The way kinetics is taught at the undergraduate level (Arrhenius and collision theory) chemical equilibrium is determined governed immensely by activation energy of the reaction. According to thermodynamics, however equilibrium is a function of free energy change. In a way, thermodynamics and kinetics seem to contradict each other. What insight am I missing. Is the energy in energy vs reaction coordinates supposed to be Gibbs free energy instead of enthalpy? Would this imply activation energy changes as a reaction proceeds? Is the way kinetics is taught wrong?
"...chemical equilibrium is governed immensely by activation energy of the reaction."
This sentence is incorrect. The relative amounts of species in a chemical equilibrium are not at all influenced by the kinetics; if anything, kinetics are influenced by the chemical equilibrium (such as in the Marcus theory of electron transfer reactions). A rigorous derivation of the expression for an equilibrium constant can be made with no direct reference to reaction kinetics.
Chemical equilibria are what happens to systems after they've been left without interference for an infinite time; any system must reach a steady state in finite time, otherwise it would essentially be a perpetual motion device. In other words, for a chemical equilibrium to be achieved, the system is given more than enough time for any finite activation energy barrier to be overcome.
A great deal of chemistry is determined by the interaction of kinetics and thermodynamics. The world would be a dull place if we didn't have both.
While thermodynamics determines what directions a reaction can go, the kinetics often determines whether the reaction can happen. Take a simple example: diamond is not the stable from of the element carbon at room temperature; graphite is. If the world were purely determined by thermodynamics all carbon would be graphite and men would be unable to impress women with the expensive shiny jewellery they buy them. But this reaction doesn't happen.
The reason it doesn't is because to convert diamond into graphite we would have to have to radically restructure the chemical bonding in diamond from a tetrahedral network breaking at least one bond for every carbon atom. Breaking all those bonds requires a huge input of energy to kick off the process (thermodynamics says you will get that energy plus a little more back if you can make this happen). You can think of diamond with a whole bunch of broken carbon-carbon bonds as an intermediate structure on the path from diamond to graphite. But that intermediate structure needs a great deal of energy to create and there simply isn't enough energy at room temperature to get there. The reaction will proceed at very high temperatures (and, if the pressure is high enough, diamond will be the thermodynamically favoured form). This is how the earth (or industry) creates diamonds: at high temperatures there is enough energy to break the carbon-carbon bonds and the equilibrium can be established (and, at high pressures, will favour diamond and at lower pressures graphite).
The point of all this is that, in chemistry, what happens involves the interplay of thermodynamics and kinetics. If it takes a lot of energy to get to an intermediate, the thermodynamic products cannot be reached. So being able to achieve an equilibrium depends not just on the start and end products of a reaction but the intermediate structures or compounds that you have to pass though to interconvert them. If those structures are hard to get to because you don't have enough energy around then thermodynamics is irrelevant to what will happen. you always have to think through the mechanism behind the reaction and understand the energy required to create the intermediate states along the reaction path.
The energy referred to on an energy diagram is enthalpy. Think of the reaction A = B, an exothermic reaction involving two gases. The forward reaction has lower activation energy so it is faster. In order for the reaction to be in equilibrium there has to be more B than A so that the forward and reverse rates can be equal.
Look at the reaction A = B + C, an exothermic reaction where all the species involved are gases. The reaction will produce more products for 2 reasons. One, the forward reaction is easier in terms of activation energy than the reverse reaction. Two, the reverse reaction requires 2 species coming together rather than 1. Both have to be at the correct angle etc. That could be thought of as a way that entropy affects the equilibrium position.
I am not sure how to deal with the entropy change in a reaction like A = B. I suppose if B has greater entropy than A this will slow the reverse reaction due to a lower frequency factor A as it is more difficult for B to be in the correct configuration to react.