How to Solve Half-Life Calculation Problems

CSEC Physics: Master the Calculations

Essential Understanding: Half-life problems involve finding one of three quantities: remaining amount, original amount, or time elapsed. The key is to first determine how many half-lives have passed, then apply the halving pattern. This article provides step-by-step strategies and worked examples for all types of half-life calculations.

🔑 Key Skill: Count Half-Lives

📈 Three Variables: M, m₀, t, T½

🎯 Learning Goal: Problem-Solving Strategy

The Three Types of Half-Life Problems

Every half-life problem involves finding one of these three quantities. Learn to identify which type you're dealing with:

📉 Type 1: Find Remaining Amount

Given: Original amount (m₀), half-life (T½), time (t)

Find: Remaining amount (m)

Example: "A sample has 100g. After 3 half-lives, how much remains?"

📅 Type 2: Find Time Elapsed

Given: Original amount (m₀), remaining amount (m), half-life (T½)

Find: Time (t)

Example: "50g decays to 6.25g. How much time passed?"

🔢 Type 3: Find Original Amount

Given: Remaining amount (m), half-life (T½), time (t)

Find: Original amount (m₀)

Example: "After 2 half-lives, 25g remains. What was the original mass?"

The Universal Formula

All half-life calculations use this fundamental formula:

\[ m = m_0 \times \left(\frac{1}{2}\right)^n \]

Where:

m = remaining mass (or activity)
m₀ = original mass (or initial activity)
n = number of half-lives passed = t ÷ T½
t = time elapsed
T½ = half-life

            💡 Pro Tip: You can rearrange the formula to solve for any variable. Remember that you need to find n (number of half-lives) first, then use it in the main formula.
        

Step-by-Step Method

Identify the knowns: Write down what you know (m₀, m, t, T½)
Identify what you're solving for: Is it m, m₀, or t?
Find the number of half-lives (n): n = t ÷ T½ (if you know t and T½)
Apply the formula: Use m = m₀ × (½)ⁿ or rearrange as needed
Calculate and verify: Check if your answer makes sense (remaining < original for decay)

Worked Examples

Easy

Example 1: Finding Remaining Mass

Problem: A radioactive sample has an initial mass of 80 g. The half-life is 5 years. What mass remains after 15 years?

Step 1: Identify knowns

m₀ = 80 g, T½ = 5 years, t = 15 years

Step 2: Calculate number of half-lives

n = t ÷ T½ = 15 ÷ 5 = 3 half-lives

Step 3: Apply the formula

m = m₀ × (½)ⁿ = 80 × (½)³

m = 80 × ⅛ = 10 g

Answer: 10 g remains

Easy

Example 2: Finding Number of Half-Lives

Problem: A sample decays from 200 g to 25 g. How many half-lives passed?

Step 1: Identify the pattern

200 g → 100 g (1 half-life) → 50 g (2 half-lives) → 25 g (3 half-lives)

Step 2: Verify with formula

m = m₀ × (½)ⁿ

25 = 200 × (½)ⁿ

(½)ⁿ = 25/200 = 1/8 = (½)³

Therefore n = 3

Answer: 3 half-lives passed

Medium

Example 3: Finding Time Elapsed

Problem: A sample has a half-life of 12 hours. If the activity drops from 800 Bq to 100 Bq, how much time has passed?

Step 1: Find how many half-lives

800 → 400 (1) → 200 (2) → 100 (3)

n = 3 half-lives

Step 2: Calculate time

t = n × T½ = 3 × 12 = 36 hours

Answer: 36 hours have passed

Medium

Example 4: Finding Original Mass

Problem: After 2 half-lives, a radioactive sample has a mass of 30 g. What was the original mass?

Step 1: Work backwards

After 2 half-lives: 30 g represents ¼ of original

Step 2: Calculate original mass

m₀ = m ÷ (½)ⁿ = 30 ÷ (¼) = 30 × 4 = 120 g

Alternative method:

m = m₀ × (½)ⁿ

30 = m₀ × (½)²

30 = m₀ × ¼

m₀ = 30 × 4 = 120 g

Answer: Original mass was 120 g

Hard

Example 5: Mixed Problem

Problem: A sample of iodine-131 (half-life = 8 days) has an initial activity of 1600 Bq. After 24 days, what is the remaining activity?

Step 1: Calculate number of half-lives

n = t ÷ T½ = 24 ÷ 8 = 3 half-lives

Step 2: Apply formula

A = A₀ × (½)ⁿ

A = 1600 × (½)³

A = 1600 × ⅛ = 200 Bq

Answer: Remaining activity is 200 Bq

🧮 Interactive Half-Life Calculator

Enter your values to calculate the missing quantity. The visualizer shows how the mass changes with each half-life!

Initial Mass (g)

Half-Life (years)

Time Elapsed (years)

Quick Reference Table

Memorize this pattern to quickly solve problems without calculations:

Half-Lives Passed	Fraction Remaining	Percentage Remaining	Decayed
0	1/1	100%	0%
1	1/2	50%	50%
2	1/4	25%	75%
3	1/8	12.5%	87.5%
4	1/16	6.25%	93.75%
5	1/32	3.125%	96.875%
6	1/64	1.56%	98.44%
7	1/128	0.78%	99.22%
8	1/256	0.39%	99.61%

            💡 Tip for Exams: If you see that the remaining amount is exactly 50%, 25%, 12.5%, etc., you can instantly tell how many half-lives have passed without doing any division!
        

Practice Problems

Practice 1: Carbon Dating Application

Problem: A wooden artifact from an ancient burial site has a Carbon-14 activity of 25% of a living tree. Carbon-14 has a half-life of 5730 years. How old is the artifact?

Solution:

25% remaining = 1/4 remaining

1/4 = (½)², so 2 half-lives have passed

Age = 2 × 5730 = 11,460 years

Answer: The artifact is approximately 11,460 years old.

Practice 2: Medical Application

Problem: A patient is given a radioactive tracer with Technetium-99m (half-life = 6 hours). The initial activity is 800 mCi. What will be the activity after 24 hours?