Half Life Sample Problems With Answers

Half-Life Sample Problems with Answers: A Comprehensive Guide

Understanding half-life is crucial in various fields, from nuclear physics and medicine to environmental science and archaeology. This comprehensive guide will walk you through a range of half-life problems with detailed explanations and answers, equipping you with the knowledge to confidently tackle similar challenges. We'll explore different scenarios, focusing on practical applications and building a strong conceptual understanding of this fundamental concept.

Understanding Half-Life

Before diving into the problems, let's solidify our understanding of half-life. Half-life (t_1/2) is the time it takes for half of a radioactive substance to decay. This decay is a random process; we can't predict which specific atom will decay next, but we can predict the overall rate of decay for a large sample. The half-life is a constant for a given radioactive isotope, meaning it remains the same regardless of the initial amount of the substance.

Key Concepts:

• Radioactive Decay: The spontaneous breakdown of an unstable atomic nucleus, resulting in the emission of radiation (alpha, beta, or gamma).
• Isotope: Atoms of the same element with the same number of protons but different numbers of neutrons. Some isotopes are radioactive, while others are stable.
• Exponential Decay: Half-life follows an exponential decay pattern, meaning the amount of substance remaining decreases by half with each successive half-life.

Types of Half-Life Problems

Half-life problems often fall into these categories:

• Calculating remaining amount: Determining how much of a substance remains after a specific time.
• Determining the time elapsed: Calculating the time that has passed based on the initial and remaining amounts.
• Finding the number of half-lives: Calculating how many half-lives have occurred given the initial and final amounts or the time elapsed.
• Complex scenarios: Problems involving multiple isotopes or decay chains.

Sample Problems and Solutions

Let's work through several examples, increasing in complexity.

Problem 1: Simple Calculation of Remaining Amount

A sample of 100 grams of Carbon-14 (¹⁴C) has a half-life of 5730 years. How much ¹⁴C will remain after 11460 years?

Solution:

1. Determine the number of half-lives: 11460 years / 5730 years/half-life = 2 half-lives.
2. Calculate the remaining amount: After one half-life, 50 grams remain (100g * 0.5 = 50g). After two half-lives, 25 grams remain (50g * 0.5 = 25g).

Answer: 25 grams of ¹⁴C will remain after 11460 years.

Problem 2: Determining Time Elapsed

A sample of Iodine-131 (¹³¹I), which has a half-life of 8 days, initially contains 100 mg. How long will it take for the sample to decay to 12.5 mg?

Solution:

1. Determine the number of half-lives: The sample needs to decay to 1/8 of its initial amount (100mg -> 12.5mg). This represents 3 half-lives (100mg -> 50mg -> 25mg -> 12.5mg).
2. Calculate the elapsed time: 3 half-lives * 8 days/half-life = 24 days.

Answer: It will take 24 days for the sample to decay to 12.5 mg.

Problem 3: Using the Exponential Decay Formula

A radioactive substance has a half-life of 20 minutes. If you start with 1000 atoms, how many atoms will remain after 60 minutes? Use the exponential decay formula: N(t) = N₀ * (1/2)^(t/t_1/2), where:

• N(t) is the amount remaining after time t
• N₀ is the initial amount
• t is the elapsed time
• t_1/2 is the half-life

Solution:

1. Substitute the values: N(t) = 1000 * (1/2)^(60min/20min)
2. Calculate: N(t) = 1000 * (1/2)³ = 1000 * (1/8) = 125

Answer: 125 atoms will remain after 60 minutes.

Problem 4: A More Complex Scenario - Two Isotopes

A sample contains two radioactive isotopes, A and B. Isotope A has a half-life of 10 years and initially comprises 80% of the sample. Isotope B has a half-life of 20 years and initially comprises 20% of the sample. What percentage of the sample will be Isotope A after 20 years?

Solution:

1. Decay of Isotope A: After 20 years (2 half-lives), the amount of Isotope A remaining is 20% of its initial amount (80% * 0.5 * 0.5 = 20%).
2. Decay of Isotope B: After 20 years (1 half-life), the amount of Isotope B remaining is 10% of its initial amount (20% * 0.5 = 10%).
3. Total remaining: The total amount remaining is 20% (Isotope A) + 10% (Isotope B) = 30%.
4. Percentage of Isotope A: (20%/30%) * 100% = 66.67% (approximately).

Answer: Approximately 66.67% of the sample will be Isotope A after 20 years.

Problem 5: Half-life and Carbon Dating

A wooden artifact is found to contain 12.5% of its original ¹⁴C. Given that the half-life of ¹⁴C is 5730 years, how old is the artifact?

Solution:

1. Determine the number of half-lives: The artifact contains 1/8 of its original ¹⁴C, which means 3 half-lives have passed.
2. Calculate the age: 3 half-lives * 5730 years/half-life = 17190 years.

Answer: The artifact is approximately 17190 years old.

Problem 6: Dealing with Fractions of Half-lives

A sample of Strontium-90 (⁹⁰Sr) has a half-life of 29 years. If you start with 500 grams, how much will remain after 10 years?

Solution:

This problem requires using the exponential decay formula since we're dealing with a fraction of a half-life.

1. Substitute the values: N(t) = 500 * (1/2)^(10 years / 29 years)
2. Calculate: N(t) ≈ 500 * 0.78 ≈ 390 grams. (Using a calculator to find the fractional exponent)

Answer: Approximately 390 grams of ⁹⁰Sr will remain after 10 years.

Advanced Concepts and Further Exploration

While the above examples cover the basics, more advanced problems might involve:

• Decay chains: One isotope decays into another, which then decays further.
• Activity: Problems involving the activity (decay rate) of a sample, measured in Becquerels (Bq) or Curies (Ci).
• Specific activity: Activity per unit mass or volume.
• Simultaneous decay: Involving multiple isotopes decaying concurrently.

Mastering half-life requires a solid understanding of exponential decay and the ability to manipulate mathematical equations. By working through various problems, gradually increasing in difficulty, you will build your confidence and competence in solving half-life problems across diverse scientific disciplines. Remember to always clearly define your variables and use appropriate units throughout your calculations. Continuous practice is key to mastering this important concept.