A sample of rock from the moon is analyzed and found to contain an exactly equal number of atoms of — Nuclear Chemistry and Radioactivity Chemistry Question
Question
A sample of rock from the moon is analyzed and found to contain an exactly equal number of atoms of Uranium (${}^{238}U$) and Lead (${}^{206}Pb$). Assuming all the Lead came purely from the decay of Uranium, and given the half-life of Uranium-238 is $4.5 \times 10^9\text{ years}$, calculate the age of the rock. Enter your answer as the coefficient $x$, where the age is $x \times 10^9\text{ years}$.
💡 Solution & Explanation
Since the number of Uranium atoms equals the number of Lead atoms, the current number of Uranium atoms $N_U = N_{Pb}$. The original number of Uranium atoms at the time of the rock's formation was $N_0 = N_U + N_{Pb} = 2N_U$. This implies that the ratio of Uranium remaining to the initial amount is $N_U / N_0 = 1/2$. By definition, the time required for a substance to decay to exactly half its initial value is one half-life. Therefore, the age of the rock is exactly $1$ half-life, which equals $4.5 \times 10^9\text{ years}$. Thus, $x = 4.5$.