Laws of Radioactive Decay and Half Life - UNSOLVED PRACTICE SET
Chapter: Nuclei | Topic: Laws of Radioactive Decay and Half Life
LAWS OF RADIOACTIVE DECAY AND HALF LIFE - UNSOLVED PRACTICE SET
Topic: Laws of Radioactive Decay and Half Life
Laws of Radioactive Decay and Half-Life
Q1. Radioactive decay follows:
- Zero-order kinetics
- First-order kinetics
- Second-order kinetics
- Third-order kinetics
Q2. The radioactive decay law is given by:
- N = N₀e^(λt)
- N = N₀e^(-λt)
- N = N₀λt
- N = N₀/λt
Q3. The half-life (T₁/₂) of a radioactive substance is related to the decay constant (λ) by:
- T₁/₂ = λ/0.693
- T₁/₂ = 0.693/λ
- T₁/₂ = λ × 0.693
- T₁/₂ = 1/λ
Q4. The mean life (τ) of a radioactive substance is:
- Equal to the half-life
- Greater than the half-life
- Less than the half-life
- Unrelated to the half-life
Q5. The activity of a radioactive sample is defined as:
- The total number of atoms in the sample
- The rate of decay of the sample
- The energy released per decay
- The half-life of the sample
Q6. After n half-lives, the fraction of a radioactive sample remaining is:
- 1/n
- (1/2)ⁿ
- 2ⁿ
- n/2
Short Answer Questions
Q7. State the law of radioactive decay. Write the mathematical expression and explain each term.
8. Define half-life and mean life. How are they related to the decay constant?
9. A radioactive substance has a half-life of 10 days. What fraction of the sample will remain after 30 days?
10. Why is radioactive decay considered a spontaneous process? Can the rate of decay be altered by chemical or physical means?
11. The activity of a radioactive sample decreases from 800 Bq to 100 Bq in 12 hours. Calculate the half-life.
12. What is the unit of activity? Define 1 Becquerel and 1 Curie.
Long Answer Questions
Q13. Derive the radioactive decay law N = N₀e^(-λt). Define decay constant and explain its physical significance.
Q14. Derive the relationship between half-life and decay constant. Show that T₁/₂ = 0.693/λ. Also derive the expression for mean life and show that τ = 1/λ.
Q15. Explain the concept of activity of a radioactive sample. Derive the expression A = A₀e^(-λt) and discuss the units of activity.
Numerical & Application-based Problems
Q16. A radioactive isotope has a half-life of 5 years. Initially, there are 10²⁴ atoms in the sample.
(a) Calculate the decay constant.
(b) Calculate the number of atoms remaining after 15 years.
(c) Calculate the activity of the sample initially and after 15 years.
(d) Calculate the time required for the activity to reduce to 1/16th of its initial value.
Q17. Carbon-14 has a half-life of 5730 years. A wooden artifact from an archaeological site has an activity of 5.2 disintegrations per minute per gram of carbon, compared to 15.6 for fresh wood.
(a) Calculate the decay constant of carbon-14.
(b) Calculate the age of the artifact.
(c) Calculate the activity of the artifact after another 5730 years.
Q18. In your school's physics lab, a student is studying radioactive decay using a simulation.
(a) She starts with 1000 atoms of a radioactive substance with a half-life of 2 hours. Calculate the number of atoms remaining after 8 hours and the number of atoms that have decayed.
(b) The student then considers a sample of iodine-131 used in thyroid treatment, which has a half-life of 8 days. A patient is given a dose containing 10¹⁵ atoms. Calculate the activity of the sample in Becquerels and in Curies.
(c) The doctor wants the activity to drop below 10⁹ Bq before the patient can be discharged. Calculate how many days the patient must wait.
(d) A classmate suggests storing radioactive waste until it is completely safe. Explain why this is practically impossible for substances with very long half-lives, and calculate how many years it would take for plutonium-239 (half-life 24,100 years) to decay to 0.1% of its original activity.
(e) In India, the Bhabha Atomic Research Centre manages radioactive waste. Explain why understanding half-life is crucial for deciding storage strategies for different types of nuclear waste.