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The order of the reaction, n, can tell us a lot about the type of reaction we are dealing with. For example, if n=0, a zeroth order reaction, we know that the reaction proceed at a constant rate regardless of the time that has passed--this is often the case when an enzyme is fully saturated. But, if we are just given the constant k, then we wouldn't be able to tell from k alone that this is a zeroth order reaction.
The order of a reactant, n, explains more about the reaction mechanism because it directly describes how the concentration of a reactant affects the rate of the reaction. From this, the rate law can be used to calculate the reaction rate using known concentrations of reactants.
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