7093
J. Phys. Chem. 1988, 92, 7093-7097
A New Method for the Estimation of Interaction Energies in Lennard-Jones Pure Liquids and Solutions A. Dejaegere,t+sM. Chessens,*M. Luhmer,? M. Bardiaux,t and J. Reisse*Tt Laboratoire de Chimie Organique E.P. (CP 165), and Unitd de Conformation de Macromoldcules Biologiques (CP 160), Universitd Libre de Bruxelles, 50, avenue F. D. Roosevelt, 1050 Bruxelles, Belgium (Received: July 16, 1987; In Final Form: May 5, 1988)
A new theoretical model is proposed for the estimation of interaction energies in pure liquids as well as in dilute binary solutions, with particular emphasis on the London and Pauli contributions to the interaction energy. Basically, interaction energies are calculated as the sum of pair interactions between two molecules, with the average over all pair interactions being effected with the aid of a simplified pair distribution function. The method is tested on a large ensemble of pure liquids and dilute organic solutions.
1. Introduction Interaction energies between solute and solvent molecules play a key role in solution chemistry. If we take into account that these energies are commonly between 5 and 15 kcal/mol for organic molecules, it is obvious that the influence of the solvent on the behavior of the solute is far from negligible. Solvent effects on reaction rates, equilibria, and spectroscopical parameters have been observed for over a century. Theoretical models for solutions have been developed’ over the same period of time. Nevertheless, it remains true that, even now, the quantitative influence of solvents on solutes is difficult to predict or even to explain in a general and convenient way when solvent and solute are “complex molecules”. By complex molecules we mean organic solutes and organic solvents. Over the past 20 years much progress has been made in the understanding of simple liquids through the development of computer simulation techniques (Monte Carlo and molecular dynamics)2-3and these have promising applications in the field of solution ~ h e m i s t r y . ~However, .~ at the present time, the sophistication of these techniques prevents their routine use in many common solute-solvent interaction problems and there is thus still a need for simpler solution models. Recently,6 we briefly described such a model which is particularly aimed at the estimation of the London-Pauli contribution to the intermolecular interaction energies in the liquid state. The salient features of the method are summarized in the next section. Numerical tests of the method are presented in section 3, and section 4 is devoted to conclusions. 2. Description of the Method This model, called PISA (pair interaction structureless approximation), is very efficient for liquids and solutions in which the intermolecular interactions are dominated by the London-Pauli contribution. It means that the interaction energy between two molecules can be described as a sum of Lennard-Jones 6-12 contributions between the i atoms (or sites) of molecule 1 and the j atoms (or sites) of molecule 27
As is usual for a Lennard-Jones liquid, the interaction energy of N molecules is considered to be pairwise additivee2 The true originality of the PISA method is related to the choice of a simplified form for ?(r,Q), the pair distribution function which determines the probability of finding two molecules in a given relative configuration. This simplified function is the following ?(r,Q) = 0
if E12(r,Q)> 0
where Norm = 3 2 n 3 is a normalization factor. H(ro) is a Heaviside step function (H(ro) = 0 if r < ro;H(r,J = 1 if r 3 ro). ro is an empirical parameter defined in the following way r~ =
rav
+ k(rmax - rmin)
(3)
where rayis the average radius of the molecules calculated from their density in the liquid state. For binary solutions, r,, is given by [rav(solute) rav(solvent)]/2. r& is the shortest intermolecular separation at which an attractive configuration occurs. rmaxis the shortest distance at which all configurations are attractive, irrespective of their orientations. Equation 3 is a series limited to its first term. rminand rmaXare calculated by using the PISA method itself, and k = 0.125 is determined by a best-fit procedure on eight nonspherical molecules (trans- and cis-decalin, toluene, o-xylene, m-xylene, p-xylene, mesitylene, and naphthalene). The k value of 0.125 is the one which minimizes the difference between the internal energy of the pure liquid (Eht)calculated by the PISA method and its experimental value deduced from its vaporization enthalpy (cf. section 3.2). As far as E12(r,Q)and ?(r,Q) can be estimated, Eint is simply given by8
+
Eint = (1/16n2)pNa”Jdr GJdQ E d r , Q ) &,Q)
(4)
where N,, is the Avogadro number, p is the number density taken where the r,Q notation stresses the radial and angular dependance of the intermolecular potential. On the other hand, the interatomic (or intersite) Lennard-Jones potentials are chosen to be spherically symmetrical. Moreover, the potential parameters cij and gi, for interaction between unlike atoms i and j are related to the parameters of like atoms by the usual mixing rules:7 e.. = ( c . . c . . ) l / 2 IJ
11
uij
=
(aii
+
Ojj)/2
(1b)
‘Laboratoire de Chimie Organique. *Unit€de Conformation de Macromoltcules Biologiques. 4 Aspirant FNRS.
0022-3654/88/2092-7093$01.50/0
(1) Claverie, P. In Intermolecular Interactions: j r o m Diatomics to Biopolymers, Pullman, B., Ed.; Wiley: New York, 1978. ( 2 ) Hansen, J. P.; McDonald, I. R. Theory Liquids; Academic: . of . Simple . . London, 1976. (3) Barker, J. A.; Henderson, D. Reu. Mod. Phys. 1976, 48, 587. (4) Fois. E. S.: Gamba, A,: Morosi. G.: Demontis, P.:Suffriti, G. B. Mol. Phys.’ 1986, 58, 65. (5) Jorgensen, W. L. J. Phys. Chern. 1983, 87, 5304. (6) Dejaegere, A.; Claessens, M.; Bardiaux, M.; Reisse, J. Bull. Soc. Chim. Belg. 1988, 97(5), 313. (7) Maitland, G. C.; Rigby, M.; Smith, E. 8.; Wakeham, W. A. Inrermolecular Forces; Oxford University Press: London, 1981. ( 8 ) Street, W. B.; Gubbins, K. E. Annu. Reu. Phys. Chem. 1977,28,373.
0 1988 American Chemical Society
7094
TABLE I: Comparison of the Internal Configurational Energies Calculated by the P E A Metbod (E-, kcal/mol) with the Values Obtained by Computer Simulations (EMraMlk kcal/m~l)~
liquid Ar
Dejaegere et al.
The Journal of Physical Chemistry, Vol. 92, No. 25, 1988
T K E M C ~ M DEPISA(2)
97 CH4 112 N2 120 F2 132 460 Cl2 Br2 556 cos 280 185 C2H6 C3H8 231 220 so2 298 cs2 CsHlo 298 neo-C5HI2 298 283 cc14 C6H6 298
-1.41 -1.73 -0.70 -0.88 -3.21 -4.02 -2.41 -3.15 -3.72 -4.82 -6.06 -6.15 -4.56 -7.80 -7.46
-1.05 f 0.02 -1.33 f 0.02 -0.54 f 0.02 -0.80 f 0.04 -3.0 f 0.1 -3.8 f 0.2 -2.3 f 0.2 -2.9 i 0.1 -3.7 f 0.1 -5.2 f 0.2 -5.9 f 0.3 -6.2 f 0.3 -3.8 f 0.2 -6.8 i 0.4 -7.6 f 0.3
BPlSA(5) ref -1.05 f 0.02 3 -1.33 f 0.02 14 -0.54 f 0.02 15 -0.80 f 0.04 15 -3.0 f 0.1 15 -3.9 f 0.2 15 -2.5 f 0.2 15 -2.9 f 0.1 16 -3.7 f 0.1 16 -5.2 f 0.2 17 -6.1 f 0.3 18 -6.3 f 0.3 16 -3.8 f 0.2 16 -6.8 f 0.4 19 -7.6 f 0.3 20
“Eplsa(2) and EplsA(5) respectively refer to calculated values obtained by using eq 2 and 5 for &$).
here as its experimental value9 r,Q are the radial and angular variables which give the relative position of two molecules. This equation is valid in the most general case of polyatomic molecules without symmetry axes. In this case, one needs five Eulerian angles’O to specify the relative orientation of two molecules; JdO stands for
r is the distance between the centers of mass of the two molecules. These centers of mass are taken as the origin of the reference frames in which the Eulerian angles are defined. Equation 4 is valid for a mole of pure liquid. Its generalization to an infinitely diluted solution of 1 mol of solute in a very large excess of solvent is straightforward. Finally, it is worth pointing out that, since see eq 2 ) is only the normalization integral (JdO e-E12(r-*)IkT: carried out for angular variables, our approximation is markedly different from a traditional Monte Carlo sampling procedure. We assume explicitly that the distribution of the molecule centers” is uniform but we do not consider that polyatomic molecules interact through a spherical p ~ t e n t i a l . ’ ~ ~ ’ ~
3. Testing the Method 3.1. Comparison ofthe PISA Results with Computer Simulation (Monte Carlo or Molecular Dynamics) Results. The method described above is intended to estimate interaction energies between molecules in pure liquids and dilute binary solutions. To begin with, we decided to analyze pure liquids which had already been studied by computer simulations (MC and MD). Provided that the interaction potential used in the computer experiments and in our method were the same, a comparison of the results obtained by using the two methods allowed a direct evaluation of the incidence of the choice of ?(r,O) on calculated interaction energies. For each system, we used the same pair potential as that used in the simulations. A comparison of the computed internal energy with the value obtained via our (hereafter referred as ElvICorMD) calculation procedure provided a direct test of the method. During this work, no liquids were retained for which an explicit elec(9) Riddick, J. A,; Bunger, W . B. Organic Soluents; Wiley: New York, 1970; Vol. 11. (IO) Margenau, H.; Murphy, G. M. The Mathematics of Physics and Chemistry; Van Nostrand: New York, 1956. (1 1 ) The correlation function for molecular centers of mass, irrespective of orientations, is obtained by integrating $(r,a) with respect to the angular variables:* ?,(r) = (1/32113)J-dn&r,O). In our model, ?,(r) reduces to a Heaviside step function. (12) Sinanoglu, 0. Ado. Chem. Phys. 1967, 12, 283. (13) Beveridge, D. L.; Kelly, M.M.; Radna, R. J. J . Am. Chem. Soc. 1974, 96,3769. (14) Verlet, L.; Weis, J.-J. Mol. Phys. 1972, 24, 1013. (15) Wojcik, M.; Gubbins, K. E.; Powles, J. G. Mol. Phys. 1982, 45, 1209.
TABLE II: Values of the Site-Site Lennard-JonesParameters Used To Build the Pair Potentials of the Molecules Listed in the Second Column u, 8, origin ref group molecule e, K 16 CH (sp3) trans-, cis-decalin 40.2 3.850 isobutane CH2 (sp3) cyclohexane 59.3 3.905 cyclopentane 16 trans-, cis-decalin 16 CH3 (sp3) toluene 80.4 3.910 isobutane 0-,m-, p-xylene
mesitylene C (arom)
toluene
50.5 3.670 isobutene’
16
55.3 3.720 benzene
20
0-,m-, p-xylene
mesitylene CH (arorn) toluene 0-, m-, p-xylene mesitylene naphthalene
“According to ref 16, the published values of e(C sp2) and u(C sp2) used in the pair potential of isobutene have been slightly modified for use in the case of an aromatic carbon. trostatic contribution had been taken into account in the simulation. Only systems where the potential law was expressed as a sum of Lennard-Jones interactions were considered, and the usual combination rules (see eq lb) were used to compute the potential energy between the i site and t h e j site of two interacting
molecule^.^ Table I summarizes the data obtained for a number of mono-, di-, and polyatomic liquids. It should be noted that for small and relatively globular molecules-so far the most frequently studied by M C and M D methods-it is not necessary to introduce the minimum approach distance ro (cf. eq 3): the use of eq 5 instead of eq 2 for &r$) does not affect the results as can be seen in Table I.
g2(r,Q) = O
if E i 2 ( r , 0 )< O
e-Elz(r,*)lkT
g?(r,Q)= Nn+m
-
if F . J r 0)
3 0 (5)
J
An analysis of the results shows that the overall agreement between M C or MD and PISA energies is good: the slope of the regression line is 0.99 with a standard error of 0.04, and this means that the probability that the slope is 1 lies between 80%and 90‘3q:I the intercept at x = 0 is -0.14 with a standard error of 0.19, and the probability that this intercept equals 0 lies between 40% and 50%; the correlation coefficient is 0.99. Since the differences between the PISA and the M C or MD energies can only be attributed to the basic approximations of our model, we can conclude that the PISA method provides a valuable estimation of the internal energy of pure liquids. However, it appears that the results obtained in the case of liquids composed of spherical molecules like argon and methane (considered here as a one-site molecule), or molecules characterized by a tetrahedrical shape like carbon tetrachloride and neopentane, are not as good as for other polyatomic liquids. 3.2. Application of the PISA Method to Pure Liquids Not Yet Studied via MC or MD Methods. In the preceding section it has been shown that, when a reliable pair potential is used, PISA permits good estimations of interaction energies in liquids. By “reliable pair potential”, we mean a potential which has been separately tested in fluid simulations (i.e., MD or MC) and which gives good energy results. In order to extend our method to new systems, we had to build suitable pair potentials for them. To build new pair potentials, we assumed that parameters t and u determined for an atom or group of atoms in a given molecule were to some extent transferable to other molecular systems. This transferability approximation is not new16,22and has been successfully used in computer simulation studies. However, since (16) Jorgensen, W. L.; Madura, J. D.; Swenson, C . J. J . Am. Chem. SOC. 1984, 106, 6638. (17) Sokolic, F.; Guissani, Y.; Guillot, B. Mol. Phys. 1985, 56, 239.
Interaction Energies in Lennard-Jones Liquids and Solutions 12 3
3 r
.. 4
W
IO
8
6 6
IO
8
12
Evap (kcalimol)
Figure 1. Comparison of internal configurational energies calculated via the PISA method (EpIsA)for different pure liquids, with the corresponding vaporization energies (Emp).The x = y straight line is only given as a reference. (1) Toluene (384 K); (2) cyclohexane (298 K); (3) p-xylene (411 K); (4) m-xylene (412 K); (5) o-xylene (418 K); (6) toluene (298 K); (7) p-xylene (298 K); (8) naphthalene (491 K); (9) rn-xylene (298 K): (10) o-xylene (298 K); (1 1) mesitylene (298 K); (12) tramdecalin (298 K); (13) cis-decalin (298 K). TABLE III: Comparison of Internal Configurational Energies for Toluene, and the Xylenes Calculated via the P E A Method (EPISA) at Their Boiling Point, with the Corresponding Vaporization Energies
O W ), liauid toluene o-xylene m-xylene p-xylene
T.K
E,"
384 418 412 41 1
7.2 8.0 7.9 7.8
-EPIPA 7.0 i 0.2 7.7 f 0.2 7.6 f 0.2 7.4 i 0.2
OExperimental vaporization enthalpies are taken from ref 9. All energies are given in kilocalories/mole. the criteria we used to recognize entities such as atoms in molecules are based on chemical intuition rather than on a rigorous and unique definition of an "atom in a molecule", this hypothesis had to be used cautiously. Whenever possible, we made an additional check of our potentials by calculating second virial coefficients. The calculated and experimental values of B2(T ) for some representative systems are given in Appendix I. The origin and values of the potential parameters used in our calculations are detailed in Table 11. Once the pair potential had been chosen, the interaction energies could be readily calculated by the PISA method. They could be compared to experimental interaction energies deduced from vaporization enthalpies by the relation In this relation it is implicitly assumed that the sum of the kinetic and vibrational energies for the liquid and the gas are the same and that the gas is ideal.ls Figure 1 present results obtained for different pure liquids which, to the best of our knowledge, have not been studied through computer experiments. From the results summarized in Figure 1, it appears that calculated values are in good agreement with the experimental vaporization energy of the pure liquid. Table 111displays some calculated values of .Empat temperatures different from the temperatures used in the best-fit procedure (see eq 3). ~
(18) Tildesley, D. J.; Madden, P. A. Mol. Phys. 1981, 42, 1137. (19) McDonald, I. R.; Bounds, D. G.; Klein, M. L. Mol. Phys. 1982, 45, 521. (20) Claessens, M.; Ferrario, M.; Ryckaert, J. P. Mol. Phys. 1983,50,217. (21) Draper, N.; Smith, H. Applied Regression Analysis; Wiley: New York, 1966. (22) Williams, D. E. J. Chem. Phys. 1967,47,4680. (23) Eley, D. D. Trans. Faraday Soc. 1939, 35, 1421.
The Journal of Physical Chemistry, Vol. 92, No. 25, 1988 7095 The agreement between calculated and experimental values is comparable to the agreement observed for the values included in the fit. 3.3. Application of the PISA Method to Binary Solutionr. In this section, the numerical results obtained on various infinitely diluted binary solutions are presented. The PISA method is designed to calculate interaction energies in solutions with the same ease as those in pure liquids. Once the site-site potentials for pure liquids A and B determined, combination rules made it easy to use the PISA equation in order to compute the interaction energy of a molecule A in a solvent composed of molecules B and vice versa. For obvious reasons, we only considered infinitely diluted binary solutions of solute A in solvent B. In order to test the PISA method, the calculated interaction energies needed to be compared with the corresponding experimental values. However, such experimental values cannot be obtained as easily as in the case of pure liquids, and we preferred to compare calculated and experimental values of the interaction enthalpy (at infinite dilution) for various solute/solvent pairs. We adopted the following prccedure to obtain experimental interaction enthalpies in the case of diluted binary solutions. It is well-known that the heat released by the solution of a gaseous solute A in a given solvent B can be formally described as the sum of two different contrib~tions.~~ The first contribution is associated with the creation in the solvent of a cavity with the same volume as the solute molecule: this term (the so-called cavity term) is related to the loss of solvent-solvent interactions and is therefore endoenergetic. The second step corresponds to the entry of the solute into the cavity, thus enabling the solute to interact with the solvent. The corresponding term associated to solute-solvent interactions is exoenergetic. Therefore, the heat of solution of 1 mol of the gaseous solute A in a large excess of solvent B (Hdi,(A-B)) is the sum of the cavity term (Hav) and the solute-solvent interaction term (Hh,). Hdiss(A-B) = -k Hint (7) In the case of a solute A which is liquid under standard conditions, Hdh(A-B) is simply obtained by subtracting the heat of vaporization from the heat of dissolution of the liquid solute A (the two being measured at the same temperature). Ha, values were calculated via the superficial tension r n e t h ~ d .This ~ ~ ~method ~~ is based on the assumption that the energy cost required to create cavities with a volume equal to that of the solvent molecules is equal to the vaporization energy of the pure solvent. For a solute molecule dissolved in a given solvent, the calculation of the cavity term requires the estimation of the volume of the cavity occupied by each solute in the solvent. In our previous publications it was demonstrated that this volume could safely be considered to be the same in all solvents, and equal to the molar volume of the pure solute in the liquid Nevertheless, when excess volumes are important, partial molar volumes must be used. The superficial tension model has been separately tested on many l i q ~ i d s and ~ ~ vgives ~ ~ reliable Ha, values. Moreover, it is based on a macroscopic approach which has nothing in common with the PISA model. We thetefore estimated experimental interaction enthalpies as Hint(expt1) = Hdiss (8) (24) Halicioglu, T.; Sinanoglu, 0. Ann. N.Y. Acad. Sci. 1969, 158, 308. (25) Moura Ramos, J. J.; Lemmers, M.; Ottinger, R.; Stien, M.-L.; Reisse, J. J . Chem. Res., Synop. 1971, 56; J . Chem. Res., Miniprint 1971, 658.
(26) Stien, M. L.; Claessens, M.; Lopez, A,; Reisse, J. J . Am. Chem. Soc.
1982, 104, 5902. (27) Moura Ramos, J. J.; Stien, M.-L.; Reisse, J. Chem. Phys. Lett. 1976, 42, 373. (28) Christensen, J. J.; Hanks, R. W.; Izatt, R. M. Handbook of Hears of Mixing; Wiley: New York, 1982. (29) Solomonov, B. N.; Konovalov, A. I.; Novikov, V. B.; Verdenikov, A. N.; Borisover, M. D.; Gorbachuk, V. V.;Antipin, I. S. Zh. Obshch. Khim. 1984, 54, 1622. (30) Moura Ramos, J. J. Ph.D. Thesis, Brussels, 1977. (31) Solomonov, B. N.; Antipin, I. S.; Novikov, V. B.; Konovalov, A. I. Zh. Obshch. Khim. 1982, 52, 2681. (32) Dumont, L.Ph.D. Thesis, Brussels, 1980.
7096 The Journal of Physical Chemistry, Vol. 92, No. 25, 1988 TABLE IV: Cakulated ( I f w f d ) ) and Experimental (H,,,)) Interaction Entealpies for Different Infinitely Diluted Binary
Dejaegere et al. TABLE V Lennard-Jones Parameters for Three Model Potentials of o-Xylene
solutions"
model
solute
solvent H,,
cs2
8.7 5.5 C6H12 CCl4 6.2 toluene 6.8 C6H 12 cs2 10.5 8.9 C6H6 1.8 cc14 toluene 8.3 naphthalene C6H6 10.8 8.6 C6H12 trans-decalin CS2 14.9 13.0 C6H6 10.5 C6H12 cis-decalin C6H6 12.6 10.2 C6H12
C6H6
HdL Hint(uptl) Hint(dd) -7.5 -16.2 -16.2 f 0.7 -12.9 -13.4 f 0.5 -7.4 -13.5 f 0.5 -14.2 -8.0 0.6 -14.6 -14.8 -8.0 -7.4 -17.5 f 0.7 -17.9 -7.1 -16.3 f 0.7 -16.0 -1 5.5 -14.7 f 0.7 -7.7 -7.4 -16.0 f 0.7 -15.7 -13.2 -23.1 f 1.1 -24.0 -11.9 -20.9 f 1.0 -20.5 -11.6 -27.2 f 1.2 -26.5 -12.0 -25.4 f 1.2 -25.0 -10.9 -23.0 f 1.0 -21.4 -23.9 -11.3 -24.6 f 1.9 -12.3 -21.8 f 1.5 -22.5
*
ref
group
28 28 29 29 30 30 30 30 29 31 32 32 28 28 28
CH-CH
c-c CH3-CH3
no. 1 2 3 1 2 3 1 2 3
K 55.3 55.3 55.3 50.5 50.5 50.5 72.9 80.4 104.1 z,
u, A
origin benzene benzene benzene isobutene isobutene isobutene neopentane isobutane ethane
3.72 3.72 3.72 3.67 3.67 3.67 3.96 3.91 3.77
L /
+.
Expcr?' hlodel I
+
hlodClZ Modcl3
Q
'The experimental values equal the difference between the dissolution enthalpy (Hdm,taken from the references) and the enthalpy of cavity formation (Ha"; see text); and the calculated ones were computed by the PISA method. All calculations were performed at 298 K. Enthalpies are expressed in kilocalories/mole.
Y = -0.66 t 1 . 0 4 ~ R = 0.99
ref 20 20 20 16 16 16 16 16 16
*
360
380
403
420
440
Temperalure ('K)
Figure 3. The second virial coefficients of o-xylene.
I 10 10
that this intercept equals 0 lies between 40% and 50%; the correlation coefficient is 0.99. Of course, the so-called experimental values of the interaction enthalpy include a cavity formation term which is, in fact, a calculated term. But the approaches used in the PISA and in the superficial tension model are so different that the results presented here can really be considered as being a positive test of both methods. For all the systems studied in Table IV, the potential law did not take into account an explicit electrostatic interaction term.
/ 20
50
- Hint exp. (Kcal/rnol)
Figure 2. Calculated (Hhttepled)) and experimental (H,t(u,,) interaction enthalpies at infinite dilution for different binary systems at 298 K. Owing to the difficulty in making reliable error estimates on the cavity term, no errors are mentioned on the abscissa. However, it should be noted that there is one, and this means that an exact picture should show error squares rather than error bars. Solute/solvent: (1) trans-decalin/carbon disulfide; (2) trans-decalinlbenzene;(3) cis-decalin/benzene; (4) trans-decalin/cyclohexane;(5) naphthalene/cyclohexane; (6) naphthalene/benzene; (7) cis-decalin/cyclohexane; (8) cyclohexane/carbon disulfide; (9) benzene/carbon disulfide; (10) cyclohexane/benzene; (1 1) cyclohexane/toluene;(1 2) benzene/toluene; (1 3) benzene/cyclohexane; (14) cyclohexane/carbon tetrachloride; (1 5 ) benzene/carbon tetrachloride.
These experimental values were then compared to the calculated ones, which, as a first approximation, could be assimilated to the energy of interaction calculated by the PISA method. Equation 7 can be applied to the dissolution of a solute A in its own liquid phase: straightforward calculations show that internal energies (as calculated so far by the PISA method) and interaction energies only differ by a factor of 2. Numerical results are given in Table IV and Figure 2 (in these calculations, the density of solvent B was taken as being equal to that of pure liquid B, and this means that we disregarded the effect of excess volume). An analysis of the results displayed in Figure 2 shows that the overall agreement between "experimental" and calculated interaction enthalpies is good: the slope of the regression line is 1.04 with a standard error of 0.04, and this means that the probability that the slope is 1 lies between 30% and 40%;*' the intercept at x = 0 is -0.66 with a standard error of 0.82, and the probability
4. Conclusions In this paper we have shown different applications of a simple theoretical model recently proposed6 to estimate solute-solvent interaction energies. In view of the results presented here, we can conclude that this method allows a reliable and rapid computation of the interaction energy in pure liquids as well as in dilute binary solutions. The method is therefore well suited to study organic liquids and solutions whose complexity prevents the study by more rigorous theoretical methods. The numerical results evidently depend on the choice of the potential parameters introduced in the computation. When these parameters are lacking it is possible to select values determined for some representative molecules and to assume the transferability of these parameters to the molecule under study. In this paper, only molecules whose interactions are described as a sum of Lennard-Jones sitesite potentials have been considered. At the present stage of this work, molecular systems cannot be handled where an explicit electrostatic contribution is included in the potential law.
Acknowledgment. We would like to thank M. Prevost and D. Van Belle from the Unite de Conformation de Macromolecules Biologiques for many invaluable discussions. M.L. and A.D. acknowledge financial support from the Institut pour 1'Encouragement de la Recherche Scientifique dans 1'Industrie et 1'Agriculture (IRSIA, Belgium) and from the Fonds National de la Recherche Scientifique (FNRS, Belgium), respectively. Appendix I Calculation of Second Virial Coefficients. As we have already stated above, we calculated second virial coefficients to test the pair potentials used in our calculations. It is a well-known fact
J. Phys. Chem. 1988, 92, 7097-7102 that the second virial coefficient of a gas is directly related to the two-body intermolecular potential1*
Bz(T) =
-(
&)Jmdr 16 p 2
3 l d Q [exp(-Elz’(r,Q)/kT) - 11
(9) where N,, is Avogadro’s number, r is the distance between the origins of the references frames of molecules 1 and 2, and Sl stands for the five Eulerian angles defining the relative orientation of 1 and 2. The supersript t on Elzindicates the true pair potential rather than some effective pairwise additive potential fitted to liquid or solid data. The six-dimensional integrals were evaluated by using the same algorithm as for the internal energy computations. The program was tested on monoatomic gases and on CS2,and our calculated values were in perfect agreement with those calculated independently in the and this serves to assess the reliability of our integration scheme. W e then calculated the second virial coefficient of 0-,m-, and p-xylene and toluene. The results obtained for o-xylene with three different potentials (see Table V) are displayed in Figure 3. When comparing experimental and calculated second virial coefficients, one must keep in mind that the pairwise additivity of intermolecular potential is not strictly valid: in condensed phases, many-body forces obviously contribute to total interaction energy. However, in a first approximation, their relatively smaller
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contribution to the energy can be incorporated in a effective way in the pair potential. The dominant contribution to these forces is thought to be the repulsive Axilrod-Teller We therefore expected our effective pair potentials to give slightly overpositive second virial coefficients, especially at low temperatures.18 It was therefore clear that model 3 was not suitable. However, it was impossible to apply this criteria to discriminate between models 1 and 2. We chose to use model 2 for self-consistency with ref 2 but it must be noted that both models give internal energies which are indistinguishable within their confidence limits. The results obtained for toluene and m- and pxylene were in all respects comparable to the results obtained for o-xylene, and we therefore used model 2 in all cases. Unfortunately, the experimental values of B2(Z‘) for organic molecules > Clo are rather scarce, and we were therefore unable to test the potentials used for naphthalene, mesitylene, and the decalins. If one is primarily concerned with energy results, second virial coefficients calculations provide a first test of pair potentials which is undoubtedly useful in the case of organic molecules when one must rely on rather crude assumptions to build pair potentials, and it can only be hoped that such experimental values will become available. (33) Axilrod, B. M.; Teller, E. J . Chem. Phys. 1943, 1 1 , 299. (34) Dymond, J. H.; Smith, E. B. Virial Coefficients of Pure Gases and Mixtures; Oxford University Press: London, 1980.
Sorption and Diffusion of Alcohols in Heteropoly Oxometalates and ZSM-5 Zeolite V. S. Nayak and J. B. Moffat* Department of Chemistry and Guelph- Waterloo Centre for Graduate Work in Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada (Received: September 29, 1987; In Final Form: May 19, 1988)
The sorption and diffusion of a series of alcohols have been measured on various microporous ammonium heteropoly oxometalates and on ZSM-5 zeolite. The sorption capacities, heats of sorption, and diffusivities have been compared and contrasted, and the factors responsible for such differences have been delineated. While in ZSM-5 the pore size restriction may be dominant, it is apparent that in the larger pore heteropoly oxometalates the electric field potential plays an important role.
Introduction The world’s oil crisis in the 1970s has led to the search for fuels from alternative sources. As coal is available in huge quantity in many parts of the world, the conversion of syngas, derived from coal, to gasoline via the formation of methanol appears to offer an attractive alternative. To this date, no single catalyst has been bound that is effective for the conversion of syngas to methanol and the conversion of methanol to gasoline. Although no efficient catalyst has been developed for the syngas to methanol process, ZSM-5 zeolite first developed in 1972 by Mobil Oil Corp.1q2has been shown to be an efficient catalyst for the methanol to gasoline/hydrocarbon process. As a consequence of its shape-selective behavior, high stability, high activity, and very high resistance to deactivation due to coking, the ZSM-5 zeolite has been gaining importance in a variety of industrially important processes such as alcohols to olefins, isomerization of xylenes, disproportionation of toluene, and alkylation of benzene and toluene. Earlier work from this laboratory has shown that 12-tungstophosphoric acid (H3PW120.& 12-tungstosilicic acid (H4SiWI2Ode), and their ammonium salts are active and selective in the conversion of methanol to c2-c6 hydrocarbon^'^ and 12(1) Argauer, R. J.; Landolt, G. R. U S . Patent 3702886, 1972. (2) Chang, C. D. Catal. Reu.-Sci. Eng. 1983, 25, 1 . (3) Hayashi, H.; Moffat, J. B. J. Catal. 1982, 77, 473. (4) Hayashi, H.; Moffat, J. B. J. Caral. 1983, 81, 61.
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molybdophosphoric acid (H3PMo12040)and its salts are active in the oxidation reactionse6 However, the ammnoium salt of 12-tungstophosphoric acid (HPW) produced larger quantities of hydrocarbons from methanol than the parent solid a ~ i d . ~ . ~ Furthermore, the hydrocarbons obtained with this salt consisted of substantially larger proportions of saturates, in contrast to the olefinic products with the parent acid. These results are similar to those observed with certain zeolites. The unique catalytic properties of ZSM-5 zeolite are attributed to both its strong acid sites and its unique channel structure of two intersecting tunnel systems with cross sections of 5-6 A. The crystal structure of ZSM-5 has as its characteristic feature a chain formed by fusing five-membered rings of corner-shared tetrahedra.’P8 Heteropoly oxometalates are ionic solids with discrete anions and cations, the former being high molecular weight cagelike structures. Of special interest in the present work are those with Keggin structure. In such structures the anion has a central atom such as, for example, phosphorus or silicon, surrounded by four oxygen atoms arranged tetrahedrally (Figure 1). Twelve oc(5) Hayashi, H.; Moffat, J. B. J . Catal. 1983, 83, 192. (6) Moffat, J. B.; Hayashi, H. Catalytic Conversion of Synthesis Gas and Alcohols to Chemicals; Herman, R. G., Ed.; Plenum: New York, 1984. (7) Chen, N. Y . ;Garwood, W. E. J . Catal. 1978, 52, 453. (8) Kokotailo, G. T.; Lawton, S . W.; Olson, D. H.; Meier, W. M. Nature (London) 1978, 272.
0 1988 American Chemical Society
