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. 

One may now assume thill the reaction rate is the product of the following three factors (I) the average number of activated complexes (2) the characteristic frequency of the activated complex 1 that is, the inverse of iis lifetime) and (3) )hc transmission coefficient. K. which is... 

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-... 

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.)... 

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... 

In accordance with Eq. (55), the activated complex C has a right to roll back from the top of the potential barrier into the both potential holes with the frequencies v, and v, which aren t the activated parameters. Parameters Kj and are activate, in other words, they depend on the value of the potential barrier, and determine the frequencies of the activated complexes formation from the initial and frnal substances, respectively. 

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 . ... 

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... 

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 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,... 

In transition-state theory, the absolute rate of a reaction is directly proportional to the concentration of the activated complex at a given temperature and pressure. The rate of the reaction is equal to the concentration of the activated complex times the average frequency with which a complex moves across the potential energy surface to the product side. If one assumes that the activated complex is in equilibrium with the unactivated reactants, the calculation of the concentration of this complex is greatly simplified. Except in the cases of extremely fast reactions, this equilibrium can be treated with standard thermodynamics or statistical mechanics . The case of... 

The enthalpy change A H° is related to the energy change in going from reactants to the transition state, that is, to the activation energy. The frequency v of breakup of the activated complex into products is often approximated by v = kT/h, where k and h are the Boltzmann and Planck constants, respectively. Comparison of Eq. (P) to the rate equation shows that... 

The first exponential term in (B.7) is related to the frequency with which the components of the activation complex assume the correct orientation. It describes, among other things, the stereospecific fit of the components A, B, and C. 

Within transition-state theory, Eq. (8.2) is an exact expression for the rate constant. We observe that the pre-exponential factor deviates from the simple interpretation, as being related to the collision frequency Zab via Z, due to the presence of internal degrees of freedom. Typically, the calculated value of Z is of the order of 1011 dm3 mol-1 s 1 10 16 m3 molecule-1 s 1 (see Example 4.1). The magnitude of the partition functions in Eq. (8.2) is typically small compared to this number. Thus, if we neglect the internal degrees of freedom of the reactants and the activated complex, except for rotational degrees of freedom of the activated complex (AB), and assume that the associated partition function can be approximated by QAB, we will get a pre-exponential factor given by Z. 

Most modern investigations of the effects of a solvent on the rate constant, where dynamical fluctuations are included, are based on a classical paper by Kramers from 1940 [1], His theory is based on the transition-state theory approach where we have identified the reaction coordinate as the normal mode of the activated complex that has an imaginary frequency. In ordinary transition-state theory, we assume that the motion in that coordinate is like a free translational motion with no recrossings. This... 

We expand the potential energy surface at the saddle point to second order in the coordinates at the top of the barrier and determine the normal modes of the activated complex one of them is the reaction coordinate y identified as the mode with an imaginary frequency. Since the other normal modes of the activated complex are not coupled to the reaction coordinate in the harmonic approximation, we do not consider them here because they are irrelevant. For the harmonic solvent, we may likewise find the normal modes S. We use these normal modes to write down the Hamiltonian, and then add a linear coupling term representing the coupling between the reaction... 

The rate of the chemical reaction per unit volume, tab, is equal to the concentration of activated complex multiplied by a frequency factor equal to A- j77//, where h is Planck s constant. The thermodynamic activity of the activated complex is equal to... 

The final transformation of the activated complex AB in product (C) is governed by oscillations of low frequencies to [6],... 

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... 

It should be noted that the changes between the structures of complexes 1, 2, and 3, are (deliberately) rather drastic, as evidenced by the large entropy changes listed in Table I. Sec. II-C,4 shows that small changes in frequency patterns or in the entropy of the activated complexes have minor effects on the magnitude of kt. 

The variation of the activated complex sum ratio with energy shown in the first line of Table X is relatively small the slightly irregular behavior that has a formal explanation in terms of the actual frequency patterns is of no concern here. 

Figure 3 indicates that the increased catalytic rates are not the mere result of changes in basicity but that also the size of the counter-ions and their number present in supercages positions determine the activity. If basicity would be the only parameter determining the turnover frequency of the active Ru-complex, the following sequence would be expected ... 

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