Partition function activated complex

Cordes discusses the magnitudes of pre-exponential terms with reference to the partition function for the activated complex in which the following cases are recognized. 

Thus, the partition function of the activated complex, with the special motion factored out, becomes... 

Quite similar equations can be formulated for AG and AH by use of the partition function f of the activated complex. It follows from equations (6) and (7) that AEp can only be evaluated if the partition functions and AEz are available from spectroscopic data or heat capacity measurements. However, if AG = AH, the entropy change AS equals zero, and if AEz also equal to zero, either AG or AH can then be identified with the potential energy change. If... 

In the following we consider a surface with adsorbed atoms or molecules that react. We will leave out the details of the internal coordinates of these adsorbed species, but note that their partition functions can be found using the schemes presented above. Let us assume that species A reacts with B to form an adsorbed product AB via an activated complex AB ... 

Note how the partition function for the transition state vanishes as a result of the equilibrium assumption and that the equilibrium constant is determined, as it should be, by the initial and final states only. This result will prove to be useful when we consider more complex reactions. If several steps are in equilibrium, and we express the overall rate in terms of partition functions, many terms cancel. However, if there is no equilibrium, we can use the above approach to estimate the rate, provided we have sufficient knowledge of the energy levels in the activated complex to determine the relevant partition functions. 

This relation indicates that the rate constant can be determined from a knowledge of the partition functions of the activated complex and the reactant species. For stable molecules or atoms... 

Thus, rate constant for a reaction can be given by equation (4.34) in terms of equilibrium constant for the formation of activated complex K which can be expressed either in terms of partition or thermodynamic functions. 

While the concentration of activated complex in terms of partition functions is given as... 

The general equation for L is Eq. (10), and the expressions to be used for C in that equation are listed in Table 1 in terms of partition functions. But the entropies of both the activated complex and the reactants, and therefore AS, can also be expressed in terms of partition functions. Therefore, C can be expressed in terms of the entropies of the activated complex and the reactants. As we shall see, it is possible to eliminate partition functions entirely. Also, in all but one case. Step 11, the entropy factor in C can be determined if one knows only the entropy of activation in those cases the entropy of a reactant or the activated complex is not needed. 

Occasionally, the rates of bimolecular reactions are observed to exhibit negative temperature dependencies, i.e., their rates decrease with increasing temperature. This counterintuitive situation can be explained via the transition state theory for reactions with no activation energy harriers that is, preexponential terms can exhibit negative temperature dependencies for polyatomic reactions as a consequence of partition function considerations (see, for example, Table 5.2 in Moore and Pearson, 1981). However, another plausible explanation involves the formation of a bound intermediate complex (Fontijn and Zellner, 1983 Mozurkewich and Benson, 1984). To... 

In terms of the vibrational partition functions Q of the reactants ( and ") and of the activated complex ( ), the Absolute Rate Theory expression for k° is... 

In theory, one can use statistical thermod3mamics to calculate the partition functions of all the species from first principles, AS, AH, and hence k. For simple systems, the calculation results are in good agreement with experimental data (e.g.. Chapter 3 in Laidler, 1987). For complicated geological systems, however, it is not possible to calculate k from first principles, but the concept of activated complexes is very useful for a microscopic understanding of the reaction... 

Note that the zero-point-energy-including difference in internal energies between A and A in the exponential term is easily computable from an electronic structure calculation (for the electronic energy) and a frequency calculation (to determine the ZPVE) for the minimum energy and TS structures corresponding to A and A, respectively. In addition, the availability of frequencies for A permits ready computation of Qa, as described in Chapter 10. Some attention needs to be paid, however, to the nature of the partition function for the activated complex, Q. ... 

Following up on the discussion in Section 15.2 about the nature of the activated complex, the TS structure should be recognized as a species that is a minimum in 3N - 7 degrees of freedom - the missing degree of freedom is the reaction coordinate. Thus, we may readily define the electronic, translational, and rotational components of the partition function associated with the TS structure in the usual way. For the vibrational component, we will separate... 

To compute the r.h.s. of Eq. (15.35), we need to define how we compute the partition function (and the ZPVE) for the non-stationary point 5. In this case, we simply continue to take advantage of our decision to treat the activated complex as a species having 3N — 7 bound degrees of freedom. In order to define this space for an arbitrary point on the MEP,... 

The activated complex partition function has contributions from translation (with total mass w,4 +mj) and from rotation of the (linear) activated molecule. Assuming that the bond length of C is the sum of the atomic radii r a and rg, the rotational partition function for the activated complex can be calculated from Eq. 8.65, the moment of inertia / = m 2(rA + re)2, where m 2 is the A-B reduced mass (Eq. 10.38). 

Introducing this into the activated complex partition function Q of Eq. (11) gives [7]... 

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