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Differential rate laws describe the change in reactant or product concentrations as a function of time. Integrated rate laws describe the actual concentrations of reactants or products as a function of time. The only 2k I can think of is for the half life equation of a zero order reaction which is [A]o/2k.
Differential rate laws express the instantaneous rate of a reaction as a function of reactant concentrations. Integrated rate laws express the concentration of reactant remaining as a function of time. You'll have 2k as a constant if you solve for the rate of consumption/formation of a particular species with a stoichiometric coefficient of 2. This results from the algebraic rearrangement of d[A]/(a dt) = k [A]^n to d[A]/dt = a*k [A]^n.
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