Chapter 16, Problem 11STP

Interpretation Introduction

Interpretation:

The rate determining step in the given reaction sequence needs to be determined.

$\begin{array}{l}{\text{H}}_{\text{3}}{\text{A}}_{\text{(aq) }}\stackrel{}{\to }{\text{ H}}_{\text{2}}{\text{A}}^{\text{-}}{{}_{\text{(aq}}}_{\text{)}}{\text{ + H}}^{\text{+}}{}_{\text{(aq)}}{\text{ Rate = k}}_{\text{1}}{\text{ [H}}_{\text{3}}{\text{A] ; k}}_{\text{1}}\text{ = 3}\text{.2 }\times {\text{ 10}}^{\text{2}}{\text{ s}}^{\text{-1}}\\ {\text{H}}_{\text{2}}{\text{A}}^{\text{-}}{{}_{\text{(aq}}}_{\text{)}}\stackrel{}{\to }{\text{ HA}}_2^{-}{{}_{\text{(aq}}}_{\text{)}}{\text{ + H}}^{\text{+}}{}_{\text{(aq)}}{\text{ Rate = k}}_{\text{2}}{\text{ [H}}_{\text{2}}{\text{A}}^{\text{-}}{{}_{\text{(aq}}}_{\text{)}}{\text{] ; k}}_{\text{2}}\text{ = 1}\text{.5 }\times {\text{ 10}}^{\text{2}}{\text{ s}}^{\text{-1}}\\ {\text{HA}}_{}^{-2}{{}_{\text{(aq}}}_{\text{)}}\stackrel{}{\to }{\text{ A}}_{}^{-3}{{}_{\text{(aq}}}_{\text{)}}{\text{ + H}}^{\text{+}}{}_{\text{(aq)}}{\text{ Rate = k}}_{\text{3}}{\text{ [HA}}_{}^{-2}{{}_{\text{(aq}}}_{\text{)}}{\text{] ; k}}_3\text{ = 0}\text{.8 }\times {\text{ 10}}^{\text{2}}{\text{ s}}^{\text{-1}}\end{array}$

Concept introduction:

Chemical kinetic is the branch of chemistry that deals with kinetic or rate of a chemical reaction. The rate of reaction can be defined as the change in the concentration of reactant and product with time. The rate of reaction mainly depends on the activation energy.

In a chemical reaction to be occurred; two most important requirements are activation energy to reactant molecule and correct orientation of reactant molecules to colloid and form product.

Answer to Problem 11STP

${\text{HA}}_{}^{\text{-2}}{{}_{\text{(aq}}}_{\text{)}}\stackrel{}{\to }{\text{ A}}_{}^{\text{-3}}{{}_{\text{(aq}}}_{\text{)}}{\text{ + H}}^{\text{+}}{}_{\text{(aq)}}$

Explanation of Solution

Given information:

$\begin{array}{l}{\text{H}}_{\text{3}}{\text{A}}_{\text{(aq) }}\stackrel{}{\to }{\text{ A}}_{}^{-3}{{}_{\text{(aq}}}_{\text{)}}{\text{ + 3 H}}^{\text{+}}{}_{\text{(aq)}}{\text{ Rate = k}}_{\text{1}}{\text{ [H}}_{\text{3}}{\text{A] ; k}}_{\text{1}}\text{ = 3}\text{.2 }\times {\text{ 10}}^{\text{2}}{\text{ s}}^{\text{-1}}\\ {\text{[H}}_{\text{3}}{\text{A}}_{\text{(aq) }}\text{]=0}\text{.100 M}\\ {\text{[H}}_{\text{2}}{\text{A}}^{\text{-}}{{}_{\text{(aq}}}_{\text{)}}\text{] = 0}\text{.500 M}\\ {\text{[HA}}_{}^{2-}{{}_{\text{(aq}}}_{\text{)}}\text{] = 0}\text{.200 M}\end{array}$

The rate determining step is the slowest step of multiple step reaction which will determine the rate of overall reaction. In the given reaction steps; calculate rate for each:

$\begin{array}{l}{\text{H}}_{\text{3}}{\text{A}}_{\text{(aq) }}\stackrel{}{\to }{\text{ H}}_{\text{2}}{\text{A}}^{\text{-}}{{}_{\text{(aq}}}_{\text{)}}{\text{ + H}}^{\text{+}}{}_{\text{(aq)}}{\text{ Rate = k}}_{\text{1}}{\text{ [H}}_{\text{3}}{\text{A] ; k}}_{\text{1}}\text{ = 3}{\text{.2 × 10}}^{\text{2}}{\text{ s}}^{\text{-1}}\\ \text{Rate = 3}{\text{.2 × 10}}^{\text{2}}{\text{ s}}^{\text{-1}}\text{ [0}\text{.100 M] }\\ \text{Rate = 32 M}{\text{.s}}^{\text{-1}}\\ {\text{H}}_{\text{2}}{\text{A}}^{\text{-}}{{}_{\text{(aq}}}_{\text{)}}\stackrel{}{\to }{\text{ HA}}_{}^{\text{-2}}{{}_{\text{(aq}}}_{\text{)}}{\text{ + H}}^{\text{+}}{}_{\text{(aq)}}\\ \text{Rate = 1}{\text{.5 × 10}}^{\text{2}}{\text{ s}}^{\text{-1}}\text{ [0}\text{.500M] }\\ \text{Rate = 75 M}{\text{.s}}^{\text{-1}}\\ {\text{HA}}_{}^{\text{-2}}{{}_{\text{(aq}}}_{\text{)}}\stackrel{}{\to }{\text{ A}}_{}^{\text{-3}}{{}_{\text{(aq}}}_{\text{)}}{\text{ + H}}^{\text{+}}{}_{\text{(aq)}}\\ {\text{Rate = k}}_{\text{3}}{\text{ [HA}}_{}^{\text{-2}}{{}_{\text{(aq}}}_{\text{)}}\text{] }\\ \text{Rate = 0}{\text{.8 × 10}}^{\text{2}}{\text{ s}}^{\text{-1}}\text{[0}\text{.200M] }\\ \text{Rate = 16M}{\text{. s}}^{\text{-1}}\end{array}$

Thus the 3^rd step must be rate determining step because the rate of 3^rd step is slowest here.

Conclusion

Rate determining step is the slowest step of a multi-step reaction.