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Looking at the rate law, the units for the rate should end up being M/s, so depending on the order of the reaction, the units of k should be different. If you write out the units of each concentration and the rate, you should be able to figure out the units for k. For example, for a first order reaction, rate = k [A] and since rate is M/s and [A] is M, k should be 1/s. Doing this same thing will give you M^-1s^-1 for second order reactions, M^-2s^-1 for third order reactions, and M/s for zeroth order reactions.
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