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Increasing the concentration of a reactant does NOT increase the rate of the reaction. But as stated above, an increase in concentration would have a greater effect on a second rate reaction than a first rate reaction due to the exponent.
the difference between first order and second order rate laws is that a first order rate law will typically have one reactant that is first order and a second order rate law will have either one reactant that is second order or two first order reactants.
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