Chemistry - Chemical Kinetics Question with Solution | TestHub

For the reaction, ${\mathrm{A}}_2\stackrel{{\mathrm{K}}_1}{\to }2\mathrm{A}$

${\mathrm{r}}_{\mathrm{f}}=-\frac{\mathrm{d}\left[{\mathrm{A}}_2\right]}{\mathrm{dt}}=\frac{1}{2}{\left(\frac{\mathrm{d}\left[\mathrm{A}\right]}{\mathrm{dt}}\right)}_1={\mathrm{K}}_1\left[{\mathrm{A}}_2\right]$

${\left(\frac{\mathrm{d}\left[\mathrm{A}\right]}{\mathrm{dt}}\right)}_1=2{\mathrm{K}}_1\left[{\mathrm{A}}_2\right]$

For the reaction, $2\mathrm{A}\stackrel{{\mathrm{k}}_{-1}}{\to } {\mathrm{A}}_2$  

${\mathrm{r}}_{\mathrm{b}}=-\frac{1}{2}{\left(\frac{\mathrm{d}\left[\mathrm{A}\right]}{\mathrm{dt}}\right)}_2={\mathrm{K}}_{-1}{\left[\mathrm{A}\right]}^2$

${\left(\frac{\mathrm{d}\left[\mathrm{A}\right]}{\mathrm{dt}}\right)}_2=-2{\mathrm{K}}_{-1}{\left[\mathrm{A}\right]}^2$

Hence, the overall rate is given by:

$\frac{\mathrm{d}\left[\mathrm{A}\right]}{\mathrm{dt}}=2{\mathrm{k}}_1\left[{\mathrm{A}}_2\right]-2{\mathrm{k}}_{-1}{\left[\mathrm{A}\right]}^2$