Half-Life Calculations Worksheet Answers Explained Clearly
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The Half-Life Calculations Worksheet is an essential tool for students and professionals working in chemistry, physics, and environmental science. Understanding half-life is crucial for analyzing radioactive decay, pharmacokinetics, and other time-dependent processes. In this article, we will break down the concept of half-life, go through calculations, and provide clear explanations of the worksheet answers. Let's dive into the fundamentals of half-life and how to approach these calculations!
What is Half-Life? 📚
Half-life is defined as the time required for half of a substance to decay or be eliminated. This concept is widely used in various scientific fields, including:
- Radioactive Decay: The time it takes for half of a radioactive substance to decay into a stable form.
- Pharmacology: The time it takes for the concentration of a drug in the bloodstream to reduce by half.
- Environmental Science: The rate at which pollutants break down in the environment.
The half-life (t₁/₂) is a constant for a given substance and does not change regardless of the quantity of the substance present.
The Half-Life Calculation Formula 🧮
The basic formula to calculate the remaining quantity of a substance after a certain number of half-lives is:
[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} ]
Where:
- ( N(t) ) = the remaining quantity of the substance after time ( t )
- ( N_0 ) = the initial quantity of the substance
- ( t ) = the total time that has elapsed
- ( t_{1/2} ) = the half-life of the substance
This formula allows you to calculate how much of a substance remains after a given time based on its half-life.
Example Half-Life Calculations 🌟
Let’s take a look at a practical example to illustrate how to use this formula effectively.
Example 1: Radioactive Decay
Problem: A radioactive isotope has a half-life of 3 years. If you start with 80 grams of the isotope, how much will remain after 9 years?
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Identify Variables:
- ( N_0 = 80 ) grams
- ( t_{1/2} = 3 ) years
- ( t = 9 ) years
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Calculate Number of Half-Lives:
- Number of half-lives ( n = \frac{t}{t_{1/2}} = \frac{9 \text{ years}}{3 \text{ years}} = 3 )
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Calculate Remaining Quantity: [ N(t) = 80 \left( \frac{1}{2} \right)^{3} = 80 \left( \frac{1}{8} \right) = 10 \text{ grams} ]
Answer: After 9 years, 10 grams of the radioactive isotope will remain.
Example 2: Pharmacokinetics
Problem: A medication has a half-life of 4 hours. If a patient receives an initial dose of 200 mg, what is the amount of the drug in the patient’s system after 12 hours?
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Identify Variables:
- ( N_0 = 200 ) mg
- ( t_{1/2} = 4 ) hours
- ( t = 12 ) hours
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Calculate Number of Half-Lives:
- Number of half-lives ( n = \frac{t}{t_{1/2}} = \frac{12 \text{ hours}}{4 \text{ hours}} = 3 )
-
Calculate Remaining Quantity: [ N(t) = 200 \left( \frac{1}{2} \right)^{3} = 200 \left( \frac{1}{8} \right) = 25 \text{ mg} ]
Answer: After 12 hours, 25 mg of the medication will remain in the patient’s system.
Table of Common Half-Lives ⏳
Here is a table that summarizes half-lives of some common substances:
<table> <tr> <th>Substance</th> <th>Half-Life</th> </tr> <tr> <td>Carbon-14</td> <td>5,730 years</td> </tr> <tr> <td>Uranium-238</td> <td>4.468 billion years</td> </tr> <tr> <td>Iodine-131</td> <td>8 days</td> </tr> <tr> <td>Radon-222</td> <td>3.8 days</td> </tr> <tr> <td>Cesium-137</td> <td>30 years</td> </tr> </table>
Important Notes 🔑
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Half-life is constant: The half-life of a substance does not change over time or depend on the amount of substance present.
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Use logarithms for complex calculations: For more complex problems, especially when dealing with large quantities or very long time periods, it may be useful to use logarithms.
[ N(t) = N_0 e^{-\lambda t} ]
Where ( \lambda = \frac{\ln(2)}{t_{1/2}} )
Conclusion
Understanding half-life calculations is crucial for various scientific fields. By mastering the formula and practicing with different examples, you can effectively predict how substances decay over time. Remember to refer to the half-life table for common substances, and don't hesitate to apply logarithmic equations for more complex scenarios. Happy calculating! 😊