Half Life Worksheet With Answers

Understanding Half-Life: A Comprehensive Worksheet with Answers

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, from radiometric dating to medical treatments using radioactive isotopes. This comprehensive worksheet will guide you through the concept of half-life, providing explanations, examples, and practice problems with detailed solutions. Whether you're a high school student tackling your chemistry homework or a university student preparing for an exam, this resource will help solidify your understanding of this essential topic.

Introduction to Half-Life

Radioactive decay is a random process where unstable atomic nuclei lose energy by emitting radiation. This process continues until the nucleus becomes stable. The rate of decay is characterized by the half-life (t_1/2), which is the time required for half of the atoms in a given sample of a radioactive substance to decay. It's important to understand that half-life is a constant for a specific isotope – it doesn't depend on the amount of the substance present. Whether you start with 1 gram or 1 kilogram of a radioactive isotope, its half-life will remain the same.

Several factors influence the rate of radioactive decay and thus indirectly affect the half-life:

• Nuclear Structure: The specific arrangement of protons and neutrons in the nucleus is a primary determinant of its stability and therefore its half-life. Nuclei with an unstable neutron-to-proton ratio are more likely to undergo radioactive decay.
• Strong Nuclear Force: This force binds protons and neutrons together in the nucleus. The strength of this force influences the stability of the nucleus and, consequently, its half-life.

Understanding Half-Life Calculations

Calculating the remaining amount of a radioactive substance after a certain time involves understanding exponential decay. The formula used is:

N_t = N₀ * (1/2)^t/t_1/2

Where:

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

This formula shows that the remaining amount decreases exponentially with time. Each half-life reduces the amount of the substance by half.

Types of Radioactive Decay

It's also important to understand the different types of radioactive decay that can occur:

• Alpha Decay (α): The nucleus emits an alpha particle (two protons and two neutrons), reducing its atomic number by 2 and its mass number by 4.
• Beta Decay (β): A neutron transforms into a proton, emitting a beta particle (an electron) and an antineutrino. The atomic number increases by 1, while the mass number remains the same.
• Gamma Decay (γ): The nucleus releases a gamma ray (high-energy photon), without changing its atomic number or mass number. This typically occurs after alpha or beta decay, as the nucleus transitions to a lower energy state.

Each type of decay has its own characteristic half-life, depending on the specific isotope involved.

Worked Examples and Practice Problems

Let's work through some examples to solidify your understanding:

Example 1:

A sample of Carbon-14 (¹⁴C), which has a half-life of 5,730 years, initially contains 100 grams. How much ¹⁴C will remain after 11,460 years?

Solution:

• N₀ = 100 grams
• t_1/2 = 5,730 years
• t = 11,460 years

N_t = 100 grams * (1/2)^11460/5730 = 100 grams * (1/2)² = 25 grams

Therefore, 25 grams of ¹⁴C will remain after 11,460 years.

Example 2:

A radioactive isotope has a half-life of 10 days. If we start with 1000 atoms, how many atoms will remain after 30 days?

Solution:

• N₀ = 1000 atoms
• t_1/2 = 10 days
• t = 30 days

N_t = 1000 atoms * (1/2)^30/10 = 1000 atoms * (1/2)³ = 125 atoms

Therefore, 125 atoms will remain after 30 days.

Practice Problems:

1. Iodine-131 (¹³¹I) has a half-life of 8 days. If you start with a 100 mg sample, how much will remain after 24 days?
2. A sample of a radioactive substance decays from 80 grams to 10 grams in 30 minutes. What is its half-life?
3. Uranium-238 (²³⁸U) has a half-life of 4.5 billion years. If a rock sample initially contained 1 kg of ²³⁸U, how much would remain after 9 billion years?
4. If a radioactive isotope has a half-life of 2 hours, and you start with 1024 grams, how many half-lives will have passed after 8 hours? How much of the isotope will remain?
5. Explain why the half-life of a radioactive substance is constant, regardless of the initial amount of the substance.

Answers to Practice Problems:

1. After 24 days (3 half-lives), 12.5 mg of ¹³¹I will remain. (100 mg * (1/2)³ = 12.5 mg)

2. The substance underwent three half-lives (80g -> 40g -> 20g -> 10g). Therefore, the half-life is 10 minutes (30 minutes / 3 half-lives = 10 minutes/half-life).

3. After 9 billion years (2 half-lives), 250 grams of ²³⁸U would remain. (1000g * (1/2)² = 250g)

4. After 8 hours (4 half-lives), 16 grams of the isotope will remain. (1024g * (1/2)⁴ = 16g)

5. The half-life is a constant because radioactive decay is a probabilistic process. The probability of a single nucleus decaying within a given time interval is constant and independent of the number of other nuclei present. Each nucleus has an equal and independent chance of decaying, regardless of its surroundings or the number of other atoms of the same isotope present.

Further Applications and Considerations

The concept of half-life extends beyond basic calculations. It's fundamental to various fields:

• Radiometric Dating: Scientists use the half-lives of radioactive isotopes like Carbon-14 and Uranium-238 to determine the age of ancient artifacts, fossils, and geological formations. By comparing the ratio of the parent isotope to its decay products, they can estimate the time elapsed since the sample was formed.
• Nuclear Medicine: Radioactive isotopes with specific half-lives are used in medical imaging and treatment. For example, Technetium-99m (^99mTc), with its short half-life, is widely used in diagnostic imaging because it allows for rapid imaging with minimal radiation exposure to the patient.
• Nuclear Power: Understanding half-life is crucial in managing nuclear waste and ensuring the safe operation of nuclear power plants. The long half-lives of some radioactive byproducts require long-term storage and careful management.

Frequently Asked Questions (FAQ)

Q: Can the half-life of a radioactive substance be changed?

A: No, the half-life of a given radioactive isotope is a fundamental physical constant. It cannot be altered by changing temperature, pressure, or any other physical or chemical means.

Q: What happens after multiple half-lives?

A: After each half-life, the amount of the radioactive substance is halved. This process continues exponentially, meaning that the amount decreases more slowly as time progresses, but it never truly reaches zero. There will always be some extremely small amount remaining.

Q: Is it possible to predict exactly when a specific atom will decay?

A: No. Radioactive decay is a random process. While we can predict the average behavior of a large number of atoms (as reflected in the half-life), it's impossible to predict the exact moment when an individual atom will decay.

Q: How is half-life measured experimentally?

A: Half-life is experimentally determined by measuring the activity (decay rate) of a sample of the radioactive substance over time. By plotting the data and fitting it to an exponential decay curve, the half-life can be extracted.

Conclusion

Understanding half-life is essential for grasping the nature of radioactive decay and its many applications. This worksheet has provided a comprehensive overview of the concept, including calculations, worked examples, practice problems, and answers. Remember that the key to mastering this topic lies in practice. Work through additional problems, and don't hesitate to consult further resources if needed. With consistent effort, you'll develop a strong understanding of this fundamental principle in nuclear science.