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Coefficients play a role in the unique rate law that states where aA --> bB, the rate = (-1/a)(dA/dt) = (1/b)(dB/dt). Because of this, you can identify how and individual product/reaction rate compares to another based on it's coefficient as shown in 15.1.
As others have mentioned, coefficients affect the unique reaction rate or the rate law for elementary steps of a reaction. In the case of the unique reaction rate, the stoichiometric coefficients are in the denominator of the rate of consumption/formation of a reactant/product to get the unique rate for the reaction (a negative is also included when using reactants so the rate becomes positive). In terms of the rate laws for elementary steps, the coefficients are used as the powers (order) corresponding to each reactant concentration. However, you can't use the coefficients to generalize a reaction's order in its rate law otherwise.
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