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The order of a reaction is just the exponents of all the reactant concentrations in the rate equation added up. So if a rate equation looked like this: rate = k[A]^2[B]^2[C] we would add 2+2+1 to get 5. So it would be a fifth order reaction.
For kinetics, the order of the reaction with respect to some sort of substance (ex. reactant, product) is going to be the exponent to which it's concentration form is raised. We use this in the rate equation and we can see that the concentration of this substance is raised to some sort of power.
To add on, the order of a reaction reveals to us what power of the concentration the rate is proportional to. For example, in a first-order reaction, its rate is proportional to the first power of the concentration. Also changing the concentration of different reactants has different effects on the rate for different orders of reactions. Doubling the concentration of a reactant in a first-order reaction doubles the reaction rate, but doing the same to a reactant in a second-order reaction increases the reaction rate by a factor of 4.
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