Activated complex vibration

One vibrational degree of freedom of activated complex is quite unstable and responsible for disrupting the activated complex (vibrating at the top of the barrier) into the products. The frequency of such vibration will be low and average energy will be of the order of khT, i.e. 

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. 

Activation Parameters. Thermal processes are commonly used to break labile initiator bonds in order to form radicals. The amount of thermal energy necessary varies with the environment, but absolute temperature, T, is usually the dominant factor. The energy barrier, the minimum amount of energy that must be suppHed, is called the activation energy, E. A third important factor, known as the frequency factor, is a measure of bond motion freedom (translational, rotational, and vibrational) in the activated complex or transition state. The relationships of yi, E and T to the initiator decomposition rate (kJ) are expressed by the Arrhenius first-order rate equation (eq. 16) where R is the gas constant, and and E are known as the activation parameters. 

At R > 400 pm the orientation of the reactants looses its importance and the energy level of the educts is calculated (ethene + nonclassical ethyl cation). For smaller values of R and a the potential energy increases rapidly. At R = 278 pm and a = 68° one finds a saddle point of the potential energy surface lying on the central barrier, which can be connected with the activated complex of the reaction (21). This connection can be derived from a vibration analysis which has already been discussed in part 2.3.3. With the assistance of the above, the movement of atoms during so-called imaginary vibrations can be calculated. It has been attempted in Fig. 14 to clarify the movement of the atoms during this vibration (the size of the components of the movement vector... 

In qualitative terms, the reaction proceeds via an activated complex, the transition state, located at the top of the energy barrier between reactants and products. Reacting molecules are activated to the transition state by collisions with surrounding molecules. Crossing the barrier is only possible in the forward direction. The reaction event is described by a single parameter, called the reaction coordinate, which is usually a vibration. The reaction can thus be visualized as a journey over a potential energy surface (a mountain landscape) where the transition state lies at the saddle point (the col of a mountain pass). 

Of course, the converse situation, in which the entropy of the transition state is lower than that of the ground state of the reactant, can also occur (Fig. 3.11). In this case, one speaks of a tight transition state tight, because rotations, vibration or motion of the activated complex are more restricted than in the ground state of the reactant. The dissociation of molecules on a surface provides an example that we shall discuss in the next section. 

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... 

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. 

These activated complexes differ from ordinary molecules in that in addition to the three normal translational degrees of freedom, they have a fourth degree of translational freedom corresponding to movement along the reaction coordinate. This degree of freedom replaces one vibrational degree of freedom that would otherwise be observed. 

Thus the effective frequency with which activated complexes are transformed into reaction products is kBT/h. At a temperature of 300 °K, this group has a value of 6 x 1012 sec-1, which is comparable in magnitude to normal molecular vibration frequencies. 

Marcus uses the Born-Oppenheimer approximation to separate electronic and nuclear motions, the only exception being at S in the case of nonadiabatic reactions. Classical equilibrium statistical mechanics is used to calculate the probability of arriving at the activated complex only vibrational quantum effects are treated approximately. The result is... 

Besides the electrically active complexes discussed above, there is indirect evidence for the existence of neutral complexes. In close analogy to the observations in silicon and several III-V materials it appears that hydrogen passivates deep and shallow acceptors. Because of the small concentrations of these neutral centers, all attempts to detect them directly with local vibrational mode (LVM) spectroscopy or electron paramagnetic resonance (EPR) have been unsuccessful. 

In activated complex, one degree of vibration has been considered of a different character from the rest, since it corresponds to a very loose vibration... 

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,... 

Thus, the steric factor may be explained with the help of entropy change. When two molecules come together to produce the activated complex, the total translational degrees of freedom are reduced (from 6 to 3) and rotational degrees of freedom also diminish. This is compensated by an increase in vibrational degrees of freedom. But the definite orientation in forming the activated complex necessarily reduced the entropy, i.e. AS is negative. This decrease in entropy is small when reaction takes place between simple atoms. The calculated value of kbT/h corresponds to collision frequency... 

The activated complex has always one degree of vibrational freedom less than a normal molecule with (NA + NR) atoms. Now, the rate constant for complex reaction... 

The formation of the activated complex may be regarded as an equilibrium process involving an almost normal molecule (almost, since it is short one mode of vibrational energy). The free energy of aetivation AG ean therefore be defined as in normal thermodynamics. 

Figure 12. Comparison of simple RRKM rate-energy curves, using three different loose activated complexes giving the same rates at the energy corresponding to about 10 s" . Calculations are shown for Eq values of 1.86 and 3.10 eV. The three transition states are (a) uniform frequency multiplier of 0.9 (—) (b) four low-frequency vibrations (—) (c) low-frequency vibration to internal rotor ( -). The corresponding values are as follows 1.86 eV, (a) = 8.2 eu, (b) = 4.9 eu, (c) = 1.6 eu 3.10 eV (a) = 6.9 eu, (b) = 4.9 eu, (c) = 2.9 eu. Also shown is the semiclassical RRK functional form...

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