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The initial rate for a reactant is the rate at which it is used up in the reaction (which is stoichiometric). Its unique rate is equal to its rate divided by its stoichiometric coefficient. All reactants and products in a reaction have the same (or opposite sign- depending on if its being used or made) unique rate.
The initial rate is the instantaneous rate at the beginning of a reaction (this is a derivative), the unique rate uses the average rates and accounts for the stoichiometric coefficients of the reactants and products (this is not a derivative).
The initial rate is the rate when there are only reactants, and it is the rate that the reactants are used up in a reaction. The unique rate is the rate a molecule is used up divided by its stoichiometric coefficient, all molecules in a reaction will have the same unique rate.
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