(a) On analysis a sample of uranium ore was found to contain `0.277g` of `._(82)Pb^(206)` and `1.667g` of `._(92)U^(238)`. The half life period of `U^(238)` is `4.51xx10^(9)` year. If all the lead were assumed to have come from decay of `._(92)U^(238)`, What is the age of earth?
(b) An ore of `._(92)U^(238)` is found to contain `._(92)U^(238)` and `._(82)Pb^(206)` in the weight ratio of `1: 0.1` The half-life period of `._(92)U^(238)` is `4.5xx10^(9)` year. Calculate the age of ore.

Explanation

Let time`=t` year
`._(92)U^(238)=1.667g=(1.667)/(238)mol`
`._(82)Pb(206)=0.277g=(0.277)/(206)mol`
`:'` All the lead have come from decay of `U`. ltbr. `:. ` Mole of `Pb` formed `=(2.77)/(206)`
Mole of `U ` decayed `=(0.277)/(206)`
`:. `Total mole of uranium before decay,
`N_(0)=(1.667)/(238)+(0.277)/(206)`
Also, `N` for `U^(238)=(1.667)/(238)`
For `U^(238),t=(2.303)/(lambda)"log"_(10) (N_(0))/(N)`
`=(2.303xx4.51xx10^(9))/(0.693)"log"_(10) ((1.667)/(238)+(0.277)/(206))/((1.667)/(238))`
`1.143xx10^(9) years`