Question

1. How long will it take for the radioisotope Rn-222 to decay to 54.1 g if the half-life of Rn-222 is 3.8235 days, starting with the following. (Isotopic mass of Rn-222 is 222.017570 g/mol.)

(a) 80.9 g Rn-222

_________ d

(b) 10.8 mol Rn-222

__________ d

Answer 1

a)

we have:

Half life = 3.8235 days

use relation between rate constant and half life of 1st order reaction

k = (ln 2) / k

= 0.693/(half life)

= 0.693/(3.8235)

= 0.1812 days-1

we have:

[Rn]o = 80.9 g

[Rn] = 54.1 g

k = 0.1812 days-1

use integrated rate law for 1st order reaction

ln[   Rn] = ln[   Rn]o - k*t

ln(54.1) = ln(80.9) - 0.1812*t

3.9908 = 4.3932 - 0.1812*t

0.1812*t = 0.4024

t = 2.22 days

Answer: 2.22 days

b)

mass of Rn = number of mol * molar mass

= 10.8 mol * 222.017570 g/mol.

= 2397.8 g

we have:

[Rn]o = 2397.8 g

[Rn] = 54.1 g

k = 0.1812 days-1

use integrated rate law for 1st order reaction

ln[Rn] = ln[Rn]o - k*t

ln(54.1) = ln(2397.8) - 0.1812*t

3.9908 = 7.7823 - 0.1812*t

0.1812*t = 3.7915

t = 20.9 days

Answer: 20.9 days