In a smaple of rock, the ration `.^(206)Pb` to `.^(238)U` nulei is found to be `0.5.` The age of the rock is (given half-life of `U^(238)` is `4.5 xx 10^(9)` years).

Options

A. `2.25 xx 10^(9)` years
B. `4.5 xx 10^(9)` ln years
C. `4.5 xx 10^(9) (ln ((3)/(2)))/(ln 2) `year
D. `2.25 xx 10^(9) ln ((3)/(2))` year

Answer

C

Explanation

Suppose an initial radionuclide I decays to a final product F with a half - life `T_(1//2)`
At any time , `N_(t) = N_(0) e^(-lambda t)`
Number of product nuclei = `N_(F) = N_(0) - N_(1)`
`(N_(F))/(N_(1)) = (N_(0) - N_(1))/(N_(I)) = ((N_(0))/(N_(1)) - I)`
`(N_(0))/(N_(1)) (1 + (N_(F))/(N_(I))) = 1 + 0.5 = 1.5`
`therefore (T_(1//2) ln (1.5))/(ln 2) = 4.5 xx 10^(9) (ln ((3)/(2)))/(ln 2) year`