Uranium ores on the earth at the present time typically have a composition consisting of 99.3% of the isotope `_92U^238` and 0.7% of the isotope `_92U^235`. The half-lives of these isotopes are `4.47xx10^9yr` and `7.04xx10^8yr`, respectively. If these isotopes were equally abundant when the earth was formed, estimate the age of the earth.

Answer

A

Explanation

Let `N_0` be number of atoms of each isotope at the time of formation of the earth `(t=0)` and `N_1` and `N_2` the number of atoms at present `(t=t)`. Then,
`N_1=N_0e^(-lambda_1t)` …(i)
and `N_2=N_0e^(-lambda_2t)` …(ii)
:. `N_1/N_2=e^((lambda_2-lambda_1)t)` ...(iii)
Further it is given that
`N_1/N_2=99.3/0.7` ...(iv)
Equating Eqs. (iii) and (iv) and taking log on both sides, we have
`(lambda_2-lambda_1)t=1n(99.3/0.7)`
:. `t=(1/(lambda_2-lambda_1))1n(99.3/0.7)`
Substituting the values, we have
`t=(1)/((0.693)/(7.04xx10^8)-(0.693)/(4.47xx10^9))1n(99.3/0.7)`
or `t=5.97xx10^9yr`