Transition state theory yield function

From the preceding paragraph, the reader will note that many assumptions are involved in transition-state theory. Alternative derivations exhibit differing hypotheses. In a quicker but perhaps less intuitive derivation, translation in the reaction coordinate is treated formally as the low-frequency limit of a vibrational mode. Expansion of the vibrational partition function given in Section A.2.3 then yields Q = Q (k T/hv), which is substituted into equation (A-24), to be used directly in equation (66), thereby producing equation (69) when v = 1/t. The decay time thus is identified as the reciprocal of the small frequency of vibration in the direction of the reaction coordinate. 

To evaluate the state numbers and densities, a structure and set of frequencies have to be chosen for the transition complex. Provided the choice is made to match the experimentally derived Q lb Qrot /Qvib Qrot (obtained from the k , measured as a function of T), the computed k ( ) turn out to be insensitive to the details of the model that is selected. It means that, to a first approximation, the lifetimes of the excited molecules and the form of the low pressure fall-off are functions only of the entropies of the parent and its transition complex and that there are no adjustable parameters. This is advantageous to those whose aim is to calculate lifetimes, but evidently comparison of theory with experiment will not, in general, yield detailed information concerning the structures of the transition state. We return to these aspects later and presently consider the problem of evaluating the state densities, supposing that the structures and frequencies are known. 

The key further assumption of RRKM theory is that there is no recrossing of the dividing surface. In other words, it is assumed that all trajectories passing through the transition state in the direction of products continue directly on to products. With this assumption, the reactivity function is approximated by a step function in the velocity through the dividing surface, yielding... 

The use of carbonate bases in conjunction with substoichiometric amounts of a pivalate source (e.g., BuCOjH) is critical to obtain the products in high yields. Density functional theory (DFT) studies suggest that the pivalate anion serves to decrease the transition state energy for C—H cleavage. In competition studies, electron-deficient arenes are arylated preferentially. For example, the reaction of /)-tolyl bromide with equimolar amounts of benzene and fluorobenzene leads to the formation of A and B in a 1 11 ratio product B is formed as a 22 3 1 mixture of o/m/p isomers (Scheme 24.5). These results implicate the importance of C—H acidity in these systems. Notably,... 

A review containing 123 references on recent mechanistic and theoretical studies of hetero-Diels-Alder reactions has been presented. The hetero-Diels-Alder reactions of homochiral 1,2-diazabuta-1,3-dienes with diethyl azodicarboxylate are accelerated by microwave irradiation to produce the corresponding functionalized 1,2,3,6-tetrahydro-l,2,3,4-tetrazines. " The transition structures for hetero-Diels-Alder reactions involving the heteroatoms O, S, and N in dienes and also in dienophiles were determined at the MP2 and the hybrid DPT levels of theory. The activation volume of the Diels-Alder reaction between dimethyl l,2,4,5-tetrazine-3,6-dicarboxylate and hex-l-ene indicates the conservation of all four nitrogen atoms in the transition state. " 4-n-Propyl-l,2,4-triazoline-3,5-dione reacts with cyclopentadienes, cyclohexadienes, and cycloheptadienes to yield 4 + 2-cycloadducts. ... 

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