91Ó°ÊÓ

The overall reactioncan be obtained by adding all

$$\begin{gathered} 2{\mathrm{H}}_2{\mathrm{SO}}_4→{\mathrm{H}}_3{\mathrm{O}}^{+}+{{\mathrm{HSO}}_4}^{-}+{\mathrm{SO}}_3 \\ {\mathrm{SO}}_3+{\mathrm{C}}_6{\mathrm{H}}_6→\mathrm{H}\left({\mathrm{C}}_6{{\mathrm{H}}_5}^{+}\right){{\mathrm{SO}}_3}^{-} \\ \mathrm{H}({\mathrm{C}}_6{\mathrm{H}}_5+){{\mathrm{SO}}_3}^{-}+{{\mathrm{HSO}}_4}^{-}→{\mathrm{C}}_6{\mathrm{H}}_5{{\mathrm{SO}}_3}^{-}+{\mathrm{H}}_2{\mathrm{SO}}_4 \\ {\mathrm{C}}_6{\mathrm{H}}_5{{\mathrm{SO}}_3}^{-}+{\mathrm{H}}_3{\mathrm{O}}^{+}→{\mathrm{C}}_6{\mathrm{H}}_5{\mathrm{SO}}_3\mathrm{H}+{\mathrm{H}}_2\mathrm{O} \\ \stackrel{¯}{{\mathrm{C}}_6{\mathrm{H}}_6+{\mathrm{H}}_2{\mathrm{SO}}_4→{\mathrm{C}}_6{\mathrm{H}}_5{\mathrm{SO}}_3\mathrm{H}+{\mathrm{H}}_2\mathrm{O}} \end{gathered}$$

Rate is expressed in terms of the product of the concentration of the reactants raised to the power of their respective stoichiometric coefficients. The actual rate law is written on the basis of the slow step:

$$\begin{gathered} {\mathrm{SO}}_3+{\mathrm{C}}_6{\mathrm{H}}_6\stackrel{{\mathrm{k}}_2}{→}\mathrm{H}\left({\mathrm{C}}_6{{\mathrm{H}}_5}^{+}\right){{\mathrm{SO}}_3}^{-} \\ {\mathrm{rate}}_2={\mathrm{k}}_2\left[{\mathrm{SO}}_3\right]\left[{\mathrm{C}}_6{\mathrm{H}}_6\right] \end{gathered}$$

Where is the rate constant for the step. Since, ${\mathrm{SO}}_3$is an intermediate, we have to substitute its concentration. Write the rate of reaction in terms of formation of ${\mathrm{SO}}_3$from the first reaction as follows:

$$\begin{gathered} 2{\mathrm{H}}_2{\mathrm{SO}}_4→{\mathrm{H}}_3{\mathrm{O}}^{+}+{{\mathrm{HSO}}_4}^{-}+{\mathrm{SO}}_3 \\ \mathrm{rate}=\frac{∆\left[{\mathrm{SO}}_3\right]}{∆\mathrm{t}} \\ {\mathrm{k}}_1{\left[{\mathrm{H}}_2{\mathrm{SO}}_4\right]}^2=\left[{\mathrm{SO}}_3\right] \end{gathered}$$

Where ${\mathrm{k}}_1$is the rate constant for the step.

Substitute the value in rate law expression to obtain the overall rate law:

$$\begin{gathered} {\mathrm{rate}}_2={\mathrm{k}}_2{\mathrm{k}}_1{\left[{\mathrm{H}}_2{\mathrm{SO}}_4\right]}^2\left[{\mathrm{C}}_6{\mathrm{H}}_6\right] \\ =\mathrm{k}{\left[{\mathrm{H}}_2{\mathrm{SO}}_4\right]}^2\left[{\mathrm{C}}_6{\mathrm{H}}_6\right] \end{gathered}$$

Where,$\mathrm{k}={\mathrm{k}}_2{\mathrm{k}}_1$

Thus, the overall rate law is${\mathrm{rate}}_2=\mathrm{k}{\left[{\mathrm{H}}_2{\mathrm{SO}}_4\right]}^2\left[{\mathrm{C}}_6{\mathrm{H}}_6\right]$