J. Phys. Chem. 1996, 100, 5781-5787
5781
Theoretical Interpretation of Kinetic Isotope Effects for the Reaction of CO2 on a Mg Surface Polona Vidmar, Antonija Lesar,* and Ivan Kobal J. Stefan Institute, P.O.B. 100, 61111 Ljubljana, SloVenia
Jozˇ e Koller Department of Chemistry and Chemical Technology, UniVersity of Ljubljana, P.O.B. 537, 61001 Ljubljana, SloVenia ReceiVed: August 29, 1995; In Final Form: January 3, 1996X
13C, 14C,
and 18O kinetic isotope effects (KIEs) in the reduction of CO2 on a Mg surface were estimated employing the Bigeleisen formalism within the framework of conventional transition-state theory. Linear and three bent four-atom configurations were considered for the transition state of the isotope-fractionation governing step of the reaction mechanism. We investigated several different decomposition modes of the transition states and the influence of diagonal force constant variations and geometry of the presumed transition state on the calculated KIEs. A comparison of the calculated results with the experimental data indicated two possible structures of the transition state. The terminal oxygen atom and the magnesium atom are in trans and cis positions, respectively, regarding the central C-O bond. The reaction coordinate motion is described by a symmetric stretching vibration in the terminal C-O and Mg-O bonds and an asymmetric stretching vibration in the central C-O bond.
Introduction Magnesium was of great interest in the past as an important reactor material. It is also known to be a potential fuel for rocket engines that could use CO2 from the atmospheres of Mars and Venus as an oxidizer. Therefore, ignition and combustion of magnesium particles in CO2 and CO2/CO mixtures were investigated.1,2 The use of solar energy for the conversion of CO2 to useful chemicals has also been investigated recently.3 Photodissociation spectroscopy studies of Mg+-CO2 ionmolecule cluster complexes have indicated that these complexes are linear,4-6 which has been confirmed also by ab initio calculations.7 Laser experiments concerning the reaction of Mg with carbon monoxide have shown that the ground-state MgO is formed from Mg(1P),8 whereas the excited-state MgO is formed from the (1S) state.9 Kinetic isotope effects (KIEs) are traditionally considered the most sensitive experimental probes for exploring the structure of the transition state for reactions at defined temperatures.10 The reaction rates of two isotopic species are different. The phenomenon is called the kinetic isotope effect. By definition it is normal when the lighter isotopic species reacts faster and the reverse in the opposite case. If the bond to the isotopesubstituted atom is formed or broken during the reaction, the related kinetic isotope effect is called a primary one, otherwise a secondary one. The present paper is concerned with the theoretical interpretation of KIEs in the reaction
CO2 + Mg f CO + MgO
(1)
and with a comparison of calculations to the available experimental data of these KIEs.11 Here, the following isotopic reactions are simultaneously running: X
Abstract published in AdVance ACS Abstracts, March 15, 1996.
0022-3654/96/20100-5781$12.00/0
k1
16
O12C16O + Mg 98 12C16O + Mg16O
16
O13C16O + Mg 98 13C16O + Mg16O
16
O14C16O + Mg 98 14C16O + Mg16O
16
O12C18O + Mg 98 12C16O + Mg18O
18
O12C16O + Mg 98 12C18O + Mg16O
k2
k3
k4
k5
(2) (3) (4) (5) (6)
where k’s stand for the overall rate constants of the isotopic reactions. The related kinetic isotope effects are denoted in this article as follows:
KIE 13C ) k1/k2
(7)
KIE 14C ) k1/k3
(8)
KIEp 18O ) k1/k4
(9)
KIEs 18O ) k1/k5
(10)
KIE 13C, KIE 14C, and intramolecular KIE 18O have been measured by the competitive technique.11 Isotope ratios of 13C and 18O and that of 14C have been determined11 by mass spectrometry and measurement of radioactivity, respectively. Of the two intramolecular oxygen isotope effects only secondary effects, KIEs 18O, have been determined. Evaluation of this effect required an account to be made of oxygen isotope exchange between CO2 and MgO observed in experiment.11 Exchange was also confirmed and extensively studied in very recent work.12 Following the Bigeleisen formalism,10,13 we calculated individual kinetic isotope effects for several assumed models of © 1996 American Chemical Society
5782 J. Phys. Chem., Vol. 100, No. 14, 1996
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TABLE 1: Normal Frequencies of CO2 Isotopic Species (cm-1) isotopic species
ω1
ω2
ω3
16O12C16O
2396.4 2328.2 2268.3 2378.6
1354.5 1354.5 1354.5 1315.4
673.0 653.6 636.7 667.6
16O13C16O 16O14C16O 16
O12C18O
the transition state and attempted to elucidate the geometry, force constants, and decomposition mode of the transition state that would satisfactorily reproduce the experimental kinetic isotope effects in the studied reaction. Theoretical Approach According to conventional transition-state theory,14,15 in the harmonic approximation, the ratio of the rate constants of two isotopic reactions is written in Bigeleisen form:10,16,17
kl ki
)
q sl siq νl,L
si sq νq l
i,L
3nq-7
∏ j)1,j*L
q q ul,j sinh(ui,j /2) q q ui,j sinh(u1,j /2)
3n-6
l,j
|GF - ΛE| ) 0
(11) sinh(ui,j/2)
(12)
where G is the Wilson matrix of kinetic energy, F is the Wilson matrix whose elements are the force constants, and E is the unit matrix. Λ is a diagonal matrix of eigenvalues with the ith element λi ) 4π2νi2, where νi is the frequency (s-1) of the ith normal mode. In solving eq 12, the condition
|F| ) 0
(13)
was imposed, thus resulting in 3nq - 7 real frequencies and one zero frequency related to the reaction coordinate. All offdiagonal elements of the F matrix were set equal to zero, with the exception of those internal coordinates involved in the reaction normal coordinate. The interaction force constant of the internal coordinates i and j was calculated by the relation
Fij ) (aij(FiiFjj)1/2
(14)
in which aij is the interaction coefficient and determines the relative magnitudes of Fij. Plus and minus signs stand for asymmetric and symmetric vibrations, respectively. Coefficients aij were derived by substitution of eq 14 into eq 13. The ratio of isotopic frequencies νqL for eq 11 was calculated by the product rule:21
q νi,L
)
( )
|Glq| 1/2 3nq-7
q νi,j
|Gq2|
q νl,j
∏
j)1,j*L
TABLE 2: Bond Lengths (pm) and Force Constants (102 Nm-1) of Reactants and Products bond
bond length
stretching force constant
bending force constant
C-O O-C-O Mg-O
113.1 116.2 174.9
18.56 15.61 3.12
0.57
ref 26 26 27
ui,j sinh(ul,j/2)
∏ j)1 u
where i ) 2, 3, 4, and 5, which refers to the heavier isotope species, while index l refers to the lighter one. Quantities marked by q refer to the transition state (TS), while those without it refer to the reactant molecules. uj is related to the wavenumber, ωj, by the relation uj ) hcωj/kBT. The symbols denote the following quantities: s, symmetry number; νqL, normal mode harmonic frequency related to the reaction coordinate; n, number of atoms in the molecule; ωj, wavenumber (cm-1) of the jth normal mode; h, Planck’s constant; c, speed of light; kB, Boltzmann’s constant; T, absolute temperature. Frequencies for CO2 isotopic molecules are collected in Table 1.18,19 Transition-state frequencies were obtained by solving the Wilson FG matrix equation:20
q νl,L
Figure 1. Models of the transition state.
(15)
The calculations were performed employing the JEDRO22 program package on a VAX 4000/VMS computer, which is Schachtschneider’s original program extensively modified.23 Modeling the Transition State The isotope-fractionation governing step of the reaction mechanism in the reduction of CO2 over a Mg surface is the step in which breaking of a C-O bond and formation of a Mg-O bond occur. Kinetic isotope effects in the overall reaction are therefore dependent on the vibrational properties of the transition state of the reaction between an Mg atom and a CO2 molecule. Thus, four models of Cs symmetry were considered for the transition state (Figure 1). Since the two oxygen atoms in the transition state are not equivalent, the oxygen atom bonded to the Mg atom is marked with an asterisk in Figure 1 and in the following text. A weak, dissociative adsorption of the CO2 molecule on the Mg surface was presumed. Regarding the products of the reaction (MgO and CO) and the assumption that the influence of other Mg surface atoms is negligible because they are more than one bond away from the reaction center,16 we employed cutoff models where only one Mg atom from the surface was included in the transition state. Configuration A in Figure 1 has its origin in the adsorption of a nearly linear CO2 molecule with its molecular axis perpendicular to the Mg surface. On the other hand, B, C, and D transition-state configurations comprise a bent CO2 molecule. The terminal oxygen and magnesium atoms are in trans and cis positions in the C and D models, respectively. Values of geometrical parameters and force constants for all models were varied in the physically reasonable range in order to simulate various possible structures for the transition state. The range of a certain parameter was estimated from its values in reactants and products. Bond lengths and force constants for reactants and products are collected in Table 2. The C-O and Mg-O bond lengths of transition states were changed by varying the corresponding stretching force constant according to the empirical relations26,27
C-O bond:
dCO ) 100(3550/fCO)1/5.79
Mg-O bond:
dMgO ) 206-63 log10(10-2fMgO)
where the lengths (di) are in picometers and the force constants (fi) in newtons/meter.
KIEs for the Reaction of CO2 on a Mg Surface
J. Phys. Chem., Vol. 100, No. 14, 1996 5783
TABLE 3: Description of Internal Coordinates and Related Force Constants coordinate index
internal coordinatea
1 2 3 4 5 6
C-O C-O* Mg-O* O-C-O* C-O*-Mg O-C-O*-Mg
s s s b b t
physical meaning
force constantb
∆(dCO) ∆(dCO*) ∆(dMgO*) ∆R ∆β ∆τ
F11 (fCO) F22 (fCO*) F33 (fMgO*) F44 (fR) F55 (fβ) F66 (fτ)
a
s, stretching; b, bending; t, torsion. b Alternative symbol in parentheses.
TABLE 4: Ranges of Force Constants (102 N m-1) Used in Calculations of KIEs for Different Models of the TS force constant
TS-A
TS-B
TS-C
TS-D
fCO FCO* fMgO* fR fβ fτ
15-18 0.01-7 0.5-3 0.1-2 0.01-0.6 0.01-0.2
12-18 1-7 0.5-3.5 0.3-3 0.1-2 0.05-0.35
12-19 1-14 0.5-3.5 0.3-5 0.1-2 0.05-0.45
12-18 0.5-7 0.3-3 0.5-3 0.1-0.8 0.1-0.3
TABLE 5: Description and Ranges of the Angles r, β, and τ (deg) for Various Models of the TS R
β
τ
models
O-C-O*
C-O*-Mg
O-C-O*-Mg
TS-A TS-B TS-C TS-D
179 100-140 100-170 100-170
179 179 100-170 100-170
0-35 0-35 90-270 0-90, 270-360
As the assumed transition states include four atoms, six internal coordinates were required for the normal coordinate analysis (the FG matrix method). The set of internal coordinates applied consisted of the three bond stretches, two angle bends, and one torsion angle. Denotation of internal coordinates, their description, and the related force constants (diagonal elements of the F matrix) are listed in Table 3. In our calculations, force constants were arbitrarily varied within the ranges given in Table 4. Both the R and β interbond angles and the dihedral angle, τ, in the transition states were varied as shown in Table 5. Using various compositions of reaction coordinates, several decomposition modes of the TS were examined. The reaction coordinate is imposed by a suitable choice of off-diagonal elements of the F matrix. They are given by eq 14, where aij is calculated from the condition |F| ) 0. Table 6 summarizes the reaction coordinates considered in this study. The descriptor contains indexes of internal coordinates involved in a certain reaction coordinate. A plus or minus sign before the index indicates an increase or decrease of the related internal coordinate with respect to its equilibrium value. Thus for example, descriptor (-1,+2,-3) represents a reaction coordinate as a simultaneous strengthening of C-O and Mg-O* bonds
and weakening of the C-O* bond. In addition, nonzero offdiagonal elements of the F matrix and aij interaction coefficients used in the calculations are presented in Table 6. Finally, rotational symmetry numbers are required for applying eq 11 in the modeling. For all isotopic transition states and the 12C16O18O reactant molecule the rotational symmetry numbers are 1. The value 2 for rotational symmetry numbers corresponds to 12C16O2, 13C16O2, and 14C16O2 reactant molecules. Results and Discussion We present here a selection of our results intended to illustrate the main conclusions drawn from these calculations. The KIEs were calculated in the temperature range 773-873 K with the exception of KIEs 18O, which was experimentally found only in the temperature range from 773 to 823 K. Then the force field, geometry, and decomposition mode of the TS that would explain the experimental data were evaluated, and the influence of geometrical parameters, force constants, and complexity of the reaction coordinate on the calculated KIEs was examined. The influence of the curvature of the potential energy surface was not studied extensively. In all calculations zero frequency was associated with motion along the reaction coordinate. However, carbon and secondary oxygen isotope effects were increased by less than 1% and decreased by less than 3% respectively, if a reaction coordinate frequency of 50 cm-1 was imposed. There was no combination of force constant values for models A and B of the TS in Table 4, which gave reasonable agreement with experiment for any of the reaction coordinates quoted in Table 6. Moreover, the KIEs obtained for these two models are inverse, which is in contradiction with experiment. On the basis of the experience from an ab initio calculation that the frequencies of CO2 for a bond angle 179° are almost the same as for a linear molecule, the same being true for the 13C and 18O isotopic shifts, one could regard model A as an approximation to a linear model. For a nearly linear molecule bending motion is no longer degenerate, and thus, in a strictly linear model this contribution to the KIEs is not accounted for. We estimated this contribution as insufficient to adjust the calculated KIEs to the experimental values. Nevertheless, further investigations are needed to clarify this point in detail. When models C or D were assumed for the transition state, better results were obtained. By definition, they differ only in their dihedral angle. In both models the influences of force field and geometry on the calculated KIEs show an expected resemblance. Variation of the reaction coordinate has the strongest effect on the calculated KIEs. The reaction coordinates used involved only stretching force constants, since their influence on KIEs is predominant. It appears from Table 6 that we generate a zero frequency associated with the reaction coordinate by setting one or two nonzero off-diagonal force constants in the F matrix. Figure 2 shows the temperature dependence of KIE 13C, KIE
TABLE 6: Reaction Coordinates Used in Modeling descriptora
abbreviationb
coefficients
description
(-1,+2) (+2,-3) (-1,+2,-3)
RC-1 RC-2 RC-3
a12 ) +1 a23 ) +1 a12 ) a23 a12 ) +1/x2 a12 ) (1/3)a23 a12 ) +1/x10 a12 ) 3a23 a12 ) +3/x10
simultaneous strengthening of the C-O bond and weakening of the C-O* bond; F12 * 0 concurrent strengthening of the Mg-O* bond and weakening of the C-O* bond; F23 * 0 simultaneous shortening of C-O and Mg-O* bonds and lengthening of the C-O* bond; different relative contribution to the reaction coordinate; F12 * 0, F23 * 0
RC-4 RC-5 a
See Table 3 for indexing of internal coordinates. b Denotations in referring to an individual reaction coordinate.
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Figure 3. Effect of simultaneous variation in angle R and β on calculated KIEs at 823 K for TS-D (τ ) 0°, reaction coordinate and values of force constants are given in Table 7). Solid lines correspond to calculated KIEs, while dashed lines, to the upper and the lower limit of the experimental data.
Figure 2. Effects of variation in reaction coordinates on calculated KIEs for TS-D: (a) KIE 13C, (b) KIE 14C, and (c) KIEs 18O (R ) β 120°, τ ) 0°, values of force constants are given in Table 7). Solid lines correspond to calculated KIEs, while dashed lines, to the upper and the lower limit of the experimental data.
and KIEs 18O for several decomposition modes of the TS model D. In these and in all the subsequent diagrams dashed lines represent the lower and the upper limit of experimental data, while solid lines stand for calculated KIEs. As seen from Figure 2, the two-element reaction coordinate RC-1 (-1,+2) gives the highest values of all the KIEs, whereas the reaction coordinate RC-2 (+2,-3) yields the lowest values for KIE 13C and KIE 14C. The former reaction coordinate consists of the C-O and C-O* internal coordinate displacements; the latter, the C-O* and Mg-O* internal coordinate displacements. Intermediate values for carbon KIEs result from the threeelement reaction coordinate (-1,+2,-3), where all three bond stretch internal coordinates are contained in it. The calculations associated with this reaction coordinate were carried out for three sets of interaction coefficients. Best results were obtained for RC-3 when the two interaction coefficients were equivalent. Dominance of the interaction force constants that couple the C-O and C-O* stretching motions in the reaction coordinate, i.e. RC-5, causes an increase of carbon KIEs. On the other hand, RC-4 involves a higher contribution of Mg-O* than the C-O contraction coupled to the C-O* stretch in the reaction coordinate. In this case, carbon KIEs become smaller. The 14C,
oxygen KIE behaves in the same way for RC-5, while RC-2, RC-3, and RC-4 give very similar values. We will not discuss the variation in reaction coordinate for model C separately because it shows a similar dependence. Thus, RC-3 reaction coordinate is appropriate for both models C and D of the transition state. Figure 3 illustrates the variation of KIEs with simultaneous change of the angles R and β for the model D transition state. We can see that angles from 110° to 150° are acceptable, so there is nothing against the “common” value of 120° for these angles. Identical values were obtained for model C. If only one of these angles (either R or β) is varied and the other is kept constant (e.g. 120°), the observed KIE dependence is similar to that in Figure 3. Values of the angle R from 115° to 130° at 823 K give suitable KIEs when β is 120°. On the other hand, β can attain any value from 100° to 130° when R is 120°. It is also evident from Figure 3 that KIEs substantially decrease when either angle increases over 150°, thus approaching the linear model A. The effect of the dihedral angle on the KIEs is very small, so a value of 180° was adopted in model C and 0° in model D. The dependence of KIEs on force constants at 823 K is presented in the diagrams of Figures 4 and 5. The transition state tested is model C, with angles R and β of 120° and the dihedral angle 180°. The decomposition mode of the transition state is described by the reaction coordinate (-1,+2,-3) with equal interaction coefficients. Diagram (a) in Figure 4 shows that the KIEs are not strongly dependent on the stretching force constant fCO. This is because the length of the C-O bond in the transition state is very close to the value in the products, so the bond slightly strengthens during decomposition. The C-O* bond in the transition state is broken during the decomposition, and thus the dependence of KIEs on the stretching force constant fCO* is most pronounced (Figure 4b). The dependence of KIEs on the stretching force constant fMgO* is also significant since the Mg-O* bond substantially shortens when the transition state splits into products (Figure 4c). It is evident from Figure 5 that bending force constants fR and fβ and the torsion force constant fτ, only slightly affect the KIEs. On the basis of the above findings model C or D may be proposed as a transition state for the investigated reaction. The force field and geometry of those TSs which most successfully describe the experimental data are reported in Table 7. The results of our calculations using these data are presented in Table 8, while Figure 6 shows the temperature dependencies of KIEs. The solid line is concerned with model C, while the dotted line stands for model D. As seen, both models successfully predict
KIEs for the Reaction of CO2 on a Mg Surface
J. Phys. Chem., Vol. 100, No. 14, 1996 5785
Figure 4. Calculated KIEs vs (a) fCO, (b) fCO*, and (c) fMgO* at 823 K for TS-C (R ) β ) 120°, τ ) 180°, reaction coordinate and values of other force constants are given in Table 7). Solid lines correspond to calculated KIEs, while dashed lines to the upper and the lower limit of the experimental data.
Figure 5. Calculated KIEs vs (a) fR, (b) fβ, and (c) fτ at 823 K for TS-C (R ) β ) 120°, τ ) 180°, reaction coordinate and values of other force constants are given in Table 7). Solid lines correspond to calculated KIEs, while dashed lines, to the upper and the lower limit of the experimental data.
KIE 13C over the whole temperature range, while small deviations from experiment at lower temperatures can be observed for KIE 14C and KIEs 18O. The disagreement of KIEs 18O is not very serious. Because of its secondary origin, it hardly deviates from unity, and therefore its determination is probably less accurate. Calculated KIEs demonstrate slight to low temperature dependencies. The temperature dependent factor of the KIE is determined by the real frequencies of the reactant molecule and those of the transition state (both products in eq 11). It seems likely that increasing the complexity of the reaction coordinate would improve the real frequencies of the TS and consequently modify the temperature dependencies of the KIEs studied. Both models C and D gave practically the same KIE 13C and KIE 14C, whereas a small discrepancy in the values of KIEs 18O exists. This difference likely originates in the different position of the terminal oxygen atom. Primary oxygen effects, KIEp 18O, are also included in Table 8. This effect is reflected in quite different values for the two models. The lack of experimental data did not allow us to check the calculated values. Nonetheless, the order of magnitude and temperature dependence of those two models are reasonable.
TABLE 7: Transition-State Parameters for Models C and D dCOa dCO* dMgO* R β τ fCOb FCO* fMgO* fR fβ fτ reaction coordinatec
TS-C
TS-D
113.6 140.3 194.9 120 120 180 17.0 5.0 1.5 1.0 0.5 0.1 RC-3
113.6 153.2 225.0 120 120 0 17.0 3.0 0.5 1.0 0.3 0.2 RC-3
a Bond lengths in picometers, angles in degrees. b Force constants in 102 N m-1. c Denotation explained in Table 6.
In our calculations, the surface was approximated by only one Mg atom because other Mg atoms are so far away from the reaction center that they do not affect the KIEs significantly. This assumption was proved to be acceptable. Namely, the effect of surrounding Mg atoms was checked through introducing the effective mass of the active Mg atom in the TS. If the
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TABLE 8: Calculated and Experimentala KIEs TS-C TS-D exptl TS-C TS-D exptl TS-C TS-D exptl
T/K
KIE 13C
KIE 14C
KIEs 18C
KIEp 18O
773
1.0221 1.0220 1.024 1.0202 1.0201 1.020 1.0186 1.0185 1.016
1.0417 1.0415 1.046 1.0382 1.0379 1.039 1.0350 1.0348 1.032
1.0029 1.0005 0.996 1.0028 1.0005 1.001 1.0028 1.0005
1.0462 1.0542
823 873
1.0445 1.0521 1.0429 1.0502
a The experimental data of ref 11 were used, and the least squares method gave the following temperature dependences of KIEs: KIE 13 C ) (1.078-7.15) × 10-5 T ((2.2 × 10-3), KIE 14C ) (1.1561.42) × 10-5 T ((3.2 × 10-3), KIEs 18O ) (0.909 + 1.12) × 10-4 T ((3.6 × 10-3).
Figure 7. Effect of variation of the dihedral angle on calculated KIEs at 823 K for the transition state with angle R and β equal to 120°, the reaction coordinate being (-1,+2,-3), a12 ) a23, and the force field of (a) TS-C and (b) TS-D from Table 7. Solid lines correspond to calculated KIEs, while dashed lines, to the upper and the lower limit of the experimental data, respectively.
bond perpendicular to the surface. From this aspect, models C and D are to a great extent similar. Both models are very product-like in character, and the same reaction coordinate was obtained in both cases. Some differences exist in the geometrical parameters and thus the vibrational properties. The C-O* and Mg-O* bond lengths are slightly longer in model D, but the two models essentially distinguish the position of the terminal oxygen atom. The C-O distance and interbond angles in models C and D are equal. This result suggests the conclusion that the energy barrier for the rotation of the terminal oxygen atom around the C-O* bond could be very small. On the basis of the KIE calculations, the above assumption was checked by calculating the KIEs as a function of the dihedral angle, which changes during the mentioned rotation. These KIE variations were found to be very small. It is evident from Figure 7 that the calculated KIEs for both configurations of the transition state lie in the limits of the experimental data at 823 K for all dihedral angles from 0° to 360°. Summary
Figure 6. Temperature dependences of (a) KIE 13C, (b) KIE 14C, and (c) KIEs 18O for TS-C (solid line) and TS-D (dotted line). Force fields and geometries of both models are given in Table 7. Dashed lines correspond to the upper and the lower limit of the experimental data, respectively.
mass of Mg was varied from 24 to 32 amu, carbon isotope effects as well the primary oxygen isotope effect were changed insignificantly, while the secondary isotope effect was increased by 0.0006. On the other hand, all the KIEs studied were changed negligibly, if another Mg atom was included in the TS. For interpretative reasons, we recall attention to the full model where the bent CO2 conformation is bound by a C-O*
Kinetic isotope effects in the reaction of CO2 with Mg were theoretically evaluated in order to provide qualitative information about the structure, vibrational properties, and decomposition mode of the transition state. The investigation presented here confirmed that the choice of reaction coordinate has the strongest influence on the calculated KIEs. A two-element reaction coordinate was too simple to reproduce simultaneously the experimental data on 13C, 14C, and 18O kinetic isotope effects in the studied reaction. KIE dependencies on the fCO* and fMgO* stretching force constants are pronounced, while variations with other force constants are smaller. A considerable dependence on angle R and β was found, too, whereas the dihedral angle hardly affects the KIEs.
KIEs for the Reaction of CO2 on a Mg Surface On the basis of classical KIE calculations we propose transition-state structures represented by model C and D; their force field and geometry are listed in Table 7. Movement along the reaction coordinate for both models is composed of all three bond stretch internal coordinate displacements. It is evident that the bond lengths and bond angles of the model C transition state are not significantly different from those of model D transition state. Nevertheless, the two structures, C and D, are essentially different regarding the position of the terminal oxygen atom. The equivalence of both models in predicting KIEs indicates that in the transition state the oxygen atom has a very small or negligible barrier for rotation around the reactive C-O* bond. At this stage one is not able to make a final selection between these two models. This feature and energetic aspects will be clarified by ab initio quantum chemical calculations, which will follow. Acknowledgment. This research was supported by Grant NO. P1-5054-0106 from the Ministry of Science and Technology, Slovenia. References and Notes (1) Shafirovich, E. Ya.; Goldshleger, U. I. Heat Mass Transfer Chem. Syst.: Proc. Int. School-Semin. 1988, 1, 37. (2) Shafirovich, E. Ya.; Goldshleger, U. I. Fiz. Goreniya VzryVa 1990, 26, 3. (3) Heminger, J. C.; Carr, R.; Somorjai, G. A. Chem. Phys. Lett. 1987, 57, 100. (4) Willey, K. F.; Yeh, C. S.; Robbins, D. L.; Duncan, M. A. Chem. Phys. Lett. 1992, 192, 179. (5) Yeh, C. S.; Willey, K. F.; Robbins, D. L.; Duncan, M. A. J. Phys. Chem. 1992, 96, 7833. (6) Yeh, C. S.; Willey, K. F.; Robbins, D. L.; Pilgrim, J. S.; Duncan, M. A. J. Chem. Phys. 1993, 98, 1867.
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