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In a first order reaction, there will be one reactant present in the rate law. For a second order reaction, you can either have a rate law with one reactant to the second order, or with two reactants both to the first order. Two examples of second order reactions are rate=k[CO2]^2 or rate=k[CO2][H2O]. The way I think of it is that if you add the exponents of all the reactants in the rate law, that determines the order.
Also, in a first order reaction, the rate of the reaction is doubled while in a second order reaction, the rate of the reaction is quadrupled.
In first order reactions, the rate is proportional to the concentration raised to the first power. In second order reactions, the rate is proportional to the concentration raised to the second power.
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