Rate constant calculation

The rate constants were calculated with the transition state theory (TST) for direct abstraction reactions and the Rice-Ramsperger-Kassel-Marcus (RRKM) theory for reactions occuring via long-lived intermediates. For reactions taking place without well-defined TS s, the Variflex [35] code and the ChemRate [36] code were used for one-well and multi-well systems, respectively, based on the variational transition-state theory approach 

For the calculation using Variflex, the number of a variational transition q uantum s tates, N ej, w as given b y t he v ariationally d etermined minimum in Nej (R), as a function of the bond length along the reaction coordinate R, which was calculated by the method developed by Wardlaw-Marcus [6, 7] and Klippenstein [8]. The basis of their methods involves a separation of modes into conserved and transitional modes. With this separation, one can evaluate the number of states by Monte Carlo integration for the convolution of the sum of vibrational quantum states for the conserved modes with the classical phase space density of states for the transitional modes. 

Based on this energy diagram and the molecular parameters obtained at the B3LYP/6-311+G(3df, 2p) level, variational TST and RRKM calculations have been carried out for the unimolecular decomposition and the reverse bimolecular association reaction rate constant of the HO-CIO3 system. 

The theoretically predicted dissociation rate constant is compared with the experimental values in Fig. 3. The figure shows that our theoretically predicted rate constant at 1 atm agrees well with those obtained from flow- 

A similar calculation has been carried out for the unimolecular decomposition of chloric acid, HOCIO2, employing the energetic parameters for the HO + OCIO reaction (vide infra). 

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For reactions between atoms, the computation needs to model only the translational energy of impact. For molecular reactions, there are internal energies to be included in the calculation. These internal energies are vibrational and rotational motions, which have quantized energy levels. Even with these corrections included, rate constant calculations tend to lose accuracy as the complexity of the molecular system and reaction mechanism increases. 

Fig. 18. Rate constant calculated with the use of (2.80a) plotted against (m/mH). The hydrogen transfer rate is assumed to be 10 s the effective symmetric vibration mass 125mH. The ratio of force constants corresponding to the intra (Kq) and intermolecular (K,) vibrations is (Ki/Ko) = 2.5 x 10 , 5 x 10 and l.Ox 10 for curves 1-3, respectively.

For Scheme XIV, and for each of the following sets of rate constants, calculate the exact relative concentration cb/ca as a function of time. Also, for each set, calculate the approximate values of cb/ca using both the equilibrium assumption and the steady-state approximation. 

RCS (.sec Rate-controlling step) Racemization, 49, 95 Rate constant calculation of. 17 composite, 161-164 diffusion-controlled, 200-201 Rate-controlling step, 9, 82-86 Rate-determining step (see Ratecontrolling step)... 

The recipe (5.58) is even more sensitive to the high-frequency dependence of kjj than similar criterion (5.53), which was used before averaging over kinetic energy of collisions E. It is a much better test for validity of microscopic rate constant calculation than the line width s j-dependence, which was checked in Fig. 5.6. Comparison of experimental and theoretical data on ZR for the Ar-N2 system presented in [191] is shown in Fig. 5.7. The maximum value Zr = 22 corresponding to point 3 at 300 K is determined from the rate constants obtained in [220],... 

Pulsed source techniques have been used to study thermal energy ion-molecule reactions. For most of the proton and H atom transfer reactions studied k thermal) /k 10.5 volts /cm.) is approximately unity in apparent agreement with predictions from the simple ion-induced dipole model. However, the rate constants calculated on this basis are considerably higher than the experimental rate constants indicating reaction channels other than the atom transfer process. Thus, in some cases at least, the relationship of k thermal) to k 10.5 volts/cm.) may be determined by the variation of the relative importance of the atom transfer process with ion energy rather than by the interaction potential between the ion and the neutral. For most of the condensation ion-molecule reactions studied k thermal) is considerably greater than k 10.5 volts/cm.). 

In Table III we compare for several reactions the experimental rate constants with rate constants calculated on the basis of ion-induced dipole interactions only from the relation (4) ... 

Twenty-three kinetics have been carried out at 25°C for pH values from 8.25 to 11.25. The rate constant, calculated as the average of all the ks, was of 27.2 9.0 mol 1 min. The pH correction according to equation (2) was not perfect, as there was a tendency to obtain higher k values at lower pH values. However, this was specially true for extreme vdues of our pH range, where the buffer capacity of ethanolamine was limited (higher pHs) or the reaction proceeded very slowly (low pHs), impairing the precision of the data. Another factor that might explain the dispersion of the data is lack of precision of pH measurement (no better than 0.02 pH units). 

In order to investigate the relationship between the surface area of skeletal copper and activity, the same sample of catalyst was tested in four successive runs. Rate constants was compared with that of another sample prepared in the same way but pretreated in 6.2 M NaOH at 473 K before use. Figure 4 shows that the first order rate constants, calculated so as to take into account the mass of catalyst relative to the volume of solution, decreased in the first three cycles but then stabilised. The surface areas, measured on small samples taken after reaction, mirrored this pattern. The rate constant, and the surface area, for the pretreated catalyst was similar to those obtained in cycles 3 and 4. It is apparent that activity and surface area are closely related for the unpromoted skeletal copper catalyst and that the pretreatment in NaOH at 473 K is approximately equivalent to three repeated reactions in terms of stabilising activity and surface area. 

Because T -> V energy transfer does not lead to complex formation and complexes are only formed by unoriented collisions, the Cl" + CH3C1 -4 Cl"—CH3C1 association rate constant calculated from the trajectories is less than that given by an ion-molecule capture model. This is shown in Table 8, where the trajectory association rate constant is compared with the predictions of various capture models.9 The microcanonical variational transition state theory (pCVTST) rate constants calculated for PES1, with the transitional modes treated as harmonic oscillators (ho) are nearly the same as the statistical adiabatic channel model (SACM),13 pCVTST,40 and trajectory capture14 rate constants based on the ion-di-pole/ion-induced dipole potential,... 

Table 4 Rate constants calculated by non-linear least square method under non-isothermal condition...

Corcelli, S.A. Rahman, J.A. Tully, J.C., Efficient thermal rate constant calculation for rare event systems, 7. Chem. Phys. 2003,118, 1085-1088... 

Using the UMIST Astrochemistry Database format for rate constants, calculate the rate constants at 20 and 100 K for the following reactions, giving the units for the rate constants in each case ... 

The algorithm for rate constant calculation has the following form ... 

Although we cannot clearly determine the reaction order from Figure 3.9, we can gain some insight from a residual plot, which depicts the difference between the predicted and experimental values of cA using the rate constants calculated from the regression analysis. Figure 3.10 shows a random distribution of residuals for a second-order reaction, but a nonrandom distribution of residuals for a first-order reaction (consistent overprediction of concentration for the first five datapoints). Consequently, based upon this analysis, it is apparent that the reaction is second-order rather than first-order, and the reaction rate constant is 0.050. Furthermore, the sum of squared residuals is much smaller for second-order kinetics than for first-order kinetics (1.28 X 10-4 versus 5.39 xl0 4). 

All the kp+ calculated by the original workers for bulk polymerisations and those obtained at high monomer concentrations are wrong because they are second-order rate-constants, calculated on the assumption that the polymerisations are second-order reactions. This is a considerable curiosity because all the kinetic curves published showed clearly that the polymerisations were of zero order with respect to the monomer concentration (m). A new set of kp+ values is given here. 

This expresses quantitatively by how much the rate-constants calculated on the basis of (40) for aryl alkenes and VE by the original investigators could be underestimated, and this effect must account-at least in part-for the great differences in the recorded in Table 1. 

Since the rate-constants calculated by Kunitake and Takarabe for styrene contain unknown contributions from at least three propagators, they cannot be included in our final table of results. It is also evident that the variation of the rate-constants with temperature cannot provide any useful information, because the relative contributions from the different propagators change with temperature. 

The carriers were impregnated and calcinated in the laboratory, and the activity was subsequently measured in the set-up shown in Fig. 9. The vanadium content was varied in the range 2-5 wt% and the molar ratios between alkali promoter and vanadium were varied in the ranges 0-4 for K/V, 0-2 for Na/V, and 0-3 for Cs/V. The sulphur content was about the same in all impregnations. The measured activities for 3 catalyst compositions A, B, and C impregnated on the same carrier and with the same vanadium content and molar ratios of K/V and Na/V are given in Table 1. The extrudates are made in the 9 mm Daisy form, which is the special 5-finned ring offered by Haldor Topsoe (Fig. 3). The observed pellet activity is reported as a pseudo-1st order rate constant calculated from... 

Kluger and Lam, 1978. Comparison with the rate constant calculated for succinamic acid at 50° gives EM = 5 x 101. Correction for the better leaving group uses the ratio of rates for maleamic and maleanilic acids measured by Aldersley et al. (1974b) at 39°... 

Rate constants calculated from the observed second order rate constant for hydroxide- ... 

Gandour et al., 1979. The reference reaction is general base catalysis of the hydrolysis of phenyl dichloroacetate at 25° by external carboxylate of the given pK,. Rate constants calculated from a two point Bronsted plot using the data of Fersht and Kirby (1967)... 

Nonequilibrium effects. In applying the various formalisms, a Boltzmann distribution over the vibrational energy levels of the initial state is assumed. The rate constant calculated on the basis of the equilibrium distribution, keq, is the maximum possible value of k. If the electron transfer is very rapid then the assumption of an equilibrium distribution over the energy levels is not valid, and it is more appropriate to treat the nuclear fluctuations in terms of a steady-state rather than an equilibrium formalism. Although a rigorous treatment of this problem has not yet appeared, intuitively it seems that since the slowest nuclear fluctuation will generally be a solvent orientational motion, ke will equal keq when vout keq and k will tend to vout when vout keq (a simple treatment gives l/kg - 1/ vout + 1/keq). These considerations are... 

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