Half Life Sample Problems With Answers

Half-Life Sample Problems with Answers: A Comprehensive Guide

Understanding half-life is crucial in various scientific fields, from nuclear physics and medicine to environmental science and archaeology. This comprehensive guide delves into the concept of half-life, providing numerous sample problems with detailed solutions to solidify your understanding. We'll cover various scenarios, including simple decay calculations, complex decay chains, and applications in real-world contexts.

What is Half-Life?

Half-life (t_1/2) is the time it takes for half of a given amount of a substance to decay. This decay process is exponential, meaning the amount remaining decreases by half repeatedly over each half-life period. The concept is crucial in understanding radioactive decay, but it also applies to other processes like chemical reactions following first-order kinetics. It's important to note that half-life is a constant for a specific substance under consistent conditions; it doesn't depend on the initial amount of the substance.

Understanding the Formula

The fundamental equation governing radioactive decay, and thus half-life calculations, 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 the substance.
• t is the elapsed time.
• t_1/2 is the half-life of the substance.

Sample Problems and Solutions

Let's work through several sample problems, starting with simple scenarios and progressing to more complex ones.

Problem 1: Simple Half-Life Calculation

Problem: 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 known values: N₀ = 100 grams, t_1/2 = 10 years, t = 30 years.
2. Plug the values into the formula: N(t) = 100 grams * (1/2)^(30 years / 10 years)
3. Calculate: N(t) = 100 grams * (1/2)^3 = 100 grams * (1/8) = 12.5 grams

Answer: After 30 years, 12.5 grams of the isotope will remain.

Problem 2: Determining Half-Life from Decay Data

Problem: 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 known values: N₀ = 500 atoms, N(t) = 125 atoms, t = 20 minutes.
2. Rearrange the formula to solve for t_1/2: t_1/2 = t * log₂(N₀/N(t))
3. Plug in the values: t_1/2 = 20 minutes * log₂(500 atoms / 125 atoms)
4. Calculate: t_1/2 = 20 minutes * log₂(4) = 20 minutes * 2 = 40 minutes

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

Problem 3: Determining Remaining Amount After Multiple Half-Lives

Problem: A sample of Carbon-14 (half-life = 5730 years) initially contains 1000 atoms. How many atoms will remain after 17190 years?

Solution:

1. Determine the number of half-lives: 17190 years / 5730 years/half-life ≈ 3 half-lives
2. Calculate the remaining amount: After each half-life, the amount is halved. Therefore, after 3 half-lives: 1000 atoms * (1/2)^3 = 125 atoms

Answer: Approximately 125 atoms of Carbon-14 will remain after 17190 years.

Problem 4: Calculating Time Elapsed Given Remaining Amount

Problem: A sample of Iodine-131 (half-life = 8 days) initially contains 200 grams. How long will it take for the sample to decay to 25 grams?

Solution:

1. Use the half-life formula: 25 grams = 200 grams * (1/2)^(t/8 days)
2. Simplify the equation: 1/8 = (1/2)^(t/8 days)
3. Take the logarithm base 1/2 of both sides: log_1/2(1/8) = t/8 days
4. Solve for t: t = 8 days * log_1/2(1/8) = 8 days * 3 = 24 days

Answer: It will take 24 days for the sample to decay to 25 grams.

Problem 5: Sequential Decay (Decay Chain)

Problem: Substance A decays into Substance B with a half-life of 20 minutes. Substance B then decays into Substance C with a half-life of 40 minutes. If you start with 1000 atoms of Substance A, how many atoms of Substance B will be present after 20 minutes? (Assume Substance B is not present initially).

Solution: This problem involves two steps.

Step 1: Decay of Substance A:

• After 20 minutes (one half-life), 500 atoms of Substance A remain. Therefore, 500 atoms of Substance A have decayed into Substance B.

Step 2: Decay of Substance B:

• We need to consider that 500 atoms of Substance B are created from the decay of Substance A and some will then decay into Substance C.
• We use the half-life formula for Substance B, with N₀ = 500 atoms, t_1/2 = 40 minutes, and t = 20 minutes:
• N(t) = 500 atoms * (1/2)^(20 minutes/40 minutes) = 500 atoms * (1/2)^0.5 ≈ 354 atoms

Answer: Approximately 354 atoms of Substance B will be present after 20 minutes.

Problem 6: Real-World Application - Carbon Dating

Problem: A fossil is found to contain 25% of the original amount of Carbon-14. Knowing the half-life of Carbon-14 is 5730 years, estimate the age of the fossil.

Solution:

1. Determine the number of half-lives: If 25% of the original Carbon-14 remains, then two half-lives have occurred (100% -> 50% -> 25%).
2. Calculate the age: 2 half-lives * 5730 years/half-life = 11460 years

Answer: The fossil is approximately 11460 years old.

Advanced Concepts and Considerations

While the examples above cover fundamental half-life calculations, several advanced considerations exist:

• Complex Decay Chains: Some isotopes undergo multiple decay steps, making calculations more intricate. These usually involve solving systems of differential equations.
• Branching Decay: Some isotopes can decay via multiple pathways, each with its own probability and half-life.
• Isotopic Abundance: Natural samples often contain a mixture of isotopes, impacting the decay rate.
• Activity: The activity of a radioactive sample is the rate at which it decays and is measured in Becquerels (Bq) or Curies (Ci). This is directly proportional to the number of radioactive atoms present.

Conclusion

Understanding half-life is essential for numerous applications. This guide provides a solid foundation, equipping you with the necessary skills to solve various half-life problems. Remember to carefully identify the given variables, select the appropriate formula, and perform the calculations accurately. Through practice, you'll master the calculations and their practical applications. By understanding these concepts and practicing the provided examples, you'll build a strong understanding of half-life and its significance in various fields. Remember that mastering these calculations requires practice; work through numerous problems to solidify your understanding and build confidence in tackling more complex scenarios.