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 very old minerals, the amount of an elemant is constant. This implies that the element exists in radioactive equilibrium. Thus, here we can use
`lambda_(U) N_(U) =lambda_(R)N_(R)`
or `(N_(U))/(T_(U))=(N_(R))/(T_(R))` or `T_(u)=(N_(u))/(N_(R)) xxT_(R)`
or `T_(U) =(m_(U) A_(R))/(m_(R)A_(U)) xx T_(R)`
or `T_(U) = (1 xx 226)/(3.32 xx 10^(-7) xx 238) xx 1600 years`
`=4.7 xx 10^(9) years` .