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In a first order reaction, the reaction rate depends on the concentration of a single reactant. In a second order reaction, the reaction rate depends on either the concentration of two different reactants raised to the first power, or the concentration of a single reactant raised to the second power. When the reactants are doubled in a first order reaction, the rate of reaction doubles, while if you double the reactants in a second order reaction, the reaction rate will quadruple.
There are many ways a reaction can become first or second order, but the main difference is how many reactants the rate depends upon. If the rate depends on one reactant, then it is first order. If the rate doesn't depend on a reactant (say it depends on a catalyst instead) then the reaction is zeroith order. If the rate depends on two reactants, then it is second order and so on. Another way a reaction can be second order is if it uses two moles of the same reactant, and that reactant changes the rate. This is where we get [a]^2 in our rate law, and is another way a reaction can be second order.
Think about it from a mathematical view, a first order means it is dependent on one reactant to the first power. A second order means either a) dependent on two reactants to the first power or one reactant to the second power, in order for reaction to proceed.
The rate of a first order reaction depends only on the concentration of one reactant. However, the rate of a second order reaction depends on the concentration of two reactants to the first power or one reactant to the 2nd power.
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