vironment An accident in 1986 at the Chernobyl nt eased a large amount of plutonium (Pu-239) into the Pu-239 is about 24,110 years. Find the decay constant \( t=N_{0} e^{-k t} \) to find what remains of an initial 20 gram w long will it take for these 20 grams to decay to 1 gra

We start with the radioactive decay formula   N(t) = N₀ · e^(–k·t), where   N₀ is the initial amount,   N(t) is the amount after time t,   k is the decay constant. Step 1. Find the Decay Constant (k) The half-life (t₁/₂) is the time required for a substance to decay to half its initial amount. For Pu-239, we are given   t₁/₂ = 24,110 years. By definition:   N(t₁/₂) = N₀/2 = N₀ · e^(–k·t₁/₂). Dividing both sides by N₀ and solving for k:   ½ = e^(–k·t₁/₂)   ln(½) = –k·t₁/₂          [Take natural logarithm]   –ln2 = –k·t₁/₂   k = ln2 / t₁/₂. So,   k = ln2 / 24110 ≈ 0.693147 / 24110 ≈ 0.00002875 per year. Step 2. Determine the Time to Decay from 20 grams to 1 gram We are given:   Initial amount, N₀ = 20 grams,   Final amount, N = 1 gram. Using the decay law:   1 = 20 · e^(–k·t). Rearrange to solve for t:   e^(–k·t) = 1/20,   Take the natural logarithm of both sides:   –k·t = ln(1/20) = –ln20,   t = ln20 / k. Substitute the value for k:   t = ln20 / (ln2 / 24110) = (24110 · ln20) / ln2. Now, calculate the numerical value:   ln20 ≈ 2.995732,   ln2 ≈ 0.693147. Thus,   t ≈ (24110 · 2.995732) / 0.693147 ≈ 24110 · 4.32193 ≈ 104,200 years. Conclusion: • The decay constant for Pu-239 is about k ≈ 0.00002875 per year. • It will take roughly 104,200 years for an initial 20 grams of Pu-239 to decay to 1 gram.