Half Life Problems And Answers Examples

Half-Life Problems and Answers: A Comprehensive Guide

Understanding half-life is crucial in various fields, from nuclear physics and medicine to archaeology and geology. This comprehensive guide explores half-life problems, providing clear explanations, step-by-step solutions, and diverse examples to solidify your understanding. We will cover various scenarios, including calculating remaining amounts, determining time elapsed, and understanding the concept of multiple half-lives.

Introduction to Half-Life

Half-life (t_1/2) is the time it takes for half of a given amount of a substance to decay or transform. This concept is primarily associated with radioactive decay, where unstable atomic nuclei lose energy by emitting radiation. However, the principle of half-life extends to other areas, such as the decay of chemical compounds or the decline of certain populations. A key characteristic of half-life is its constant rate; it's independent of the initial amount of the substance. This means that whether you start with 100 grams or 1 gram of a substance with a specific half-life, it will take the same amount of time for half of it to decay.

Understanding the Exponential Decay Formula

The decay of a substance follows an exponential decay pattern, accurately described by the following formula:

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.

This formula is the cornerstone of solving most half-life problems. Let's delve into various examples to illustrate its application.

Half-Life Problem Examples and Solutions

Problem 1: Simple Half-Life Calculation

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

Solution:

1. Identify the knowns: N₀ = 100 grams, t_1/2 = 10 years, t = 30 years.
2. Apply 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 Time Elapsed

A sample of a radioactive element with a half-life of 50 days initially contains 80 grams. After a certain time, only 20 grams remain. How long has it been?

Solution:

1. Identify the knowns: N₀ = 80 grams, N_t = 20 grams, t_1/2 = 50 days.
2. Rearrange the formula to solve for t: We can rewrite the formula as: t = t_1/2 * log_(1/2)(N_t/N₀) Alternatively, we can use a simpler approach by considering the number of half-lives:
   • After one half-life, 40 grams remain.
   • After two half-lives, 20 grams remain.
3. Calculate: Two half-lives have elapsed. Therefore, the time elapsed is 2 * 50 days = 100 days.

Answer: 100 days have passed.

Problem 3: Multiple Half-Lives and Fractional Amounts

Carbon-14 has a half-life of 5,730 years. A sample of ancient wood contains 25% of its original Carbon-14. How old is the wood?

Solution:

1. Identify the knowns: N_t/N₀ = 0.25 (25%), t_1/2 = 5730 years.
2. Determine the number of half-lives: If 25% remains, then it has undergone two half-lives (50% after one, 25% after two).
3. Calculate the age: 2 half-lives * 5730 years/half-life = 11460 years.

Answer: The wood is approximately 11,460 years old.

Problem 4: Dealing with Very Small Amounts and Large Time Scales

A radioactive substance with a half-life of 1000 years initially contains 1 kilogram. How much will remain after 5000 years?

Solution:

1. Identify the knowns: N₀ = 1 kg, t_1/2 = 1000 years, t = 5000 years.
2. Apply the formula: N_t = 1 kg * (1/2)^{5000 years/1000 years}
3. Calculate: N_t = 1 kg * (1/2)⁵ = 1 kg * (1/32) = 0.03125 kg or 31.25 grams

Answer: After 5000 years, approximately 31.25 grams will remain.

Problem 5: A More Complex Scenario Involving Decay Chains

Let's consider a slightly more complex scenario. Suppose we have a radioactive element A that decays into element B, which is also radioactive. Element A has a half-life of 10 years, and element B has a half-life of 5 years. If we start with 100 grams of A, how much of A and B will remain after 15 years? (This problem requires understanding of sequential decay, which is beyond the scope of a simple half-life calculation and would typically involve differential equations.) This scenario highlights the limitations of simple half-life calculations in complex decay systems.

Explanation of Scientific Principles

The exponential decay observed in half-life problems arises from the probabilistic nature of radioactive decay. Each nucleus has a certain probability of decaying within a given time frame. This probability is constant and doesn't depend on the age of the nucleus or the presence of other nuclei. The large number of nuclei in a sample allows for the use of statistical averages, resulting in the smooth exponential decay curve. The half-life is a measure of this average decay rate. The formula we use is a direct consequence of this probabilistic nature and the fact that the decay rate is proportional to the amount of remaining substance.

The accuracy of half-life calculations depends on various factors, including the precision of the half-life measurement and the assumption that the decay process is truly first-order kinetics (meaning the decay rate is directly proportional to the amount of remaining substance).

Frequently Asked Questions (FAQ)

• Q: Can the half-life of a substance change? A: No, the half-life of a specific radioactive isotope is a constant. It's a fundamental property of the nucleus. However, the effective half-life can change if the substance is involved in processes that remove it from the system (such as biological excretion).

• Q: What is the difference between half-life and mean life? A: Half-life is the time it takes for half the substance to decay, while mean life is the average lifespan of all the atoms in the sample. Mean life is approximately 1.44 times the half-life.

• Q: How is half-life used in carbon dating? A: Carbon dating relies on the known half-life of Carbon-14 (5730 years). By measuring the ratio of Carbon-14 to Carbon-12 in organic materials, scientists can estimate the time since the organism died.

• Q: Are there substances with very short or very long half-lives? A: Yes, the range of half-lives for radioactive isotopes is incredibly vast, from fractions of a second to billions of years.

Conclusion

Half-life problems, while seemingly straightforward, require a solid understanding of the underlying exponential decay formula and the probabilistic nature of decay processes. By mastering the techniques presented here and practicing with various examples, you can confidently tackle a wide array of half-life calculations. Remember that the key is to carefully identify the known variables and apply the appropriate formula or reasoning, particularly concerning multiple half-lives and more complex scenarios. A strong grasp of this concept opens doors to deeper understanding in numerous scientific disciplines. Continued practice and exploring more advanced decay models will further enhance your expertise in this critical area.