Question
__ A sample of radium has a weight of 1.5 mg and a half-life of approximately 6 years. 1) How much of the sample will remain after: iii) 1 year? i) 6 years? ii) 3 years?
Solution
4.3
(308 Votes)
Gianna
Professional · Tutor for 6 years
Answer
Here's how to calculate the remaining radium after a given time, using the concept of half-life:**Understanding Half-life**A half-life is the time it takes for half of a radioactive substance to decay. After one half-life, 50% remains. After two half-lives, 25% remains (half of the remaining 50%), and so on.**Formula**The formula for radioactive decay is:*N(t) = N₀ * (1/2)^(t/T)*Where:* N(t) = the amount remaining after time t* N₀ = the initial amount* t = the time that has passed* T = the half-life**Calculations**In this case, N₀ = 1.5 mg and T = 6 years.**i) 6 years (1 half-life)*** N(6) = 1.5 * (1/2)^(6/6)* N(6) = 1.5 * (1/2)^1* N(6) = 1.5 * 0.5* N(6) = 0.75 mg**ii) 3 years (0.5 half-lives)*** N(3) = 1.5 * (1/2)^(3/6)* N(3) = 1.5 * (1/2)^0.5* N(3) = 1.5 * 0.7071 (approximately)* N(3) ≈ 1.06 mg**iii) 1 year (1/6 of a half-life)*** N(1) = 1.5 * (1/2)^(1/6)* N(1) = 1.5 * 0.8909 (approximately)* N(1) ≈ 1.34 mg**Summary of Results*** After 1 year: Approximately 1.34 mg will remain.* After 3 years: Approximately 1.06 mg will remain.* After 6 years: 0.75 mg will remain.