Answer:
[tex]t=23.07\ years[/tex]
Explanation:
Half life of lead - 210 = 22.3 years
[tex]t_{1/2}=\frac {ln\ 2}{k}[/tex]
Where, k is rate constant
So,
[tex]k=\frac{\ln2}{t_{1/2}}[/tex]
[tex]k=\frac{\ln2}{22.3}\ year^{-1}[/tex]
The rate constant, k = 0.0311 year⁻¹
Using integrated rate law for first order kinetics as:
[tex][A_t]=[A_0]e^{-kt}[/tex]
Where,
[tex][A_t][/tex] is the concentration at time t
[tex][A_0][/tex] is the initial concentration
Initial concentration [tex][A_0][/tex] = 33.2 mg
Final concentration [tex][A_t][/tex] = 16.2 mg
Time = ?
Applying in the above equation, we get that:-
[tex]16.2=33.2e^{-0.0311\times t}[/tex]
[tex]332e^{-0.0311t}=162[/tex]
[tex]e^{-0.0311t}=\frac{81}{166}[/tex]
[tex]-0.0311t=\ln \left(\frac{81}{166}\right)[/tex]
[tex]t=23.07\ years[/tex]