Chapter 17, Problem 17.20P

Interpretation Introduction

Interpretation:

The rate law and the value of the rate constant for formation of hydrogen ion have 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 ${\text{CH}}_{\text{3}}{\text{COCH}}_{\text{3(aq)}}{\text{, Br}}_{\text{2(aq)}}{\text{ and H}}^{+}{}_{\text{(aq)}}$ are calculated as,

    $\begin{array}{l}\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[CH}}_{\text{3}}{\text{COCH}}_{\text{3(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{{\text{[Br}}_{\text{2(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[H}}^{\text{+}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{z}}\text{for run1}}{{\text{k[CH}}_{\text{3}}{\text{COCH}}_{\text{3(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{{\text{[Br}}_{\text{2(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[H}}^{\text{+}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{z}}\text{for run2}}\\ \\ \frac{\text{4}\text{.0 }\text{×}{\text{10}}^{\text{-3}}}{\text{7}\text{.0}\text{×}{\text{10}}^{\text{-3}}}\text{=}\frac{{\text{(1}\text{.00)}}^{\text{x}}{\text{(1}\text{.00)}}^{\text{y}}{\text{(1}\text{.00)}}^{\text{z}}}{{\text{(1}\text{.75)}}^{\text{x}}{\text{(1}\text{.00)}}^{\text{y}}{\text{(1}\text{.00)}}^{\text{z}}}\\ \\ \text{(0}\text{.571)}\text{=}{\text{(0}\text{.571)}}^{\text{x}}\\ \\ \text{x}\text{=}\text{1}\text{Order}\text{of}\text{reaction}\text{with}\text{respect}\text{to}{\text{CH}}_{\text{3}}{\text{COCH}}_{\text{3(aq)}}\text{is}\text{one}\\ \\ \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[CH}}_{\text{3}}{\text{COCH}}_{\text{3(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{{\text{[Br}}_{\text{2(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[H}}^{\text{+}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{z}}\text{for run2}}{{\text{k[CH}}_{\text{3}}{\text{COCH}}_{\text{3(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{{\text{[Br}}_{\text{2(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[H}}^{\text{+}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{z}}\text{for run3}}\\ \\ \frac{\text{7}\text{.0 }\text{×}{\text{10}}^{\text{-3}}}{\text{9}\text{.8}\text{×}{\text{10}}^{\text{-3}}}\text{=}\frac{\text{(1}\text{.75)(1}{\text{.00)}}^{\text{y}}{\text{(1}\text{.00)}}^{\text{z}}}{\text{(1}\text{.75)(1}{\text{.40)}}^{\text{y}}{\text{(1}\text{.00)}}^{\text{z}}}\\ \\ \text{(0}\text{.714)}\text{=}{\text{(0}\text{.714)}}^{\text{y}}\\ \\ \text{y}\text{=}\text{1}\text{Order}\text{of}\text{reaction}\text{with}\text{respect}\text{to}{\text{Br}}_{\text{2(aq)}}\text{is}\text{one}\\ \\ \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[CH}}_{\text{3}}{\text{COCH}}_{\text{3(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{{\text{[Br}}_{\text{2(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[H}}^{\text{+}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{z}}\text{for run2}}{{\text{k[CH}}_{\text{3}}{\text{COCH}}_{\text{3(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{{\text{[Br}}_{\text{2(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[H}}^{\text{+}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{z}}\text{for run3}}\\ \\ \frac{\text{7}\text{.0 }\text{×}{\text{10}}^{\text{-3}}}{\text{9}\text{.8}\text{×}{\text{10}}^{\text{-3}}}\text{=}\frac{\text{(1}\text{.75)(1}\text{.00)(1}{\text{.00)}}^{\text{z}}}{\text{(1}\text{.75)(1}\text{.40)(1}{\text{.00)}}^{\text{z}}}\\ \\ \frac{\text{(0}\text{.714)}}{\text{(0}\text{.714)}}\text{=}{\text{(1}\text{.00)}}^{\text{z}}\\ \\ \text{z}\text{=}\text{1}\text{Order}\text{of}\text{reaction}\text{with}\text{respect}\text{to}{\text{H}}^{\text{+}}{}_{\text{(aq)}}\text{is}\text{one}\\ \\ \text{Overall}\text{order}\text{of}\text{reaction}\text{=}\text{1}\text{+}\text{1}\text{+1=}\text{3}\end{array}$

The rate law of reaction and rate constant are given by

    $\begin{array}{l}\text{k}\text{=}\frac{{\text{(rate}\text{of}\text{reaction)}}_{\text{o}}}{{{\text{[CH}}_{\text{3}}{\text{COCH}}_{\text{3(aq)}}\text{]}}_{\text{0}}^{}{{\text{[Br}}_{\text{2(aq)}}\text{]}}_{\text{o}}^{}{{\text{[H}}^{\text{+}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{}}\text{=}\frac{\text{4}\text{.0 ×}{\text{10}}^{\text{-3}}}{\text{(1}\text{.00)(1}\text{.00)(1}\text{.00)}}\text{=}\text{4}\text{.0 ×}{\text{10}}^{\text{-3}}{\text{M}}^{\text{-1}}{\text{s}}^{\text{-1}}\\ \\ \text{Rate}\text{law}\text{:}\text{rate of reaction =}\text{4}\text{.0 ×}{\text{10}}^{\text{-3}}{\text{M}}^{\text{-1}}{\text{s}}^{\text{-1}}{\text{[CH}}_{\text{3}}{\text{COCH}}_{\text{3}}{\text{][H}}^{\text{+}}{\text{][Br}}_{\text{2}}\text{]}\end{array}$