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To determine the order of a reaction with respect to each reactant, we use the method of initial rates.
When applying the method of initial rates to a reaction involving two reactants, A and B, it is necessary to conduct two trials in which the concentration of A is held constant, and B changes, as well as two trials in which the concentration of B is held constant, and A changes.
A reaction is said to be second-order when the overall order is two.
For a reaction with the general form $aA+bB\rightarrow C$, the reaction can be second order in two possible ways.
It can be second-order in either A or B, or first-order in both A and B.
If the reaction were second-order in either reactant, it would lead to the following rate laws:
The second scenario, in which the reaction is first-order in both A and B, would yield the following rate law:
Note that on the right side of the equation, both the rate constant k and the term $(0.200)^y$ cancel.
This was done intentionally, because in order to determine the reaction order in A, we need to choose two experimental trials in which the initial concentration of A changes, but the initial concentration of B is constant, so that the concentration of B cancels.
Our equation simplifies to:
Therefore, the reaction is second-order in A.
Determining Reaction Order in B
Next, we need to determine the reaction order for B.
We do this by picking two trials in which the concentration of B changes, but the concentration of A does not.
Trials 1 and 3 will do this for us, and we set up our ratios as follows: