(a) On analysis a sample of uranium ore was found to contian `0.277g` of `._(82)Pb^(206)` and `1.667g` of `._(92)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. Caluculate the age of ore.

Explanation

(a) Let, Time `= t` year
`._(92)U^(238) = 1.667g = (1.667)/(2378)` mole
`._(82)Pb^(206) = 0.277g = (0.277)/(206)` mole
`:'` All the lead have come from decay of `U.` Therefore Mole of `Pb` formed `= (0.277)/(206)`
`:.` Mole of `U` decayed `= (0.277)/(206)`
`:.` Total moles of unranium before decay, i.e.,
`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))`
`t = 1.143xx10^(9)` year
(b) Ans. `7.097xx10^(8)` year