Question

The rate of the reaction
$2N_2{O}_5\to 4N{O}_2+O_2$
can be written in three ways:
$\frac{-d\left[N_2{O}_5\right]}{dt}=k\left[N_2{O}_5\right]$
$\frac{d\left[N{O}_2\right]}{dt}=k^{{}^{\prime }}\left[N_2{O}_5\right]$
$\frac{d\left[O_2\right]}{dt}=k^{\prime \prime }\left[N_2{O}_5\right]$
The relationship between $k$, $k^{{}^{\prime }}$ and between $k$, $k^{\prime \prime }$ are:

• A. $k^{{}^{\prime }}=2k;k^{\prime \prime }=k$
• B. $k^{{}^{\prime }}=2k;k^{\prime \prime }=\frac{k}{2}$
• C. $k^{{}^{\prime }}=2k;k^{\prime \prime }=2k$
• D. $k^{{}^{\prime }}=k;k^{\prime \prime }=k$

Answer 1

The correct option is B $k^{{}^{\prime }}=2k;k^{\prime \prime }=\frac{k}{2}$

$\text{Rate}=-\frac{1}{2}\frac{d\left[N_2{O}_5\right]}{dt}=\frac{1}{4}\frac{d\left[N{O}_2\right]}{dt}=\frac{d\left[O_2\right]}{dt}$
$\therefore \frac{1}{2}k\left[N_2{O}_5\right]=\frac{1}{4}{k}^{{}^{\prime }}\left[N_2{O}_5\right]=k^{\prime \prime }\left[N_2{O}_5\right]$
or, $\frac{k}{2}=\frac{k^{{}^{\prime }}}{4}=k^{\prime \prime }$
Hence, $k^{{}^{\prime }}=2k$ and $k^{\prime \prime }=\frac{k}{2}$