Question

The long-lived isotope of radium, Ra226, decays by α emission with a half-life of 1622 years. Calculate how long it will take for 1% of this nuclide to disappear and how long until 1% of it remains.

Answer 1

1) for 1% to disappear
Given:
Half life = 1.622*10^3 years
use relation between rate constant and half life of 1st order reaction

k = (ln 2) / k
= 0.693/(half life)
= 0.693/(1.622*10^3)
= 4.273*10^-4 years-1

we have:
[A]o = 100
[A] = 99
k = 4.273*10^-4 years-1

use integrated rate law for 1st order reaction
ln[A] = ln[A]o - k*t
ln(99) = ln(1*10^2) - 4.273*10^-4*t
4.595 = 4.605 - 4.273*10^-4*t
4.273*10^-4*t = 1.005*10^-2
t = 23.52 years
Answer: 23.5 years

2) for 1 % to remain
Given:
Half life = 1.622*10^3 years
use relation between rate constant and half life of 1st order reaction

k = (ln 2) / k
= 0.693/(half life)
= 0.693/(1.622*10^3)
= 4.273*10^-4 years-1

we have:
[A]o = 1*10^2 M
[A] = 1 M
k = 4.273*10^-4 years-1

use integrated rate law for 1st order reaction
ln[A] = ln[A]o - k*t
ln(1) = ln(1*10^2) - 4.273*10^-4*t
0 = 4.605 - 4.273*10^-4*t
4.273*10^-4*t = 4.605
t = 1.078*10^4 years
Answer: 1.08*10^4 years