The pre-exponential factor

The general relation between the rate constant of (apparent) unimolecular reactions and the microscopic dynamics is illustrated in Fig. 8.1.1. 

Based on the theoretical descriptions of Chapter 7, the pre-exponential factor of an apparent unimolecular reaction is, roughly, expected to be of the order of a vibrational frequency, i.e., 1013 to 1014 s 4. 

according to RRKM theory for an apparent unimolecular reaction, Eq. (7.58) gives the (canonical) rate constant for such an elementary reaction  

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 

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Since the reactions are very similar, we will assume that the pre-exponential factors A are the same, thus giving. 

The simplest approach to computing the pre-exponential factor is to assume that molecules are hard spheres. It is also necessary to assume that a reaction will occur when two such spheres collide in order to obtain a rate constant k for the reactants B and C as follows ... 

The overall requirement is 1.0—2.0 s for low energy waste compared to typical design standards of 2.0 s for RCRA ha2ardous waste units. The most important, ie, rate limiting steps are droplet evaporation and chemical reaction. The calculated time requirements for these steps are only approximations and subject to error. For example, formation of a skin on the evaporating droplet may inhibit evaporation compared to the theory, whereas secondary atomization may accelerate it. Errors in estimates of the activation energy can significantly alter the chemical reaction rate constant, and the pre-exponential factor from equation 36 is only approximate. Also, interactions with free-radical species may accelerate the rate of chemical reaction over that estimated solely as a result of thermal excitation therefore, measurements of the time requirements are desirable. 

The classical experiment tracks the off-gas composition as a function of temperature at fixed residence time and oxidant level. Treating feed disappearance as first order, the pre-exponential factor and activation energy, E, in the Arrhenius expression (eq. 35) can be obtained. These studies tend to confirm large activation energies typical of the bond mpture mechanism assumed earlier. However, an accelerating effect of the oxidant is also evident in some results, so that the thermal mpture mechanism probably overestimates the time requirement by as much as several orders of magnitude (39). Measurements at several levels of oxidant concentration are useful for determining how important it is to maintain spatial uniformity of oxidant concentration in the incinerator. 

Rule 7. Determine the pre-exponential term by setting all variables to center point value, where everything becomes one except the pre-exponential factor, because ... 

According to the transition state theory, the pre-exponential factor A is related to the frequency at which the reactants arrange into an adequate configuration for reaction to occur. For an homolytic bond scission, A is the vibrational frequency of the reacting bond along the reaction coordinates, which is of the order of 1013 to 1014 s 1. In reaction theory, this frequency is diffusion dependent, and therefore, should be inversely proportional to the medium viscosity. Also, since the applied stress deforms the valence geometry and changes the force constants, it is expected... 

In a detailed study on shear degradation of DNA, Adam and Zimm found a complex dependence of kc on solution viscosity. Considering that the macromolecules can rupture only after tumbling had brought them into the right configuration, these authors proposed to include solution viscosity into the pre-exponential factor A (l/r)s) [84],... 

The statistical weight (i — 1) in the pre-exponential factor accounts for the fact that the number of fracture sites under stress is proportional to the number of bonds present in the macromolecule. 

Degradation is appreciable only if K > l/tr this occurs whenever the coefficient a M e offsets the dissociation energy term — U0. It is physically unrealistic that an elementary chemical process can have negative activation energy, therefore the permissible maximum for K is controlled by the pre-exponential factor A (i — 1). 

Fig. 51. Dependence of the scission yield on strain rate and temperature. Curves with a continuous line are calculated with the pre-exponential factor A = 1012 s 1 curves with broken lines, (a) and (b), are obtained using respectively A = 10u s and 1013 s 1...

Significantly, the pre-exponential factors decrease with increasing reactivity, and this suggests that the Wheland intermediate is more nearly formed in the transition state, the more reactive the compound. Or, considered another way, the position along the reaction co-ordinate at which a given amount of carbon-halogen bond formation occurs is nearer to the ground state the more reactive the compound. 

If a data set containing k T) pairs is fitted to this equation, the values of these two parameters are obtained. They are A, the pre-exponential factor (less desirably called the frequency factor), and Ea, the Arrhenius activation energy or sometimes simply the activation energy. Both A and Ea are usually assumed to be temperature-independent in most instances, this approximation proves to be a very good one, at least over a modest temperature range. The second equation used to express the temperature dependence of a rate constant results from transition state theory (TST). Its form is... 

Here a and b are considered as fitting parameters depending on temperature. De-excitation rate constants (s < 0) are obtained from the detailed balance principle. AH fitting laws differ in the pre-exponential factor in Eq. (5.70). In the PEG model... 

On K modified Ni(100) and Ni(lll)62,63 and Pt(lll)64 the dissociative adsorption of hydrogen is almost completely inhibited for potassium coverages above 0.1. This would imply that H behaves as an electron donor. On the other hand the peaks of the hydrogen TPD spectra shift to higher temperatures with increasing alkali coverage, as shown in Fig. 2.22a for K/Ni(lll), which would imply an electron acceptor behaviour for the chemisorbed H. Furthermore, as deduced from analysis of the TPD spectra, both the pre-exponential factor and the activation energy for desorption... 

The two constants, A and Ea, are known as the Arrhenius parameters for the reaction and are found from experiment A is called the pre-exponential factor, and , is the activation energy. Both A and a are nearly independent of temperature bur have values that depend on the reaction being studied. 

This exponential temperature dependence for k is much stronger than the weak temperature dependence of the collision frequency itself. By comparing Eq. 17 with Eq. 13b, we can identify the term nvrviNA2 as the pre-exponential factor. A, and Emin as the activation energy, Ea. That is, we can conclude that... 

Arrhenius parameters The pre-exponential factor A (also called the frequency factor) and the activation energy Ea. See also Arrhenius equation. aryl group An aromatic group. Example —C6H5, phenyl. 

The pre-exponential factors were then piece wise estimated by trial and error. The model showed good agreement with the experimental data as illustrated in Figure 3. 

The rate constants are determined by fitting the PO concentrations that change with time, as shown in Fig. 1. With the rate constants at different reaction temperatures, the activation energies and the pre-exponential factors are determined by plotting In k against 1 / T. 

In the same manner, the activation energy and the pre-exponential factor of the combustion are determined Irom an Arrhenius plot. As can be seen the kinetic equation of the combustion can be expressed as ... 

We again assume that the pre-exponential factor and the entropy contributions do not depend on temperature. This assumption is not strictly correct but, as we shall see in Chapter 3, the latter dependence is much weaker than that of the energy in the exponential terms. The normalized activation energy is also shown in Fig. 2.11 as a function of mole fraction. Notice that the activation energy is not just that of the rate-limiting step. It also depends on the adsorption enthalpies of the steps prior to the rate-limiting step and the coverages. 

This chapter will present theories that are capable of predicting the rate of a reaction, in particular the value of the pre-exponential factor. In Chapter 2 we introduced the Arrhenius equation. 

Expression (109) appears to be similar to the Arrhenius expression, but there is an important difference. In the Arrhenius equation the temperature dependence is in the exponential only, whereas in collision theory we find a dependence in the pre-exponential factor. We shall see later that transition state theory predicts even stronger dependences on T. 

Figure 3.n. A tight transition state possesses lower entropy than the ground state of the reactants, therefore the pre-exponential factor will be lower than the standard value of... 

Equation (12) also contains a pre-exponential factor. In Section 3.8.4 we treated desorption kinetics in terms of transition state theory (Figure 3.14 summarizes the situations we may encounter). If the transition state of a desorbing molecule resembles the chemisorbed state, we expect pre-exponential factors on the order of ek T/h = 10 s . However, if the molecule is adsorbed in an immobilized state but desorbs via a mobile precursor, the pre-exponential factors may be two to three orders of magnitude higher than the standard value of 10 s . ... 

As the initial coverages of CO and O are known, and the surface is free of CO at the end of the temperature-programmed experiment, the actual coverages of CO and O can be calculated for any point of the TPD curves in Fig. 7.14. Hence, an Arrhenius plot of the rate of desorption divided by the coverages, against the reciprocal temperature yields the activation energy and the pre-exponential factor ... 

What is a tight or a loose transition state How can one infer the nature of a transition state from the value of the pre-exponential factor ... 

Sketch the transition state and give an order of magnitude for the pre-exponential factor for the following reactions ... 

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