Is there a error at page 566 of Atkins Physical Chemistry? [closed]








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At page 556 I found this question being analyzed and the rate law is provided.
However, when I apply steady state approximation, I can't find the result without the factor 2. 
Is it me being wrong?


[![enter image description here](https://i.stack.imgur.com/DRxSB.jpg)](https://i.stack.imgur.com/DRxSB.jpg)




The factor of 2 originates from the definition of $k\_b$ (I thought it was an error, but it turns out IUPAC guidelines lead to its presence). 


If $k\_\mathrm{a}' \gg k\_\mathrm{b}[\ce{O2}]$ and defining


$$K\_\mathrm{a} = \frac{k\_\mathrm{a}}{k\_\mathrm{a}'}$$


it follows from the equation posted in the OP that


$$\frac{\mathrm{d}[\ce{NO2}]}{\mathrm{d}t} = 2k\_\mathrm{b}K\_\mathrm{a}[\ce{NO}]^2[\ce{O2}]$$


However, starting from scratch we *could* write that


$$\frac{\mathrm{d}[\ce{NO2}]}{\mathrm{d}t} = k\_\mathrm{b}[\ce{N2O2}][\ce{O2}]$$


(this is not the standard IUPAC convention however).
Then, if


$$K\_\mathrm{a} = \frac{k\_\mathrm{a}}{k\_\mathrm{a}'} = \frac{[\ce{N2O2}]}{[\ce{NO}]^2}$$


it follows that


$$\frac{\mathrm{d}[\ce{NO2}]}{\mathrm{d}t} = k\_\mathrm{b}K\_\mathrm{a}[\ce{NO}]^2[\ce{O2}]$$


and the factor of $2$ is not there. In the fourth version of Atkins' book (which I parallel in this derivation from "scratch") the factor of 2 is missing, but in that book the analysis does not proceed by dissecting the full rate law before imposing the pre-equilibrium approximation. The OP presumably sources a later version of the book. As noted in a comment, IUPAC guidelines call for incorporation of that factor of 2. One should write 


$$\frac{\mathrm{d}[\ce{NO2}]}{\mathrm{d}t} = 2k\_\mathrm{b}[\ce{N2O2}][\ce{O2}]$$ because that step of the reaction reads $$\ce{N2O2 + O2->2NO2}$$ The stoichiometric factor of 2 means two moles of 
$\ce{NO2}$ are produced for every mole of reaction.


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