Answer:
The rock is formed [tex]1.7\times 10^9[/tex] years ago.
Explanation:
Given that:
Half life = [tex]4.5\times 10^9[/tex] years
[tex]t_{1/2}=\frac {ln\ 2}{k}[/tex]
Where, k is rate constant
So,
[tex]k=\frac {ln\ 2}{t_{1/2}}[/tex]
[tex]k=\frac{ln\ 2}{4.5\times 10^9}\ years^{-1}[/tex]
The rate constant, k = [tex]1.54\times 10^{-10}[/tex] 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 final concentration= 0.275 mg
[tex][A_0][/tex] is the initial concentration = 1 mg
Let [tex][A_t][/tex] = 1 mg
A rock contains 0.275 mg of lead-206 for each milligram of uranium-238. So,
[tex][A_0][/tex] = [tex]1+\frac{238}{206}\times 0.275[/tex] mg = 1.297 mg
Time = ?
So,
[tex]\frac{1}{1.297}=e^{-1.54\times 10^{-10}\times t}[/tex]
[tex]\ln \left(\frac{1}{1.297}\right)=-1.54\times \:10^{-10}t[/tex]
[tex]t=\frac{10^{10}\ln \left(1.297\right)}{1.54}[/tex]
[tex]t=1.7\times 10^9[/tex] years
The rock is formed [tex]1.7\times 10^9[/tex] years ago.
