B BD i ¼ 1 Ni ln e B ¼ kT B B DV @

0

∑

B BD i ¼ 1 Ni ln e B ¼ kT B B @ c

"

V Vexcl C C Λ3i C C C A

∑

3937

#1 V Vexcl þ ln C C Λ3i C C C A T

ðEi amf kðiÞ=V kðiÞE1 Þ=kT

DV

amf kðiÞ 1 1 ¼ kT Ni þ V 2 V Vexcl kT i¼1 c 1 kT Ni amf kðiÞ 2 þ ¼ V V Vexcl i¼1

∑

T

#1

#

T

ð4Þ

dx.doi.org/10.1021/jp109950h |J. Phys. Chem. B 2011, 115, 3936–3941


ARTICLE

Figure 2. Cluster distribution in methanol (A), ethanol (B), propan-1-ol (C), and butan-1-ol (D), respectively, obtained with the B3LYP method.

Figure 3. Cluster distributions at the bxv = 0 value (i.e., clusters with no volume). T = 298.15 K. The monomers are dominating especially in BuOH.

’ RESULTS AND DISCUSSION Figure 2 shows the temperature-dependence of the normalized distribution of clusters, i.e. how many monomers participate in a given cluster form in 1 mol of the liquid, formed in different primary alcohols, methanol, ethanol, propan-1-ol, butan-1-ol, as calculated by applying the QCE model with the B3LYP density functional without dispersion correction. It can be seen that cyclic cluster structures are much more important than monomers or dimers. In each case the amount of monomers of the different alcohols is increasing similarly with increasing temperature. This tendency is slightly more pronounced in the case of higher homologes, where, of course the end temperature is also higher. The temperature dependencies for the contributions of the dimers to the liquid structure are similar. Characteristically, the five and six membered rings are the main components, while the seven membered rings are still significant for ethanol to butan-1-ol; especially in the case of ethanol. The contribution of monomers to the six and seven membered rings decrease with the temperature; approximately up to 340 K the contribution to the five membered rings slightly increases, above this

Figure 4. B3LYP/6-311þþG(2d,2p) counterpoisecorrected intracluster interaction energies with respect to the monomer units for n = 57.

Figure 5. Cluster distribution in butan-1-ol within the QCE model obtained with the B3LYP-D method (with empirical dispersion correction). The monomer (1) has tgt conformation and dimer (2) consists of gtt, gþtt monomers; the four (4) and five (5) membered clusters possess the gþgt (ggþt) monomers, the six (6) membered cluster is formed by gþtt and gtt monomers. 3938



ARTICLE

Table 1. Experimental and Theoretically Calculated Constant Pressure Heat Capacities (Cp)a Cp (exp.)

Cp (theor., B3LYP)

methanol

81.56

72.3

ethanol propan-1-ol

112.68 143.96

119.0 237.3

butan-1-ol

176.69

380.6

Cp (theor., B3LYP-D)

222.9

a

The experimental values Cp (exp.) are taken from ref 58. The unit is J mol1 K1.

Figure 6. The most significant structures obtained with the B3LYP-D method. Structure A (tetramer) and B (pentamer) are with gþgt (ggþt) conformations of the monomers, while structure C (hexamer) has gþtt (gtt) monomer conformations.

temperature the tendency is toward decreasing. Figure 3 shows the importance of the bxv fitting parameter on the cluster distribution. If its value is selected to be zero, which is a limiting case without real physical meaning, mainly monomers constitute the liquid. The binding energy per monomer units (Ebind,m) is defined as the total counterpoise corrected interaction energy (Ei,clust in eq 3) divided by the number of monomers in the cluster. The trends of the various intracluster interaction energies are shown in Figure 4. The following main differences between the investigated alcohols can be seen: (i) methanol has a minimum of Ebind,m for the cluster size n = 6; (ii) ethanol, propan-1-ol, and butan-1-ol possess essentially identical interaction energies which are, in the absolute value, 11.5 kJ/mol smaller than those obtained for methanol clusters. The basis set size was found to have a moderate effect on the cluster distribution through the basis set superposition error (BSSE). The BSSE is nearly twice as much in the case of the 6-311þþG(d,p) basis compared with that resulting from applying the 6-311þþG(2d,2p) basis set; the difference between the results from these two basis sets is approximately 12 kJ/mol/monomer, thus comparable to the calculated energy differences between the clusters. The B3LYP-D result of butan-1-ol is shown in Figure 5, where only the most significant structures were included (these structures are in Figure 6). For reference at the calculation of the binding energies, the global minimum tgt monomer structure was selected; this tgt conformation in the clusters was found to be less preferred energetically. The preferred tetramer and pentamer structures are those where the alcohol molecules adopt the gþgt (ggþt) conformation, in which, thus, weak CH...O interactions are present. In the case of the hexamer structure, the alcohol molecules have gþtt (gtt) conformations and thus the largest dispersion interaction between them. With the B3LYP-D method, no considerable amount of seven membered clusters was found in the liquid phase within the model. Table S4 collects the distribution of the monomers in the different cluster forms of the chosen cluster set for B3LYP-D including also less significant ones. The geometries in Cartesian coordinates are provided in the Supporting Information. In Table 1 the experimental and calculated constant pressure heat capacities (Cp) are compared for the four different alcohols. Also included are the results for butan-1-ol obtained with the B3LYP-D method. Adequate results with the B3LYP method are only obtained for methanol and ethanol and for butan-1-ol with the B3LYP-D method. The Cp values obtained for propan-1-ol and butan-1-ol with the B3LYP method are significantly larger than the experimental values. The agreement between experimental and calculated thermodynamic properties leads to the conclusion that the B3LYP-D method and the cluster set derived therefrom gives a better description of the liquid phase within the 3939


The Journal of Physical Chemistry B QCE model than its B3LYP counterpart for longer chain molecules. For alcohols with longer alkyl chains where also dispersion interaction as well as CH...O interactions become important, inclusion of dispersion energy in the theoretical description of the cluster is necessary. Moreover, not only clusters with the same type of interaction but mainly with different topologies have to be included in the cluster set.

’ CONCLUSIONS The QCE model previously applied for liquid methanol was used here to investigate the temperature dependence of the cluster distribution in the first few members of the liquid phase of the homologous series of linear aliphatic alcohols, methanol, ethanol, propan-1-ol, and butan-1-ol. The B3LYP method was used for the whole series. In addition, for butan-1-ol the B3LYPD method was also applied to take the dispersion interaction into account. A larger cluster set for this alcohol was also investigated. In each case, cyclic cluster structures (ROH)n up to n = 7 were taken into account. At lower temperatures, i.e. close to room temperature, the five and six membered clusters were found as the dominant species and also tetramers with B3LYP-D. The seven membered clusters were found important for ethanol and propan-1-ol and a slightly smaller amount in the case of butan-1ol. When the dispersion correction was applied in the butan-1-ol case, the amount of the heptamer was found to be negligible. Weak CH...O interactions in tetramers and pentamers and dispersion interactions in hexamers were found to be significant. The calculated constant pressure heat capacities (Cp) were in relatively good agreement with experiment in the case of methanol and ethanol at the application of the B3LYP method and in the case of butan-1-ol at the application of the van der Waals correction (B3LYP-D method). Thus, consideration of the weaker dispersion interactions in cases of the alcohols with longer alkyl chains is important. The temperature dependence of the structural behavior of these liquids could play an important role in the molecular interactions between solute molecules. This is because the molecular interactions in the liquid phase assume at least partial destruction of the solvation shells,5557 which, however, highly depends on the structure of the solvents discussed here. ’ ASSOCIATED CONTENT

bS

Supporting Information. Tables for density values of the liquid phases of the alcohols; methanol, ethanol, propan-1-ol, butan-1-ol (Tables S1S2), and the van der Waals parameters of the vapor phase together with the molar volumes and standard pressure boiling temperatures (Table S3), and Cartesian coordinates of the B3LYP/6-31þG(d,p) optimized geometries of the alcohol clusters and the B3LYP-D/6-31þG(d,p) optimized geometries of the butan-1-ol clusters together with the obtained distributions (Table S4). This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*Phone: þ 36 72-503-600 (4208). Fax: þ 36 72-503-635. E-mail: [email protected]. Corresponding author address: University of Pecs, Faculty of Sciences, Department of General and Physical Chemistry, H-7624 Pecs, Ifjusag 6, Hungary.

ARTICLE

’ ACKNOWLEDGMENT Financial support by “Wissenschaftlich-technische Zusammenar€ beit Osterreich Ungarn” (HU 4/2009) is gratefully acknowledged. ’ REFERENCES (1) Lehmann, S. B. C.; Spickermann, C.; Kirchner, B. J. Chem. Theory Comput. 2009, 5, 1640–1649. (2) Lehmann, S. B. C.; Spickermann, C.; Kirchner, B. J. Chem. Theory Comput. 2009, 5, 1650–1656. (3) Ludwig, R.; Weinhold, F.; Farrar, T. C. J. Chem. Phys. 1995, 103, 6941–6950. (4) Ludwig, R.; Weinhold, F. J. Chem. Phys. 1999, 110, 508–515. (5) Ludwig, R.; Weinhold, F. Phys. Chem. Chem. Phys. 2000, 2, 1613–1619. (6) Ludwig, R. Angew. Chem., Int. Ed. 2001, 40, 1808–1827. (7) Borowski, P.; Jaroniec, J.; Janowski, T.; Wolinski, K. Mol. Phys. 2003, 101, 1413–1421. (8) Ludwig, R. ChemPhysChem 2005, 6, 1369–1375. (9) Ludwig, R. ChemPhysChem 2005, 6, 1376–1380. (10) Ludwig, R; Weinhold, F.; Farrar, T. C. Ber. Bunsen-Ges. Phys. Chem. 1998, 102, 197–204. (11) Ludwig, R.; Weinhold, F.; Farrar, T. C. Ber. Bunsen-Ges. Phys. Chem. 1998, 102, 205–212. (12) Ludwig, R; Weinhold, F.; Farrar, T. C. Mol. Phys. 1999, 97, 465–477. (13) Ludwig, R; Weinhold, F.; Farrar, T. C. Mol. Phys. 1999, 97, 479–486. (14) Huelsekopf, M.; Ludwig, R. J. Mol. Liq. 2000, 85, 105–125. (15) Huelsekopf, M.; Ludwig, R. J. Mol. Liq. 2002, 9899, 163–171. (16) Weinhold, F. J. Chem. Phys. 1998, 109, 367–372. (17) Weinhold, F. J. Chem. Phys. 1998, 109, 373–384. (18) Valdez-Gonzalez, M.; Saint-Martin, H.; Hernandez-Cobos, J.; Ayala, R.; Sanchez-Marcos, E.; Ortega-Blake, I. J. Chem. Phys. 2007, 127, 224507. (19) Handgraaf, J.; van Erp, T. S.; Meijer, E. J. Chem. Phys. Lett. 2003, 367, 617–624. (20) Matisz, G.; Fabian, W. M. F.; Kelterer, A.-M.; Kunsagi-Mate, S. J. Mol. Struct. THEOCHEM 2010, 956, 103–109. (21) Møller, C.; Plesset, M. S. Phys. Rev. 1934, 46, 618–622. (22) Becke, A. D. J. Chem. Phys. 1993, 98, 5648–5652. (23) Lee, C.; Yang, W.; Parr, R. G. Phys. Rev. B 1988, 37, 785–789. (24) Vosko, S. H.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200–1211. (25) Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. J. Phys. Chem. 1994, 98, 11623–11627. (26) Hehre, W. J.; Ditchfield, R.; Pople, J. A. J. Chem. Phys. 1972, 56, 2257–2261. (27) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. J. Chem. Phys. 1980, 72, 650–654. (28) Vansteenkiste, P.; Pauwels, E.; Speybroeck, V. Van; Waroquier, M. J. Phys. Chem. A 2005, 109, 9617–9626. 00 (29) Tasi, G.; Mizukami, F.; Palinko, I.; Csontos, J.; Gyorffy, W.; Nair, P.; Maeda, K.; Toba, M.; Niwa, S.; Kiyozumi, Y.; Kiricsi, I. J. Phys. Chem. A 1998, 102, 7698–7703. (30) Tasi, G.; Mizukami, F. J. Math. Chem. 1999, 25, 55–64. 00 (31) Tasi, G.; Mizukami, F.; Csontos, J.; Gyorffy, W.; Palinko, I. J. Math. Chem. 2000, 27, 191–199. (32) Huang, Z.; Yu, L.; Dai, Y. Struct. Chem. 2010, 21, 565–572. (33) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, Jr., J. A.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; 3940



ARTICLE

Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, N. J.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, € Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; S.; Daniels, A. D.; Farkas, O.; Fox, D. J. Gaussian 09, Revision A.02; Gaussian, Inc.: Wallingford, CT, 2009. (34) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. J. Chem. Phys. 2010, 132, 154104–154122. (35) Neese, F. ORCA an ab initio, Density Functional and Semiempirical program package, Version 2.8; University of Bonn, 2010. (36) Lemmon, E. W.; McLinden, M. O.; Friend, D. G. Thermophysical Properties of Fluid Systems; National Institute of Standards and Technology: Gaithersburg, MD 20899. http://webbook.nist.gov (retrieved January 7, 2010). (37) Kretschmer, C. B.; Nowakowska, J.; Wiebe, R. J. Am. Chem. Soc. 1948, 70, 1788. (38) Sepalowa-Michailowa Chem. Abstr. 1940, 3960. (39) Fiock, E. F.; Ginnings, D. C.; Holton, W. B. J. Res. Natl. Bur. Stand. (U. S.) 1931, 6, 881–892. (40) Winkler, L. W. Chem. Ber. 1905, 38, 3612–3616. (41) Price, T. W. J. Chem. Soc. 1915, 107, 188–198. (42) Schiff, R. Liebigs Ann. Chem. 1883, 220, 99–101. (43) Young, S.; Fortey, E. C. J. Chem. Soc. 1902, 81, 735. (44) Atkins, G.; Wallace, T. A. J. Chem. Soc. 1913, 103, 1469. (45) Timmermans, J.; Delcourt, Y. J. Chim. Phys. Phys.-Chim. Biol. 1934, 31, 89–90. (46) Crismer, L. Chem. Zentralbl. 1904, 75, 1480. (47) Brunel, R. F. J. Am. Chem. Soc. 1923, 45, 1336. (48) Arbusow, A. E. Z. Phys. Chem., Stoechiom. Verwandtschaftsl. 1928, 131, 55. (49) Timmermans, J.; Martin, F. J. Chim. Phys. Phys.-Chim. Biol. 1928, 25, 412. (50) Jones, G.; Christian, S. M. J. Am. Chem. Soc. 1939, 61, 82. (51) Rao, M. V. P.; Naidu, P. R. J. Chem. Thermodyn. 1976, 8, 73–75. (52) Reddy, K. S.; Naidu, P. R. J. Chem. Thermodyn. 1978, 10, 201–202. (53) Smyth, C. P.; Walls, W. S. J. Am. Chem. Soc. 1931, 53, 2118. (54) Speight, J. G. Lange’s Handbook of Chemistry, 16th ed.; McGraw-Hill: New York, 2004. (55) Kunsagi-Mate, S.; Ortmann, E.; Kollar, L.; Nikfardjam, M. P. J. Phys. Chem. B 2007, 111, 11750–11755. (56) Kunsagi-Mate, S.; Csok, Zs.; Tuzi, A.; Kollar, L. J. Phys. Chem. B 2008, 112, 11743–11749. (57) Kunsagi-Mate, S.; Kumar, A.; Sharma, P.; Kollar, L.; Nikfardjam, M. P. J. Phys. Chem. B 2009, 113, 7468–7473. (58) Tanaka, R.; Toyama, S.; Murakami, S. J. Chem. Thermodyn. 1986, 18, 63–73.

3941


Application of the Quantum Cluster Equilibrium (QCE) Model for the

Recommend Documents