Chapter 17, Problem 17.25P

Interpretation Introduction

Interpretation:

A graph of time versus $\text{ln}\left({\text{[C}}_{\text{2}}{\text{H}}_{\text{4}}\text{O]/}\text{M}\right)$ has to be drawn to prove the first order reaction.  Also, the value of rate constant of the reaction has to be determined.

Concept Introduction:

For the first-order decomposition of a single reactant A, the rate expression is

    $\begin{array}{l}\text{rate}\text{of}\text{reaction}\text{=}\text{-}\frac{\text{Δ[A]}}{\text{Δt}}\text{=}\text{k}\text{[A]}\\ \\ \text{By using elementary calculus, the concentration}\\ \text{of A at any time}\text{. The result is}\\ \\ \text{ln}\frac{\left[\text{A}\right]}{\left[{\text{A}}_{\text{°}}\right]}\text{=}\text{-kt}\end{array}$

By comparing the above equation to the equation $\text{y=mx+b}$, it is clear that a plot of

$\text{ln}\left[\text{A}\right]$ versus time, t, is a straight line whose slope is the rate constant, $\text{-k}$.