For a reaction, \(\mathrm{A}+\mathrm{B} \rightarrow\) products, the rate of the reaction at various concentrations are given below



The rate law for the above reaction is

Let order of reaction wrt A and \(\mathrm{B}\) is \(\mathrm{m}\) and \(\mathrm{n}\) respectively. Then
\[
\begin{aligned}
\text { rate } &=\mathrm{k}[\mathrm{A}]^{\mathrm{m}}[\mathrm{B}]^{\mathrm{n}} \\
2 &=\mathrm{k}[0.2]^{\mathrm{m}}[0.2]^{\mathrm{n}}
\end{aligned}
\]
\[
4=k[0.2]^{m}[0.4]^{n}
\]
\[
36=\mathrm{k}[0.6]^{\mathrm{m}}[0.4]^{\mathrm{n}}
\]
On comparing Eqs. (i) and (ii), we get
\[
\begin{aligned}
& \frac{4}{2} &=\frac{[0.4]^{\mathrm{n}}}{[0.2]^{\mathrm{n}}} \\
\Rightarrow & 2 &=2^{\mathrm{n}} \\
\Rightarrow & \mathrm{n} &=1
\end{aligned}
\]
Again on comparing Eqs. (ii) and (iii), we get
\[
\begin{aligned}
& & \frac{36}{4} &=\frac{[0.6]^{\mathrm{m}}}{[0.2]^{\mathrm{m}}} \\
\Rightarrow & & 9=3^{2}=3^{\mathrm{m}} \\
\Rightarrow & & \mathrm{m}=2
\end{aligned}
\]
\(\therefore\) Rate law for the given reaction will be
\[
\mathrm{r}=\mathrm{k}[\mathrm{A}]^{2}[\mathrm{~B}]
\]