Half Life Problems And Answers Examples

Half-Life Problems and Answers: A Comprehensive Guide

Half-life is a fundamental concept in various scientific fields, most prominently in nuclear physics and chemistry. Understanding half-life is crucial for comprehending radioactive decay, carbon dating, and numerous other applications. This comprehensive guide delves into half-life problems, providing explanations, examples, and step-by-step solutions to equip you with a strong understanding of this essential concept.

What is Half-Life?

Half-life (t_1/2) is the time required for half of the atoms in a radioactive sample to decay. It's a characteristic property of each radioactive isotope, meaning that 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 scientists to use half-life to determine the age of materials, predict future decay, and understand the behavior of radioactive substances. The decay follows a first-order kinetics process, meaning the rate of decay is directly proportional to the amount of radioactive material present.

Understanding the Half-Life Equation

The fundamental equation governing half-life is often expressed in different forms, depending on the context. The most common forms are:

• 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.

• ln(N_t/N₀) = -kt

Where:

• k is the decay constant, related to half-life by the equation: k = ln(2)/t_1/2

Types of Half-Life Problems

Half-life problems can be categorized into several types, each requiring a slightly different approach to solving:

1. Determining the remaining amount after a given time: These problems provide the initial amount, half-life, and elapsed time, and ask you to calculate the remaining amount of the substance.

2. Determining the time elapsed given the initial and remaining amounts: Here, you know the initial and remaining amounts and the half-life, and you need to find the time that has passed.

3. Determining the half-life given the initial and remaining amounts and elapsed time: This type of problem requires you to calculate the half-life given the other three parameters.

4. Problems involving multiple half-lives: These problems involve calculating the remaining amount after several half-lives have passed.

5. Problems involving complex decay chains: These involve more than one radioactive isotope decaying into other isotopes, each with its own half-life.

Solved Examples of Half-Life Problems

Let's work through several examples illustrating the different types of half-life problems:

Example 1: Determining the remaining amount after a given time.

Problem: A sample of radioactive iodine-131 has an initial mass of 100 grams and a half-life of 8.02 days. What mass of iodine-131 remains after 24.06 days?

Solution:

We use the equation: N_t = N₀ (1/2)^t/t_1/2

• N₀ = 100 grams
• t = 24.06 days
• t_1/2 = 8.02 days

N_t = 100 grams * (1/2)^{24.06 days / 8.02 days} = 100 grams * (1/2)³ = 12.5 grams

Answer: After 24.06 days, 12.5 grams of iodine-131 remain.

Example 2: Determining the time elapsed given the initial and remaining amounts.

Problem: A sample of carbon-14, with a half-life of 5730 years, initially contains 100 grams. After a certain time, only 25 grams remain. How long has it been?

Solution: Again we use the half-life equation: N_t = N₀ (1/2)^t/t_1/2

• N₀ = 100 grams
• N_t = 25 grams
• t_1/2 = 5730 years

25 = 100 * (1/2)^t/5730

0.25 = (1/2)^t/5730

Taking the logarithm of both sides (base 1/2 or using base 10 and the change of base rule):

log_1/2(0.25) = t/5730

2 = t/5730

t = 2 * 5730 years = 11460 years

Answer: 11460 years have passed.

Example 3: Determining the half-life given the initial and remaining amounts and elapsed time.

Problem: A radioactive sample initially contains 200 grams. After 10 years, only 50 grams remain. What is the half-life of the sample?

Solution: We again use the half-life equation:

50 = 200 * (1/2)^10/t_1/2

0.25 = (1/2)^10/t_1/2

Taking the logarithm (base 1/2):

log_1/2(0.25) = 10/t_1/2

2 = 10/t_1/2

t_1/2 = 10/2 = 5 years

Answer: The half-life of the sample is 5 years.

Example 4: Problems involving multiple half-lives.

Problem: A substance has a half-life of 20 minutes. If we start with 1 kg, how much will remain after 1 hour (60 minutes)?

Solution:

First, determine how many half-lives have occurred: 60 minutes / 20 minutes/half-life = 3 half-lives

After each half-life, the amount is halved. Therefore:

• After 1 half-life: 1 kg / 2 = 0.5 kg
• After 2 half-lives: 0.5 kg / 2 = 0.25 kg
• After 3 half-lives: 0.25 kg / 2 = 0.125 kg

Answer: 0.125 kg (or 125 grams) will remain after 1 hour.

Example 5: A more complex decay chain scenario (Conceptual)

Let's consider a simplified scenario where Uranium-238 (with a very long half-life) decays into Thorium-234 (with a shorter half-life), which then decays into Protactinium-234 (with a yet shorter half-life), and so on. Solving these problems requires a more advanced mathematical approach involving coupled differential equations, reflecting the interconnected decay rates of each isotope. Each isotope's concentration at a given time is dependent on its own decay rate and the decay rate of its precursor isotope. These calculations are beyond the scope of this introductory guide but illustrate the complexity that can arise in more realistic scenarios.

Applications of Half-Life

The concept of half-life has broad applications across various scientific disciplines:

• Radioactive dating: Carbon-14 dating is a prominent example, used to determine the age of organic materials. By measuring the ratio of carbon-14 to carbon-12, scientists can estimate the time elapsed since the organism died. Similar techniques using other radioactive isotopes are used to date geological formations and artifacts.

• Nuclear medicine: Radioactive isotopes with specific half-lives are used in diagnostic and therapeutic procedures. The short half-life ensures that the radiation exposure to the patient is minimized after the procedure.

• Nuclear power: Understanding the half-lives of radioactive materials is crucial in the design, operation, and safety of nuclear reactors and the management of nuclear waste.

• Environmental science: Half-life is used to track the movement and persistence of radioactive pollutants in the environment.

• Industrial applications: Radioactive isotopes with specific half-lives are used in various industrial applications, including gauging the thickness of materials and tracing the flow of fluids in pipelines.

Conclusion

Understanding half-life is critical for numerous applications across science and technology. This guide provides a foundation for solving various half-life problems, emphasizing the importance of the fundamental half-life equation and different problem-solving approaches. While simple half-life problems can be solved with straightforward calculations, more complex scenarios may require more sophisticated mathematical methods. Nevertheless, a strong grasp of the core principles discussed here will enable you to tackle a wide range of half-life related questions. Remember to always carefully identify the given variables and the unknown you need to solve for, and choose the appropriate formula accordingly. Practice is key to mastering this essential concept!