Find the half-life of uranium, given that `3.32 xx 10^(7) g` radium is found per gram of uranium in old minerals. The atomic weights of uranium and radium are `238` and `226` and half-life of radium is `1600` years (Avogadro number is `6.023 xx 10^(23)//g -"atom"`).

Explanation

In radioactive equilibrium
`lambda_(1)N_(1) = lambda_(2)N_(2)`
`lambda_(U)N_(U) = lambda_(Ra)N_(Ra)`
`(ln2)/(T_(1//2))_(U).((m)/(M))_(U)=(ln2)/((T_(1//2))_(Ra))((m)/(M))_(Ra)`
`(1)/(T_(1//2))_(U).((1)/(235))=(1)/(1600)((3.32xx10^(-7))/(226))`
`(T_(1//2))_(U) = ((226)/(235)) xx (1600)/(3.32) xx 10^(7)`
`= 4.63xx10^(9)`year`