Answer:
7.0 × 10⁸ years
Explanation:
U-238 decays following first-order kinetics.
If we know the half-life (t1/2), we can calculate the rate constant (k).
k = ln2 / t1/2 = ln2 / 4.5 × 10⁹ y = 1.5 × 10⁻¹⁰ y⁻¹
When 10% of the U-238 atoms decay, the remaining concentration is 90% of the initial one. We can find the time required (t) using the following expression.
ln ([U] / [U]₀) = - k × t
ln (0.9[U]₀ / [U]₀) = - k × t
t = ln 0.9 / -k
t = ln 0.9 / -1.5 10⁻¹⁰ y⁻¹
t = 7.0 × 10⁸ y
