A substance has a half-life of 2.045 minutes. If the initial amount of the substance was 132.8 grams, how many half-lives will have passed before the substance decays to 8.3 grams? What is the total time of decay?
Answer
Radioactive Decay: Half-Life and Total Decay Time
When a radioactive substance undergoes decay, its mass decreases by half after each half-life. We can calculate how many half-lives are needed for the substance to reduce from its initial amount to a given remaining mass.
🔢 Given:
Initial mass (M0) = 132.8 grams
Final mass (M) = 8.3 grams
Half-life (t½) = 2.045 minutes
Final mass (M) = 8.3 grams
Half-life (t½) = 2.045 minutes
📘 Step 1: Use Half-Life Formula
M = M0 × (1/2)n
Where:
- M is the remaining mass
- M0 is the initial mass
- n is the number of half-lives
➡️ Step 2: Plug in the Values
8.3 = 132.8 × (1/2)n
Divide both sides by 132.8:
(1/2)n = 8.3 / 132.8 ≈ 0.0625
Now, take logarithm on both sides:
n × log(1/2) = log(0.0625)
n = log(0.0625) / log(0.5)
n = (-1.2041) / (-0.3010) ≈ 4 half-lives
✅ Number of Half-Lives:
4 half-lives are required for the substance to decay from 132.8g to 8.3g.
⏱️ Step 3: Calculate Total Time of Decay
Total Time = n × t½ = 4 × 2.045 = 8.18 minutes
✅ Final Answer:
🔹 Number of half-lives: 4
🔹 Total decay time: 8.18 minutes
🔹 Total decay time: 8.18 minutes
Note: The half-life formula is based on exponential decay and is commonly used in physics, chemistry, archaeology (carbon dating), and nuclear science.