This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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Question: A sample of a radioactive isotope has an initial mass of 200 g. After 10 days, 50 g of the isotope remains. What is the half-life of the isotope? Answer: The formula relating the remaining mass of a radioactive isotope to its half-life is: N(t) = N₀ * (1/2)^(t / T) where: N(t) is the mass remaining after time t N₀ is the initial mass t is the time elapsed T is the half-life Step 1: Identify the given values. N₀ = 200 g N(t) = 50 g t = 10 days Step 2: Substitute the values into the formula. 50 g = 200 g * (1/2)^(10 days / T) Step 3: Isolate the term with the half-life. (50 g) / (200 g) = (1/2)^(10 days / T) (1/4) = (1/2)^(10 days / T) Step 4: Express both sides with the same base. (1/2)² = (1/2)^(10 days / T) Step 5: Equate the exponents. 2 = (10 days) / T Step 6: Solve for T. T = (10 days) / 2 T = 5 days The half-life of the isotope is 5 days. Send me the next one 📸