Vibration frequency of the activated complex

Evaluate whether the difference between kexp and kxsx can be attributed to an uncertainty of 100 cm in the vibrational frequencies of the activated complex. 

Consider a collinear reaction of the form A + BC —> AB + C, i.e., all atoms are assumed to move along the same line. Imagine that a calculation of the (realvalued) vibrational frequency of the activated complex, at two different levels of accuracy, gives z>i cm 1 and z>2 cm 1, respectively, and Pi > p2-... 

We see that the pre-exponential factor in k(T), roughly, corresponds to a typical vibrational frequency, since kp T/h = 6.25 x 1012 s 1 at T = 300 K. We can also understand why the pre-exponential factors can be somewhat larger than typical vibrational frequencies, because very often > Q. This situation will arise when the (product of the) vibrational frequencies of the activated complex are smaller than the vibrational frequencies of the reactant. Furthermore, the rotational contribution in the... 

Fortunately, the first aspect, although conceptually very important, is not serious for the practical numerical evaluation of kt. It has been shown that as long as the entropy of activation requirements are roughly satisfied,9 18 20 the magnitude of the specific rate constant will not be particularly sensitive to details in structure or vibrational frequencies of the activated complex. Furthermore, by using various empirical aids21 some of the remaining ad hoc aspect can be removed.9 16... 

The product of xZ - which is related to the vibrational frequency of the activated complex - determines the upper limit of k, which is less than 10 cm s . ... 

To calculate the vibration frequency of the activated complex, we use the surfaces of potential energies. The method used is based on the theory of small vibrations applied to the stationary points of a potential surface, which is either a maximum or minimum of a system at equilibrium whereas the coordinate variation does not affect the potential energy. 

We thus obtain n linear equations of this type when k varies between one and n, n being the number of normal vibration frequencies of the active complex. Since these movements are vibrations, the solutions of this system will be written as ... 

In order to find the TST rate-coefficient expression one must have the partition function of the activated complex. While the translational part of this partition function is known exactly, the vibrational and rotational parts can only be found by using estimates of the structures and vibrational frequencies of the activated complex. These depend on the electronic energy of the activated complex as a function of intramolecular coordinates, the so-called potential energy hypersurface. 

It can be difficult to estimate theoretically the bond lengths and vibrational frequencies for the activated complex and the energy barrier for its formation. It is of interest to assess how the uncertainty in these parameters affect the rate constant predicted from transition state theory (TST). For the exchange reaction... 

The degeneracies of the vibrational states are given in parentheses after the frequencies. The imaginary frequency of the activated complex is not included. The vibrational frequencies of CH3CI correspond to the data in G. Herzberg, Electronic spectra of polyatomic molecules. 

The potential energy surface for the reaction has been calculated and the classical barrier height associated with the activated complex is Ec = 90.8 kJ/mol. The relevant vibrational frequencies are given in the table below. (Note that the imaginary frequency of the activated complex is not included in the table.)... 

For an interpretation of our results we performed statistical RRKM calculations of the individual decay rate constants of all four competing decay channels at low threshold energy. The latter have been experimentally extracted from the directly measured total decay rate constant (see Fig. 4) and the simultaneously measured branching ratios of the relevant fragment ions /16/. For different isotopically labelled species a good simulation of experimental results is obtained with a single set of parameters for the determination of the frequencies of the activated complex ( solid line in Fig. ). Isotope shifts of the vibrational frequencies were obtained by use of the Teller-Redlich product rule. This points to a high reliability of the set of parameters used and yields detailed information on the structure of the activated complex for the four decay channels under consideration /16/. In... 

In the previous section only internal motion had been considered, i.e. the description had been restricted to vanishing total angular momentum (J=0). Then the number of relevant vibrational states of the activated complex was found to be small for typical temperatures since the vibrational frequencies are usually much larger than the thermal energy. This situation changes dramatically if rotational motion is considered. Rotational spacings are typically much smaller than thermal energies. Thus, if rotation is included explicitly in the calculation, an enormous amount of rovibrational states has to be considered. 

One of the vibrational modes of the activated complex leads to the A" B bond cleavage (in the above H2/HD example asymmetric stretching of H- H -D complex). The frequency of this vibration is closely related to the (AB)t decay frequency ... 

In order to see how Eq. (3.60) approaches its high-temperature limiting value one must assign a set of vibrational frequencies to the activated complex. In Fig. 12 a comparison of an exact calculation using the activated complex vibrational frequencies proposed by Smith and Zellner (1974) to the prediction of Eq. (3.63) is shown as expected, the approach to the high-temperature limit is slower than the simple formula predicts. 

The rate at which the complex breaks up into the products, i.e. the rate of reaction depends on two factors (i) concentration of the activated complex and (ii) frequency of vibration of activated complex. Hence,... 

Theoretical calculations on the cycloaddition reactions of a range of 1,3-dipoles to ethene in the gas phase have been carried out (85) with optimization of the structures of these precursor complexes and the transition states for the reactions at the B3LYP/6-31G level. Calculated vibration frequencies for the orientation complexes revealed that they are true minima on the potential energy surface. The dipole-alkene bond lengths in the complexes were found to be about twice that in the final products and binding was relatively weak with energies <2 kcal mol . Calculations on the cycloaddition reactions of nitrilium and diazonium betaines to ethene indicate that the former have smaller activation energies and are more exothermic. 

Once the activated complex has formed (i.e., the critical bond contains sufficient energy for reaction), C is assumed to react very quickly. The reaction takes place within the first vibrational period after formation of C. The rate constant is usually assumed to be on the order of the vibrational frequency of the critical bond. A steady-state analysis of reaction set 10.136-10.138 yields... 

The results of the DFT calculations for various stable C2H.V species and transitions states on Pt(lll) and Pt(211) are summarized in Table V, which also shows entropy changes for the various steps, as estimated from DFT calculations of the vibrational frequencies of the various adsorbed C2H species and transition states on 10-atom platinum clusters (55). Table V also includes estimates of the standard Gibbs free energy changes for the formation of stable C2H surface species and activated complexes responsible for C-C bond cleavage at 623 K. These estimates were made by combining... 

This is the same as Bronsted s theory which was designed particularly for solutions. The concentration of the activated complex can be expressed in terms of the reactants and the equilibrium constant K. Also the heat of the reaction, AH, to give the activated complex, can be calculated approximately from the quantum theory or from the Arrhenius theory. Since AF= —RT In K and AF = AII — TAS, and since K can, in some cases, be calculated from known, fundamental constants, the entropy term remains the only unknown. Rodebush has long pointed out that the unknown quantity 5 in the formula k = se E/RT is related to an entropy term. As a first approximation it has been related to a collision frequency in bimolecular reactions and to a vibration frequency in unimolecu-lar reactions. Combining the two thermodynamic equations23... 

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