Half Life Problems And Answers Worksheet

Half-Life Problems and Answers: A Comprehensive Worksheet

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Half-life is a fundamental concept in nuclear chemistry and physics, describing the time it takes for half of a radioactive substance to decay. Understanding half-life is crucial for various applications, including radioactive dating, medical treatments, and nuclear safety. This comprehensive worksheet provides a series of problems with detailed answers to solidify your understanding of this essential concept.

Understanding Half-Life

Before diving into the problems, let's review the key principles of half-life.

What is Half-Life?

Half-life (t_1/2) is the time required for half the atoms of a radioactive material to decay into a different, more stable form. This decay is a random process; we can't predict when a specific atom will decay, but we can predict the behavior of a large number of atoms using the concept of half-life.

Exponential Decay

Radioactive decay follows an exponential decay pattern. This means the amount of radioactive material remaining decreases exponentially over time. The formula governing this decay is:

N_t = N₀ * (1/2)^t/t_1/2

Where:

• N_t is the amount of the substance remaining after time t.
• N₀ is the initial amount of the substance.
• t is the elapsed time.
• t_1/2 is the half-life of the substance.

Importance of Half-Life

Understanding half-life is vital in various fields:

• Radioactive Dating: Determining the age of artifacts and geological formations using the decay of radioactive isotopes like carbon-14.
• Nuclear Medicine: Utilizing radioactive isotopes in diagnostic and therapeutic procedures, carefully managing their decay to ensure safety and effectiveness.
• Nuclear Safety: Assessing the risk associated with radioactive waste and ensuring safe storage and disposal practices.
• Nuclear Power: Monitoring the decay of radioactive materials in nuclear reactors and managing spent fuel.

Half-Life Problems and Solutions

Now, let's tackle some practice problems to build your proficiency in half-life calculations.

Problem 1: Simple Half-Life Calculation

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

Solution:

1. Identify the given values: N₀ = 100 grams, t_1/2 = 10 years, t = 30 years.
2. Apply the half-life formula: N_t = N₀ * (1/2)^t/t_1/2
3. Substitute the values: N_t = 100 grams * (1/2)^{30 years/10 years}
4. Calculate: N_t = 100 grams * (1/2)³ = 100 grams * (1/8) = 12.5 grams

Answer: 12.5 grams will remain after 30 years.

Problem 2: Determining Half-Life from Decay Data

A sample of a radioactive substance initially contains 500 atoms. After 20 minutes, only 125 atoms remain. What is the half-life of this substance?

Solution:

1. Identify the given values: N₀ = 500 atoms, N_t = 125 atoms, t = 20 minutes.
2. Use the half-life formula: 125 atoms = 500 atoms * (1/2)^{20 minutes/t_1/2}
3. Simplify the equation: (1/4) = (1/2)^{20 minutes/t_1/2}
4. Since (1/4) = (1/2)², we can write: 2 = 20 minutes/t_1/2
5. Solve for t_1/2: t_1/2 = 20 minutes / 2 = 10 minutes

Answer: The half-life of the substance is 10 minutes.

Problem 3: Multiple Half-Lives

A sample of strontium-90 has a half-life of 29 years. If a sample initially contains 1 kilogram of strontium-90, how much will remain after 116 years?

Solution:

1. Determine the number of half-lives: 116 years / 29 years/half-life = 4 half-lives
2. Calculate the remaining amount: After each half-life, the amount is halved. Therefore, after 4 half-lives: 1 kg * (1/2) * (1/2) * (1/2) * (1/2) = 1 kg * (1/16) = 0.0625 kg

Answer: 0.0625 kg (or 62.5 grams) will remain after 116 years.

Problem 4: Working Backwards from Remaining Amount

A radioactive isotope has a half-life of 5730 years. A sample contains 25 grams of this isotope. If the original sample contained 100 grams, how old is the sample?

Solution:

1. Determine the fraction remaining: 25 grams / 100 grams = 0.25
2. Express the fraction as a power of 1/2: 0.25 = (1/2)²
3. This indicates that two half-lives have passed.
4. Calculate the age: 2 half-lives * 5730 years/half-life = 11460 years

Answer: The sample is approximately 11460 years old.

Problem 5: Complex Scenario Involving Multiple Isotopes (Advanced)

A rock sample contains two radioactive isotopes: Isotope A with a half-life of 1000 years and Isotope B with a half-life of 5000 years. Initially, the rock contains equal amounts of both isotopes. After 10,000 years, the ratio of Isotope A to Isotope B is 1:8. What was the initial amount of each isotope? (Assume the decay of one isotope doesn't affect the other.)

Solution:

This problem requires a more sophisticated approach.

1. Isotope A Decay: After 10,000 years (10 half-lives), the amount of Isotope A remaining is: (1/2)^10 = 1/1024 of the initial amount.
2. Isotope B Decay: After 10,000 years (2 half-lives), the amount of Isotope B remaining is: (1/2)^2 = 1/4 of the initial amount.
3. Ratio: The ratio of Isotope A to Isotope B is given as 1:8. Let x be the initial amount of each isotope. Then: (x/1024) / (x/4) = 1/8
4. Solve for x: This simplifies to 4/1024 = 1/8, which is consistent. This means our initial assumption that the ratio is 1:8 after 10,000 years is correct. However, we cannot uniquely determine the initial amount (x) based on the provided ratio. More information is needed to solve this completely.

Answer: Further information is required to determine the initial amount of each isotope. This problem highlights the complexity of radioactive decay scenarios involving multiple isotopes.

Further Exploration

These problems provide a solid foundation in half-life calculations. To further enhance your understanding, consider exploring:

• Carbon Dating: Investigate how carbon-14 dating is used to determine the age of organic materials.
• Uranium-Lead Dating: Learn about the principles behind uranium-lead dating, a technique used to date rocks and minerals.
• Potassium-Argon Dating: Understand the application of potassium-argon dating in geological dating.
• More complex decay schemes: Explore scenarios involving decay chains where one isotope decays into another radioactive isotope.

By practicing various problems and delving deeper into the applications of half-life, you'll master this fundamental concept in nuclear science. Remember to always carefully note the units of time and amount in your calculations. Consistent practice is key to building your confidence and expertise.