The half-life for the radioactive decay of U-238 is 4.5 billion years and is independent of initial concentration. How long will it take for 10% of the U-238 atoms in a sample of U-238 to decay

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