I know that activation energy for a reaction is the extra energy given to the reactants to reach the threshold energy so that they can collide and react. But then, why is it said that the activation energy does not depend on the temperature?
If the temperature is higher, the reactant molecules have a higher kinetic energy, and thus the energy difference between the threshold energy and the energy of the molecules will decrease, that is, the activation energy will decrease.
Answer
There is a simple (some might say simplistic) way to get an intuition about this and it involves thinking at the molecular or atomic level rather than about the bulk properties of the reaction (the thermodynamic view).
For a simple reaction where molecule A has to bang into molecule B to create molecule C, the reaction will only happen if the amount of energy involved in the collision is large enough. The amount of energy involved depends on the nature of the reaction which is determined by the electronic structures of A and B. Within limits, those electronic structures don't vary with temperature and a given reaction will only occur when there is enough energy in the collision for the reaction to happen.
What changes with temperature isn't the energy required for the reaction to occur, it is the number of molecules that have enough energy to overcome the barrier. At higher temperatures more molecules have that energy, but the amount of energy required doesn't change. The amount of energy require to initiate the reaction is roughly equivalent to the activation energy. It can be calculated by looking at the way the overall rate of the reaction changes with temperature. So the rate of reaction changes with temperature but not the barrier to the reaction happening.
This view is a little simplified but it gives the right intuition about the problem.
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