Half-Life Calculations: Step-by-Step Examples for Chemistry Students

📚 Quick Study Guide

• ⚛️ Half-Life Definition: The time required for half of the radioactive nuclei in a sample to decay.
• ➗ Formula: $N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}}$, where:
  • $N(t)$ = amount remaining after time $t$
  • $N_0$ = initial amount
  • $t$ = elapsed time
  • $t_{1/2}$ = half-life
• 📏 Units: Ensure time units for $t$ and $t_{1/2}$ are consistent.
• 📝 Applications: Radioactive dating, medical treatments, and nuclear chemistry.
• 💡 Tip: If you know how many half-lives have passed, you can easily find the remaining amount by repeatedly dividing by 2.

🧪 Practice Quiz

1. Question 1: A radioactive substance has a half-life of 10 years. If you start with 100 grams, how much will remain after 30 years?
   1. 25 grams
   2. 12.5 grams
   3. 50 grams
   4. 3.125 grams
2. Question 2: Carbon-14 has a half-life of 5730 years. An ancient wooden tool contains 25% of its original Carbon-14. Approximately how old is the tool?
   1. 2865 years
   2. 5730 years
   3. 11460 years
   4. 17190 years
3. Question 3: A sample of a radioactive isotope has an initial activity of 200 mCi. If the activity is 25 mCi after 12 hours, what is the half-life of the isotope?
   1. 3 hours
   2. 4 hours
   3. 6 hours
   4. 8 hours
4. Question 4: Polonium-210 has a half-life of 138 days. What fraction of a sample of Polonium-210 will remain after 414 days?
   1. 1/2
   2. 1/4
   3. 1/8
   4. 1/16
5. Question 5: A radioactive tracer used in medical imaging has a half-life of 6 hours. If a patient is injected with 32 mCi of the tracer, how much will remain after 24 hours?
   1. 1 mCi
   2. 2 mCi
   3. 4 mCi
   4. 8 mCi
6. Question 6: The half-life of a certain isotope is 1.5 hours. If you have 800 grams of the isotope, how much will remain after 6 hours?
   1. 25 grams
   2. 50 grams
   3. 100 grams
   4. 200 grams
7. Question 7: A rock sample is found to contain Uranium-238 and its decay product Lead-206 in a ratio of 1:3. Given that the half-life of Uranium-238 is 4.5 billion years, approximately how old is the rock?
   1. 2.25 billion years
   2. 4.5 billion years
   3. 9.0 billion years
   4. 13.5 billion years