In the Arrhenius equation, the term activation energy (E) is used to describe the energy required to reach the transition state, and the exponential relationship  holds.  In transition state theory, a more sophisticated model of the relationship between reaction rates and the transition state, a superficially similar mathematical relationship, the Eyring equation, is used to describe the rate constant of a reaction: .  However, instead of modeling the temperature dependence of reaction rate phenomenologically, the Eyring equation models individual elementary steps of a reaction.  Thus, for a multistep process, there is no straightforward relationship between the two models.  Nevertheless, the functional forms of the Arrhenius and Eyring equations are similar, and for a one-step process, simple and chemically meaningful correspondences can be drawn between Arrhenius and Eyring parameters.
Instead of also using E, the Eyring equation uses the concept of Gibbs energy and the symbol ΔG to denote the Gibbs energy of activation to achieve the transition state. In the equation, k and h are the Boltzmann and Planck constants, respectively. Although the equations look similar, it is important to note that the Gibbs energy contains an entropic term in addition to the enthalpic one.  In the Arrhenius equation, this entropic term is accounted for by the pre-exponential factor A.  More specifically, we can write the Gibbs free energy of activation in terms of enthalpy and entropy of activation: .  Then, for a unimolecular, one-step reaction, the approximate relationships  and  hold.  Note, however, that in Arrhenius theory proper, A is temperature independent, while here, there is a linear dependence on T.  For a one-step unimolecular process whose half-life at room temperature is about 2 hours, ΔG is approximately 23 kcal/mol.  This is also the roughly the magnitude of E for a reaction that proceeds over several hours at room temperature.  Due to the relatively small magnitude of TΔS and RT at ordinary temperatures for most reactions, in sloppy discourse,  E, ΔG, and ΔH are often conflated and all referred to as the "activation energy".
The enthalpy, entropy and Gibbs energy of activation are more correctly written as ΔH, ΔS and ΔG respectively, where the o indicates a quantity evaluated between standard states. However, some authors omit the o in order to simplify the notation.
The total free energy change of a reaction is independent of the activation energy however. Physical and chemical reactions can be either exergonic or endergonic, but the activation energy is not related to the spontaneity of a reaction. The overall reaction energy change is not altered by the activation energy.