# Question d01e3

Apr 14, 2017

The time required is 8.499 s.

#### Explanation:

Whenever you see a kinetics problem that asks something like "What time is needed …", you can be pretty sure that you will have to use an integrated rate law.

The integrated rate law for a first-order reaction is

color(blue)(|bar(ul(color(white)(a/a)ln(["A"]_0/(["A"]_t)) = ktcolor(white)(a/a)|)))" "

In this problem,

${\left[\text{A"]_0 = ["A}\right]}_{0}$
${\left[\text{A"]_t = 2/3 ["A}\right]}_{0}$
$k = \text{0.047 71 s"^"-1}$

ln(stackrelcolor(blue)(1)(color(red)(cancel(color(black)(["A"]_0))))/(2/3color(red)(cancel(color(black)(["A"]_0))))) = ln(3/2) = 0.4055 = "0.047 71" t color(white)(l)"s"^"-1"#

$t = \frac{0.4055}{\text{0.047 71 s"^"-1" = "8.499 s}}$