Worksheet titled "Radioactive Half-Life Worksheet with Answers" containing six questions related to radioactive decay, half-life calculations, and understanding of radioactive materials.

Here is the complete solution and explanation for each question on the “Radioactive Half-life Worksheet”:

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1. What is the meaning of the term ‘half-life’ with regard to radioactivity?

✔ Answer:
The half-life of a radioactive substance is the time required for half of the radioactive atoms in a sample to decay into a different element or isotope.

➡️ *Explanation:*
Radioactive decay is a random process, but statistically, we can predict that after one half-life, 50% of the original radioactive material will remain. After two half-lives, 25% remains, and so on. It’s a constant for each radioactive isotope, regardless of sample size or physical conditions (like temperature or pressure).

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2. Why is it necessary to know the half-life of radioactive materials?

✔ Answer:
Knowing the half-life is essential for:
- Safety: Determining how long a radioactive material remains hazardous.
- Medical use: Calculating proper dosage and timing for diagnostic or therapeutic procedures.
- Dating artifacts/geological samples: Using radiometric dating (e.g., carbon-14 dating).
- Nuclear waste management: Planning storage and disposal based on how long materials stay radioactive.
- Scientific research: Understanding nuclear reactions and decay chains.

➡️ *Explanation:*
Half-life tells us the rate of decay — whether a material decays quickly (seconds/minutes) or slowly (thousands/millions of years). This directly affects risk assessment, usage, and environmental impact.

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3. What is the half-life of a radioactive sample if it decays from 100 grams to 12.5 grams in 27.9 hours?

✔ Answer:
9.3 hours

➡️ *Step-by-step Solution:*

We start with 100 g → ends with 12.5 g.

Let’s track the decay by half-lives:

- After 1 half-life: 100 → 50 g
- After 2 half-lives: 50 → 25 g
- After 3 half-lives: 25 → 12.5 g ✔

So, 3 half-lives = 27.9 hours

Therefore, one half-life = 27.9 ÷ 3 = 9.3 hours

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4. How many half-lives have passed if you started with 200 grams of a radioactive substance and now have 25 grams?

✔ Answer:
3 half-lives

➡️ *Step-by-step Solution:*

Start: 200 g

- After 1 half-life: 200 → 100 g
- After 2 half-lives: 100 → 50 g
- After 3 half-lives: 50 → 25 g ✔

So, 3 half-lives have passed.

*Alternative method using formula:*

Remaining amount = Initial × (1/2)^n
25 = 200 × (1/2)^n
→ 25/200 = (1/2)^n
→ 1/8 = (1/2)^n
→ (1/2)^3 = (1/2)^n → n = 3

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5. If we start with 480 grams of a substance, how much is left after 3 half-lives?

✔ Answer:
60 grams

➡️ *Step-by-step Solution:*

Start: 480 g

- After 1 half-life: 480 ÷ 2 = 240 g
- After 2 half-lives: 240 ÷ 2 = 120 g
- After 3 half-lives: 120 ÷ 2 = 60 g

*Using formula:*

Remaining = 480 × (1/2)^3 = 480 × 1/8 = 60 g

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6. What is the half-life of a radioactive isotope that decays from 600 grams to 75 grams in 6.9 days?

✔ Answer:
2.3 days

➡️ *Step-by-step Solution:*

Start: 600 g → End: 75 g

Track decay by half-lives:

- After 1 half-life: 600 → 300 g
- After 2 half-lives: 300 → 150 g
- After 3 half-lives: 150 → 75 g ✔

So, 3 half-lives = 6.9 days

Therefore, one half-life = 6.9 ÷ 3 = 2.3 days

*Formula check:*

75 = 600 × (1/2)^n
→ 75/600 = 1/8 = (1/2)^n → n = 3
→ t½ = total time / n = 6.9 / 3 = 2.3 days

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✔ Summary of Answers:

1. Half-life = time for half the radioactive atoms to decay.
2. Necessary for safety, medicine, dating, waste management, etc.
3. 9.3 hours
4. 3 half-lives
5. 60 grams
6. 2.3 days

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