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.

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