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The correct difference between first- and second-order reactions is that
the rate of a first-order reaction does dependon reactant concentrations; the rate of a second-order reaction does not depend on reactant concentrations
a first-order reaction can be catalyzed; a second-order reaction cannot be catalyzed
the half-life of a first-order reaction does not depend on $$[A]_{0}$$ the half-life of a second-order reaction does depend on $$[A]_{0}$$
the rate of a first-order reaction does not depend on reactant concentrations; the rate of a second-order reaction does depend on reactant concentrations
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