Assuming 1μg of trace radioactive element X with a half life of 30 years is absorbed by a growing tree. The amount of X remaining in the tree after 100 years is ______ × 10–1μg. [Given : ln 10 = 2.303; log 2 = 0.30]
To determine the remaining amount of radioactive element X after 100 years, we use the formula for radioactive decay: N(t) = N0 × (1/2)^(t/T1/2), where N(t) is the remaining quantity, N0 is the initial quantity, t is the elapsed time, and T1/2 is the half-life. Here, N0 = 1 μg, T1/2 = 30 years, and t = 100 years. First, calculate n, the number of half-lives: n = t / T1/2 = 100 / 30 ≈ 3.3333. The amount remaining is calculated as: N(t) = 1 × (1/2)^(3.3333). To solve for (1/2)^3.3333, use logarithms: log(N(t)) = log(1) + 3.3333 × log(1/2) Given log 2 = 0.30, log(1/2) = -0.30. Calculate: log(N(t)) = 0 + 3.3333 × (-0.30) = -1.0. Thus, N(t) = 10^(-1.0) = 0.1 μg. Expressed as a power of 10–1: 0.1 = 1 × 10–1 μg. Checking against the range 1,1 confirms N(t) is in the expected range. Therefore, the correct solution is: 1 × 10–1 μg.
