Answer: The sample is 13414 years old.
Step-by-step explanation:
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
t = age of sample
a = let initial amount of the reactant= 100
a - x = amount left after decay process= [tex]\frac{20}{100}\times 100=20[/tex]
a) for completion of half life:
Half life is the amount of time taken by a radioactive material to decay to half of its original value.
[tex]t_{\frac{1}{2}}=\frac{0.693}{k}[/tex]
[tex]k=\frac{0.693}{5730}=0.00012years^{-1}[/tex]
b) for completion of 20% of reaction
[tex]t=\frac{2.303}{0.00012}\log\frac{100}{20}[/tex]
[tex]t=13414years[/tex]
The sample is 13414 years old.
