Answer :
Answer:
11552.45 years
Explanation:
Given that:
Half life = 5730 years
[tex]t_{1/2}=\frac{\ln2}{k}[/tex]
Where, k is rate constant
So,
[tex]k=\frac{\ln2}{t_{1/2}}[/tex]
[tex]k=\frac{\ln2}{5730}\ years^{-1}[/tex]
The rate constant, k = 0.00012 years⁻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
Given that:
The rate constant, k = 0.00012 years⁻¹
Initial concentration [tex][A_0][/tex] = 160.0 counts/min
Final concentration [tex][A_t][/tex] = 40.0 counts/min
Time = ?
Applying in the above equation, we get that:-
[tex]40.0=160.0e^{-0.00012\times t}[/tex]
[tex]e^{-0.00012t}=\frac{1}{4}[/tex]
[tex]-0.00012t=\ln \left(\frac{1}{4}\right)[/tex]
[tex]t=11552.45\ years[/tex]