`.^(238)U` decays with a half-life of `4.5 xx10^(9)` years, the decay series eventaully ending at `.^(206)Pb`, which is stable. `A` rock sample analysis shows that the ratio of the number of atoms of `.^(206)Pb` to `.^(238)U` is 0.0058. Assuming that all the `.^(206)Pb` is prodduced by the decay of `.^(238)U` and that all other half-lives on the chain are negligilbe, the age of the rock sample is `(1n 1.0058 =5.78 xx10^(-3))`.

Options

A. `38 xx 10^(8) years`
B. `38 xx 10^(6) years`
C. `19 xx 10^(8) years`
D. `19 xx 10^(6) years`

Answer

b

Explanation

After a time 't, a sample of `.^(238)U` originally consisting of N atoms will have decayed to `Ne^(-lambda t)` . The number of `.^(206)Pb` atoms,
`N_(pb)=N(1-e^(-alphat))`
`:. (N_(Pb))/(N_(u))=N(1-e^(-lambdat))`
`e^(lambda t) -1 =0.0058 rArr e^(lambda t)=1.0058`
`:. t=(1)/(lambda) In(1.0058)=((4.5 xx10^(9) years))/(In 2) 1n (1.0058)`
`=0.0376 xx 10^(9)years =38 xx10^(6)years`.