# Is Gibbs Free Energy of formation equal to Gibbs Free Energy of transition state?

Is activation energy equivalent to Gibbs Free Energy of transition state as related by Eyring equation? $$E_a=\Delta ^\ddagger G \, \, ?$$

Is Gibbs Free Energy of transition state defined by the Gibbs Free Energy of formation? $$\Delta ^\ddagger G :=\Delta _\mathrm{f} G \, \, ?$$ If not, which are the conditions for which such an approximation (second equation) would apply?

The Arrhenius equation

$$k=A\exp\left(-\frac{E_a}{RT}\right)$$ places all of the $$T$$-dependence in the exponential factor. The pre-exponential factor is not assumed to be temperature-dependent. By contrast, the Eyring equation

$$k=\frac{\kappa k_\mathrm{B}T}{h}\exp\left(-\frac{\Delta ^\ddagger G^⦵}{RT}\right)$$

which is justified on the basis of statistical-mechanical principles, displays a a temperature-dependence in the pre-exponential factor. So the exponential terms are not expected to be equal, at least theoretically, unless the pre-exponential factor in the Eyring equation becomes T-independent by some coincidental cancellation of terms.

As regards the second equation, I would define the free energy of the transition state more accurately as

$$\Delta ^\ddagger G :=\Delta _\mathrm{f} G(\mathrm{ts})-\Delta _\mathrm{f} G(\mathrm{ini})$$

• Tech detail: please note that using a circled minus symbol (⊖, \ominus) for the standard state isn't correct. IUPAC suggests either a superscript circle, or a plimsoll symbol. Since it looks like you intended to use plimsoll, you want to stick with a Unicode symbol ⦵ as getting it right with MathJax (or LaTeX, for that matter) is rather tricky. Commented Jul 6, 2019 at 8:46
• @andselisk That's not the standard state, that's the transition state, and I guess I wanted plimsoll but picked the nearest thing based on detexify, but technically I guess you're right, I didn't check the with IUPAC. I remember the post not so long ago about plimsoll, that was a good one. Commented Jul 6, 2019 at 17:03
• Either way, a transition state must have a reference point, which is normally the standard state, so there is no contradiction. $Δ^\ddagger G^⦵$ is perfectly fine. Some books and even a Wikipedia article on transition state are using a circled minus — probably due to typographic difficulties in getting the plimsoll symbol right. Here we don't have this problem, hence my edit to comply with IUPAC rules. Commented Jul 6, 2019 at 17:37