How to find the kinetic equation for reverse reactions

This problem, put forward independently by Horiuti (1939) [41] and Bores-kov (1945) [42], can be formulated as follows to find a kinetic equation for a complex reaction in the reverse direction from the known similar expression for the direct reaction rate and applying only thermodynamic relationship for the brutto-reaction. In other words it is necessary to answer the question, in what cases is the equation 

For this case the Horiuti-Boreskov concept is always valid and does not require an assumption about the rate-determining step. 

Let us note that in eqn. (59) the expressions f+ (c) and f (c) are the kinetic dependences that are written according to the law of mass action for the natural brutto-reaction, i.e. for the reaction obtained by a simple addition of all cycle steps, and K fT) is the equilibrium constant for this reaction. However, as we mentioned above for the reaction of catalytic isomerization, the natural brutto-equation should not necessarily have integer-valued coefficients. For the mechanism 

In eqns. (60) and (61), Keq m is the equilibrium constant for the brutto-reaction with the minimum integer-valued coefficients. 

Equation (62) is an analog of that obtained by Boreskov [42] but in contrast to it, it does not require the assumption of the existence of a rate-determining step. 

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