Chapter 17, Problem 17.53P

Interpretation Introduction

Interpretation:

The reaction rate law for formation of chloride ion has to be calculated.

Concept Introduction:

Rate law:  The mathematical equation that relates the concentrations of the reactants to

the rate of reaction is called the rate law of a reaction.  An example of rate law of thermal decomposition of ${\text{N}}_{\text{2}}{\text{O}}_{\text{5}}$ is given by

    ${\text{rate of reaction = k[N}}_{\text{2}}{\text{O}}_{\text{5}}{\text{]}}^{\text{x}}$

The proportionality constant, k, in a rate law is called the rate constant of the reaction.

Explanation of Solution

The order of reaction with respect to ${\text{ClO}}_{\text{3}}{{}^{\text{-}}}_{\text{(aq)}}\text{,}{\text{I}}^{\text{-}}{}_{\text{(aq)}}\text{,}\text{and}{\text{H}}^{\text{+}}{}_{\text{(aq)}}$ are calculated as,

    $\begin{array}{l}\frac{{\text{(rate of reaction)}}_{\text{0}}\text{ for run 2}}{{\text{(rate of reaction)}}_{\text{0}}\text{ for run 3}}\text{=}\frac{{\text{k[ClO}}_{\text{3}}{{}^{\text{-}}}_{\text{(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{\text{[}\text{,}{\text{I}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[H}}^{+}{}_{\text{(aq)}}\text{]}}_o^z\text{for run2}}{{\text{k[ClO}}_{\text{3}}{{}^{\text{-}}}_{\text{(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{\text{[}\text{,}{\text{I}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[H}}^{+}{}_{\text{(aq)}}\text{]}}_o^z\text{for run3}}\\ \\ \frac{\text{1}\text{.70 }\text{×}{\text{10}}^{-3}}{\text{4}\text{.42}\text{×}{\text{10}}^{-3}}\text{=}\frac{{\text{(0}\text{.10)}}^{\text{x}}{\text{(0}\text{.17)}}^{\text{y}}{(\text{0}\text{.10})}^z}{{\text{(0}\text{.26)}}^{\text{x}}{\text{(0}\text{.17)}}^{\text{y}}{(\text{0}\text{.10})}^z}\\ \\ \text{(0}\text{.385)}\text{=}{\text{(0}\text{.385)}}^{\text{x}}\\ \\ \text{x}\text{=}\text{1}\text{Order}\text{of}\text{reaction}\text{with}\text{respect}\text{to}{\text{ClO}}_{\text{3}}{{}^{\text{-}}}_{\text{(aq)}}\text{is}\text{one}\\ \\ \frac{{\text{(rate of reaction)}}_{\text{0}}\text{ for run 1}}{{\text{(rate of reaction)}}_{\text{0}}\text{ for run 2}}\text{=}\frac{{\text{k[ClO}}_{\text{3}}{{}^{\text{-}}}_{\text{(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{\text{[}\text{,}{\text{I}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[H}}^{+}{}_{\text{(aq)}}\text{]}}_o^z\text{for run}1}{{\text{k[ClO}}_{\text{3}}{{}^{\text{-}}}_{\text{(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{\text{[}\text{,}{\text{I}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[H}}^{+}{}_{\text{(aq)}}\text{]}}_o^z\text{for run}2}\\ \\ \frac{1.00\text{×}{\text{10}}^{-3}}{\text{1}\text{.70}\text{×}{\text{10}}^{-3}}\text{=}\frac{{\text{(0}\text{.10)}}^{\text{x}}{\text{(0}\text{.10)}}^{\text{y}}{(\text{0}\text{.10})}^z}{{\text{(0}\text{.10)}}^{\text{x}}{\text{(0}\text{.17)}}^{\text{y}}{(\text{0}\text{.10})}^z}\\ \\ \text{(0}\text{.588)}\text{=}{\text{(0}\text{.588)}}^{\text{y}}\\ \\ \text{y}\text{=}\text{1}\text{Order}\text{of}\text{reaction}\text{with}\text{respect}\text{to}{\text{I}}^{\text{-}}{}_{\text{(aq)}}\text{is}\text{one}\\ \\ \frac{{\text{(rate of reaction)}}_{\text{0}}\text{ for run 3}}{{\text{(rate of reaction)}}_{\text{0}}\text{ for run 4}}\text{=}\frac{{\text{k[ClO}}_{\text{3}}{{}^{\text{-}}}_{\text{(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{\text{[}\text{,}{\text{I}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[H}}^{+}{}_{\text{(aq)}}\text{]}}_o^z\text{for run}3}{{\text{k[ClO}}_{\text{3}}{{}^{\text{-}}}_{\text{(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{\text{[}\text{,}{\text{I}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[H}}^{+}{}_{\text{(aq)}}\text{]}}_o^z\text{for run}4}\\ \\ \frac{\text{4}\text{.42}\text{×}{\text{10}}^{-3}}{\text{1}\text{.13}\text{×}{\text{10}}^{-2}}\text{=}\frac{\text{(0}\text{.26)(0}\text{.17)}{(\text{0}\text{.10})}^z}{\text{(0}\text{.26)(0}\text{.17)}{(\text{0}\text{.16})}^z}\\ \\ \text{(0}\text{.391)}\text{=}{\text{(0}\text{.625)}}^{\text{y}}\\ \\ z\text{=}2\text{Order}\text{of}\text{reaction}\text{with}\text{respect}\text{to}{\text{H}}^{+}{}_{\text{(aq)}}\text{is}\text{two}\\ \\ \text{Overall}\text{order}\text{of}\text{reaction}\text{=}\text{1}\text{+}\text{1}+2\text{=}4\end{array}$

The rate law of reaction and rate constant are given by

    $\begin{array}{l}\text{k}\text{=}\frac{{\text{(rate}\text{of}\text{reaction)}}_{\text{o}}}{{{\text{[ClO}}_{\text{3}}{{}^{\text{-}}}_{\text{(aq)}}\text{]}}_{\text{0}}^{\text{x}}{\text{[}\text{,}{\text{I}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[H}}^{+}{}_{\text{(aq)}}\text{]}}_o^z}\text{=}\frac{\text{1}\text{.0 ×}{\text{10}}^{\text{-3}}}{\text{(0}\text{.10)(0}\text{.10)(0}{\text{.10)}}^2}\text{=}1.\text{0 ×}{\text{10}}^{\text{-1}}{\text{M}}^{\text{-1}}{\text{s}}^{\text{-1}}\\ \\ Ratelaw:\text{rate of reaction =}1.\text{0 ×}{\text{10}}^{\text{-1}}{\text{M}}^{\text{-1}}{\text{s}}^{\text{-1}}{{\text{[ClO}}_{\text{3}}}_{{{}^{\text{-}}}_{\text{(aq)}}}^{\text{]}}{\text{[}}_{}^{\text{,}}{{\text{[H}}^{+}}_{{}_{\text{(aq)}}}^{\text{]}}\end{array}$