Q:

Proof for Arrhenius equation

The Arrhenius equation states that:
$$k=A\mathrm e^{\frac{-E_\mathrm a}{R T}}$$
What is the mathematical proof for this equation?

A:

In science, theories, hypothesis and particular mathematical models cannot be proven to be right.
It is the principal limitation. They can be proven right only formally as mathematical construct, proving consistency. In relation to reality, there can be just confirmation or refutation of their agreement with experimental data.
The Arrhenius equation is based on the model that supposes:
There is a frequency A of geometrically  favourable molecular collisions, that might lead to their reaction. As simplification, it is  implied A does not depend on T. But we know it does, as molecules move faster at higher T.
The exponential factor is based on the Boltzmann distribution, predicting the probability molecules would have enough thermal energy to pass the energy barrier - the reaction activation energy.
The confirmation can be done by measuring dependency of reaction kinetics and equilibrium constants on temperature. The latter follows  van't Hoff equation,  that is derived from Arrhenius equation, applying it to both forward and reversed reaction.

