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The intermediate in a series of reactions can be identified by seeing which element is both in the products of one reaction and reactants of another (it will cancel out as they are on opposite sides). Since it is not written in the overall reaction (as it is cancelled out), it is not included in the rate law equation.
Intermediates are easily recognizable in a reaction mechanism, which is the sequence of elementary reactions to go from reactants to products. The molecules that are formed and consumed in these elementary reactions are intermediates
Intermediates ultimately get canceled out when you write out chemical equations in the full, so they will not show up in the rate law because only the initial reactants and final products have an influence on the rate law.
An intermediate shows up in the products of step one but the reactants of step two and therefore is used up in the second reaction so none will remain after the whole thing is over. The reason this does not affect the rate law is because the rate law is determined by the reactants of the slow step and the intermediate has no effect on this.
The intermediate cancels out in the rate determining step (before the final overall equation), and is not present in the overall reaction --> as its presence in the reactant of one reaction and the product of another reaction, makes itself cancel out.
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