Assuming of trace radioactive element X with a half life of 30 years is absorbed by a growing tree. — Chemical Kinetics Chemistry Question

💡 Solution & Explanation

**Step 1: Identify the radioactive decay formula** Use the decay equation: N(t) = N₀ × (1/2)^(t/t₁/₂) Where: - N(t) = amount remaining after time t - N₀ = initial amount - t = elapsed time (100 years) - t₁/₂ = half-life (30 years) **Step 2: Calculate the number of half-lives** Number of half-lives = t/t₁/₂ = 100/30 = 10/3 ≈ 3.33 **Step 3: Set up the decay calculation** N(t) = N₀ × (1/2)^(10/3) **Step 4: Simplify using logarithms** (1/2)^(10/3) = 2^(-10/3) Taking log: log[2^(-10/3)] = (-10/3) × log 2 = (-10/3) × 0.30 = -1.00 **Step 5: Convert back to find the fraction remaining** If log(fraction) = -1.00, then fraction = 10^(-1.00) = 10^(-1) = 0.1 **Step 6: Express in the required format** Remaining amount = 0.1 × N₀ = 1.00 × 10^(-1) × N₀ Therefore, the answer is 1.00.