One of the nuclides in spent nuclear fuel is U-235, an alpha emitter with a half-life of 703 million years. How long will it take for an amount of U-235 to reach 23.0% of its initial amount

1.49 × 10⁹ years

Explanation:

Step 1: Calculate the rate constant (k) for the nuclear decay of U-235

The decay follows first-order kinetics with a half-life (t1/2) of 703 × 10⁶ years. We can calculate "k" using the following expression.

k = ln2/ t1/2 = ln2 / 703 × 10⁶ y = 9.86 × 10⁻¹⁰ y⁻¹

Step 2: Calculate the time elapsed (t) so that the final amount ([U]) is 23.0% of the initial amount ([U]₀)

For first order kinetics, we will use the following expression.

ln ([U]/[U]₀) = -k × t

ln (0.230[U]₀/[U]₀) = -9.86 × 10⁻¹⁰ y⁻¹ × t

ln 0.230 = -9.86 × 10⁻¹⁰ y⁻¹ × t

t = 1.49 × 10⁹ y