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In my book it is written that if we have an reaction equation such that $$\ce{a A (g) + b B(g) + c C(g) -> product}$$ The rate expression of the reaction will be $$R = k[\ce{A}]^a [\ce{B}]^b [\ce{C}]^c.$$

But, when I researched on expression of rate law, I understood that $a$, $b$ and $c$ are not coefficients of the reaction. But, in most part of my book, the coefficients are taken as the powers $a$, $b$ and $c$.

Is my book wrong or is there something that I don't know about rate expression?

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But when i researched on rate of expression i undertood that a, b and c are not coefficients of the reaction

You are correct. $a,b$ and $c$ summed together represent the order of the reaction. Note that this is very different from the molecularity of a reaction.

But in most part of my book the coefficients are taken as the powers a b and c

Without knowing the reactions for which this is done, it is impossible to comment further. However, assuming that the book is correct, we can say, that in those specific cases, your book was dealing with elementary reactions.

Recall that molecularity of an elementary reaction is the same as its order. However, most natural reactions occur through complex mechanisms, and in those cases, molecularity is rarely equal to its order.

For more details on the difference between molecularity and order of a reaction, have a look at this question.

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    $\begingroup$ The reaction is $Pb(aq)+2I(aq)⟹PbI₂(s)$ will as this reaction occur in one step we can say that it is elementary so we can write rate of reaction as $R=[Pb][I]^2$ but if the reaction occur in several step it's Rate expression would be the rate expression of its slowest step am I right? $\endgroup$ – Marva Jami Feb 19 '18 at 10:53
  • $\begingroup$ @MarvaJami Yes you are absolutely correct Marva. Well done! :) $\endgroup$ – Gaurang Tandon Feb 19 '18 at 10:54

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