Efficient Ethanol Fuel Production: A Study on Ethanol Decomposition Reactions, Study notes of Chemical Kinetics

Download Efficient Ethanol Fuel Production: A Study on Ethanol Decomposition Reactions and more Study notes Chemical Kinetics in PDF only on Docsity! 1 Experimental and Numerical Studies of Ethanol Decomposition Reactions Juan Li, Andrei Kazakov, and Frederick L. Dryer* Department of Mechanical & Aerospace Engineering Princeton University Princeton, NJ 08544 juanli@princeton.edu ABSTRACT. Ethanol pyrolysis experiments at 1.7 – 3.0 atm and 1045 – 1080 K were performed in the presence of radical trappers using a variable pressure flow reactor. Stable species time histories were determined using continuous sampling, on-line Fourier Transform Infrared Spectrometry, and off-line Gas Chromatography. The rate constant k1 of the molecular decomposition reaction, C2H5OH → C2H4 + H2O (R1), was determined experimentally. The obtained result agrees very well with extrapolation of the recent shock tube data of Herzler et al.1 The multi-channel unimolecular decomposition of ethanol was also investigated theoretically based upon RRKM/master equation calculations. The effects of the hindered rotations in C2H5OH and quantum tunneling on the molecular decomposition reaction were taken into account. The reaction (R1) was found to be strongly dependent on temperature and the dominant channel over the range of temperatures from 300 to 2500 K at 1 atm. The calculated k1 is in excellent agreement with the recent theoretical work of Tsang2 as well as with the experimental measurements of Herzler et al.1 and the present data. The influence of tunneling on the shape of the falloff is discussed. In addition, the RRKM/master 2 equation results were fit to modified Arrhenius expressions to facilitate chemical kinetic modeling applications of the results. INTRODUCTION. Ethanol (C2H5OH) is a very important energy carrier that can be produced from renewable energy resources. It can be used as a fuel extender, octane enhancer, and oxygen- additive in, or as an alternative, neat fuel to replace reformulated gasoline. Ethanol also has potential as a hydrogen carrier for fuel cell applications. The 1990 Clean Air Act Amendments3 presently require the addition of oxygenates to reformulated gasoline, with seasonal adjustments, on the premise that oxygen content decreases automotive emissions, particularly smog generation participants and CO. Ethanol is favored to replace methyl tertiary butyl ether (MTBE), another widely used oxygenate additive that has become unpopular based upon ground water contamination and human health effects. While most ethanol is currently generated by fermentation (grain alcohol), recent developments suggest that ethanol fuel can be derived more efficiently from other biomass, thus offering potential to reduce dependence on fossil fuel energy resources. The chemical kinetics of ethanol related to combustion has been extensively studied in many previous works, with the most recent detailed modeling studies being those of Marinov.4 His work emphasized the high sensitivity of experimentally measured ignition delay during shock-induced decomposition of rich ethanol mixtures to the rate constants of ethanol decomposition reactions. Moreover, his analyses showed that high temperature ethanol oxidation is strongly sensitive to the falloff kinetics of the ethanol decomposition process, and to the branching ratio assignments among the ethanol abstraction reactions. Unfortunately, there were few ethanol pyrolysis data available for comparison at the time of this modeling work. While our recent ethanol pyrolysis experiments5 using the same variable pressure flow reactor employed here showed that H2O and C2H4 are the major products of ethanol thermal decomposition, we found that Marinov's model underestimated their production rate as well as the overall ethanol consumption rate. We also confirmed that ethanol pyrolysis is very sensitive to the decomposition reactions: C2H5OH → C2H4 + H2O, (R1), 5 Based upon these results, ethanol pyrolysis experiments with the addition of toluene were performed in a variable pressure flow reactor (VPFR). Herzler et al.1 used a similar technique in their shock tube pyrolysis studies, with 1,3,5-trimethylbenzene as the radical trapping species. We also performed flow reactor studies with this radical trapper. The flow reactor is a continuous flow device, where the reactants are highly diluted in an inert carrier gas, and the total flow rate is very large. The diluted reactions are sufficiently slow such that the reaction zone itself occurs over a large physical distance in the reacting flow. Thus, the flow reactor provides an experimental means to directly measure the chemical kinetics by determining the species-distance profiles and interpreting them as species time history information. A schematic of the flow reactor is shown in Figure 2. The entire reactor is enclosed in a carbon steel pressure vessel rated for operation from full vacuum to 30 atm pressure, permitting experiments to be carried out over a wide range of ambient pressures. The reactor is maintained at pressures above atmospheric through control of a backpressure valve at the reactor exit. Carrier gas (N2 in this study) is heated by a pair of electrical resistance heaters and directed into the 10.16 cm diameter quartz reactor duct in which the reaction zone is to be stabilized. The flow enters the tube and then passes around an 8.9 cm baffle plate and radially inward into a 0.64 cm gap which serves as the entrance to a silica foam diffuser (Figure 3). The liquid fuel (mixed with radical trappers in this study) is vaporized as an aerosol suspended in heated nitrogen inside a 300 cm3 stainless steel cylinder maintained at temperatures above the saturation point of the local mixture fraction. The nitrogen/vapor mixture is delivered through the center passage of a fuel injector probe to the location of the inward-directed carrier flow, and exits the fuel injector through a large number of orifices as opposed jets. Additional N2 is introduced via an annular passage in the fuel injector probe to prevent excessive heating of the fuel vapor/nitrogen flow inside the fuel injector probe. The buffering flow exits the fuel injector probe through a second set of injector orifices at the mixing location of carrier gas and fuel vapor/nitrogen flows. 6 After fuel vapor injection and mixing, the reacting mixture then flows through the 5-degree half angle diffuser into a constant area test section. At different distances from the injector location a hot- water cooled, stainless steel sampling probe is positioned on the flow centerline and convectively quenches and continuously extracts a small portion of the reacting flow. The gas extracted and quenched sample stream flows through heated Teflon lines and a multi-port sampling valve. The (heated) multi-port sampling valve can be used to trap and store up to sixteen individual volumes of sample flow for subsequent off-line analyses using a Hewlett Packard 6890 gas chromatograph equipped with a hydrogenation catalyst and flame ionization detection. This ex-situ analysis allows for the identification and quantification of the stable species related to radical trappers, such as toluene, benzene, ethylbenzene, and 1,3,5-trimethylbenzene. The sampled flow directed through the multi-port valve is then directed through the following on-line analyzers: 1) a Nicolet Magna IR 560 Fourier transform infrared spectrometer (FTIR) for measurement of the majority of the stable species of interest (C2H5OH, H2O, C2H4, CH4, CH3CHO, CH2O, etc.); 2) a pair of Horiba Model PIR-2000 non-dispersive infrared analyzers to measure CO and CO2 concentrations; 3) a DELTA F Type A Plus electrochemical analyzer to monitor the initial trace O2 concentrations present in the nitrogen carrier supply. The measurement uncertainties for the data reported here are: CO - ±2%; C2H5OH - ±2%; H2O - ±6%; C2H4 - ±2%; CH4 - ±3%; CH3CHO - ±4% of reading. The temperature of the reacting mixture is measured locally at the point of sampling using a silica-coated type R thermocouple accurate to ±3 K.9 The distance between the point of fuel injection and the sampling position is varied by moving the injector with attached mixer/diffuser assembly relative to the fixed sampling location by means of a slide table driven by a computer-controlled stepper motor. Mean velocity measurements along the centerline of the reactor are used to correlate distance with residence time. Experimental conditions are chosen to produce reaction zones in which one centimeter of reaction distance corresponds to a reaction time between 10-4 and 10-2 seconds. In the present study, data points were taken at 5 cm intervals. 7 The high degree of dilution of the C2H5OH/radical trappers/N2 mixture used in this study ensures that the maximum chemical enthalpy change due to reaction is small. Five individually controlled electrical resistance heaters maintain the local wall temperature of the quartz reactor duct at the initial gas temperature, establishing a nearly adiabatic condition at the reactor walls. As a result of these two conditions, the local gas temperature variation from the initial reaction temperature is due solely to chemical enthalpy changes. A series of pyrolysis experiments were conducted at 1.7 – 3.0 atm and 1045 – 1080 K with ethanol and radical trappers of equal initial concentrations. Figure 4 shows the mole fraction profiles of stable species related to ethanol pyrolysis in the VPFR test section for an experiment at 1050 K and 3 atm. Temperature is not shown here because, for pyrolysis, it remains nearly constant (within about 4 K). As shown in Figure 4, H2O and C2H4 are the major products of ethanol thermal decomposition. In the present work, measured C2H5OH and C2H4 concentrations were used to estimate the rate constant of reaction (R1), k1, according to the standard rate equation: ]OHHC[]HC[ 521 42 ⋅= k td d , (E1), where [X] is the concentration of species X, and t is the reaction time. The experimentally determined rate constant k1 is presented in Figure 5. The excellent agreement between the experimental data using different kinds of radical trappers further supports the present experimental methodology. Experimental uncertainties are also reported in Figure 5. The uncertainty in the temperature measurements, shown horizontally, is less than ±0.4%. At each fixed temperature, the total uncertainty in the determined rate, shown vertically, is about ±15%, including those from the experimental measurements and the methodology itself. The experimental values of k1 obtained here are in a very good agreement with extrapolation of the shock-tube pyrolysis measurements of Herzler et al.1 taken at higher temperatures (see Figure 8). THEORETIAL CALCULATION OF RATE CONSTANTS. The multi-channel unimolecular decomposition reactions of ethanol were also investigated theoretically based on RRKM/master 10 here is based on the assumptions of the Gorin model for loose transition state.21 The vibrational frequencies for this transition state were assigned by using those of the two fragments, CH3 and CH2OH, and its standard heat of formation was taken as the sum of the heats of the two fragments. As can be shown,21 the quantitative information regarding the remaining rotational degrees of freedom (i.e., rotational constants, symmetry, simple steric hindrance) enters the density of states and, therefore, the microscopic rate coefficient in the form of constant multipliers. Thus, one can simply evaluate the microscopic rate coefficient using arbitrarily assigned rotational constants and then scale it to match the prescribed value of k∞ at given temperature. Obviously, the resulting scaling factor will also be a function of temperature. The prescribed value of k2∞ was derived from the equilibrium constant and k-2∞, the high-pressure-limit rate constant of the reverse (recombination) reaction of (R2). In this study, k-2∞ was fixed at 1.2×1013 cm3/mol/s. This is a suggested value from the literature22 and is in good agreement with available experimental data for similar reactions,23 which lie between 1.0×1013 and 3.0×1013 cm3/mol/s. The tunneling effect was considered for reaction (R1), since it involves the transfer of a light atom, H.24 In the present study, the Marcus-Miller quantum approach25 with a one-dimensional unsymmetrical Eckart potential26 was employed to account for the hydrogen tunneling effect. The imaginary frequency of C2H4-H2O, a parameter in the Eckart function, is nearly the same at the levels of theory considered here (MP2(FC)/6-311G(d,p) and B3LYB/6-31G(d,p)), 1926 and 1928 cm-1, respectively. The energy increment was fixed at 1 cm-1 in all sum and density-of-states computations. The standard form of the exponential-down model was used for collision energy transfer. In the absence of reliable measurements in the falloff region that are normally used to calibrate the collision model, the model parameters have to be assigned a priori. Specifically, following recommendations of Knyazev et al.27 based on their extensive comparisons of theoretical predictions with the large body of experimental data, we have accounted for both temperature and energy dependencies of the main 11 collision model parameter, energy transfer per downward collision, downE∆ , using the following expression: 2/1 down ETAE =∆ , (E2), where T is the temperature, E is the internal energy, and A is an adjustable constant taken as 3.3×10-3 cm-1/2K-1 in the present study, which yields about 500 cm-1 at 1000 K and the energy corresponding to the barrier of reaction (R1). The influence of downE∆ on the calculated rate constants is discussed below. The collision frequency of ethanol with the bath gas, nitrogen, was estimated from the Lennard-Jones parameters adopted from the reference 28. Because C2H5OH is a polar molecule, we have also investigated the influence of ethanol dipole moment on its collision frequency with the bath gas molecules using the expressions described in reference 28. This effect was found to be unimportant (below 0.6 %) over the entire temperature range of 300 – 2500 K. An energy grain size of 10 cm-1 was used in the master equation solutions, and the resulting matrix size was 7410×7410. This grain size provided numerically convergent results for all temperatures and pressures considered in this study. RESULTS AND DISCUSSION. The calculated rate constants of reactions (R1) and (R2) over the temperature range 300 – 2500 K at 1 atm are presented in Figure 7. The H2O elimination reaction (R1) is observed to be the dominant decomposition channel at 1 atm over the entire temperature range. Figure 8 demonstrates that the calculated k1 is in excellent agreement with both the shock tube measurements of Herzler et al.1 and the theoretical work of Tsang.2 The result also agrees with that of Marinov4 within 45%, but is about 3.2 times higher than the theoretical value of Park et al.7 at 1100 K and 1 atm. The present theoretical extrapolation gives values consistently higher than the experimental data reported by Park et al.6, with a difference of about a factor of 4.5 for the low- temperature (820-860K) data set, and a factor of 3 for the high-temperature data set (Figure 8). Figure 9 shows that the computed k2 is about 2.4 times lower than that of Tsang2 at 1100 K and 1 atm. This is caused partly by the choice of k-2∞, which is 3.0×1013 cm3/mol/s in the reference 2. If 12 this value for k-2∞ is used, the computed k2 would agree with the result of Tsang2 within 8%. The present theoretical value is also about a factor of 3.8 times lower than the data reported by Park et al.6 The pressure dependencies of rate constants k1 and k2 at 1100 K are plotted in Figure 10. The figure also includes the results when the reactions (R1) and (R2) are treated independently (i.e., ignoring the other channel). The value of k1 obtained in the multi-channel calculation is nearly the same as that of the isolated reaction (R1). The value of k2 obtained in both the multi-channel and single-channel calculation approaches k2∞ at pressures higher than about 100 atm. However, at lower pressures, the value of k2 in the multi-channel calculation is much smaller than that derived from the single-channel calculations, consistent with the conclusion of Tsang.2 The channel (R1) is observed to exhibit the classical falloff shape and reaches the high-pressure limit smoothly with increasing pressure. Channel (R2), on the other hand, shows a more complex falloff shape. In the falloff range, k2 obtained from the multi-channel calculation increases much faster than that in the single-channel calculation, effectively exhibiting an order of reaction that is higher than 2, and finally reaches the pressure-independent (high-pressure limiting) asymptote. This shape is a result of the complex evolution of the reactant energy level population in the multi-channel systems that can be captured only with the full solution of master equation and cannot be observed using modified strong-collision model treatment.29-30 In addition to uncertainties yielded from those of k-2∞, the theoretical result is also affected by the choice of downE∆ . Figure 11 shows the influence of downE∆ on the calculated values of k1 and k2 at 1100 K with the coefficient A in the equation (E2) changing from 3.3×10-3 to 1.7×10-3 cm-1/2K-1 (a factor of 2 reduction). The calculated k1 is much less sensitive to variation in downE∆ than k2. At low pressures (~ 10-3 atm), k2 changes by a factor of 2 to 3 with decreasing downE∆ . As expected, the influence of downE∆ decreases very rapidly with pressure. For example, at 10 atm, k2 changes only 20% with a doubling of downE∆ . 15 APPENDIX A. Evaluation of Rotational Potentials for Hindered Internal Rotations in C2H5OH The rotational potentials for two internal rotors in C2H5OH molecule used in the present calculations were estimated in two steps. First, the electronic energies were computed at B3LYP/6- 31G(d) level as functions of the corresponding torsion angles, φ. All coordinates except for the fixed torsion angle were fully optimized. Next, the extrema (stationary) points identified in the previous step were optimized with respect to all coordinates to find both torsion angles and electronic energies at these conditions at B3LYP/6-31G(d) level, followed by (stationary) G3B3 energy calculations. Finally, the portions of B3LYP/6-31G(d) rotational potential between the extrema points were linearly scaled to match the differences between the extrema energy values computed at the G3B3 level. The results for internal rotations around C-C and C-O bonds are presented in Figures 13 and 14, respectively. The rotation around C-C bond has a three-fold symmetry. The present B3LYP/6-31G(d) and G3B3 computations agree very well at the maximum point, and the overall potential is in good agreement with the computations of Sun and Bozzelli31 who also used B3LYP with a larger, 6-31G(d,p) basis set. In addition to the starting trans configuration, the rotation around the C-O bond reveals two more gauche rotational conformers at about 118° and 242° (0° corresponds to the trans conformer) with the overall potential exhibiting a mirror symmetry about φ = 180° (Figure 15). Calculations at B3LYP/6-31G(d) level (in agreement with the calculations of Sun and Bozzelli31) also indicate that the gauche conformers are about 0.29 kcal/mol more stable than the trans conformer. The present G3B3 calculations, however, suggest the opposite, i.e., that the trans conformer is about 0.11 kcal/mol more stable than the gauche conformers. A similar result (0.1 kcal/mol) was reported in the G2 study of Curtiss et al.32 A more recent detailed study of Weibel et al.33 presents a series of calculations at different levels of theory and also summarizes the results of prior investigations of the energy difference between the trans and gauche configurations of C2H5OH. Weibel et al.33 16 concluded that the energy difference between the two configurations is very small (±0.15 kcal/mol) making the conformers essentially degenerate. The present G3B3 calculations are consistent with conclusions of Weibel et al.33, and the rotational potential scaled in the manner described above was chosen for further use. Reduced moments of inertia for C-C and C-O rotors were found to remain nearly constant during the corresponding rotations, and were taken at ground-state configuration obtained at B3LYP/6- 31G(d) level (2.907 and 0.839 amu×Å2 for C-C and C-O rotations, respectively). APPENDIX B. Density of States Inclusive of Hindered Rotations In the absence of internal hindered rotors, the densities of states needed for evaluation of the RRKM microscopic rate coefficients were computed following the conventional method of Astholz et al.34 Inclusion of internal hindered rotations into the density of states calculations requires special consideration. Recently, a number of analytical approximations for the density of states of one- dimensional quantum hindered internal rotor have been proposed.35-36 These approximations, however, involve two major assumptions: (1) separability of internal rotational degrees of freedom (ignoring the coupling between the external and internal rotational degrees of freedom), and (2) simple sinusoidal (single harmonic) form of rotational potential. Knyazev36 has concluded that it is assumption (2) that is typically a major source of error in the analytical expressions for the density of states for complex shapes of rotational potentials. While having an analytical expression certainly presents an advantage, it is possible to evaluate the density of states numerically by using only assumption (1), thus avoiding the restrictions (2) and substantially increasing the fidelity of the final result. This direct numerical evaluation can also be performed without significant modification of existing numerical algorithms34 or noticeable computational penalty, as discussed next. To obtain the energy levels nE of an isolated one-dimensional hindered rotor, one needs to solve the Schrödinger equation: 17 )()()()( 2 2 22 φψ=φψφ+ φ∂ φψ∂ − nnn n EV I h , (B.1), where h is the Plank’s constant, I the reduced moment of inertia, φ the torsion angle, V(φ) the rotational potential, and )(φψ n the wavefunction corresponding to the energy level nE . The numerical solution of equation B.1 for an arbitrary rotational potential can be accomplished by37 - 40 (1) decomposing the sought wavefunctions )(φψ n in the basis of the free-rotor eigenfunctions π=ψ φ 20 in n e , and (2) decomposing the rotational potential V(φ) into truncated Fourier series, ( )∑ = φ+φ+=φ M m mm mbmacV 1 )sin()cos()( , (B.2). The problem of finding the discrete energy levels En is then reduced to the diagonalization of banded complex Hermitian matrix. The size of the matrix will be defined by N, the number of energy levels to be determined, and the matrix bandwidth is defined by M, the number of Fourier coefficients used to approximate the rotational potential in equation B.2. This problem has well-established efficient solution algorithms41, and does not present any difficulties even for moderately large number of levels considered using very modest computational facilities (such as a desktop computer). In the present work, we have developed an in-house computer program which performs the entire task of finding the required number of energy levels given the reduced moment of inertia I and an arbitrary rotational potential defined on a set of discrete points for a φ covering the range from 0 to π2 . The number of energy levels determined for C-C and C-O rotors considered in this study was set to 2001, which covered more than adequate energy range for the density of states calculations. The rotational potentials used in these calculations are described in Appendix A. Having energy levels En available, the density of states inclusive of internal hindered rotor(s) can be computed via a trivial extension of Astholz et al.’s method. In its original implementation34, this method involves the initialization of the density of states vector with the analytically evaluated rotational density of states followed by application of the Beyer-Swinehart direct count42 for harmonic vibrations to the initialized vector. Here, before the application of the Beyer-Swinehart 20 REFERENCES. (1) Herzler, J.; Tsang, W.; Manion, J.A., J. Phys. Chem. 1997, 101, 5500. (2) Tsang, W., 2nd Joint U.S. Sectional Meeting of the Combustion Institute, Oakland, CA, 2001, Paper 92. 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Phys. Chem. A 1998, 102, 3916. (37) Marston, C.C.; Balint-Kuri, G.G. J. Chem. Phys. 1989, 91, 3571. (38) Lay, T.H.; Krasnoperov, L.N.; Venanzi, C.A.; Bozzelli, J.W.; Shokhirev, N.V., J. Phys. Chem. 1996, 100, 8240. (39) Van Speybroeck, V.; Van Neck, D.; Waroquier, M.; Wauters, S.; Saeys, M.; Marin, G.B., J. Phys. Chem. A 2000, 104, 10939. (40) Dimeo, R.M., in DAVE: Data Analysis and Visualization Environment, NIST, http://www.ncnr.nist.gov/dave/documentation/methylcalc.pdf, 2002. (41) Anderson, E.; Bai, Z.; Bischof, C.; Blackford, S.; Demmel, J.; Dongarra, J.; Du Croz, J.; Greenbaum, A.; Hammarling, S.; McKenney, A.; Sorensen, D., LAPACK Users’ Guide, Third Edition, SIAM, Philadelphia, PA, 1999. (42) Beyer, T.; Swinehart, D.F., Comm. Assoc. Comput. Machines. 1973, 16, 379. 25 TABLE 3. Recommended Expressions for Rate Constants (unit: s-1) at Different Pressures. Pressure (atm) C2H5OH → C2H4 + H2O C2H5OH → CH3 + CH2OH 1×10-10 3.77×1043 T-11.92 exp(-31527/T) 6.41×1034 T-9.16 exp(-64751/T) 0.01 1.61×1045 T-9.69 exp(-39199/T) 5.58×1050 T-11.45 exp(-49616/T) 0.1 4.27×1036 T-6.95 exp(-37855/T) 2.59×1054 T-11.99 exp(-50576/T) 1 8.80×1025 T-3.68 exp(-35627/T) 1.26×1051 T-10.59 exp(-50759/T) 10 1.41×1016 T-0.74 exp(-33322/T) 1.39×1042 T-7.71 exp(-49327/T) 100 2.66×109 T1.25 exp(-31645/T) 1.07×1032 T-4.63 exp(-47122/T) ∞ 1.32×105 T2.52 exp(-30530/T) 9.20×1022 T-1.93 exp(-44890/T) 26 30 50 70 90 0 1 2 3 4 5 6 C on tri bu tio n (% ) Initial Mole Fraction Ratio of Toluene : Ethanol contribution to C2H4 production from C2H5OH → C2H4 + H2O contribution to C2H5OH consumption from its decomposition reactions C on tri bu tio n (% ) Figure 1. Integrated contributions of specified reactions to ethanol consumption and C2H4 yield in a pyrolysis as a function of the toluene/ethanol mole fraction in the initial mixture. Model: ethanol (Marinov4) and toluene (Emdee et al.8). Initial conditions: T = 1050 K, P = 3 atm, C2H5OH = 0.15% with corresponding toluene and balance N2. 27 Fuel Injector Mixer/Diffuser Sample Probe Wall Heaters Electric Resistance Heater Oxygen Inlet Slide Table Fuel Evaporator Main N2 Carrier Gas Figure 2. Schematic of the Variable Pressure Flow Reactor (VPFR). 30 0.7 0.8 0.9 1 2 3 0.92 0.93 0.94 0.95 0.96 k (s -1 ) 1000/T (K-1) Figure 5. Rate constant of the reaction C2H5OH → C2H4 + H2O determined in the present flow reactor experiments. Open symbols represent results by using toluene as a radical trapper, closed symbols by using 1,3,5-trimethylbenzene as a radical trapper. 31 C2H5OH C2H4 + H2O CH3 + CH2OH TS Figure 6. Schematic diagram of the potential energy surface of the ethanol decomposition process. 32 C2H5OH → C2H4 + H2O C2H5OH → CH3 + CH2OH -50 -40 -30 -20 -10 0 10 0 0.5 1 1.5 2 2.5 3 3.5 lo gk (s -1 ) 1000/T (K-1) lo gk (s -1 ) Figure 7. Rate constants of the reactions (R1) and (R2) at 1 atm. 35 10-6 10-4 10-2 100 10-3 10-1 101 103 105 k (s -1 ) P (atm) k 2 k 1 Figure 10. Rate constants of the reactions (R1) and (R2) at 1100 K. Solid lines: muti-channel calculations, dashed lines with symbols: only the reaction (R1) or (R2) is considered. 36 10-9 10-7 10-5 10-3 10-1 101 10-3 10-1 101 103 105 k (s -1 ) P (atm) k 2 k 1 Figure 11. Influence of downE∆ on the rate constants of the reactions (R1) and (R2) at 1100 K. Solid lines: A = 3.3×10-3 cm-1/2K-1 in the equation (E2), dashed lines: A = 1.7×10-3 cm-1/2K-1. 37 10-27 10-21 10-15 10-9 10-3 10-10 10-5 100 105 1010 k (s -1 ) P (atm) k 2 k 1 Figure 12. Rate constants of the reactions (R1) and (R2) at 800 K. Solid lines: with tunneling for (R1), dashed lines: without tunneling for (R1). 40 -0.5 0 0.5 1 1.5 0 40 80 120 160 V (k ca l/m ol ) φ (degree) trans gauche Figure 15. Rotational potential for internal rotation around C-O bond in C2H5OH (the portion from 180° to 360° is omitted due to mirror symmetry about φ = 180° axis). The symbol and line markings are the same as in Figure 14. 41 10-1 101 103 105 107 109 0 0.5 1 1.5 2 D en si ty o f S ta te s (c m ) E/104 (cm-1) Harmonic Vibrations Hindered Rotors Free Rotors Figure 16. Density-of-states of C2H5OH computed with different treatments of two internal rotations.