1546
J. Phys. Chem. 1981, 85, 1546-1548
Calculation of the Entropy of Solid Toluene and Hexamethylbenzene A. J. Pertsin, Yu. P. Ivanov, and A. I. Kitaigorodsky” Institute of Organo-Element Compounds, Academy of Sciences of the (Received: December 9, 1980)
USSR,Moscow 117813, GSP-1, V-3 12, Vavilov 28, USSR
The intermolecular contributions to the entropies of solid toluene and hexamethylbenzene are calculated by using the cell model as a statistical-mechanicalbasis and the atom-atom potential method as an approximation to intermolecular potential. For toluene, the intramolecular contribution is estimated by using experimental internal mode vibration frequencies and an internal rotation barrier ranging from 0 to 0.75 kcal mol-’. The resulting total entropies show good agreement with the measured third-law entropy. For hexamethylbenzene, three different estimates of the intramolecular contribution to the entropy are reported. The best agreement with experiment is observed for the model of independent hindered rotations of the six methyls with a barrier of 2.2 kcal mol-l,
Introduction Recently a new method has been suggested for calculating the thermodynamic functions of organic so1ids.l The method is based on the so-called cell model2intended originally for simple fluids and then applied to rare-gas crystal^.^ The most serious assumption of the cell model is that the motions of individual particles in a system are assumed to be uncorrelated. As explained by Barker? this may be well justified for crystals at high themperatures where a major contribution to their thermodynamic functions is accounted for by high vibration frequencies close to that corresponding to independent motion of particles. With the above assumption the configurational integral of a crystal is reduced to the form of eq 1,2where
N is the number of particles in the crystal, uo is the total static energy, the vector q describes displacements of a particle from its equilibrium position (q = 0), and u(q) is the potential energy of interaction of the particle at q with its neighbors at their equilibrium sites. The cell model is easily extended to crystals composed of polyatomic molecules, providing that a rule for calculating the intermolecular potential is available. One more assumption should however be made in this case-that the total free energy of the crystal can be separated into its intramolecular and intermolecular (or “crystalline”) parts (eq 2). The cell model itself provides a means for evaluFtot = F m o l + Fcryst (2) ating Fcryst only while Fmol is to be estimated from other considerations. In ref 4-9 such an approach has been employed for a number of organic crystals composed of rigid molecules. The intermolecular potential was calcu(1) A. J. Pertsin, V. V. Nauchitel, and A. I. Kitaigorodsky, Mol. Cryst. Liq. Cryst., 31, 205 (1976). (2) J. A. Barker, “Lattice Theories of the Liquid State”, Pergamon Press, Oxford, 1963. (3) A.C. Holt, W. G. Hoover, S. G. Gray, and D. R. Shortley, Physica (Utrecht),49,61 (1970). (4) A. J. Pertsin and A. I. Kitaigorodsky, Kristallografiya, 21, 587 (1976). (5) A. J. Pertain and A. I. Kitaigorodsky, Mol. Phys., 32,1781 (1976). (6) Yu. P. Ivanov, M. Yu. Antipin, A. J. Pertsin, and Yu. T. Struchkov, Mol. Cryst. Liq. Cryst., in press; (7) A. J. Pertsin, Yu. P. Ivanov, and A. I. Kitaigorodsky, Kristallografiya, 26, 115 (1981). (8)A. J. Pertsin, Yu. P. Ivanov, and A. I. Kitaigorodsky, Acta Crystallogr., in press. (9) Yu. P. Ivanov and A. J. Pertsin, presented at the 5th All-Union Symposium on Intermolecular Interactions and Conformations of Molecules, Alma-Ata, 1980, Collected Abstracts. 0022-3654/81/2085-1546$01.25/0
lated as a sum of atom-atom interactionslO of the form &) = -Ar4 B exp(-Cr) (3) It should be emphasised that the empirical parameters A, B , and C in eq 3 were taken from the literature but not specially adjusted to obtain the best agreement between calculated and experimental data. Thus, the only information needed to compute Fcryst was that on the crystal structure at the given temperature. The 6-fold integrals appearing in the expressions for thermodynamic functions (such as the integral in eq 1) were computed by using the important sampling (Monte Carlo) procedure described in full detail in ref 1, 7, and 8. The intramolecular contributions to the thermodynamic quantities were estimated from internal mode vibration frequencies taken from spectroscopic experiments. The most striking results were obtained for entropy. Table I presents the mean and maximum deviations of the calculated entropies from the corresponding observed values for a number of organic crystals examined. It is seen that the calculation reproduces the observed entropies with an average accuracy of the order of 0.1 eu. It should be also noted that the higher the temperature, the more often are correct values usually obtained, which is not surprising as the cell model is essentially a high-temperature model of the crystalline state.2 In the present paper we attempt to extend the cell-model calculations to cover molecules which contain rotating methyl groups. Such an extension is justified by the fact that the crystal field only slightly affects the internal rotation barrier peculiar to the molecule alone. This fact has been verified in this work by a direct calculation using the atom-atom potential method, which showed the crystalfield effect to be not greater than 0.1 kcal mol-l. Such a low value allows one to neglect the correlations between the internal rotation and the motion of the molecule as a whole. That is, formula 2 remains approximately valid in this case too. The possibility of evaluating Scryst,proceeding from the crystal-structure data alone is, in our opinion, of great importance. Indeed, if a calorimetric third-law value of entropy, Sbt, is available, one can easily estimate Smol as Stot- S,, The knowledge of Smol, obtained in such an independent way, may provide, in its turn, important information about internal rotation barriers, unobserved vibration frequencies, and so on.
+
(10) A. I. Kitaigorodsky, “Molecular Crystals and Molecules”, Academic Press, New York, 1973.
0 1981 American Chemical Soclety
The Journal of Physical Chemistry, Vol. 85, No. 11, 198 1
Entropy of Solid Toluene and Hexamethylbenzene
1547
TABLE I: Mean and Maximum Deviations of Calculated Entropies from Corresponding Observed Values for a Number of Organic Crystals Composed of Rigid Molecules compd
temp, K
benzene naphthalene anthracene pyrene phenantrene acenaphtene hexamethylenetetramine hexachlorobenzene
13 8-270 78-296 78-300 133-293 129-293 133-294 100- 294 124-293
sobsd
- Scalcd 1m e a n , eu
TABLE 11: Calculated and Observed Entropies of the Toluene Crystal at 165 K (eu) SCIYSt
Svib Sint rot Stot
Stat, observed
20.0 2.2 3.0‘ 25.2 24.7
Calculated for the case of free rotation. for V , = 0.75 kcal mol-‘. a
lsobsd
- Scalcd 1maxs eu
0.1 0.3 0.15 0.4 0.5 0.4 0.2 0.1
TABLE 111: Calculated and Observed Entropies of Solid Hexamethylbenzene at 323.15 K (eu) Scryst
2.5b 24.7 Calculated
Results Toluene. The cell-model calculations of the crystalline contribution to the entropy were carried out by using the crystal-structure data of Anderson et al.I1 obtained at 165 K. The empirical parameters A, B, and C for H...H, H...C, and C-.-C nonbonded atom-atom interactions were taken from ref 12. The calculations resulted in ScWt= 20.0 eu with a statistical error of 0.2 eu. The intramolecular contribution to the entropy, Smol, was calculated as the sum of a vibrational component, Svib, and that due to internal rotation, Sintrot. The former was evaluated by using the fundamental frequencies from ref 13. As to Sintrot, two limiting values were considered corresponding to the lowest and highest internal rotation barriers reported in the literature for toluene: Vo= 014and Vo = 0.75 kcal/mol.16 The corresponding Sintrot values were taken from Pitzer’s tables for the entropy of hindered rotation.16 The value of Qf;l was estimated to be 0.374. The calculation results are summarized in Table 11. As the observed value for S,, we present a third-law entropy measured in a low-temperature calorimetric study by Scott et al.14 As seen from Table 11, the calculated and observed values are in very good agreement. Hexamethylbenzene. Quite reliable calorimetric data on hexamethylbenzene (HMB) in the temperature range 13-340 K have been reported by Frankosky and Ast0n.l’ At 323.15 K the total entropy of the solid phase, Sht, was found to be 76.6 eu. The authors combined their results with the vapor pressures and heats of sublimationla to evaluate the standard entropy of the HMB vapor, S, at the same temperature. Two different values for S,, were presented corresponding to two different estimates of the heat of sublimation: AH = 17.8 and 19.6 kcal mol-’ at 323.15 K. The values of S,, along with spectroscopic data (11) M. Anderson, L. Bosio, J. Bruneaux-Poulle, and R. Fourme, J. Chin. P h y ~Phy~.-Chim. . B i ~ l .74, , 68 (1977). (12)D.E.Williams, J. Chem. Phys., 47,4680 (1967). (13)C. La Lau and R. G. Snyder, Spectrochim. Acta, Part A , 27,2073 (1971). ~~-- -, (14)D.W. Scott, G. B. Guthrie, J. F. Messerly, S. S. Todd, W. T. Berg, I. A. Hossenlopp, and J. P. McCullough, J. Phys. Chen., 66,911 (1962). (15)S. H. Hustings and D. E. Nicholson, J. Phys. Chem., 61, 730 (1957). (16)D.R. Stull, E. F. Westrum, Jr., and G. C. Sinke, “The Chemical Thermodynamics of Organic Compounds”, Wiley, New York, 1969. (17)M.Frankosky and J. G. Aston, J. Phys. Chem., 69,3126(1965). (18)S.Seki and H. Chihara, Collect. Pap. Fac. Sci., Osaka Imp. Uniu., Ser. C, 11, 1 (1950).
ref 1 4, 7 4, 7 6 7 9 9 8
0.2 0.7 0.2 0.8 1.6 0.7 0.5 0.3
smo1 Stot
Stat, observed
26.8 40.4-45.5‘ 67.2-72.3 76.6
48.0b 74.8
49.7’ 76.5
’Calculated by using A H =
17.8 and 19.6 kcal/mol.” Calculated by using A H = 20.2 kcal/mol (this work). Calculated as S a + Sintrotusing fundamental frequencies from ref 17 and V, = 2.2 kcal/mol.
were used to evaluate the barrier hindering internal rotation, V,. The limiting values found for Vowere 2.985 and 7.870 kcal mol-l. The intramolecular contribution to the entropy of solid HMB is deduced directly from S,, by subtracting the contribution to the entropy from rotational and translational motion of the molecule as a whole. The result of such a calculation is given in Table I11 (marked by footnote a).
To verify the consistency of Frankosky and Aston’s data for Stotand Smol, we have calculated the crystalline conIn deriving the unit-cell tribution to the entropy, Scryst. parameters of the crystal a t 323.15 K, we used the roomtemperature crystal-structure data reported by Brockway and Robertsonl9 and the thermal-expansion parameters measured by Woodward.20 The coordinates of hydrogen atoms in the HMB molecule were found by minimization of the potential energy of an isolated molecule, uo,as a function of six torsion angles describing orientations of the six methyl groups. The energy uo included nonbonded methyl-methyl interactions only, and no terms related to the methyl group alone. Such a model was well justified in view of the low value of the internal rotation barrier observed for t01uene.l~ The nonbonded methyl-methyl interactions were computed by using the atom-atom potential method.1° The minimum-energy conformation of the HMB molecule, found in such a way, may be depicted as e*
(for convenience, the benzene ring is shown as opened). Note that the molecule has S6symmetry, which is in accord with spectroscopic evidence.21 Using the above geometry for the HMB molecule, we performed the cell-model calculations to give S, = 26.8 eu. Addition of this value to Smol derived from 8; Frankosky and Aston datal7 resulted in Sbt much lower than the measured third-law entropy (see Table 111). The observed discrepancy stems, in our opinion, from a low accuracy of obtaining Smol by using the vapor-pressure and (19)L. 0.Brockway and J. M. Robertson, J. Chem. Soc., 24, 1324 (1939). (20)I. Woodward, Acta Crystallogr., 11,441 (1958). (21)0.Schnepp and D. S. McClure, J. Chem. Phys., 26,83 (1957).
1548
The Journal of Physical Chemistry, Vol. 85, No. 11, 1981
heat-of-sublimation data. Indeed, at 323.15 K even an error of 1-2 kcal mol-', usual for determinations of the heat of sublimation, produces an uncertainty in Smol equal to 3-6 eu. The heats of sublimation derived by Frankosky and Aston for HMBl'seem to be underestimated. Evaluation of AH,performed in this work via the cell-model calculation of the internal energy of the HMB crystal, resulted in a value of 20.2 kcal mol-', which, when used to estimate Smol, appreciably improved the results (see Table 111, marked by footnote b). The third estimate of Smol (marked by footnote c in Table 111) was obtained in a traditional way, by representing the intramolecular entropy as Svib + Sintrot. The vibrational component S,, was evaulated by using internal mode vibration frequencies listed in ref 17, except those corresponding to torsion vibrations of the methyl groups. As to evaluation of Sht rot, the situation proved to be much more complicated. For small displacements of the methyl groups from their equilibrium positions, the torsion vibration frequencies may be calculated following an approach similar to that applied by Born and KarmanZ2to vibrations of an infinite atomic chain. The following are two modifications that one should make here: (1) to replace the equation for forces by that for torques and (2) to take into account a torque depending on the displacements of the given methyl alone and not associated with the methyl-methyl interactions (this torque may arise both from the own torsion potential and from the crystal-field effect; the neglect of this torque would result in a zero vibration frequency). The final expression for torsion vibration frequencies has the form of eq 4,where CY and 0are the force constants due w = [(4asin2 aS
+
(4)
to the methyl-methyl nonbonded interaction and to the own torsion potential, respectively, I is the moment of inertia of the methyl group, and S is the wavenumber, -'I2
e s I 'I2.
Equation 4 holds for an opened infinite chain of the methyl groups, such as that shown above for the equilib(22) M. Born and Th. von Karman, Phys. Z., 13, 297 (1912).
Pertsin et al.
rium conformation of the HMB molecule. The "closure" of the chain is accomplished via introduction of periodic boundary conditions according to which the displacements of the nth and (n 6)th methyls are to be coincident. This gives rise to the following values allowed for the wavenumber: S = 0, &'I6, &'I3,lI2. The harmonic model suggested above may be in principle used to evaluate Sht rot at low temperatures, since the force constants, at least a and the contribution to /3 from the crystal field, can be estimated by using the atom-atom potentials or another method. A direct calculation of the internal rotation barrier, V,, performed by the atom-atom potential method, shows however that this model is hardly applicable to HMB a t 323.15 K. For the methyl group rotating in the field of its neighbors fixed at their equilibrium positions, the calculation yields V, = 2.2 kcal mol-'. This implies that at 323.15 K the probabilities of occupying the highest and lowest energy levels are related as exp[-Vo/(RT)]N 0.33. Clearly, with such a high probability of a 60" rotation, the displacements of the methyl from its equilibrium position cannot be regarded as being small. Instead of the harmonic model, the model of independent hindered rotation of the methyl in the field of its fixed neighbors may be accepted as an alternative. The latter model probably may be justified just in the same terms as the cell model has been, which makes a similar assumption with respect to molecular motions. Using Pitzer's tabled6 for the entropy of hindered rotation and assuming Vo = 2.2 kcal mol-l, we estimated the contribution to the entropy from the six methyls to be 16.8 eu. The resulting value for Smol is given in Table 111. The yields the total entropy almost addition of it to Scryet coincident with the measured third-law value" (see Table 111). Thus the cell model2and atom-atom potential approximation,', combined with the assumption of independent hindered rotations of the methyl groups with V, 2.2 kcal mol-l, are shown to provide a consistent high-temperature model for calculating the entropy of solid HMB. It is of interest to extend the model to other methyl-substituted benzenes. This will be the subject of our future study.
+
Acknowledgment. We thank Professor V. T. Aleksanyan for helpful discussion.
