In the graph of the reaction co-ordinate, is the $y$-axis that is used $∆H,$ $∆U$ or $∆G$?
Single molecule reaction
Some diagrams refer to a single molecule reacting. If that is the case, several quantities are not defined or ill-defined:
- entropy of mixing
In those diagrams, the best label would be potential energy, and the energy difference between reactant and transition state should be labeled activation energy.
If the diagram refers to an ensemble reacting (reaction mixture with sufficient number of molecules to have macroscopic properties), the y-axis could correspond to Gibbs energy, enthalpy or internal energy.
If the y-axis represents Gibbs energy, the energy difference between reactant and transition state should be labeled $\Delta G^\ddagger$. Those diagrams make sense when discussing how equilibrium constants remain unchanged but reaction rates change when a catalyst is added.
If the y-axis represents enthalpy or internal energy, the energy difference between reactant and transition state should be labeled $E_a$ for activation energy. Those diagrams make sense when discussing reaction rates in the context of collision theory.
What is most popular?
Looking at about 50 images of reaction energy diagrams found in a google image search, I found a wide variety, and lots of mixing and matching (i.e. incompatible quantities in a single diagram). This tells you that the diagram is conceptual, and the concepts are not always clearly articulated in diagrams.
Following is an example of Reaction Coordinate Diagram. I labelled all necessary details for your convenient. Please note that the Blue path is for the same reaction when catalyst is involved. The difference is the activation energy, which is lower in the presence of a catalyst.