Half Life Problems Worksheet And Answers

Half-Life Problems: A Comprehensive Worksheet and Solutions Guide

Understanding half-life is crucial in various scientific fields, from nuclear physics and chemistry to medicine and archaeology. This comprehensive worksheet and answer guide will walk you through various half-life problems, equipping you with the skills to tackle any challenge. We'll explore different approaches, emphasizing the underlying principles and providing detailed explanations for each solution. Mastering half-life calculations will enhance your problem-solving abilities and deepen your understanding of radioactive decay.

What is Half-Life?

Half-life is the time it takes for half of a radioactive substance to decay into a more stable form. This decay is a random process; we can't predict when a specific atom will decay, but we can accurately predict the behavior of a large number of atoms over time. The half-life of a particular radioactive isotope is constant and doesn't depend on the initial amount of the substance. This constant nature allows us to use half-life to calculate the age of objects or predict the amount of a radioactive substance remaining after a certain time.

Key Concepts and Formulas

Before diving into the problems, let's review the essential formulas and concepts:

• N(t) = N₀(1/2)^(t/T): This is the primary equation for half-life calculations. Where:

  • N(t) is the amount of substance remaining after time t.
  • N₀ is the initial amount of the substance.
  • t is the elapsed time.
  • T is the half-life of the substance.

• Understanding Exponential Decay: Half-life problems involve exponential decay. The amount of substance remaining decreases exponentially with time, halving with each half-life period.

• Number of Half-Lives: To simplify calculations, determining the number of half-lives that have elapsed (t/T) is often helpful.

• Graphical Representation: Plotting the remaining amount against time on a graph provides a visual representation of the exponential decay.

Half-Life Problems Worksheet

Now, let's tackle a series of problems of varying difficulty:

Problem 1: Simple Half-Life Calculation

A radioactive isotope has a half-life of 10 years. If we start with 100 grams, how much will remain after 30 years?

Solution:

1. Identify the knowns: N₀ = 100 grams, T = 10 years, t = 30 years.
2. Calculate the number of half-lives: 30 years / 10 years/half-life = 3 half-lives.
3. Apply the half-life formula: N(t) = 100 grams * (1/2)^3 = 12.5 grams.

Therefore, 12.5 grams will remain after 30 years.

Problem 2: Finding the Half-Life

A sample of a radioactive substance initially contains 256 grams. After 40 days, only 8 grams remain. What is the half-life of this substance?

Solution:

1. Identify the knowns: N₀ = 256 grams, N(t) = 8 grams, t = 40 days.
2. Determine the number of half-lives: The substance has gone through 5 half-lives (256 -> 128 -> 64 -> 32 -> 16 -> 8).
3. Calculate the half-life: 40 days / 5 half-lives = 8 days/half-life.

Therefore, the half-life of the substance is 8 days.

Problem 3: Determining Remaining Amount After a Fraction of a Half-Life

A radioactive isotope with a half-life of 20 minutes has an initial amount of 500 grams. How much will remain after 15 minutes?

Solution:

1. Identify the knowns: N₀ = 500 grams, T = 20 minutes, t = 15 minutes.
2. Use the half-life formula: N(t) = 500 grams * (1/2)^(15 minutes / 20 minutes) ≈ 353.6 grams.

Therefore, approximately 353.6 grams will remain after 15 minutes.

Problem 4: Working Backwards – Determining Initial Amount

After 60 years, 10 grams of a radioactive isotope remain. The half-life of this isotope is 20 years. What was the initial amount of the isotope?

Solution:

1. Identify the knowns: N(t) = 10 grams, T = 20 years, t = 60 years.
2. Calculate the number of half-lives: 60 years / 20 years/half-life = 3 half-lives.
3. Work backwards from the remaining amount: 10 grams * 2 * 2 * 2 = 80 grams.

Therefore, the initial amount of the isotope was 80 grams.

Problem 5: A More Complex Scenario – Multiple Isotopes

A sample contains two radioactive isotopes, Isotope A and Isotope B. Isotope A has a half-life of 10 years, and Isotope B has a half-life of 20 years. Initially, the sample contains 100 grams of Isotope A and 50 grams of Isotope B. How much of each isotope will remain after 30 years?

Solution:

This problem requires separate calculations for each isotope:

Isotope A:

1. Number of half-lives: 30 years / 10 years/half-life = 3 half-lives.
2. Remaining amount: 100 grams * (1/2)^3 = 12.5 grams.

Isotope B:

1. Number of half-lives: 30 years / 20 years/half-life = 1.5 half-lives.
2. Remaining amount: 50 grams * (1/2)^1.5 ≈ 17.7 grams.

Therefore, after 30 years, approximately 12.5 grams of Isotope A and 17.7 grams of Isotope B will remain.

Problem 6: Applications of Half-Life – Carbon Dating

A piece of wood is found to contain 1/8 the amount of Carbon-14 found in living trees. The half-life of Carbon-14 is approximately 5730 years. How old is the piece of wood?

Solution:

1. Determine the number of half-lives: 1/8 represents 3 half-lives (1/2 * 1/2 * 1/2 = 1/8).
2. Calculate the age: 3 half-lives * 5730 years/half-life = 17190 years.

Therefore, the piece of wood is approximately 17190 years old.

Advanced Concepts and Applications

While the worksheet covered fundamental half-life problems, several more advanced concepts deserve mention:

• Decay Series: Some isotopes decay into other radioactive isotopes, forming a decay chain. Analyzing these series requires understanding the half-lives of each isotope in the chain.

• Radioactive Equilibrium: In a decay series, secular equilibrium can occur where the amount of a daughter isotope remains relatively constant.

• Applications in Medicine: Radioisotopes are used in medical imaging (PET scans) and treatments (radiotherapy). Understanding half-life is crucial for determining appropriate dosages and minimizing radiation exposure.

• Geological Dating: Radioactive isotopes with very long half-lives are used to date rocks and geological formations, providing insights into the Earth's history.

Conclusion

This comprehensive guide provided a thorough introduction to half-life problems, along with detailed solutions and explanations. Mastering these concepts will significantly enhance your understanding of radioactive decay and its applications across various scientific disciplines. Remember to practice consistently and explore advanced concepts to further refine your skills. The key to success lies in a solid grasp of the fundamental formula and its application to different scenarios. Don't hesitate to revisit these problems and try variations to solidify your understanding. The more you practice, the more confident you will become in tackling even the most challenging half-life calculations.