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An example I like to think of is having to reactants, A and B, and a few different experiments changing the concentrations of each, and analyzing the change in the rate. When A is increased, let's say that the reaction shows no change. You do the same for B, still no change. I think of zero order as any increase or decrease of the reactant with zero order, there is ZERO change to rate.
a zero order reaction is one whose rate is independent of the concentration. An example is when nitrous oxide decomposes to nitrogen and oxygen. In the presence of a catalyst, platinum, we see that changing concentration of N2O has no effect on the rate of decomposition so we know the rate only depends on the rate constant, k. More precisely, as long as there is sufficient n2o to react with platinum the rate is not determined by concentration of n2o.
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