# For thermal decomposition of ozone to oxygen:Step (i):$O_3\;\;\;\overset{\overset{k_1}{\rightleftharpoons}}{k_{-1}}\;\;\; O_2+O$.Step (ii):$O_3+O\;\;\;\underrightarrow{k_2}\;\;\;2O_2$ then rate will be $\{k_{-1}[O_2] > > k_2[O_3]\}$

$\begin{array}{1 1}(a)\;r=-k[O_3][O_2]\\(b)\;r=-k[O_3]/[O_2]\\(c)\;r=-k[O_3]^2/[O_2]\\(d)\;r=-k[O_3]^2[O_2]\end{array}$

Ozone decomposes in steps 1 & 2 and is formed in reverse of step 1.
So $r=\large\frac{d[O_3]}{dt}$$=-k_1[O_3]+k_{-1}[O_2][O]-k_2[O_3][O] Using steady state approximation for O atom which is intermediate . \large\frac{d[O]}{dt}=$$k_1[O_3]-k_{-1}[O_2][O]-k_2[O_3][O]=0$
$k_1[O_3]=\{k_{-1}[O_2]+k_2[O_3]\}[O]$
$[O]=\large\frac{k_1[O_3]}{k_{-1}[O_2]+k_2[O_3]}$
Substituting [O]
$r=-k_1[O]+\large\frac{kk_{-1}[O_2][O_3]}{k_{-1}[O_2]+k_2[O_3]}-\frac{k_1k_2[O_3]^2}{k_{-1}[O_2]+k_2[O_3]}$
$\;\;\;=\large\frac{-k_1k_{-1}[O_2][O_3]-k_1k_2[O_3]^2+k_1k_{-1}[O_2][O_2]-k_1k_2[O_3]^2}{k_{-1}[O_2]+k_2[O_3]}$
$r=\large\frac{-2k_1k_2[O_3]^2}{k_{-1}[O_2]+k_2[O_3]}$
$r=-k\large\frac{[O_3]^2}{[O_2]}$
Hence (c) is the correct answer.