Question

Radon-222 ( 222/86 Rn) is a radioactive gas with a half-life of 3.82 days. A gas sample contains 4.1 e 8 radon atoms initially.

a) Determine how many radon atoms will remain after 12 days.

b) Determine how many radon nuclei will have decayed by this time.

Answer 1

(a) The number of radon atoms will remain after 12 days is,
`4.67* 10^7`

(b) The number of radon nuclei have decayed by this time will be,
`3.6* 10^8`

Explanation :

For part (a) :

Half-life = 3.82 days

First we have to calculate the rate constant, we use the formula :

`k=(0.693)/(t_(1/2))`

`k=\frac{0.693}{3.82\text{ days}}`

`k=1.81* 10^(-1)\text{ days}^(-1)`

Now we have to calculate the number of radon atoms will remain after 12 days.

Expression for rate law for first order kinetics is given by:

`t=(2.303)/(k)\log(a)/(a-x)`

where,

k = rate constant =
`1.81* 10^(-1)\text{ days}^(-1)`

t = time passed by the sample = 12 days

a = initially number of radon atoms =
`4.1* 10^8`

a - x = number of radon atoms left = ?

Now put all the given values in above equation, we get

`12=(2.303)/(1.81* 10^(-1))\log(4.1* 10^8)/(a-x)`

`a-x=4.67* 10^7`

Thus, the number of radon atoms will remain after 12 days is,
`4.67* 10^7`

For part (b) :

Now we have to calculate the number of radon nuclei will have decayed by this time.

The number of radon nuclei have decayed = Initial number of radon atoms - Number of radon atoms left

The number of radon nuclei have decayed =
`(4.1* 10^8)-(4.67* 10^7)`

The number of radon nuclei have decayed =
`3.6* 10^8`

Thus, the number of radon nuclei have decayed by this time will be,
`3.6* 10^8`