During the kinetic study of the reaction, $2 A + B \rightarrow C + D$, following results were obtained $…
During the kinetic study of the reaction, $2 A + B \rightarrow C + D$, following results were obtained
$
\begin{array}{|c|c|c|c|}
\hline \text{Run} & [A] / \text{mol L}^{-1} & [B] / \text{mol L}^{-1} & \text{Initial rate of formation of} D / \text{mol L}^{-1} \text{min}^{-1} \\
\hline \text{I} & 0.1 & 0.1 & 6.0 \times 10^{-3} \\
\hline \text{II} & 0.3 & 0.2 & 7.2 \times 10^{-2} \\
\hline \text{III} & 0.3 & 0.4 & 2.88 \times 10^{-1} \\
\hline \text{IV} & 0.4 & 0.1 & 2.40 \times 10^{-2} \\
\hline
\end{array}
$
Based on the above data which one of the following is correct?
rate $=\mathrm{k}[\mathrm{A}]^2[\mathrm{~B}]$
rate $=k[A][B]$
rate $=\mathrm{k}[\mathrm{A}]^2[\mathrm{~B}]^2$
rate $=\mathrm{k}[\mathrm{A}][\mathrm{B}]^2$
Solution
Let the order of reaction with respect to $\mathrm{A}$ is $x$ and with respect to $B$ is $y$. Thus,
$\text {rate }=\mathrm{k}[\mathrm{A}]^{\mathrm{x}}[\mathrm{B}]^{\mathrm{y}}$
For the given cases,
(I) rate $=\mathrm{k}(0.1)^{\mathrm{x}}(0.1)^{\mathrm{y}}=6.0 \times 10^{-3}$
(II) rate $=\mathrm{k}(0.3)^{\mathrm{x}}(0.2)^{\mathrm{y}}=7.2 \times 10^{-2}$
(III) rate
$=\mathrm{k}(0.3)^{\mathrm{x}}(0.40)^{\mathrm{y}}=2.88 \times 10^{-1}$
(IV) rate $=\mathrm{k}(0.4)^{\mathrm{x}}(0.1)^{\mathrm{y}}=2.40 \times 10^{-2}$
On dividing Eq. (I) by (IV), we get
$\left(\frac{0.1}{0.4}\right)^x\left(\frac{0.1}{0.1}\right)^y=\frac{6.0 \times 10^{-3}}{2.4 \times 10^{-2}}$
or
$\left(\frac{1}{4}\right)^x=\left(\frac{1}{4}\right)^1$
$\therefore \quad \mathrm{x}=1$
On dividing Eq. (II) by (III), we get
$\begin{aligned}
& & \left(\frac{0.3}{0.3}\right)^x\left(\frac{0.2}{0.4}\right)^y & =\frac{7.2 \times 10^{-2}}{2.88 \times 10^{-1}} \\
\text { or } & & \left(\frac{1}{2}\right)^y & =\frac{1}{4} \\
\text { or } & & \left(\frac{1}{2}\right)^y & =\left(\frac{1}{2}\right)^2 \\
\therefore & & y & =2
\end{aligned}$
$\therefore$ Thus, rate law is,
$\text {rate }=\mathrm{k}[\mathrm{A}]^1[\mathrm{~B}]^2 \text { or }=\mathrm{k}[\mathrm{A}][\mathrm{B}]^2$