Chapter 17, Problem 17.75P

Interpretation Introduction

Interpretation:

The rate law and the value of the rate constant for formation of ${\text{OI}}^{-}$ 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 ${\text{OCl}}^{-}{}_{\text{(aq)}}{\text{, I}}^{-}{}_{\text{(aq)}}{\text{ and OH}}^{-}{}_{\text{(aq)}}$ 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[OCl}}^{\text{-}}{}_{\text{(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{{\text{[ I}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[OH}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{z}}\text{for run1}}{{\text{k[OCl}}^{\text{-}}{}_{\text{(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{{\text{[ I}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[OH}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{z}}\text{for run2}}\\ \\ \frac{\text{3}\text{.06 }\text{×}{\text{10}}^{\text{-4}}}{\text{5}\text{.44}\text{×}{\text{10}}^{\text{-4}}}\text{=}\frac{{\text{(1}\text{.62}\text{×}{\text{10}}^{\text{-3}}\text{)}}^{\text{x}}{\text{(1}\text{.62}\text{×}{\text{10}}^{\text{-3}}\text{)}}^{\text{y}}{\text{(0}\text{.52)}}^{\text{z}}}{{\text{(1}\text{.62}\text{×}{\text{10}}^{\text{-3}}\text{)}}^{\text{x}}{\text{(2}\text{.88}\text{×}{\text{10}}^{\text{-3}}\text{)}}^{\text{y}}{\text{(0}\text{.52)}}^{\text{z}}}\\ \\ \text{(0}\text{.563)}\text{=}{\text{(0}\text{.563)}}^{\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 2}}{{\text{(rate of reaction)}}_{\text{0}}\text{ for run 4}}\text{=}\frac{{\text{k[OCl}}^{\text{-}}{}_{\text{(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{{\text{[ I}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[OH}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{z}}\text{for run2}}{{\text{k[OCl}}^{\text{-}}{}_{\text{(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{{\text{[ I}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[OH}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{z}}\text{for run4}}\\ \\ \frac{\text{5}\text{.44}\text{×}{\text{10}}^{\text{-4}}}{\text{3}\text{.11}\text{×}{\text{10}}^{\text{-4}}}\text{=}\frac{{\text{(1}\text{.62}\text{×}{\text{10}}^{\text{-3}}\text{)}}^{\text{x}}{\text{(2}\text{.88}\text{×}{\text{10}}^{\text{-3}}\text{)}}^{\text{y}}{\text{(0}\text{.52)}}^{\text{z}}}{{\text{(1}\text{.62}\text{×}{\text{10}}^{\text{-3}}\text{)}}^{\text{x}}{\text{(2}\text{.88}\text{×}{\text{10}}^{\text{-3}}\text{)}}^{\text{y}}{\text{(0}\text{.91)}}^{\text{z}}}\\ \\ \text{(1}\text{.749)}\text{=}{\text{(0}\text{.571)}}^{\text{z}}\\ \\ \text{z}\text{=}\text{1}\text{Order}\text{of}\text{reaction}\text{with}\text{respect}\text{to}{\text{OH}}^{\text{-}}{}_{\text{(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[OCl}}^{\text{-}}{}_{\text{(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{{\text{[ I}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[OH}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{z}}\text{for run2}}{{\text{k[OCl}}^{\text{-}}{}_{\text{(aq)}}{\text{]}}_{\text{0}}^{\text{x}}{{\text{[ I}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{y}}{{\text{[OH}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{\text{z}}\text{for run3}}\\ \\ \frac{\text{5}\text{.44}\text{×}{\text{10}}^{\text{-4}}}{\text{3}\text{.16}\text{×}{\text{10}}^{\text{-4}}}\text{=}\frac{{\text{(1}\text{.62}\text{×}{\text{10}}^{\text{-3}}\text{)}}^{\text{x}}\text{(2}\text{.88}\text{×}{\text{10}}^{\text{-3}}\text{)(0}\text{.52)}}{{\text{(2}\text{.71}\text{×}{\text{10}}^{\text{-3}}\text{)}}^{\text{x}}\text{(1}\text{.62}\text{×}{\text{10}}^{\text{-3}}\text{)(0}\text{.84)}}\\ \\ \text{(1}\text{.65)=}{\text{(0}\text{.60)}}^{\text{x}}\\ \\ \text{x}\text{=}\text{1}\text{Order}\text{of}\text{reaction}\text{with}\text{respect}\text{to}{\text{OCl}}^{\text{-}}{}_{\text{(aq)}}\text{is}\text{one}\end{array}$

The rate law of reaction and rate constant are given by

    $\begin{array}{l}\text{k}\text{=}\frac{{\text{(rate}\text{of}\text{reaction)}}_{\text{o}}}{{{\text{[OCl}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{0}}^{}{{\text{[ I}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{}{{\text{[OH}}^{\text{-}}{}_{\text{(aq)}}\text{]}}_{\text{o}}^{}}\text{=}\frac{\text{5}\text{.44 ×}{\text{10}}^{\text{-4}}}{\text{(1}\text{.62}\text{×}{\text{10}}^{\text{-3}}\text{)(1}\text{.62}\text{×}{\text{10}}^{\text{-3}}\text{)(0}\text{.52)}}\text{=}\text{4}\text{.0 ×}{\text{10}}^2{\text{M}}^{\text{-1}}{\text{s}}^{\text{-1}}\\ \\ Ratelaw:\text{rate of reaction =}\text{4}\text{.0 ×}{\text{10}}^2{\text{M}}^{\text{-1}}{\text{s}}^{\text{-1}}{{\text{[OCl}}^{\text{-}}}_{{}_{\text{(aq)}}}^{\text{]}}{{\text{[ I}}^{\text{-}}}_{{}_{\text{(aq)}}}^{\text{]}}{{\text{[OH}}^{\text{-}}}_{{}_{\text{(aq)}}}^{\text{]}}\end{array}$