Gibbs Free Energy and Activation Energy for the Reactions of XH with Y and Y’

Question:
For the reaction XH + Y → X + HY , assume that the potential energy of the reactants is given by E = x2 , while for the products it is E = (x−a)2 + a. Here x is the reaction coordinate, and a is a parameter. For the reaction XH + Y’ → X + HY’ the energy of reactants is E = x2 − ∆E, while for the products the potential energy did not change. Calculate the coefficient B between the free energy change and the activation energy change for two reactions in the relation ∆G = B∆Ea , assuming that ∆E is small and a > 1.

Solution:
Calculation of activation energy and Gibbs free energy for first reaction:
Potential energy of reactants and products are given by,

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