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.

Options

A. `4.78xx10^(9)yrs`
B. `5.97xx10^(9)yrs`
C. `6.78xx10^(7)yrs`
D. `7.29xx10^(8)yrs`

Answer

B

Explanation

Let `N_(0)` be the number of atoms of each isotope at t=0 and `N_(1) and N_(2)` be the number of atoms of each isotope at t=t.
Then `N_(1)=N_(0)e^(-lambda_(1)t)`
and `N_(2)=N_(0)e^(-lambda_(2)t)`
`:. (N_(1))/(N_(2))=e^((lambda_(2)-lambda_(1))t)`
or `(lambda_(2)-lambda_(1))t=ln ((N_(1))/(N_(2)))`
or `t=1/(lambda_(2)-lambda_(1))ln (N_(1))/(N_(2))`
or `t=1/(lambda_(2)-lambda_(1)) ln (99.3/0.7).......(i)`
`lambda_(1)=0.693/(4.47xx10^(9)) and lambda_(2)=0.693/(7.04xx10^(8))`
Putting values in (i)
`t=1/(0.693/(7.04xx10^(8))-0.693/(4.47xx10^(9))) ln (99.3/0.7)`
or `t=5.97xx10^(9)yrs`