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The value of the rate constant K varies depending on the temperature and activation energy which can be seen through the Arrhenius equation. For example, raising the temperature of the reaction will increase the value of the rate constant K and therefore increase the speed at which the reaction is occurring.
Essentially, at different temperatures and activation energies, K changes. You calculate it the same way, using concentrations, but you're going to get a different value at different states. We don't have a way to calculate new K's based on old ones yet, but we might learn one.
K constant just describes how fast a reaction can occur. The only way to speed up a reaction is to change the amount of energy input as in raise its temperature or lower the activation energy needed, I. E. Add a catalyst perhaps.
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