Answer :
(a) The number of radon atoms will remain after 12 days is, [tex]4.67\times 10^7[/tex]
(b) The number of radon nuclei have decayed by this time will be, [tex]3.6\times 10^8[/tex]
Explanation :
For part (a) :
Half-life = 3.82 days
First we have to calculate the rate constant, we use the formula :
[tex]k=\frac{0.693}{t_{1/2}}[/tex]
[tex]k=\frac{0.693}{3.82\text{ days}}[/tex]
[tex]k=1.81\times 10^{-1}\text{ days}^{-1}[/tex]
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:
[tex]t=\frac{2.303}{k}\log\frac{a}{a-x}[/tex]
where,
k = rate constant = [tex]1.81\times 10^{-1}\text{ days}^{-1}[/tex]
t = time passed by the sample = 12 days
a = initially number of radon atoms = [tex]4.1\times 10^8[/tex]
a - x = number of radon atoms left = ?
Now put all the given values in above equation, we get
[tex]12=\frac{2.303}{1.81\times 10^{-1}}\log\frac{4.1\times 10^8}{a-x}[/tex]
[tex]a-x=4.67\times 10^7[/tex]
Thus, the number of radon atoms will remain after 12 days is, [tex]4.67\times 10^7[/tex]
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 = [tex](4.1\times 10^8)-(4.67\times 10^7)[/tex]
The number of radon nuclei have decayed = [tex]3.6\times 10^8[/tex]
Thus, the number of radon nuclei have decayed by this time will be, [tex]3.6\times 10^8[/tex]
