Clockwise rate coefficients

The product of the "clockwise" rate coefficients in a closed loop must equal the product of the "counter-clockwise" rate coefficients. 

Self-consistency of postulated forward and reverse rate equations can be tested with the principles of thermodynamic consistency and so-called microscopic reversibility. The former invokes the fact that forward and reverse rates must be equal at equilibrium the latter is for loops in networks and can be stated as requiring that the products of the clockwise and counter-clockwise rate coefficients of the loop must be equal, or, for catalytic cycles, that the product of the forward coefficients must equal that of the reverse coefficients multiplied with the equilibrium constant of the catalyzed reaction. 

Self-consistency of postulated forward and reverse rate equations and their coefficients can be tested with the principles of thermodynamic consistency and so-called microscopic reversibility. The former invokes the fact that forward and reverse rates must be equal at equilibrium. The latter is for loops of parallel pathways and for catalytic cycles. Thermodynamic consistency allows the reverse rate equation to be constructed from the forward one if at least one of its reaction orders is known, and requires the ratio of the products of the forward and reverse rate coefficients to be equal to the thermodynamic equilibrium constant. Microscopic reversibility leads to several useful conclusions The products of the clockwise and counter-clockwise rate coefficients of a loop must be equal the product of the forward rate coefficients of a catalytic cycle must be equal that of the reverse rate coefficients multiplied with the equilibrium constant of the catalyzed reaction forward and reverse reaction must occur along the same pathway and the ratio of the products of forward and reverse rate coefficients must be the same along all parallel pathways from same reactants to same products. The latter two rules apply regardless of whether or not any of the reactions are catalytic. 

Derivation. In any loop of two parallel pathways, clockwise is forward along one pathway and reverse along the other. Thus, the clockwise-to-counter clockwise product ratio of the rate coefficients is a forward-to-reverse product ratio in one pathway and a reverse-to-forward one in the other. The equality of the total clockwise and counter-clockwise products of the loop therefore requires the forward-to-reverse product ratios to be the same along the two pathways. 

The rate at which the complex phase changes from one AO Xn to the next AO is proportional to k. It jumps from one vertex of the phase polygon to the -th, counterclockwise for t and clockwise for 0 i. This procedure is illustrated for the HOMOs and LUMOs of benzene in Figure 2.12a. The products of AO coefficients needed for evaluating the overlap densities are easily obtained from these diagrams by simply adding the complex phases. Thus,... 

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