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The rate constant changes with temperature because the rate of a reaction is dependent on temperature. This has to do with activation energy and the likeliness of a reaction happening because a greater presence of reactants (concentration) or a higher temperature, meaning more kinetic energy, which means molecules are more likely to collide in the right orientation to make the reaction happen. Normally, at higher temperatures, the rate constant increases.
I think under different temperature conditions, a reaction can only proceed under a high temperature. The process may be spontaneous but very slow so an increase in temperature will speed up the process. The example given in class was the car to diamond reaction I believe.
Another way of explaining this is that at higher temperatures, the particles have much more energy in them and will move more and theoretically "clash" with particles of the other reactant, thus accelerating the speed of the reaction.
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