How long will it take for 20% of the u-238 atoms in a sample of u-238 to decay?

Answer is: it takes 1,448 billion years.

The half-life for the radioactive decay of U-238 is 4.5 billion years and is independent of initial concentration.

c₀ - initial concentration of U-238.

c - concentration of U-238 remaining at time.

t = 4,5 ·10⁹ y.

First calculate the radioactive decay rate constant λ:

λ = 0,693 : t = 0,693 : 4,5·10⁹ y = 1,54·10⁻¹⁰ 1/y.

ln (c/c₀) = - λ·t₁.

ln (0,8/1) = - 1,54·10⁻¹⁰ 1/y · t₁.

t₁ = 1,448·10⁹ y.