Half Life Problems Worksheet With Answers

Half-Life Problems Worksheet with Answers: A Comprehensive Guide

Understanding half-life is crucial in various scientific fields, from nuclear physics and chemistry to medicine and geology. This comprehensive worksheet and answer key will guide you through a range of half-life problems, building your understanding from basic concepts to more complex scenarios. We'll cover everything from simple calculations to problems involving multiple half-lives and decay curves. Prepare to master this fundamental concept!

Understanding Half-Life

Before diving into the problems, let's refresh our understanding of half-life. Half-life is the time it takes for half of a radioactive substance to decay into a more stable form. This decay process is exponential, meaning it doesn't happen at a constant rate. Instead, the amount of substance remaining decreases by half with each passing half-life.

Key Concepts to Remember:

• Radioactive decay: The spontaneous breakdown of an unstable atomic nucleus, resulting in the emission of radiation.
• Parent isotope: The original radioactive isotope.
• Daughter isotope: The stable isotope formed after decay.
• Exponential decay: The characteristic pattern of radioactive decay, where the rate of decay is proportional to the amount of radioactive material remaining.

Half-Life Problems: A Step-by-Step Approach

Let's tackle a variety of half-life problems, progressing in difficulty. Each problem will be followed by a detailed solution.

Problem 1: Basic Half-Life Calculation

A sample of a radioactive isotope has a half-life of 10 years. If you start with 100 grams, how much will remain after 20 years?

Solution:

• Step 1: Determine the number of half-lives. 20 years / 10 years/half-life = 2 half-lives.
• Step 2: Calculate the remaining amount. After the first half-life (10 years), 50 grams remain (100g / 2 = 50g). After the second half-life (another 10 years), 25 grams remain (50g / 2 = 25g).

Answer: 25 grams will remain after 20 years.

Problem 2: Determining Half-Life from Decay Data

A sample of a radioactive isotope initially contains 80 grams. After 30 minutes, only 20 grams remain. What is the half-life of this isotope?

Solution:

• Step 1: Determine how many times the sample has halved. The sample has gone from 80 grams to 20 grams, meaning it halved twice (80g -> 40g -> 20g).
• Step 2: Calculate the total time elapsed. 30 minutes.
• Step 3: Determine the time for one half-life. Since it took 30 minutes for two half-lives, one half-life is 15 minutes (30 minutes / 2 half-lives = 15 minutes/half-life).

Answer: The half-life of the isotope is 15 minutes.

Problem 3: Calculating Remaining Amount After Multiple Half-Lives

A radioactive substance with a half-life of 5 days starts with 1 kilogram. How much remains after 20 days?

Solution:

• Step 1: Determine the number of half-lives. 20 days / 5 days/half-life = 4 half-lives.
• Step 2: Calculate the remaining amount. After each half-life, the amount is halved:
  • After 1 half-life: 500 grams (1000g / 2)
  • After 2 half-lives: 250 grams (500g / 2)
  • After 3 half-lives: 125 grams (250g / 2)
  • After 4 half-lives: 62.5 grams (125g / 2)

Answer: 62.5 grams remain after 20 days.

Problem 4: Working Backwards from Remaining Amount

A sample of carbon-14 (half-life = 5730 years) initially contained 100 grams. If only 25 grams remain, how old is the sample?

Solution:

• Step 1: Determine how many times the sample halved. The sample went from 100 grams to 25 grams, meaning it halved twice (100g -> 50g -> 25g).
• Step 2: Calculate the age. Since two half-lives have passed, and each half-life is 5730 years, the sample is 11460 years old (2 half-lives * 5730 years/half-life).

Answer: The sample is approximately 11460 years old.

Problem 5: Using a Decay Curve

(Imagine a graph here showing the exponential decay of a substance. The y-axis represents the amount of substance remaining, and the x-axis represents time.)

Using the provided decay curve for a substance with a half-life of 20 minutes, determine approximately how much substance remains after 60 minutes.

Solution:

This problem requires visual interpretation of the graph. Locate 60 minutes on the x-axis, and trace vertically to the decay curve. Then trace horizontally to the y-axis to determine the approximate amount remaining. (The exact answer will depend on the specific values shown on the graph). The student should visually estimate the amount based on the graph.

Answer: The answer will depend on the decay curve provided.

Problem 6: More Complex Scenario – Multiple Isotopes

A sample contains two radioactive isotopes, A and B. Isotope A has a half-life of 10 years, and isotope B has a half-life of 5 years. If we start with 100 grams of each isotope, how much of each will remain after 15 years?

Solution:

We need to calculate the remaining amounts separately for each isotope:

Isotope A:

• Number of half-lives: 15 years / 10 years/half-life = 1.5 half-lives
• Since we can't have half a half-life, we need to use exponential decay formula: Amount remaining = Initial amount * (1/2)^(number of half-lives)
• Amount remaining = 100g * (1/2)^1.5 ≈ 35.4g

Isotope B:

• Number of half-lives: 15 years / 5 years/half-life = 3 half-lives
• Amount remaining = 100g * (1/2)^3 = 12.5g

Answer: Approximately 35.4 grams of isotope A and 12.5 grams of isotope B will remain after 15 years.

Advanced Half-Life Problems and Concepts

These problems introduce more challenging aspects of half-life calculations:

Problem 7: Considering Decay Products

A radioactive isotope X decays into a stable isotope Y with a half-life of 20 years. If we start with 100 grams of X, how much of X and Y will remain after 40 years?

Solution:

• After 20 years (1 half-life): 50 grams of X and 50 grams of Y.
• After 40 years (2 half-lives): 25 grams of X and 75 grams of Y (100g initial - 25g remaining X = 75g Y).

Answer: After 40 years, 25 grams of X and 75 grams of Y will remain.

Problem 8: Using the Decay Constant (λ)

The decay constant (λ) is related to the half-life (t_1/2) by the equation: λ = ln(2) / t_1/2. If an isotope has a half-life of 10 years, what is its decay constant?

Solution:

λ = ln(2) / 10 years ≈ 0.0693 years^-1

Answer: The decay constant is approximately 0.0693 years^-1.

Problem 9: Applications in Carbon Dating

A piece of ancient wood contains only 12.5% of the original carbon-14. Given that the half-life of carbon-14 is 5730 years, approximately how old is the wood?

Solution:

• Determine how many times the carbon-14 halved. Since only 12.5% remains, it halved three times (100% -> 50% -> 25% -> 12.5%).
• Calculate the age: 3 half-lives * 5730 years/half-life = 17190 years.

Answer: The wood is approximately 17190 years old.

Conclusion

This worksheet provides a comprehensive overview of half-life problems, ranging from basic calculations to more complex scenarios. Understanding half-life is vital in numerous scientific fields, and mastering these problems will strengthen your understanding of this fundamental concept. Remember to always approach problems systematically, paying close attention to units and conversions. Consistent practice is key to building proficiency in solving half-life problems. By working through these examples and similar exercises, you will confidently tackle any half-life problem you encounter.