J. Phys. Chem. 1984,88, 221-232 as Figure 10 indicates, but Table V shows that the contribution of such collisions to the overall cross section for u = 1 is small. Only the rotational distributions associated with higher u such as 5 or 6 will be sensitive to C O H formation. IV. Conclusion In this paper we have used the quasiclassical trajectory method with a realistic potential energy surface to study collisional excitation in H C O and D + CO. Comparison of our results with experiment has been very good for the most part, and this has enabled us to assess in detail just what features of the potential energy surface the measured results are sensitive to. Probably the most important conclusion of this analysis is that the high u portion of the vibrational distribution is due to formation of a C O H complex which lives for a few vibrational periods. The barrier for forming this complex directly determines how large are the high u cross sections, and thus one can use the data to infer (at least crudely) the barrier height. In the present case we found
+
221
that the 1.72-eV barrier predicted by Dunning is about 0.2 eV too high. Other features of the measured vibrational and rotational distributions were found to be less sensitive to the presence of barriers or wells. For example, the average vibrational energy transfer ( AE)is largely controlled by u = 1 and 2 excitation, which primarily comes from impulsive collisions of H with C or 0. The rotational distributions were found to be decomposable into low and high j components, with much of the low j component due to collisions of H with 0, and much of the high j component from collisions of H with C.
Acknowledgment. Helpful discussions with T. H. Dunning, S. R. Leone, C. A. Wight, and R. E. Weston are gratefully acknowledged. This research was supported by NSF Grant CHE-8 115109. Registry No. Atomic hydrogen, 12385-13-6; carbon monoxide, 63008-0; COH, 71080-92-7.
A Theoretical Study of Deuterium Isotope Effects in the Reactions H,
+ CH, and H 4-
CH4 George C. Schatz,+Albert F. Wagner,* and Thomas H. Dunning, Jr. Theoretical Chemistry Group, Chemistry Division, Argonne National Laboratory, Argonne, Illinois 60439 (Received: February 3, 1983; In Final Form: June 13, 1983)
This paper presents ab initio potential surface parameters and transition state theory (TST) rate constants for the reaction Hz CH3 H + CH4, its reverse, and all the deuterium isotopic counterpartsassociated with it and its reverse. The potential surface parameters are derived from accurate POLCI calculations and include vibrational frequencies, moments of inertia, and other quantities for CH3, H2, CHI, and the H-H-CH, saddle point. TST rate constants are calculated from standard expressions and the Wigner tunneling correction. For H2 CH3 and H2 + CD,, agreement of the rate constant with experiment is good over a broad temperature range, suggesting that the calculated 10.7 kcal/mol barrier is accurate to within about 0.5 kcal/mol. Agreement with experiment for H + CH, using the calculated 13.5 kcal/mol reverse reaction barrier is poorer; a 12.5 kcal/mol barrier is found to provide a more reasonable estimate of the true barrier. Primary isotope effects for the deuterated analogues of Hz + CH, are found to be correct in magnitude at high temperature, but with a weaker temperature dependence than experiment. The calculated secondary isotope effects also appear to be weaker functions of temperature than experiment, although a large uncertainty in the experimental results precludes a quantitative assessment of errors. Our analysis of isotope effects in the H + CH4 reaction is restricted to examining the branching ratios between H and D atom abstraction in the reaction of H with the mixed species CH,D, CH2D2,and CHD,. A combination of reaction path multiplicity, favorable zero point energy shifts, and a greater likelihood of tunneling causes H atom abstraction to predominate over D atom abstraction in H + CH3D and H + CH2DZ,but for H + CHD,, we find that the H atom and D atom abstraction rate constants cross near 700 K, with H atom abstraction dominating at low temperatures and D atom at high.
+
-+
+
-
I. Introduction The reaction H2 CH3 H + CH4and its reverse have long played an important role in the theoretical and experimental development of chemical kinetics. Early studies of the potential surface for this system were done in the 1930’s,’ and by the mid-l95O’s, a detailed comparison between theory and experiment for H2 + CH, and seven isotopic counterparts had already been completed.2 Reflecting general advances in experimental methods, the rate constants for H 2 + CH3 and H CH4 have been remeasured several times in the past few years. Recent reviews have summarized the current status of the measured results for both H2 + CH33and H + CH4.4 Generally speaking, there is good agreement between the rate constant measured by different groups within the past 20 years, although it has been noted that the ratio of forward to reverse rate constant is somewhat at variance with the equilibrium constant a t low temperature^.^
+
+
Consultant. Permanent address: Department of Chemistry, Northwestern University, Evanston, IL 60201. Alfred P. Sloan Research Fellow and Camille and Henry Dreyfus Teacher-Scholar.
0022-3654/84/2088-0221$01.50/0
In this paper we present a detailed theoretical analysis of H 2 H CHI, its reverse, and their deuterated isotopic counterparts. This analysis is based on a potential surface which was recently determined by accurate configuration interaction technique^.^ An earlier analysis of the rate constant for just the H + CH4 reaction using this surface6 suggested that the barrier for H CH4 was accurate to within 1 kcal/mol. In this paper we present new a b initio results for the CH, force field which provide the information necessary to evaluate the H, CH, rate constant and isotope effects. Since the barrier height tends to
+ CH3
-+
+
+
+
(1) Gorin, E.; Kauzmann, W.; Walter, J.; Eyring, H. J. Chem. Phys. 1939, 7, 633. (2) Polanyi, J. C. J. Chem. Phys. 1955, 23, 1505. (3) Kerr, J. A.; Parsonage, M. J. “Evaluated Kinetic Data on Gas Phase Hydrogen Transfer Reactions of Methyl Radicals”; Butterworths: London,
1916.
(4) Shaw, R. J . Phys. Chem. ReJ Data 1978, 7, 1179. (5) Walch, S. P. J . Chem. Phys. 1980, 72, 4932. (6) Schatz, G.C.; Walch, S. P.; Wagner, A. F. J. Chem. Phys. 1980, 73, 4536.
0 1984 American Chemical Society
222 The Journal of Physical Chemistry, Vol. 88, No. 2, 1984
dominate absolute rate constant comparisons, our focus upon isotope effects should enable the assessment of the accuracy of other features of the surface such as saddle point frequencies and geometry. In this respect, H 2 CH3 is ideal because rate constants for many of the 16 possible deuterium isotopic combinations have been measured. Isotope effects for H CH4 are less well characterized, but we include an analysis of them here since some of the mixed deuterated systems show unusual branching ratios which should be amenable to experimental verification. These studies of isotope effects should also enable us to expand upon our earlier studies6of the influence of different vibrational modes on zero point energy shifts. Of key interest here is the relative importance of “reactive” vs. “nonreactive” modes in determining isotopically related zero point shifts. The present work will determine rate constants by using transition state theory (TST). One important question which we will address by this study is the ability of TST to describe isotope effects quantitatively. In this regard, it should be noted that recent TST studies of chlorinated hydrocarbon reactions’ using empirical potential surfaces have suggested that TST is unable to describe substituent effects on rate constants quantitatively. Furthermore, quantitative errors in calculated isotopic effects have been found in TST studies of O H H 2 (D2)s and H (D) C2H2(C2D2)9 with a b initio surfaces. Also, comparisons of TST, variational TST, and exact quantum rate constantsloindicate that TST isotope effects can be very inaccurate unless variational effects are included. Although Polanyi’s early work2 indicated that TST was effective in describing the H2 + CH3 isotope effects, more recent work by Shapiro and Weston” was less successful. Thus it will be interesting to reexamine the application of TST to this system to see how the use of more accurate potential surface information and better experimental data changes these conclusions. As mentioned, there are a total of 16 deuterium isotope variants of H2 CH,. The numbering system we will use to label them is as follows:
+
+
+
+
+
+ CH3 -% H + CH4 kl H2 + CD, H + CHD3 H2
4
HD
+ CH3
k3 +
D
+ CH4
-k H + CH3D
+ CD3
HD
k5 .-L
k6
+H D2
+ CH3
D2
D
+ CHD,
+ CD4 ki D + CH3D
-
+ CD3
ks +
D
HD
+ CD4
k14 ---.*
-
D
k13
H
+ CHD,
+ CHID2
D2 + CH2D -% D
+ CH2D2
D2 + CHD2 -% D
+ CHD3
Although rate constants for all 16 reactions and their reverses (k-l to k-16) will be presented, experimental results exist only for kl-ks, k-,, and k-,, so our analysis will concentrate mainly on these. The primary experimental results of relevance to isotope effects in CH, + H 2 are due to Shapiro and Weston” and Steacie and coworker~.’~-’~ Measurements have also been made by o t h e r ~ . ’ ~ J ~ Since almost all of these results refer to measurement of the ratio of the rate constants of reactions 1-8 to rate constants for other reactions, most of our comparisons with experiment refer to rate cosntant ratios in which the effect of the reference reaction has been factored out. This also removes the potential surface barrier from influencing the comparison between theory and experiment, and emphasizes transition state force constant information. Previous theoretical studies of H 2 CH3 H CH4 have mainly concentrated on the reverse reactions and along with the tritium analogues T + CH4 and T + CD4 (of relevance to hot atom studies). These studies include several potential energy surface calculations (reviewed and compared in ref 5 and 6) as well as trajectoryI8 and transition stateIg dynamical studies. The only previous studies of H2 CH3 include the TST work by Polanyi2 and Shapiro and Weston” and a trajectory study due to Chapman and Bunker.20 Polanyi used LEP surfaces for the transition state (based on the even earlier work of Gorin et al.’) with the methyl radical frequencies assumed not to change in going from the reagents to the saddle point. Rather poor isotope effects were obtained with his initial surface because of an unphysical well (“Lake Eyring”) present in the LEP surface. Much better results were obtained when the surface was adjusted to remove the well. Shapiro and Weston” used both LEPS and BEBO surfaces in their study, again assuming that the CH3 group force constants were identical with those in the reagents. Their resulting isotope effects were strong functions of the H-C-H (methyl) bending force constant, with certain choices of this force constant leading to ratios k l / k 7 and k3/k4 which match experiment approximately as long as tunneling was ignored. None of the surfaces developed were able to explain the measured secondary isotope effect ratios such as (ki/k7)/(kZ/ks). The trajectory study of Chapman and Bunker used an empirical surface which was not optimized to produce accurate thermal rate constants. Instead, they studied the influence of H2 vibrational excitation and CH, out-of-plane bending excitation on the H2 CH3 cross section at 25 kcal/mol translational energy. To summarize the rest of this paper, in section I1 we present ab initio results for the CH3 force field, along with the frequencies,
+
+
+
k9
---.+
+ CH2D -kH + CH2D2
-% D + CH3D Jeong, K.-M.; Kaufman, F. J . Phys. Chem. 1982,86, 1816. Isaacson, A. D.; Truhlar, D. G. J . Chem. Phys. 1982, 76, 1380. ( 9 ) Harding, L. B.; Wagner, A. F.; Bowman, J. M.; Schatz, G. C.; Christofel, K. J . Phys. Chem. 1982, 86, 4312. (10) Garrett, B. C.; Truhlar, D. G. J . Am. Chem. SOC.1980, 102,2559; Garrett, B. C.; Truhlar, D. G.; Grev, R. S. In “Potential Energy Surfaces and Dynamics Calculations”; Truhlar, D. G., Ed.; Plenum: New York, 1981; p (7) (8)
587. (11)
+ CHD,
+
+ CH2D H + CH3D H2 + CHD2 -% H + CH2D2 H2
HD
Schatz et al.
Shapiro, J. S.; Weston, R. E. J . Phys. Chem. 1972, 76,
1669.
(12) Majury, T. G.; Steacie, E. W. R. Can. J . Chem. 1952, 30, 800. (13) Whittle, E.; Steacie, E. W. R.J . Chem. Phys. 1953, 21, 933. (14) Gesser, H.; Steacie, E. W. R. Can. J . Chem. 1956, 34, 113. (15) Rebbert, R. E.; Steacie, E. W. R. Can. J . Chem. 1954, 32, 113. (16) (a) Marshall, R.M.; Shahkar, G. J . Chem. SOC.,Faraday Trans. 1 1981, 77, 2271. (b) Kobrinsky, P. C.; Pacey, P. D. Can. J . Chem. 1974, 52, 3665. (c) Clark, T. C.; Dove, J. E. Ibid. 1973, 51, 2115. (d) Benson, S.W.; Jain, D. V . S. J . Chem. Phys. 1959,31, 1008. (e) Gowenlock, B. G.; Polanyi, J. C.; Warhurst, E. Proc. R. SOC.London, Ser. A 1953, 218, 269. (17) (a) Pratt, G.; Rogers, D. J . Chem. Soc., Faraday Trans. 1 1976,72, 2769. (b) McNesby, J. R.; Gordon, A. S.; Smith, S.R. J . Am. Chem. SOC. 1956, 78, 1287. Gordon, A. S.; Smith, S. R.; McNesby, J. R., Symp. ( h i . ) Combust., [Proc.],7ih 1958, 317. (18) )!( Raff, L. M. J. Chem. Phys. 1974,60,2220. (b) Bunker, D. L.; Patterngill, M. D. Ibid. 1970. 53, 3041. Valencich, T.; Bunker, D. L. Ibid. 1974, 61, 21. (19) (a) Kurylo, M. J.; Hollinden, G. A.; Timmons, R. B. J . Chem Phys. 1970,52, 1773. (b) Carsky, P.; Zahradnik, R. In?. J . Quantum Chem. 1979, 16, 243. J . Mol. Siruci. 1979, 54, 247. Carsky, P. Collect. Czech. Chem. Commun. 1979,44, 3452. (20) Chapman, S.;Bunker, D. L. J . Chem. Phys. 1975, 62, 2890.
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984 223
Theoretical Study of Deuterium Isotope Effects TABLE I: Hartree-Fock (HF) and POLCI Total Energies for CH, (X'A,)
energy, au RcH,,
a
RcH,,
1.08 1.03 1.13 1.08 1.08 1.08
a
RcH,, a 1.08 1.08 1.08 1.08 1.08 1.08
1.08 1.08 1.08 1.08 1.08 1.08
Piz,
deg
P , 3 , deg
P,,, deg
a,deg
120 120 120 120 120 130
120 120 120 120 120 120
0 0 0
i 20 120 120 120 120 110
10 20 0
HF
-39.561 -39.566 -39.565 -39.561 -39.561 -39.564
POLCI
655 194 510 628 396 330
-39.581 941 -39.580 435 -39.519 921 -39.581 803 -39.581 265 -39.518636
and Po = 2*/3 radians or 120'. The values for kR, k,, ka, k,,, and k, are, respectively, 0.396 au (6.17 mdyn/A), 0.217 (0.946 0.00887 au (0.0387 mdyn A), 0.0191 au (0.0833 mdyn mdyn A), and -0.0510 au (-0.222 mdyn A). This force field is the minimum necessary to describe the calculated points in Table I and several comments regarding its form are important. First, the PlZB13 cross term replaces an expected [P23 - $I2 term to make explicit the constraint that by definition PI2 Pi3 P 2 3 = 27r rad (see Figure 1). Second, k,,, a quartic force constant, is necessary to fit the energies at the two different values of a listed in Table I. Experimental evidence to be discussed shortly suggests a strong quartic contribution to out-of-plane bends (Le., changing a). Third and last, the k,, term is required by the following argument. If the Hz-C-H3 angle is fixed at 120' and the C-HI bond is tilted out of the H2-C-H3 plane, the change in energy must by symmetry be the same as that produced by fixing the H3-C-H1 angle at 120' and tilting the C-Hz bond out of the H3-C-H1 plane. In the first case, the resulting H2-C-H, and H3-C-Hl angles are the same resulting in angles projected on the H2-C-H3 plane, Le., Plz and P13, that are the same and unchanged at 120". However in the second case, the resulting H2-C-H2 and H3-C-H1 angles are not the same and hence P12and P i 3 must be different and changed from 120'. Similarly, a, measured with respect to the H2-C-H3 plane, also varies between the two cases. The cross term is necessary to make the energy change equal for the two cases. The term must be at least quadratic in a because tilting the bond above or below the plane produces, by symmetry, equivalent changes in energy. The quadratic portion of the force field has been used to derive the normal-mode vibrational frequencies, zero point energies, and moments of inertia for CH,, CH2D, CHD,, and CH3. The results are listed in Table 11. The value for the umbrella mode v2 depends only on k, and is decoupled (in the normal-mode approximation) from all in-plane motion. k, in turn is primarily dependent on the a = 10' POLCI energy in Table I and that energy is only 0.000 144 au from the energy for a = 0'. Because a It 0.00010 au uncertainty from round-off and convergence errors is possible for the POLCI calculation, the consequent uncertainty in the table values of v2 are about f25 cm-'. For all the other calculated values, possible errors in frequencies due to round-off and convergence effects are insignificant. In addition to errors in k,, the large size of k,, and k,, suggest strong anharmonic contributions to v2 from both cubic and quartic terms. More calculations would be necessary to accurately determine these terms, especially k,, and the other cubic term kR,. The experimental frequencies for C H 3 are also listed in Table I1 for comparison to the calculations. The experimental information is quite incomplete and requires some discussion before the comparison to theory can be evaluated. v l , the symmetric stretch frequency, has never been directly observed but was estimated by S n e l ~ o from n ~ ~ empirical ~ considerations and his value is given in the table. The frequency of the umbrella mode, v2, has been observed by manyz2-23and the range of values is listed
w),
+ +
W Figure 1. Internal coordinates used in the force field for CH3.
geometries, and other properties associated with the reagents, saddle points, and products of the 16 reactions listed above. The resulting rate constants are presented and analyzed in section I11 for both the forward and reverse reactions, and the results are summarized in section IV. 11. Potential Energy Surface Properties Details of the a b initio calculations are described extensively in ref 5, and the resulting force fields for CH4 and CH5 are presented in ref 6. The only new a b initio calculations needed for applications to Hz CH3involve generating a force field for C H 3 and H2using the same basis set (double { plus polarization) and type of configuration interaction5 (POLCI) used previously. These are presented in subsections 1I.A and 1I.B. In order to calculate TST rate constants, we used the force fields for CH,, Hz, CH4,and CH5 to determine vibrational frequencies, moments of inertia, and zero point energies for the species appropriate to the 16 reactions listed in section 1. These are summarized in the subsections below, along with comparisons of the calculated frequencies with experiment. A . CH3. As the nonreactive C H bond orbitals in the CHI and CH5calculations5 were taken to be doubly occupied, the reference configuration for the POLCI calculations on C H 3 is just the Hartree-Fock configuration
+
1a1'22a1'z1e'4 1a / symmetry). The POLCI wave function then includes all single and double excitations relative to this configuration with the restriction that the core orbital ( lal') remain doubly occupied and that no more than one electron be in a virtual orbital (Le., an orbital other than 2al', le', and l a r ) . Calculations on CH3were done at six geometries to determine a force field. These geometries and resulting Hartree-Fock and POLCI energies are summarized in Table I. The internal coordinates used are defined in Figure 1. In the figure, PI, represents the bending angle between bonds as projected on the plane containing H2, C, and H3 and a represents the bending angle of the C-H, bond out of that plane. The POLCI energies were used to define the following force field: 3
VCH,= -39.581974
+ ~ ~ ~ R X ( R-C&H'H)' , 1-1
%k,d[P12 - Polz + [Pis - Pol2 + [PIZ- $][Pis - Pol) + '/zk,a2 + 1/2k,,a4 + '/zk,,9[Pl2 + P13 - 2P01a2 where RCHo= 2.0341 a. (compared to 2.039 experimentally2') (21)
Herzberg, G. Proc. R. SOC.London, Ser. A
1961, 262, 291.
(22) (a) Snelson, A. J . Phys. Chem. 1970, 74, 537. (b) Pacansky, J.; Bargon, J. J. Am. Chem. SOC.197597,6896. (c) Milligan, D. E.; Jacox, M. E. J . Chem. Phys. 1967,47,5146. (d) Jacox, M. E. J. Mol. Spectrosc. 1977, 66, 212.
(23) (a) Riveros, J. J. Chem. Phys. 1969,5Z, 1269. (b) Tan, L. Y.; Winer, A. M.; Pimentel, G. C. Ibid. 1972, 57, 4028. (c) Yamada, C.; Hirota, E.; Kawaguchi, K. Ibid. 1981, 75, 5256. (d) Ellison, G. B.; Engelking, P. C.; Lineberger, W. C. J. Am. Chem. SOC.1978, ZOO, 2556. (e) Hermann, H. W.; Leone, S . R. J . Chem. Phys. 1982, 76, 4166.
224
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984
Schatz et al.
TABLE 11: POLCI Harmonic Frequencies, ExperimentallyObserved Frequencies,Experimentally Derived Harmonic Frequencies, Associated Zero Point Energies (ZPE), and POLCI Moments of Inertia for CH, and Its Isotopic Variants' POLCI u , (A')
u,(A") ~3 (E)
v,(E) ZPE, kcal/mol I , , arm A'
CH, expt
cxpth (HAR)
POLCI
CH,D expt (HAR)
POLCI
CHD, expt (HAR)
CD, POLCI expt (HAR)
2361 2278 3220 3044' 3270 2456 459 630-617'-g 495-545e-gvi 427 460-507e-g9i 393 424-467e-g9i 356 3423 3162; 3150b 3297,3285 (B) 3422 (A) 3367 2561 (A13302 (B) 2563 1513 1396u 1385b 1447,1436 (B) 1503 (A) 1381 1109 (A) 1266 (B) 1118 15.99 14.26 19.37 18.2 18.9 17.69
ex])? WAR)
2153' 2269 453-463a-e 383-4 22e-g2 2381: 236gb 2436, 2424 1026,' 1029& 1056, 1059 13.5
13.8
1.75
1.75 2.27 3.50 2.85 3.50 3.50 4.60 5.77 7.01 ' Reference 22a; u 1 was not observed but calcukdted from a force field derived from the directly observed frequencies u,, u,, and u,; u 2 = 617 (CH,) and 463 (CD,). Reference 22b; u , = 612 (CH,) and 461 (CD,). Reference 22c; u , = 611 (CH,), 566 (CH,D), 518 (CHD,), and 463 (CD,). Reference 22d; u2 = 603 (CH,), 557 (CH,D), 508 (CHD,), 463 (CD,). e Reference 23b: v 2 = 607 (CH,), 561 (CH,D), and 461 (CD,); u,(HAR)= 538 (CH,), 501 (CH,D), 461 (CHD,), and 417 (CD,). Reference 23c; uz = 606 (CH,), v,(HAR)= 495 (CH,), 460 (CH D), 424 (CHD,), and 383 (CD,). Reference 23e; u , = 607 (CH,); u,(HAR) = 521 (CH,), 485 (CH,D), 446 (CHD,), dnd 404 Obtained by subtraction from the prebious column the same anharmonic correction found in ref 24 for the analogous frequency in (CD,). CH, (u2 is handled separately). Reference 23a which analyzed the experiments of ref 22c, u,(HAR) = 545 (CH,), 507 (CH,D), 467 (CHD,), and 422 (CD,). Frequencies are in cm-'.
I,, amu A' 1.75 I,, amu A* 3.50
in the table. The asymmetric stretch frequency v3 and the inplane-bend frequency v4 have been reported by Snelson22aand Pacansky et for CH, and CD3. All these experimental values contain anharmonic contributions absent in the calculations. Only in the case of v2 have these anharmonic contributions been measured24and force fields including quartic but not cubic terms have been empirically derived. The resulting range of harmonic frequencies are listed in the table with the label (HAR). Because cubic force terms were neglected in the analysis but appear to be large in the theoretical calculations, these derived empirical values may be erroneous. For vl, v3, and v4 no empirical measurements of the anharmonic contributions are available. To illustrate the possible size of the effect, we used the known anharmonic contribution~,~ in the analogous frequencies in methane to transform the experimental frequencies into the frequencies labeled (HAR) in Table 11. As Table I1 shows, the estimated harmonic values of vl, v3, and v4 are higher than the experimental value while the measured harmonic value of v, is lower than the experimental value. The agreement between theory and the harmonic experimental frequencies in Table I1 is good with a maximum difference of 130 cm-I. For the most directly measured harmonic frequency, v2, the theoretical value is low by -50 cm-' but as mentioned above the empirical results may be unreliable. It appears that the calculated values for v j and v4 are systematically too high, resulting in a zero point energy that is about 0.5 kcal/mol higher than the empirical estimates. B. H,. For consistency, a POLCI calculation on Hz was performed with the same hydrogen basis set, Le., (4slp) contracted to (2slp), used in the CH3, CH4,596and CH55,6calculations. In Table I l l we summarize the calculated and e ~ p e r i m e n t a harl~~ monic frequencies, zero point energies, and moments of inertia of H,, HD, and D,. The calculated results are little different from a similar calculation with a slightly larger basis set.26 Note that unlike the case for CH,, the calculated zero point energy is a little less than the experimental value. The maximum error in the calculated frequencies, -260 cm-', is doubIe that in CH3. This difference between theory and experiment can be significantly altered by increasing the number of calculated potential energies used to determine the fundamental frequency. As detailed in Table 111, the calculated frequency was determined from the calculated energies at three internuclear distances spaced 0.1 a,, apart and spanning the equilibrium distance. When energies are calculated at two additional distances spaced 0.1 a. beyond the
-
(24) Jones, L. H.; McDowell, R. S. J . Mol. Specrrosc. 1959, 3, 632. (25) Huber, K.; Herzberg, G. "Constants of Diatomic Molecules"; Van Nostrand: New York, 1979. (26) Schatz, G. C.; Walch, S. P. J . Chem. Phys. 1980, 72, 776.
TABLE 111: POLCI and Experimental Harmonic Frequencies, Zero Point Energies (ZPE), and Moments of Inertia for H, and Its Isotopic Variantsa,& HD D, expt expt expt POLCI (HAR) POLCI (HAR) POLCI (HAR) H2
w,,cm'' 4140 4401 3585 3813 2928 3116 ZPE, 5.92 6.29 5.13 5.45 4.19 4.45 kcal/mol I, amu A' 0.290 0.387 0.580 'The calculated potential energies in au relative to separate atoms are (-0.149441 -0.150548, -0.148 387) for distances in au (1.35, 1.45, 1.55) Experimental information is from ref 25.
'
extremes of the original three distances, then a fourth-order polynomial fit to the five calculated energies produces a fundamental frequency 140 cm-' higher than that listed in Table 111. If this fourth-order polynomial is assumed to be the exact potential energy curve, then it is easy to show that a three-energy determination of the fundamental frequency, or equivalently the quadratic force constant, for this curve is contaminated by the higher order cubic and quartic force constants. Furthermore it is also easy to show that the contamination is not just a consequence of too large a distance increment in the finite difference approximation to the quadratic force constant. For example, if the distance increment of 0.1 a. used in Table I11 were systematically reduced holding the middle distance fixed, the fundamental frequency derived from the resulting three-energy calculation would converge on a value -20 cm-' lower than that in Table I11 and even further away from the fundamental frequency of the assumed exact fourth-order polynomial. On the other hand, with the distance increment held fixed, if the middle distance were shifted to the equilibrium position of the assumed exact fourthorder polynomial (a shift of -0.02 ao), then the resulting three-energy calculation would produce a fundamental frequency 160 cm-l higher than that in Table 111and only -20 cm-I higher than the assumed exact frequency. Thus in this case anharmonic effects on a three-energy determination of the fundamental frequency is more controlled by placement, rather than increment size, of the three-point grid of distances. Despite their anharmonic contamination the results in Table I11 are used in the rest of this paper because they are the most consistent with the other calculations on CH,, CH4, and CH5. For these polyatomic systems, the calculation analogous to the fiveenergy study discussed above for H2 would be too involved and small shifts, relative expensive. However, from the analysis on Hz, to the true equilibrium, of the center of the grid of distances at which energies are calculated can result in 100-cm-I errors in the
-
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984 225
Theoretical Study of Deuterium Isotope Effects
TABLE IV: POLCI and Experimental Harmonic Frequencies, Zero Point Energies (ZPE), and Moments of Inertia for CH, and Is0topic Variant su CH, D, expt (HAR) POLCI expt (HAR) 2306 2217 2279 v i (A) 1512 (A,)1490 1477 v,(E) (A,) 1362 1363 v3 (TI 3147 3154 (A,)3029 3146 (A,)3073 3149 (E)3146 3154 (B,)3146 3153 (B,) 2350 2333 v,(T) 1405 1357 (A,)1388 1352 (A,)1082 1061 (E) 1232 1197 (B,) 1165 1126 (B,) 1309 1278 ZPE, kcal/mol 28.28 28.34 26.44 26.49 24.58 24.62 I,, ahu 8 2 3.22 3.22 3.93 3.22 4.35 4.83 I,, amu A 2 I,, amu AZ 3.22 4.35 5.54 Frequencies are in cm-’. Experimental information is from ref 24. CH,D
CH4
POLCI expt (HAR) POLCI 2977 3143 2281 1573 1573 1521
larger fundamental frequencies due to anharmonic effects. Thus although the results on H,, CH,, CH4, and CHS presented in this paper are consistent, 100-cm-l variations in the accuracy of the larger fundamental frequencies are possible due to essentially random shifts of the grid center from the unknown true equilibrium geometry. C. CH4. Table IV summarizes the harmonic frequencies, zero point energies, and moments of inertia for CH4, CH3D, CH2Dz, CHD,, and CD,. These results were derived with a force field developed in ref 6. For these molecules Jones and McDowellZ4 have analyzed the experimental gas-phase spectra in detail to isolate anharmonic effects and derive accurate harmonic frequencies. Their results are compared with ours in Table IV, and we see excellent agreement (better than 50 cm-I) between theory and experiment for all isotopes and for all the frequencies except the symmetric stretch. For that frequency the theory is low by as much as 166 cm-I. Nevertheless, the difference in zero point energies is no more than 0.1 kcal/mol. D. CH,. The saddle point of H, CH, or H C H 4 has the configuration H-H-CH, wherein the H-H-C atoms are collinear. Using a force field developed in ref 6, we have calculated the frequencies, zero point energies, and moments of inertia which are summarized in Table V. Also included is the change in zero point energy (AZPE) in going from either set of reactants (H2 + CH3 or H + CH4) to the saddle point. All 16 isotopic variants are included. An analysis of the normal-mode characteristics of the nondeuterated species and a comparison with the results of other calculations have been given previously. The calculated transition state frequencies in Table V can be correlated with those in Tables 11-IV for reactants and products. The resulting correlation is graphically presented in Figure 2 for the fully hydrogenated system H2 + CH, e H + CH4. Since the calculations for H2, CH,, CH,, and C H S are all mutually consistent, the correlations in Figure 2 are more meaningful than those given in ref 6 where only the experimental frequencies of CH, were available. Figure 2 shows that the symmetric and asymmetric stretch frequencies v, and v, in CH, evolve with only slight change (less than 20 cm-I) into v3 and v, in CHS. These two in turn drop by more than 200 cm-I to v, and u3 in CH4. The in-plane bend mode v4 in CH, increases by 20 cm-I to v6 in CH5 and by 40 cm-I more to v 2 in CH,. The out-of-plane bend u2 in CH, increases by over 500 cm-I to v I in C H 5 which in turn increases by over 400 cm-I to v4 in CH4. In addition, CH, has three modes which do not correlate to bound vibrational motion in CH,. There is a stretch mode v2 which correlates with we for H2and with the asymmetric stretch u3 in CH,. There is a methyl bending mode vs which correlates with v4 in CH4. Finally there is a bending mode u4 which evolves into free motion for either products or reactants. Correlations involving deuterated species are analogous to those in Figure 2, with specific exceptions that will be noted later (section VI). The correlation of vibrational frequencies is relevant to the question of uniform accuracy in the theoretical description of CH,,
+
+
CHD, CD4 POLCI expt (HAR) POLCI expt (HAR) 2159 2251 2105 2224 1339 1327 1112 1113 (A,)3111 (E)2349
3151 2333
2347
2333
(A,)1068 (E) 1080
1035 1063
1053
1027
22.69 5.16 5.16 6.43
22.71
20.77 6.43 6.43 6.43
20.77
48 4200 0 -
3600 0 -
5-
’ E
v
30000-
55
24000-
L i
18000-
CI
p: Fr
1200 0 -
600 0 -
00
~
I
H+CH4 H-CH4 H2tCH3 Figure 2. Correlation of calculated harmonic frequencies for CH,, Ha 4 H 4 , and CH3 + H2. H,, CH,, and CH4. The calculated and measured zero point energy of CH, are the same while for CH, the calculated energy is 0.5 kcal/mol higher and for H2 the calculated energy is 0.4 kcal/mol lower. The correlations in Figure 2 show that, in the symmetric and asymmetric stretches and in the in-plane bend, CH5 is much more like C H 3 than CH,. From these modes, the calculated zero point energy of C H S would be expected to be several tenths of a kcal/mol higher than that for the “true” transition state at the saddle point. The remaining frequencies in CH, are relatively different from analogous frequencies in reactant and products and so their contribution to any error in the zero point energy of the transition state is hard to estimate. This is especially true of v, which correlates to the vibrational frequency of Hz but is only half that frequency’s size. The zero point energy error in H2 cancels most of that of CH, but its effect in CH, cannot be estimated. Although incomplete, this analysis would suggest, if anything, that the transition state, like Hz + CH3 and unlike CH4 + H , has a calculated zero point energy that is too high by several tenths of kcal/mol. The calculated activation energy involves the difference in zero point energy between transition state and reactants. The analysis then suggests a partial cancellation of errors in the calculated activation energy for H, + CH3 H + CH4 but no cancellation and an overestimation of the activation energy for H C H 4 H2 CH,. This discussion has ignored the large effects of anharmonicity. Anharmonicity has two effects. First, as explained in regards to H2, the POLCI fundamental frequencies listed in Tables 11-V are in error from the true POLCI fundamental frequencies because the minimal grid of potential energy calculations used to determine them cannot exclude contamination by anharmonic terms. The degree of contamination is controlled not only by the size of the
-
+
-
+
226 The Journal of Physical Chemistry, Vol. 88, No. 2, 1984
Schatz et al. anharmonic force constants but also by the location of the grid of energies relative to the true equilibrium position (as calculated in the presence of anharmonic effects). Such effects in Hz produce errors on the order of 0.1-0.2 kcal/mol. Second, given true POLCI fundamental frequencies, the anharmonic constants themselves can significantly alter the observed frequencies and zero point energies. For this reaction system, such effects of anharmonicity are especially complicated for they uniformly lower the observed frequencies below the harmonic ones in CH4 but increase the observed umbrella mode frequency above the harmonic one in CH3. The use of normal modes presumes that all these anharmonic effects are the same from reactant to transition state to product, an assumption that could easily produce errors in zero point energy differences of the size of tenths of kcal/mol. Unfortunately, to do more than a normal-mode analysis on any one species in the reaction would require, for consistency, a similar description for all other species, a project beyond the scope of this current effort. To a lesser extent, the same could be said for the effect of off-diagonal quadratic terms in the force field which have not been included for any of the reaction species. Such terms may change individual normal-mode frequencies by as much as 50-100 cm-’. In going from reactants to transition state to products, the errors in the frequencies induced by the lack of off-diagonal terms will undoubtably be similar but not indentical and resulting zero point energy differences could again be tenths of kcal/mol. 111. Rate Constant Calculations Rate constants have been calculated by using the parameters in Tables 11-V and conventional transition state theory.27 Included in this evaluation is a tunneling factor which is taken to be the often used but poorly justified Wigner e x p r e s s i ~ n .For ~~ other reactions,26-28such calculations on a b initio su’rfaces have produced reasonably accurate thermal rate constants. However, the agreement has not been especially in isotope effects.8-10Unfortunately, improvements to this dynamical theory, such as variational transition state theory with Marcus-Coltrin tunneling’O or reduced dimensionality calculation^,^^ are bath more difficult to apply and require more of the potential energy surface than has been calculated. In view of the simplicity of both the dynamical theory and the description of the reactants, products, and saddle point (no anharmonic terms or harmonic cross terms in the force field), this study represents a minimal POLCI-level calculation of the rate constants for this reaction. The comparison to experimental results in this section will suggest if and where further effort a t the POLCI level would be beneficial. For the evaluation of absolute rate constants, a crucial parameter in transition state theory is the barrier height p. For H z CH3, there are two possible ways to estimate f l based on the POLCI results of ref 5. The most straightforward is simply to subtract the total electronic energy of the reactants at equilibrium from the saddle point at equilibrium. This gives p = 10.7 kcal/mol. The second method involves using the experimental reaction exoergicity to convert the calculated barrier for H CH4 into one for H2 CH3. The experimental exoergicity is 2.6 kcal/mo15 (compared to 5.2 kcal/mol actually calculated). The barrier for H CH4 was studied in ref 6 where it was concluded that although the calculated barrier was 16.9 kcal/mol, 13.5 kcal/mol is consistent with extrapolations based on basis set convergence studies. Combining these results gives = 10.9 kcal/mol. This is very close to 10.7 kcal/mol, suggesting good internal consistency, but it may be somewhat accidental given the 2.6 kcal/mol error in the reaction exoergicity. As discussed in ref 5 , it appears that this fortunate cancellation of errors is due to the fact that the error in the reaction exoergicity and that in the calculated H CH, barrier are about the same. This is
+
+
+
+
+
(27) Johnston, H. S. “Gas Phase Reaction Theory”; Ronald: New York, 1966; Chapter 8. (28) Walch, S. P.; Wagner, A. F.; Dunning, T. H.; Schatz, G. C. J. Chem.
e G h
v
a
a U
Phys. 1980, 72, 2894. (29) Lee, K. T.; Bowman, J. M.; Wagner, A. F.; Schatz, G. C. J . Chem. Phys. 1982, 76,3563, 3583.
The Journal of Physical Chemistry, Vol. 88. No. 2, 1984 227
Theoretical Study of Deuterium Isotope Effects TABLE VI: Rate Constants for H,
+ CH,
-+
H + CH, and Isotopic Variantsa In k = A ' + B' In T - C ' / T = In A ( T ) - E,(T)/kT
reaction
8.345 H, + CH, H + CH, H, + CD, -+H + CHD, 6.145 6.256 HD + CH, +D + CH, 6.407 HD + CH, -+H + CH3D 4.639 HD + CD, +D + CHD, 4.731 HD + CD, +H + CD, 7 5.895 D, + CH, -+D + CH,D 4.209 8 D, + CD, +D + CD, H, + CH,D -+ H + CH,D 7.825 9 7.285 10 H, + CHD, H + CH,D, 5.859 11 HD + CH,D H + CH,D, 5.128 HD + CH,D -+ D + CH3D 12 5.304 13 HD + CHD, -+H + CHD, 5.193 14 HD + CHD, D + CH,D, 5.323 15 D, + CH,D + D + CH,D, D, + CHD, -+ D + CHD, 4.803 16 a All rate constants are in L/(mol s), and temperatures in K. 1 2 3 4 5 6
B'
A'
-+
--f
-f
-f
1.828 2.020 1.999 1.955 2.193 2.155 2.106 2.308 1.890 1.956 2.020 2.062 2.086 2.126 2.112 2.238
5240 4908 5162 5368 4826 5027 5361 5018 5132 5018 5251 5053 5144 4942 5249 5135 Activation energy at 500 K
- C'/T
kXY+CH,/kXY+CD,
=
--
~DD+CH,/~DD+CD,
103
103 103
103 103 103
104 104 103 103 103 103 103
103
io9
109 lo8 lo8
108 lo8
109 109 109 109 108
10'
10' lo8
109 109
in kcal/mol.
10 ' 1o6 Q
kXY+CH,
p x y = --~DD+CH, k7 SXY
2.3 x 1.8 x 9.5 x 8.1 X 7.7 x 6.4 X 1.5 x 1.2 x 2.1 x 2.0 x 7.5 x 8.9 X 6.9 X 8.3 X 1.3 x 1.3 x
12.2 11.8 12.2 12.6 11.8 12.1 12.7 12.3 12.1 11.9 12.5 12.1 12.3 11.9 12.6 12.4
0
+
-
104 104
A(500)
h
with the parameters A', B', and C'given. The Arrhenius parameters at 500 K are also tabulated. For reactions 1-8, several comparisons with experiment are possible. Reference 3 presents a review of the experimental results, the most pertinent of which are in ref 11-17. Almost all these experiments3 involve measuring rates of reactions 1-8 relative to the rates of specific reference reactions of C H 3 (such as CH3 CH3COCH3 or isotopic variations). Thus the experimental evaluation of the absolute rate constants for reactions 1-8 is subject to error in both the experimental relative rate measurement and the reference rate measurement. While the available rate constants for the reference reactions have been compiled in ref 3, the effects of a reference reaction can be eliminated from the relative rates by taking a ratio. Since for experimental reasons the same reference reaction cannot be used for every one of reactions 1-8, only certain ratios can be assembled from the experimental data. Those ratios are of the form kXY+CH3
1.0 x 1.3 x 4.2 x 2.5 x 5.5 x 3.3 x 3.8 x 5.0 x 1.1 x 1.2 x 2.7 x 4.6 x 3.0 x 5.0 x 4.1 x 4.6 x
E,(500)&
1o8
consistent with the fact that the calculated transition state normal-mode frequencies and structure are more similar to CH, than they are to CH4. In all the calculations presented below, the 10.7 kcal/mol barrier is used. A. Rate Constants for H2 + CH, and Isotopic Variants. Table VI summarizes the calculated rate constants k for reactions 1-16. Included in the table is a fit of k to the function In k = A ' + B'ln T
k(500)
C'
kXY+CH3/kXY+CD, k7/k8
where XY = H H , HD, and DH. Pxy measures the magnitude of the primary isotope effect while S x y is a ratio of measures of the secondary isotope effect. In general, the two or more relative rates used in the construction of Pxy or S x y are measured at different temperatures. Thus to obtain Pxy or Sxuat specific temperatures, some interpolation of measured relative rates is required. The interpolation we used assumed an Arrhenius form for the rate constant between the two measurements closest to the desired temperature. The experimental errors associated with Pxu and Sxywere also determined by interpolation of the errors in all the measured rates involved. A fuller description of the reduction of the available experimental data to Pxy and Sxy,and also to k 7 / k 8discussed below, can be found in Appendix I. Four different collections of experimental measurements will be used in the comparison with theory: k l and k7, Pxy, k 7 / k 8 , and S X ~For . k , , k7, and k 7 / k 8 ,the directly measured relative rate constants have been multiplied by the rate constants for the reference reactions as compiled in ref 3. Since no error bars for
7 io5
i
2
1 io4 v
3
io3 16'
ld
05
1.0
1.5
2.0
2.5
IOOOK/T
.O
.,
Figure 3. Comparison of calculated and experimental reaction rate constants vs. 1/T for CH3 + H2 and CH3 + D,. For CH, + H,: +, ref
11; V, ref 14; A, ref 16a; *, ref 16b; 0,ref 16d;
For CH3 + D,: 0, ref 13; 0,ref 12; A, ref 15;
X,
ref 16c; 0, ref 16e. ref 17b; 0 , ref 17a.
the reference rate constants are listed in ref 3, the errors associated with k l , k7, and k 7 / k 8 come only from the relative rate measurements and thus are underestimated. From k l , k7, and Pxy, one can obtain absolute values for k3 and k4. From k 7 / k s and Sxy,one can obtained kXY+CH3/kXY+CD3 or a direct measure of the secondary isotope effect. With the secondary isotope effect and k l , k3, k4, and k7, one can obtain absolute values for k2, k5, k6, and k 8 . Thus reactions 1-8 will all be sampled in the comparison between experiment and theory. We begin by presenting comparisons of theoretical and experimental absolute rate constants for H2 CH, and D2 CH, or k l and k7. These are displayed in an Arrhenius plot in Figure 3. Over the range of temperatures indicated the agreement between theory and experiment is good for both isotopes although, at low temperatures, the theory is slightly low, especially for H 2 + CH,. This good agreement suggests that the estimated 10.7 kcal/mol barrier is probably accurate or perhaps up to 0.5 kcal/mol too high. Let us now consider the ratio Pxy. This is plotted in Figure 4 for XY = HH, DH, and HD, corresponding to the ratios k l / k 7 , k3/k7,and k4/k7,respectively, from Table VI. The figure indicates PDH, and PHDare 2.6, that the magnitudes of the calculated PHH, 1.3, and 0.7, respectively, at 500 K, and are slowly increasing
+
+
228
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984
Schatz et al.
fin.
1.0 A 0 4
5.0-
HH .......... DH ...................HD
A
0.9
h
T
T
I 4.0 -
i
?j0.8 0 +
Y
2
m \
2.0 l,o
f o
1
to
- I
B 0 +
3n
,
150
0s '$......04.. .........@.,, ...
0 0
0.5
,,9.d,,, . . . . . . .........n..o,?... 0
I
0.6
8
_ _ _ _ ~ _ _ _ _ _ _ ~ _ _ _ _ _ ~ _ _.._..._..._..._..._..._..._..._.... - g - - - _ . - - . n
...I ......................
0.7
I
.iI
1
STEACIE ET AL SHAPIRO ET AL
I
6'"""'
0.4 165
180
195
210
225
240
j0
5
1.65
180
Figure 4. Comparison of calculated and experimental values of Pxuvs. 1/ T. Experimental values are from ref 1 1 ( 0, 0, A), ref 13 (0 , 0),and ref 12 ( 0 ) (see Appendix I).
functions of 1/ T. The experimental results have a fair amount of uncertainty associated with them, but at 500 K, Pxyroughly equals 3.3, 1.2, and 0.7 for HH, DH, and HD, in good agreement with theory. The temperature dependence of PHD(expt) shows no significant deviations from PHD(TST), but PDH(expt)appears to increase more rapidly than PDH(TST), and P"(expt) is definitely a more rapidly increasing function of T and P"(TST). Indeed, at T = 385 K, P"(expt) N 5 while P"(TST) I! 3.2. This indicates that although the TST isotope ratio Pxyis qualitatively of the right order of magnitude, it does not have the right temperature dependence when H is the abstracted atom. The present TST results are in slightly better agreement with experiment than those of Shapiro and Weston," but the overall conclusion concerning the quantitative accuracy is the same. Note that Shapiro and Weston found that the inclusion of tunneling into TST leads to less accurate isotope effects than when it was omitted. In our calculations the inclusion of tunneling improves the agreement for XY = HH and DH, and has no effect for HD. Polanyi's calculations2 (which ignore tunneling) give Pxy = 4.2, 1.0, and 1.2 for XY = H H , DH, and HD, respectively, at 400 K. Figure 4 indicates that the HH and D H values are quite close to experiment, but that for H D is about a factor of 2 too high. The inverse primary isotope effect displayed by P H D is mainly a consequence of the reduced reaction path degeneracy in going from D2 to H D as a reactant. This is not offset by changes in zero point energy because the reactive stretch mode v2 in the transition state drops in going from H-.D-.CH3 to D.-D-CH3 by close to the same amount that we changes in going from H D to D2 (see Tables 111and V). When other frequency changes are included, AZPE is the same for reactions 4 and 7 (see Table V). Thus the activation energy for the two reactions, as measured by C'or E , (500) in Table VI, are quite close. Both P D H and PHH display regular isotope effects because, from Table V, v2 is dominated by motion of the unabstracted atom in the original diatomic molecule. Hence the change in v2 in going from D.-H--CH, to D-.D--CH3 is small resulting in an activation energy advantage for D H that, along with tunneling, overcomes the reaction path degeneracy disadvantage. The change in v2 in going from HeH-CH, to D-D-CH, is large but not large enough to overcome the change in we in going from H2 to D2. Furthermore, there is an advantage to H2 in tunneling and no disadvantage in reaction path degeneracy. Thus the isotope effects displayed by both PDH and P" are regular. In order to study secondary isotope effects, we now consider = k7/ks. Our TST value the rate constant ratio kD2+CH,/kD2+CDa of this is plotted in Figure 5, along with the experimental results. The figure does indicate that the TST isotope ratio is generally less than one, and approximately equals experiment at 500 K. In addition, the TST ratio decreases somewhat more slowly with
1.95
2.10
225
240
2.55
1000/T
1000/T
Comparison of calculated and experimental values for k 7 / k svs. 1/T. Experimental values are from ref 12 (0)and from ref 11 (V) (see Appendix I).
Figure 5.
kDD+CH3/kDD+CDj
*
rnx
075-
0 50
f
A 0 0 I
__________ HH DH
0 I
I
I
I
HD I
55
Figure 6. Comparison of calculated and experimental values of Sxyvs. 1/ T. Experimental values are from ref 1 1 ( 0 , 0, A where error bars are drawn) and from ref 12 and 13 ( 0 , 0, A) (see Appendix I).
increasing 1/ T than experiment. The more accurately known ratio of ratios Sxyis considered in Figure 6 for XY = H H , DH, and HH (corresponding to the ratios (ki / k2) / (k7/ ks) (k3/ k d / (k7/ks) and (k4/k6) / (k,/ks) respectively). The figure indicates that the TST values of SXY are very close to unity over the entire temperature range considered for all three isotopes. The experimental Sxy's show considerable scatter, with values centered around unity at high T, but showing a perceptible increase with decreasing T . Indeed, at 400 K, Sxy is 3040% above unity for all three isotopes. This inability of TST to explain the temperature dependence of Sxuis identical with what Shapiro and Weston concluded,' although our calculated ratio in Figure 4 differs from unity by more than what they indicate. The calculated secondary isotope effects are all controlled by the lower three frequencies of the transition state (uI,v4, and u 5 ) and the lowest frequency of the methyl radical (u2). These four frequencies dominate the change in AZPE in going from a C H 3 to a CD, reactant and hence control the activation energy changes. From Table V, that change is the same, a uniform drop of 0.7 kcal/mol favoring CD, over CH,, independent of isotopic substitution in H2. From Figure 2, the three other frequencies of the methyl radical hardly change in evolving into three analogous frequencies in the transition state and so their contribution to 9
9
9
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984 229
Theoretical Study of Deuterium Isotope Effects TABLE VII: Rate Constants for H
+ CH,
-+
H + CH, and Isotopic Variantsa Ink = A ' + B' In T - C ' / T = In A(n- Ea(n/kT
reaction H + CH, -+H, + CH, H+CD,+HD+CD, -6 D + CH, + HD + CH, -3 D+CD4-+D,+CD, -8 H + CH,D + H, + CH,D -9 -4 H + CH,D -+HD +CH, D + CH,D -+ HD + CH,D -12 -7 D + CH,D D, + CH, H t CH,D, + H, + CHD, -10 H t CH,D, -+ HD t CH,D -11 D + CH,D, + HD t CHD, -14 D, + CH,D D t CH,D, -15 H t CHD, H, + CD, -2 H + CHD, HD t CHD, -13 D + CHD, -+ IID + CD, -5 D + CHD, -+ D, + CHD, -16 * Using V* = 13.5 kcal/mol;all rate constants are in -1
-+
-+
-+
-+
A' B' C' 10.91 1.974 5640 13.10 1.692 6462 9.21 5157 2.169 11.31 1.898 5963 10.81 1.951 5646 11.0 1.781 6422 2.146 9.103 5162 9.217 1.986 5926 1.928 10.59 5648 1.752 11.92 6434 2.123 8.886 5167 10.14 5936 1.958 10.10 1.903 5655 12.56 1.723 6446 2.100 8.382 5169 1.929 10.78 5948 L/(mol s), and temperatures in K.
AZPE is minimal. The diatomic frequency evolves into a much different transition state frequency but what that frequency is is largely controlled by the diatomic molecule and not whether the methyl radical is CH3 or CD, (see v2 in Table V). From Figure 2, v4 and v5 transition state frequencies do not correlate with any bound motion in the reactants. Both frequencies involve bending motion of the methyl fragment bonds and either the atom being abstracted (v5) or the atom not being abstracted (v4). Thus changing from either C H 3 to CD3 or from H2to H D to D2 decreases both v4 and v5. From Table V, the changes are such that the difference in v4 v5 upon deuteration of the methyl fragment is unaffected by the diatomic reactant. On the other hand, from Figure 2, v I in the transition state correlates with v2 in the methyl radical and involves out-of-plane bends of the bonds in the methyl fragment. From Table V, v 1 involves very little motion of either atom in the diatomic molecule. In going from CH, to CD,, both v 1 and v2 decrease by the same relative amounts and hence the difference between them also decreases. This difference is unaffected by the diatomic reactant. Along with a similar effect in v4 v5, this unchanging difference with diatomic reactant produces changes in AZPE and in activation energies that also do not change with diatomic reactant resulting in nearly identical theoretical values of Sxu. The difference between the theoretical and experimental values of Sxymay be due to an incomplete theoretical description of the force field. Within the harmonic description, insufficient calculations were performed to determine the quadratic correlation between bends of the partial reactive bond and bends of the unreactive bonds of the methyl fragment. Such a term could perhaps through v4 and us introduce a net dependency of AZPE on the deuteration of both the methyl fragment and diatomic molecule. On the anharmonic level, the substantial quartic (and perhaps cubic) effects on v2 of CH, may in v l of the transition state be dependent on the diatomic molecule, again resulting in a AZPE dependent on the deuteration of the methyl fragment and the diatomic molecule. Further calculations will be required to understand these effects on the secondary isotope effect. B. Rate Constantsfor H CH4 and Isotopic Variants. Table VI1 presents the calculated TST rate constants for the 16 isotopic variants of H + C H 4 (the reverse of reactions 1-16) using the 13.5 kcal/mol barrier estimated previ~usly.~ We have been able to find only two experimental s t ~ d i e s ' ~of~ ,deuterium ~" isotope effects in H + CH4 and they refer to D + CH4. Both measurements are severely complicated by either unknown stoichiometric factors3" or poorly known reference rate ons st ants.^^^*^^ Their most reliable conclusion is that in going from H to D the activation energy over 400 to 700 K drops 0.5 f 0.3 kcal/mol, consistent with the results in Table VII. Since extensive com-
+
+
+
(30) Klein, R.; McNesby, J. R.; Scheer, M. D.; Schoen, L. J. J. Chem.
Phys. 1959, 30, 5 8 .
k(500) 1.5 x 105 4.4 x 104 2.4 x 105 7.2 x i o 4 1.1 x 105
E,(500)b A(500) 13.2 8.4 X 1O'O 14.5 9.8 X 10'O 12.4 6.3 X 1O'O 13.7 7.2 X 1Olo 13.2 6.4 X 1O'O 1.0 x 104 14.5 2.3 X 1O'O 1.8 x 105 12.4 4.8 x 1010 1.6 x 104 13.7 1.7 X 1 O l o 7.9 x 104 13.1 4.4 x 10'0 2.1 x 104 14.5 4.6 X 1 O ' O 1.3 x 105 12.4 3.2 X 1 O l o 3.4 x 104 13.7 3.5 x 10'0 4.1 x 104 13.1 2.2 x 10'0 3.2 x 104 14.5 7.1 X 10'O 6.6 X l o 4 12.4 1.7 X 1O'O 5.3 x 104 13.7 5.3 x 10'0 Activation energy at 500 K in kcal/mol
iog
z loa a, v)
I
2
10'.
-
lo6
4
1 41
io5
io4 io3
0.5
1.5
1.0
2.0
2.5
.o
~OOOK/T Figure 7. Comparison of calculated and experimental reaction rate constants vs. 1/T for H t CH4. The experimental values are from Table 111 of ref 4 (0)or from ref 31 (0).
+
parison of theoretical and experimental isotope ratios for H CH4 is not possible, we will instead consider two issues. First, how well does the TST rate constant for H CH4 compare with experiment? Second, what are the H and D atom abstraction branching ratios for the mixed systems H CH3D, CH2D2,and CHD,? The first issue will help us to assess the accuracy of the 13.5 kcal/mol barrier, while the second will explore isotope effects on TST frequency factors and zero point shifts. The comparison of the H CH4 TST rate constant with experiment is presented in Figure 7 . The experimental results in this case are those recommended by Shaw4and those of Sepehrad et aL3I The TST results have been calculated with both the Vt = 13.5 kcal/mol barrier and an alternative barrier f l = 12.5 kcal/mol. The lower barrier has been considered because the Vt = 13.5 kcal/mol results are in good agreement with experiment at high temperature, but are clearly too low at low temperature. Since the least accurate parameter in the POLCI saddle point calculations tends to be p,it is reasonable to assume that only
+
+
+
(31) Sepehrad, A.; Marshall, R. M.; Purnell, H. J . Chem. SOC.,Faraday Trans. 1 1978, 75, 835.
230 The Journal of Physical Chemistry, Vol. 88, No. 2, 1984
Schatz et al.
0.7
0.6
0.5
/---------
3
r?c
I
z\
0.4
r?c
I
0.3
0.2
0.1
0.0
I
5
1.0
I
1.5
I
I
2.5
2.0
I
3.0
0.5
1000K/T
(e-),
(-e-).
fl needs to be shifted to match experiment, and we find that fl = 12.5 kcal/mol improves the agreement substantially at low T . At high T , the fl = 12.5 kcal/mol result is slightly higher than experiment, but this may be rationalized by noting that recrossing effects tend to make TST estimates too high at high T. Thus the = 12.5 kcal/mol barrier estimate seems to do well in explaining the experimental results within the context of the simple TST presented here. If we assume that the 10.7 kcal/mol barrier that we used for H2 CH3 is correct, the fl = 12.5 kcal/mol estimate for H + C H 4 implies that the reaction exoergicity AE for H2 + C H 3 H CH4 is 1.8 kcal/mol. This compares reasonably well with the experimental estimate of 2.6 kcal/m01,~certainly within the uncertainty of our fits of rate constants to experiment. Shaw4 has suggested that the experimental value of bE might be too high by 1.2 kcal/mol since this shift is needed to make the ratio of rate constants kl/k4 agree with the equilibrium constant for this reaction. Reducing 2.6 kcal/mol by 1.2 gives 1.4 kcal/mol, which is in better agreement with our estimate of 1.8 kcal/mol. The uncertainties in our comparisons with experiment are, however, too large to enable us to distinguish with any confidence between the two experimental values of AE. Let us now consider isotope effects for the reactions H + CH,D, CH2D2,and CHD3. Ratios of the TST rate constants for these reactions to that for H CH4 are plotted in Figure 8. For each reaction there are two branches, corresponding to H or D abstraction. The ratios plotted thus correspond to k_g/k-land k4/k-1 (for H CH,D H2or HD, respectively), k-lo/k-l and k - l l / k - l (for H CHzD2 Hz or HD), and k-z/k-l and k-13/k-1(for H + CHD3 Hzor HD). Figure 8 indicates that the reactions producing the H, product have a ratio which is nearly temperature independent and simply proportional to the ratio of hydrogens on the deuterated methane to methane (Le., 3/4 for CH3D, for CH2Dz,and for CHD,). This indicates that the secondary isotope effects are determined by the reaction path degeneracy. The reactions producing H D as products have a ratio which increases with increasing temperature, with a magnitude at each temperature which is also propqrtional to the reaction path degeneracy. The results in Figure 8 are readily explained by inspection of Tables IV and V. That inspection indicates that for the sequence of reactions X + YCH,, X + YCH2D, X YCHD,, and X YCD,, AZPE drops sequentially by no more than 0.07 kcal/mol
-
+ +
+
-
1.5
2.0
2.5
0
1000K/T
Figure 8. Calculated reaction rate constants, relative to the reaction rate constant for H + CH4, vs. 1/T. Reactions represented are k4 (-), k-ll (--), k-13 k9(-*-), k-,,, (---), and k-2
+ +
1.0
--
+
+
Figure 9. Calculated reaction rate constants for H
+ CHD, vs. 1/T.
for a total drop from the beginning to the end of the sequence of no more than 0.21 kcal/mol. At the same time, the zero point energy of the sequence YCH,, YCHzD, YCHDz, and YCD3 drops sequentially by about 2 kcal/mol. Hence the shift in zero point energy associated with isotopic substitution of the methyl fragment is very nearly the same in the transition state as it is in the reactant. Changing Y from H to D causes AZPE to increase by about 1.1 kcal/mol. The results in Figure 8 are a consequence of these trends. Perhaps the most interesting result in Figure 8 refers to the branching between H atom abstraction and D atom abstraction. Clearly the reaction path degeneracy factor favors H atom abstraction for CH3D, and D atom abstraction for CHD,. At the same time, the zero point energy shifts associated with H atom abstraction is more negative than that for D atom abstraction, and this favors H atom abstraction at low temperature for all the isotopes. Tunneling similarly favors H atom abstraction. The net result of these two trends is that H atom abstraction is favored for reaction with CH,D and CHzD2between 300 and 2000 K, while for CHD3 the H and D atom abstraction rate constants cross at about 600-700 K. Figure 9 shows the Arrhenius plots for H CHD3, indicating both the H and D atom abstraction rate constants and their sum. Since the crossover between k-2 and k-13 occurs at an experimentally convenient temperature, and since the crossover temperature is a rather sensitive function of the saddle point frequencies, experimental studies of the branching ratios for H + CHD3 would provide an important test of the validity of our potential surface, and of transition state theory. The branching ratios in Figures 8 and 9 and the associated results in Tables IV and V lend themselves to further discussion. Consider the following three facts derived from the two tables. First, successive deuteration drops the calculated zero point energy of methane by about 1.84 to 1.92 kcal/mol per D atom. In other words, the drop is a constant amount, independent of the deuteration state of methane, to within 0.05 kcal/mol. Second, for partially deuterated methanes under hydrogen atom attack, the calculated change in zero point energy for H atom abstraction is lower than that for D atom abstraction by about 1.15 to 1.19 kcal/mol. In other words, H atom abstraction is favored by an amount essentially independent of deuteration. Third, the calculated change in zero point energy for H atom abstraction in H + CH4 is 0.97 kcal/mol lower than that for H CD4. This difference is only 0.2 kcal/mol different from that observed in partially deuterated methanes.
+
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The Journal of Physical Chemistry, Vol. 88, No. 2, 1984 231
Theoretical Study of Deuterium Isotope Effects These three aspects of the calculated zero point energetics for the abstraction reaction are essentially unchanged if D, rather than H , is the attacking atom. The energetics suggest that partially deuterated methanes can be thought of as a collection of independent C-D or C-H oscillators with essentially fixed zero point energy properties (at least to within 0.2 to 0.1 kcal/mol). Successive deuteration merely changes a C-H oscillator into a C-D oscillator. H or D atom attack will break either a C-H or C-D oscillator with a change in zero point energy characteristic of the oscillator attacked. Zero point energy changes favor attack of the C-H oscillator for the standard reasons given for a primary isotope effect: the drop in zero point energy from a C-H to a C-D oscillator is much larger in the strong bonds of methane than in the weak bond at the transition state, making the C-H bond in effect shallower and easier to break. In addition to describing the overall zero point energetics of abstraction, the above model is also useful in understanding how individual frequencies change during reaction. In particular consider the symmetric stretch, vl, in methane. From Table IV, upon deuteration to CH3D, that frequency drops about 700 cm-'. Further deuteration all the way to CD4 causes an additional drop of only 175 cm-'. In contrast, the triply degenerate symmetric stretch, v3, drops by no more than 130 cm-' upon deuteration to CH3D. Only upon subsequent deuteration do relatively lowfrequency asymmetric stretches appear. What has happened is that a large amplitude C-D motion can leave unchanged the center-of-mass of the partially deuterated methane when in phase with large amplitude C-H motion. Thus the symmetric stretch of all deuterated methanes contains large amplitude C-D motion and correspondingly much lower frequencies than that in CH4. However, until there are at least two C-D bonds, large amplitude C-D motion out ofphase with large amplitude C-H motion will not leave unchanged the center-of-mass. Thus the asymmetric stretch in CH3D can contain no large amplitude D motion and thus its frequency remains quite similar to that of CH,. An identical analysis can be made for C H 3 with the results in Table 11. The behavior of the symmetric and asymmetric stretch in CH3D directly bears on the abstraction dynamics. Upon H atom abstraction, one of the components of the asymmetric stretch will evolve into the reactive stretch u2 a t the transition state and on into the H2vibrational stretch in the products. The symmetric stretch will evolve into a very similar large C-D amplitude, lowfrequency symmetric stretch in the CH2D product. This is in complete accord with the frequency correlation diagram of Figure 2 for H CH4. Upon D atom abstraction, however, the correlation must change. The reactive stretch must evolve from the only reactant stretch involving large C-D motion, namely, the symmetric stretch. Both the symmetric and asymmetric stretches in the CH3 product must evolve from the reactant stretches involving little C-D motion, namely, the asymmetric stretches. Thus the two lines which cross between reactant and transition state in Figure 2 for H + CHI would have their reactant origin switched in a corresponding figure for H CH3D H D CH,. Both the calculated zero point energy changes and the dependence of the frequency correlation diagram on the branch of the abstraction suggests that a local-, rather than normal-, mode description of methane and its deuterated analogues would be a more convenient way to understand the abstraction dynamics. Such a local-mode description would characterize methane as a collection of C-H and C-D stretches and H-C-H, H-C-D, and D-C-D bends and would trace the properties of these stretches and bends as a function of successive deuteration or abstraction. While the construction of useful local-mode models is beyond the scope of this paper, the analysis presented so far does suggest several interesting experiments. Because the frequency correlation diagram changes in CH3D H depending on the branch of the abstraction reaction, vibrationally exciting the asymmetric or the symmetric stretch of CH3D would enhance only the H atom or the D atom abstraction, respectively, and thus alter the branching ratio in opposite directions.
+
+
+
-
+
IV. Conclusion The main conclusions of this paper may be summarized as follows: (a) Within the context of conventional TST, the 10.7 kcal/mol POLCI barrier for H2 + C H 3 seems accurate to within 0.5 kcal/mol, while the 13.5 kcal/mol barrier for H + CH, seems high by about 1 kcal/mol. A more accurate theoretical description of zero point energy could account for some of this overestimation. For both H2 C H 3 and H CH,, the POLCI potential parameters are accurate enough to predict rate constants with conventional TST that are of chemical accuracy. (b) The TST primary isotope effects for H2 + CH, have the right order of magnitude, but too weak a dependence on temperature. The TST secondary isotope effects also have a weaker dependence on temperature than is seen experimentally. These conclusions concerning isotope effects are similar to those previously inferred by Shapiro and Weston," although our results are in somewhat better agreement with experiment. (c) The most interesting isotope effect in the H CH, reaction CHD3, where the H atom abstraction rate is found for H constant equals the D atom abstraction rate constant at 600-700 K. This occurs because of the offsetting contributions of reaction path degeneracy factors (which favor H D formation) and zero point energy shifts and tunneling (which favor H2 formation). None of the other isotopic variants of H CH,, nor any of those of H2 CH,, show branching ratios which equal unity at temperatures easily accessible experimentally. The calculated abstraction dynamics suggest that partially deuterated methanes can be modeled as a collection of largely independent C-D and C-H oscillators, Le., a collection of local-, rather than normal-, modes. This analysis also suggests that vibrational excitation of certain frequencies in partially deuterated methanes will exclusively enhance only one abstraction branch. Our conclusions concerning isotope effects in H2 + C H 3 bear on an important and long-debated question, namely, are the errors in calculated isotope effects due to errors in the potential energy surface or in transition state theory? Previously, Shapiro and Weston analyzed errors in isotope effects based on errors in potential surface parameters. Errors in the temperature dependence of secondary isotope effects could be explained only by requiring significant coupling between vibrations in the H-H-C and CH3 parts of the transition state. However, no combination of force constants was found which could simultaneously explain both primary and secondary isotope effects. The present POLCI results do not indicate that coupling between the H-H-C and CH3 parts of the transition state is strong, but the calculated harmonic force field is not complete and anharmonic effects may be important. Further theoretical work is required. The alternative explanation of errors in the CH, + H2 isotope effects, that they are inherent in TST, has some support from other calculations. Variational transition state theory calculations,8*10 for example, indicate that the location of the optimum dividing surface often varies with isotope, leading to important contributions to isotope effects which are not included in ordinary TST. Unfortunately, it is not obvious that such variational effects will be important to secondary isotope effects when the coupling between reaction coordinates motions and CH, vibrations is weak. Tunneling also plays an important role in determining isotope effects, and it is likely that the Wigner expression used here is not quantitative. However, it is again not clear how tunneling would influence secondary isotope effects. Finally, vibrational nonadiabaticity in the reaction dynamics (which allows for reaction to occur with other than zero point energy in the modes perpendicular to the reaction coordinate) would be isotope dependent (even secondary isotope dependent). At this point it is not clear how important nonadiabatic effects are, but the present results do indicate that a treatment of the reaction dynamics at a level which does include them (e.g., classical trajectories) would be useful to understanding the H2 and CH3 reaction kinetics in better detail.
+
+
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+
+
+
Acknowledgment. George C. Schatz thanks NSF Grant
J. Phys. Chem. 1984,88, 232-238
232
CHE81-15109 for partial support. This work was performed under the auspices of the Office of Basic Energy Sciences, Division of Chemical Sciences, US.Department of Energy, under Contract W-3 1-109-Eng-38.
Appendix I As discussed in section IIIA, the experimental values of Pxu, Sxy,and k7/ks are constructed from two or more measured relative rates. The purpose of this appendix is to specifically enumerate the relative rates involved. These directly measured rates will be denoted below in brackets. As mentioned in the text, for each construction, interpolation based on the Arrhenius form was used to obtain relative rates (and their associated errors) at a common set of temperatures. The error bars on the constructed value were made from the relative error which was assumed to be the sum of the relative errors of each rate in the construction. Only ref 11 provided the necessary relative error information. For Pxy,the experimental results obtained by construction are marked by symbols in Figure 4. Those due to ref 11, denoted by symbols with error bars, are constructed as follows: P" from [kl/k7] (Le., directly measured), PDH from [k3/k4]/ ([kCH3+CH3COCH / k 4 I [kCH3+CH3COCH,Ik71 1 7 and PHD from fk4/ kC€l +CH,COCH,] ikCH3+CH3COCH3/k71 Those due to ref l 2 and 3, [ ~ ~ / ~ c H ~ + c [H~ ~~ ]/ /~ c H ~ + c by H ~Symbols ]. without error bars, are constructed as follows: P" from [kl/kCH3+CH3]/ [k7/k~H3+CH3], P D H from [kfkCH,+CH3] / [k7/kCH3+CH31 and PHD from Lk4/ ~ C H , + C H ~/] [k 7 / kcH3+CH,]. Reference 12 was used alone to provide 9
provide P" while ref 13 was used alone to provide P D H and PHD. For k7/k8,the experimental results obtained by construction are indicated by symbols in Figure 5. Those denoted by symbols with error bars are from ref 11 according to [kCD3+cD3cocD,/ Those denoted by symbols k21/ Ik8/k21/ [kCH3+CH$OCH,/k71* without error bars are from ref 12 according to [ k c ~ , + c ~ , / k + / [ k c ~ , + c ~ ~ / k 7 As ] . mentioned in the text, the reference reaction rate constants needed to reduce these constructions to the values in Figure 5 are all taken from ref 3. For Sxy,the experimental results obtained by construction are indicated by symbols in Figure 6 . Those denoted by symbols with error bars are from ref 11 and are obtained as follows. For S H D and SDH, the required ratios (see text for the definition of Sxy) k3/k59 k 7 / k 8 9 and k l / k 2 are from ([k3/k41/[kCH,+CH3COCH 1 k411[kCD3+CD3COCDj/k51? ([kCD,+cD,COCD,/kzl/[k8/kzlf/ [kCH3+CH,COCH,/k71, and ( [ k C D , + C D , C O C D , / k 5 1 / [ k 6 / k 5 1 ) /
[ ~ c H ~ ~ c H , c o c Hrespectively. ,/~~], For S", a more direct construction is available, from [kl/k7][k8/k2]. The experimental results denoted by symbols without error bars are from ref 12 and 13 and are obtained as follows. Reference 12 was used alone to obtain SHH from [kl/kCH3+CH,l/[k2/kCD3+CD31/([k7/ kCH3+CH,l[k8/kCD3+CD,I). For SHD and S D H , ref l 2 provided [ k s / k c ~ ~ + cwhile ~ ~ ] ref 13 provided all the remaining relative rates in the form [k7/kCH3+CH,l [k3/kCH3+CH,], [k5/kCD3+CD31 [ k 4 / kCH3+CH31, and [k6/kCDj+CD31. Registry No. Hydrogen, 1333-74-0;methyl, 2229-07-4;deuterium, 9
9
7782-39-0.
Molecular Statlstical Theory of Adsorption for Hydrocarbons on Graphite. Effect of Polarizability Anisotropy in Adsorption Potential Calculations Claire Vidal-Madjar* and Erika Bekassy-Molnart Laboratoire de Chimie Analytique Physique, Ecole Polytechnique, 91 128 Palaiseau Cedex, France (Received: November 29, 1982; In Final Form: June 21, 1983)
The molecular statistical theory of adsorption taking into account the high anisotropic polarizability tensor of graphite has been applied to the calculation of the second adsorbate/surface virial coefficient and the isosteric heat of adsorption of various hydrocarbons on graphite for zero surface coverage. The anisotropic adsorption potential model carried out for methane and benzene adsorbed on the basal (0001) graphite surface predicts values which are in good agreement with the results of gas-chromatographicand static-adsorptionmeasurements on graphitized carbon blacks. For methane, the C..C distance and the vibration frequencies perpendicular to the graphite surface, calculated at 0 K, are close to the structural data of the adsorbed layer as measured from neutron scattering techniques at low temperature. The major role of the adsorbate polarizability in adsorption potential calculations is demonstrated in the prediction of naphthalene, phenanthrene, and anthracene relative adsorption on graphite.
Introduction The development of the molecular statistical theory of adsorption based on the atom-atom approximation for the potential energy permits the prediction of the thermodynamic functions of adsorption on Comparison with adsorption experiments obtained on graphitized carbon black is valid as its fairly homogeneous surface exposes the basal face of graphite. It is the dispersion forces which account for most of the interaction energy of nonpolar hydrocarbon molecules on graphite and the models consider a n adsorption potential which is an additive function of the adsorbate-adsorbent interactions. The methods based on a priori c a l c ~ l a t i o n s ~determine -~ the attractive constant from the Kirkwood-Muller equations-' or from combination rules derived from the self-interaction atom-atom potentials.lO-ll As too large values for the attractive constant are Present address: Department of Chemical Engineering, Technical University of Budapest, 1521 Budapest, Hungary.
0022-3654/84/2088-0232$01.50/0
~ b t a i n e d ~from - ~ a priori calculations, more empirical models are which adjust the constants of the adsorption potential (1) W.A. Steele,"The Interaction of Gases with Solid Surfaces"; Pergamon Press: Oxford, 1974. (2) D. P. Poshkus, Discuss. Faraday Soc., 40, 195 (1965). (3) A. V. Kiselev and D. P. Poshkus, J. Chem. Soc., Faraday Trans. 2,72, 950 (1976). (4) A. V. Kiselev and D. P. Poshkus, Tram. Faraday Soc., 59,428 (1963). ( 5 ) A. V. Kiselev, D. P. Poshkus, and A. Y . Afreimovich, Rum. J . Phys. Chem. (Engl. Transl.), 42, 1345, 1348 (1968). (6) C . Vidal-Madjar and G. Guiochon, Bull. SOC.Chim. Fr., 3105,3110 (1971). (7) C. Vidal-Madjar, M. F. Gonnord, and G. Guiochon, J . Colloid Interface Sci., 52, 102 (1975). ( 8 ) J. G. Kirkwood, Phys. Z . , 33, 57 (1932). (9) A. Muller, Proc. R. SOC.London, Ser. A , 154,624 (1936). (10)M. La1 and D. Spencer, J . Chem. SOC.,Faraday Trans. 2,70, 910 (1974). (1 1) L.Battezzati, C.Pisani, and F. Ricca, J. Chem. Soc., Faraday Trans. 2,71, 1629 (1975).
0 1984 American Chemical Society
