Question

A sample of pitchblende contains 50 kg uranium. The half-life of uranium is 4.5×10⁹ yr. The atomic weight of uranium is 238.4. The total number of uranium atoms in the beginning is 1.269×10²⁶ atoms. Calculate the age of the sample

Answer 1

Let's first find the number of atoms in a sample of pitchblende:

$$N=\dfrac{mN_A}{M_m},$$ $$N=\dfrac{50\ kg\times6.022\times10^{23}\ mol^{-1}}{238.4\times10^{-3}\ \dfrac{kg}{mol}}=1.263\times10^{26}\ atoms.$$

Let's use the formula for radioactive decay:

$$N=N_0e^{-\lambda t},$$ $$\dfrac{N}{N_0}=e^{-\lambda t},$$ $$ln(\dfrac{N}{N_0})=ln(e^{-\dfrac{0.693}{T_{1/2}}t}),$$ $$ln(\dfrac{N}{N_0})=-\dfrac{0.693}{T_{1/2}}t,$$ $$t=(\dfrac{ln(\dfrac{N}{N_0})}{-0.693})T_{1/2},$$ $$t=(\dfrac{ln(\dfrac{1.263\times10^{26}\ atoms}{1.263\times10^{26}\ atoms})}{-0.693})\times4.5\times10^9\ yr=3.1\times10^7\ yr.$$