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To me it makes sense that the activation energy $E_\mathrm a$, as given by the Arrhenius equation, doesn't depend on the temperature as the needed energy (potential energy) to start a reaction has nothing to do with the temperature (kinetic energy) of the molecules. And a higher temperature simply means that more molecules have sufficient energy to proceed in the reaction $E\gt E_\mathrm a$.

So I don't understand why activation energy would depend on temperature, but this equation shows that it does $E_\mathrm a=\Delta H^{\ddagger}+RT$.

Can someone explain why?

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Although the question has been already answered, I will like to add a bit more of detail.

The expression you are quoting come from the thermodynamic formulation of the TST that expresses the dependence of the rate constant of temperature as

$$k(T)=\frac{k_\mathrm bT}h\exp\left(-\frac{-\Delta G^\ddagger_0}{RT}\right)$$

and can be partitioned in an entropic and enthalpic contributions

$$k(T)=\frac{k_\mathrm bT}h\exp\left(-\frac{-\Delta S^\ddagger_0}R\right)\exp \left(-\frac{-\Delta H^\ddagger_0}{RT}\right)$$

From this equation you can obtain the activation energy $E_\mathrm a$ using the relation

$$\frac{\mathrm d\ln k(t)}{\mathrm dT}=-\frac{E_\mathrm a}{RT^2}$$

Using this equation you will obtain

$$E_\mathrm a=\Delta U^\ddagger_0+RT$$

Since $H=U+pV$

$$\Delta H^\ddagger_0=\Delta U^\ddagger_0+p\,\Delta V^\ddagger_0+RT$$

For a condensed phase reaction $\Delta V^\ddagger_0\approx0$ and

$$E_\mathrm a=\Delta H^\ddagger_0+RT$$

For a gas phase reaction, $p\Delta V^\ddagger_0=\Delta n^\ddagger RT$.

For a gas phase unimolecular reaction $\Delta n^\ddagger=0$, the result is equal to that the obtained by a condensed phase reaction, while for a second-order reaction $\Delta n^\ddagger=-1$, and

$$E_\mathrm a=\Delta H^\ddagger_0+2RT$$

You can find a more detailed discussion in Keith Laidler, Chemical kinetics 3rd. ed., Harper Collins Publishers, (1987); Jeffrey I. Steinfeld, Joseph S. Francisco, William L. Hase, Chemical Kinetics and Dynamics, Prentice Hall (1989); or Paul L. Houston, Chemical Kinetics and Reaction Dynamics, McGraw-Hill (2001).

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