# Question f9471

Jun 18, 2017

The mass of the original sample was 60.1 g.

#### Explanation:

Radioactive decay is a first order process.

Calculate the first-order rate constant

t_½ = ln2/k

k = ln2/t_½ = ln2/"2.62 h" = 0.2645 color(red)(cancel(color(black)("h"^"-1"))) × (1 color(red)(cancel(color(black)("h"))))/"60 min" = 4.409 × 10^"-3"color(white)(l) "min"^"-1"

Calculate the original mass

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)|)))" "

where

${\text{A}}_{0}$ and ${\text{A}}_{t}$ are the amounts at time $t = 0$ and at time $t$ and

$k$ is the rate constant

$\text{A"_t = "47.1 g}$

k = 4.409 × 10^"-3"color(white)(l) "min"^"-1"

$t = \text{55.2 min}$

ln("A"_0/"A"_t) = 4.409 × 10^"-3" color(red)(cancel(color(black)("min"^"-1"))) × 55.2 color(red)(cancel(color(black)("min"))) = 0.2434#

${\text{A"_0/"A}}_{t} = {e}^{0.2434} = 1.276$

$\text{A"_0 = 1.276"A"_t = "1.276 × 47.1 g = 60.1 g}$

The original mass was 60.1 g.