14C has a half-life of 5760 years; 100 mg of a sample of 14C will completely disintegrate in:
(1) 23,040 years
(2) 1440 years
(3) 11,520 years
(4) infinite time

📘 Problem Statement

The isotope 14C has a half-life of 5760 years. If you start with 100 mg of 14C, how long will it take for the entire sample to completely disintegrate?

Options:

1. 23,040 years

2. 1440 years

3. 11,520 years

4. Infinite time

🔬 Understanding Half-Life and Radioactive Decay

The half-life of a radioactive substance is the time it takes for half of the substance to decay. In this case, the half-life of 14C is 5760 years, which means after 5760 years, half of the 14C will have decayed, and half will remain.

It’s important to note that radioactive decay is not linear, which means that as the substance decays, it never truly “completely” disintegrates in finite time. Instead, it approaches zero over time. However, it’s often assumed that the material is essentially gone after 10 half-lives, or when only a very tiny fraction remains.

🧮 Step-by-Step Breakdown

• After 1 half-life (5760 years), 50% of the original 100 mg will have decayed, leaving 50 mg of 14C.

• After 2 half-lives (11,520 years), 50% of the remaining 50 mg will decay, leaving 25 mg of 14C.

• After 3 half-lives (17,280 years), 50% of the remaining 25 mg will decay, leaving 12.5 mg of 14C.

• This process continues indefinitely.

However, the concept of “complete disintegration” in radioactive decay means that, in practical terms, the sample never fully decays. There will always be a tiny amount of the substance left, even after countless half-lives.

✅ Conclusion

From the choices provided, none accurately represents the point at which 100 mg of 14C would completely disintegrate, because complete disintegration takes infinite time. In reality, the remaining 14C gets smaller and smaller with each half-life but never truly reaches zero.

Thus, the correct answer is:

4. Infinite time

💡 Why This Concept Is Important

Understanding half-life is crucial in fields such as:

• Radiocarbon dating: Used to determine the age of ancient artifacts and fossils.

• Nuclear chemistry: Involves understanding the decay rates of different isotopes.

• Medical imaging: Techniques like PET scans use radioactive isotopes to track biological processes.

In everyday life, the half-life concept helps us understand how substances decay over time, influencing everything from dating fossils to managing nuclear waste.

✅ Key Takeaways

• The half-life of a substance is the time it takes for half of it to decay.

• In the case of 14C, the material never completely disintegrates in finite time; it takes infinite time for complete decay.

• Radioactive decay is exponential, meaning the remaining material decreases by half in each successive half-life.