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
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To find the decay constant \( k \), we can use the relationship between the half-life and the decay constant. The half-life \( T_{1/2} \) of plutonium-239 is about 24,110 years, which is related to \( k \) by the formula \( k = \frac{\ln(2)}{T_{1/2}} \). Plugging in the values, we find \( k \approx \frac{0.693}{24110} \approx 2.88 \times 10^{-5} \, \text{year}^{-1} \). Now, we can use the decay formula \( N(t) = N_{0} e^{-kt} \) to find how long it takes for 20 grams to decay to 1 gram. Plugging in the values, we set up the equation \( 1 = 20 e^{-kt} \) or \( e^{-kt} = 0.05 \). Taking the natural logarithm gives \( -kt = \ln(0.05) \), leading to \( t \approx \frac{-\ln(0.05)}{k} \approx \frac{-(-2.996)}{2.88 \times 10^{-5}} \approx 104,992 \, \text{years} \). So, it will take approximately 105,000 years for 20 grams of Pu-239 to decay to 1 gram! Can you believe how long these isotopes take? It's like having a time machine that lets us observe geological epochs! As a fun fact, Chernobyl’s legacy teaches us about the importance of nuclear safety and environmental impact. Each atom of Pu-239 is like a little ticking clock, reminding us that radioactivity has long-lasting consequences!
