Chapter 16, Problem 16.96P

(a)

Interpretation Introduction

Interpretation:

The rate constant and half-life of the reaction has to be calculated.

Concept introduction:

First order reaction:

$\begin{array}{l}\text{Rate}\text{Equation:}\text{-Δ[R]/Δt}\text{=}{\text{k[R]}}^{\text{1}}\\ \\ \text{Integrated}\text{rate equation: ln}\left({\text{[R]}}_{\text{t}}{\text{/[R]}}_{\text{0}}\right)\text{= - kt}\end{array}$

Half-life: The time required for the concentration of a reactant to decrease to one half of its initial value.

$\begin{array}{l}{\text{[R]}}_{\text{t}}\text{=}\frac{{\text{[R]}}_{\text{0}}}{\text{2}}\text{,}\\ \\ \text{First}\text{order}\text{reaction:}\\ \\ {\text{ln[R]}}_{\text{t}}\text{=}\text{- kt}\text{+}{\text{ln[R]}}_{\text{0}}\\ \\ \text{ln}\frac{{\text{[R]}}_{\text{t}}}{{\text{[R]}}_{\text{0}}}\text{=}\text{- kt}\left(\text{or}\right)\text{ln}\text{2 = k}{\text{t}}_{\text{1/2}}\\ \\ {\text{t}}_{\text{1/2}}\text{=}\frac{\text{0}\text{.693}}{\text{k}}\end{array}$

(b)

Interpretation Introduction

Interpretation:

The time interval for $40\%$ of a sample of acetone to decompose has to be calculated.

Concept introduction:

First order reaction:

$\begin{array}{l}\text{Rate}\text{Equation:}\text{-Δ[R]/Δt}\text{=}{\text{k[R]}}^{\text{1}}\\ \\ \text{Integrated}\text{rate equation: ln}\left({\text{[R]}}_{\text{t}}{\text{/[R]}}_{\text{0}}\right)\text{= - kt}\end{array}$

Half-life: The time required for the concentration of a reactant to decrease to one half of its initial value.

$\begin{array}{l}{\text{[R]}}_{\text{t}}\text{=}\frac{{\text{[R]}}_{\text{0}}}{\text{2}}\text{,}\\ \\ \text{First}\text{order}\text{reaction:}\\ \\ {\text{ln[R]}}_{\text{t}}\text{=}\text{- kt}\text{+}{\text{ln[R]}}_{\text{0}}\\ \\ \text{ln}\frac{{\text{[R]}}_{\text{t}}}{{\text{[R]}}_{\text{0}}}\text{=}\text{- kt}\left(\text{or}\right)\text{ln}\text{2 = k}{\text{t}}_{\text{1/2}}\\ \\ {\text{t}}_{\text{1/2}}\text{=}\frac{\text{0}\text{.693}}{\text{k}}\end{array}$

(c)

Interpretation Introduction

Interpretation:

The time interval for $90\%$ of a sample of acetone to decompose has to be calculated.

Concept introduction:

First order reaction:

$\begin{array}{l}\text{Rate}\text{Equation:}\text{-Δ[R]/Δt}\text{=}{\text{k[R]}}^{\text{1}}\\ \\ \text{Integrated}\text{rate equation: ln}\left({\text{[R]}}_{\text{t}}{\text{/[R]}}_{\text{0}}\right)\text{= - kt}\end{array}$

Half-life: The time required for the concentration of a reactant to decrease to one half of its initial value.

$\begin{array}{l}{\text{[R]}}_{\text{t}}\text{=}\frac{{\text{[R]}}_{\text{0}}}{\text{2}}\text{,}\\ \\ \text{First}\text{order}\text{reaction:}\\ \\ {\text{ln[R]}}_{\text{t}}\text{=}\text{- kt}\text{+}{\text{ln[R]}}_{\text{0}}\\ \\ \text{ln}\frac{{\text{[R]}}_{\text{t}}}{{\text{[R]}}_{\text{0}}}\text{=}\text{- kt}\left(\text{or}\right)\text{ln}\text{2 = k}{\text{t}}_{\text{1/2}}\\ \\ {\text{t}}_{\text{1/2}}\text{=}\frac{\text{0}\text{.693}}{\text{k}}\end{array}$