Half Life Practice Problems Answer Key

Half-Life Practice Problems: Answers and Detailed Explanations

Understanding half-life is crucial in various scientific fields, from nuclear physics and chemistry to medicine and geology. This comprehensive guide provides a detailed explanation of half-life concepts, followed by a series of practice problems with complete, step-by-step solutions. We'll cover different scenarios and complexities to solidify your understanding of this fundamental concept.

What is Half-Life?

Half-life (t_1/2) is the time it takes for one-half of the atoms of a radioactive material to disintegrate. It's a characteristic property of each radioactive isotope, meaning each radioactive element has its own unique half-life, which can range from fractions of a second to billions of years. This constant decay rate allows us to predict the amount of radioactive material remaining after a specific time.

Key Concepts:

• Radioactive Decay: The spontaneous transformation of an unstable atomic nucleus into a more stable one, often accompanied by the emission of particles and/or energy.
• Exponential Decay: The rate of decay is proportional to the amount of radioactive material present. This means the decay isn't linear; it slows down as the amount of material decreases.
• Parent Isotope: The original radioactive isotope undergoing decay.
• Daughter Isotope: The stable isotope produced after the decay process.

Half-Life Formula and Calculations

The most common formula used to calculate the remaining amount of a radioactive substance after a certain number of half-lives is:

N_t = N₀ * (1/2)ⁿ

Where:

• N_t is the amount of the substance remaining after time t.
• N₀ is the initial amount of the substance.
• n is the number of half-lives that have passed. This can be calculated as: n = t / t_1/2 where 't' is the elapsed time and 't_1/2' is the half-life.

Practice Problems with Detailed Solutions

Let's tackle a series of problems to master half-life calculations.

Problem 1:

A sample of radioactive iodine-131 has a half-life of 8 days. If you start with 100 grams of iodine-131, how much will remain after 24 days?

Solution:

1. Find the number of half-lives: n = t / t_1/2 = 24 days / 8 days = 3 half-lives

2. Use the half-life formula: N_t = N₀ * (1/2)ⁿ = 100 grams * (1/2)³ = 100 grams * (1/8) = 12.5 grams

Therefore, 12.5 grams of iodine-131 will remain after 24 days.

Problem 2:

Carbon-14 has a half-life of 5,730 years. A sample of wood initially contained 10 mg of Carbon-14. After a certain amount of time, only 2.5 mg remains. How many years have passed?

Solution:

1. Determine the number of half-lives: The sample has decayed to 1/4 of its original amount, meaning two half-lives have passed (10mg -> 5mg -> 2.5mg).

2. Calculate the elapsed time: t = n * t_1/2 = 2 half-lives * 5730 years/half-life = 11460 years

Therefore, 11,460 years have passed.

Problem 3:

A radioactive isotope has a half-life of 15 minutes. If you start with 200 g of the isotope, how much will remain after 45 minutes? How much will remain after 1 hour?

Solution:

• 45 minutes:

  1. Number of half-lives: 45 minutes / 15 minutes/half-life = 3 half-lives
  2. Amount remaining: 200g * (1/2)³ = 25g

• 1 hour (60 minutes):

  1. Number of half-lives: 60 minutes / 15 minutes/half-life = 4 half-lives
  2. Amount remaining: 200g * (1/2)⁴ = 12.5g

Therefore, 25g will remain after 45 minutes, and 12.5g will remain after 1 hour.

Problem 4:

A sample of Uranium-238, which has a half-life of 4.5 billion years, is found to contain 25% of its original amount. How old is the sample?

Solution:

1. Determine the number of half-lives: If 25% remains, then three half-lives have passed (100% -> 50% -> 25%).

2. Calculate the age: t = n * t_1/2 = 3 half-lives * 4.5 billion years/half-life = 13.5 billion years

Therefore, the sample is approximately 13.5 billion years old.

Problem 5 (More Advanced):

A substance decays from 80 grams to 5 grams in 100 days. What is its half-life?

Solution:

This problem requires a slightly different approach. We need to use logarithms to solve for the half-life.

1. Find the fraction remaining: 5 grams / 80 grams = 0.0625

2. Use the half-life formula and solve for n: 0.0625 = (1/2)^n

   Taking the logarithm of both sides (base 1/2): n = log_(1/2)(0.0625) = 4 half-lives

3. Calculate the half-life: t_1/2 = t / n = 100 days / 4 = 25 days

Therefore, the half-life of the substance is 25 days.

Understanding Decay Curves

Plotting the remaining amount of a radioactive substance against time results in an exponential decay curve. The curve illustrates how the decay rate slows down over time. Understanding these curves can help visualize the concept of half-life and predict future decay.

Applications of Half-Life

Half-life has numerous practical applications:

• Radioactive Dating: Determining the age of artifacts, fossils, and geological formations using isotopes like Carbon-14 and Uranium-238.
• Medical Applications: Radioactive isotopes are used in diagnosis (e.g., medical imaging) and treatment (e.g., radiation therapy). The half-life dictates how long the isotope remains active in the body.
• Nuclear Power: Understanding the half-life of radioactive waste is crucial for safe storage and disposal.
• Industrial Applications: Radioactive isotopes are used in various industrial processes, such as gauging thickness and detecting leaks.

Conclusion

Mastering half-life calculations is essential for understanding radioactive decay. By working through these practice problems and understanding the underlying concepts, you can confidently tackle more complex scenarios involving radioactive materials and their applications in diverse scientific fields. Remember to always clearly define your variables and follow the steps systematically. Consistent practice will solidify your understanding and improve your problem-solving skills. This knowledge is invaluable for various scientific and technical applications.