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The unique rate equation sets the rate of change of product concentration and reactant concentration over time equal. The equation takes into account the stoichiometric coefficients in the reaction. For example if one product has a stoichiometric coefficient of 2 and all other stoichiometric coefficients are 1, the unique rate equation makes it 1/2(d[P]/dt). Since this product's concentration rate of change is twice as fast as the other components, a coefficient of 1/2 basically gives the concentration rate of change as if the stoichiometric coefficient was 1.
The unique rate equation tells us that there is a ratio of products and reactants that is equal in terms of the rate in which the reaction is proceeding. Coefficients in the rate law relate to the coefficients of the balanced equation and shows the ratio between them.
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