Calculate the Number of Nuclei Decayed in a Radioactive Sample
Question:
The radioactive isotope ⁸²Sr has a half-life of 25.4 days. A sample containing this isotope has an initial activity at
The radioactive isotope ⁸²Sr has a half-life of 25.4 days. A sample containing this isotope has an initial activity at
t = 0 of 4.50 × 108 Bq. Calculate the number of nuclei that will decay in the time interval between t₁ = 34.0 hours and t₂ = 70.0 hours.
🧮 Step-by-Step Solution:
Step 1: Convert hours to days
t₁ = 34.0 hours = 34 / 24 = 17/12 ≈ 1.4167 dayst₂ = 70.0 hours = 70 / 24 = 35/12 ≈ 2.9167 days
Step 2: Calculate the decay constant (λ)
λ = ln(2) / half-life = 0.69314718 / 25.4 ≈ 0.02728925 day⁻¹
Step 3: Use the decay formula
Activity at time t:
At t₁:
At t₂:
Activity at time t:
A(t) = A₀ × e-λtAt t₁:
A(t₁) = 4.5 × 10⁸ × e−(0.02728925 × 17/12) ≈ 432935091.21 BqAt t₂:
A(t₂) = 4.5 × 10⁸ × e−(0.02728925 × 35/12) ≈ 415571190.65 Bq
Step 4: Find the difference in activity to get decayed nuclei
Decayed = A(t₁) − A(t₂)Decayed = 432935091.21 − 415571190.65 ≈ 17363900.55
✅ Final Answer:
The number of nuclei that decayed between
1.74 × 10⁷ nuclei
The number of nuclei that decayed between
t₁ = 34.0 hours and t₂ = 70.0 hours is approximately:1.74 × 10⁷ nuclei