Table of contents

1. Matter and Measurements4h 31m
2. Atoms and the Periodic Table5h 23m
3. Ionic Compounds2h 18m
4. Molecular Compounds2h 18m
5. Classification & Balancing of Chemical Reactions3h 17m
6. Chemical Reactions & Quantities2h 27m
7. Energy, Rate and Equilibrium3h 46m
8. Gases, Liquids and Solids3h 25m
9. Solutions4h 10m
10. Acids and Bases3h 29m
11. Nuclear Chemistry56m
BONUS: Lab Techniques and Procedures1h 38m
BONUS: Mathematical Operations and Functions47m
12. Introduction to Organic Chemistry1h 34m
13. Alkenes, Alkynes, and Aromatic Compounds2h 12m
14. Compounds with Oxygen or Sulfur1h 6m
15. Aldehydes and Ketones1h 1m
16. Carboxylic Acids and Their Derivatives1h 11m
17. Amines39m
18. Amino Acids and Proteins1h 51m
19. Enzymes1h 37m
20. Carbohydrates1h 41m
21. The Generation of Biochemical Energy2h 8m
22. Carbohydrate Metabolism2h 22m
23. Lipids2h 26m
24. Lipid Metabolism1h 45m
25. Protein and Amino Acid Metabolism1h 37m
26. Nucleic Acids and Protein Synthesis2h 54m

11. Nuclear Chemistry

Radioactive Half-Life

GOB Chemistry

11. Nuclear Chemistry

Radioactive Half-Life

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4:31 minutes

Problem 86

Textbook Question

The half-life for the radioactive decay of Ce-141 is 32.5 days. If a sample has an activity of 4.0 µCi after 130 days have elapsed, what was the initial activity, in microcuries, of the sample?

Verified step by step guidance

Understand the problem: The half-life of Ce-141 is 32.5 days, and the activity of the sample after 130 days is 4.0 µCi. We need to calculate the initial activity of the sample using the radioactive decay formula.

Use the radioactive decay formula: \( A = A_0 \cdot e^{-kt} \), where \( A \) is the activity at time \( t \), \( A_0 \) is the initial activity, \( k \) is the decay constant, and \( t \) is the elapsed time.

Calculate the decay constant \( k \) using the relationship between half-life and \( k \): \( k = \frac{\ln(2)}{t_{1/2}} \). Substitute \( t_{1/2} = 32.5 \; \text{days} \) into the equation to find \( k \).

Rearrange the decay formula to solve for \( A_0 \): \( A_0 = \frac{A}{e^{-kt}} \). Substitute \( A = 4.0 \; \mu\text{Ci} \), \( k \) (calculated in the previous step), and \( t = 130 \; \text{days} \) into the equation.

Simplify the expression to calculate \( A_0 \), which represents the initial activity of the sample in microcuries.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Half-Life

Half-life is the time required for half of the radioactive nuclei in a sample to decay. It is a crucial concept in understanding radioactive decay processes, as it allows us to predict how long it will take for a given amount of a radioactive substance to reduce to a specific fraction of its original quantity. In this case, the half-life of Ce-141 is 32.5 days.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This process results in the transformation of the original element into a different element or isotope. The activity of a radioactive sample, measured in microcuries (µCi), indicates the rate of decay and is directly related to the number of radioactive atoms present in the sample.

Exponential Decay Formula

The exponential decay formula describes how the quantity of a radioactive substance decreases over time. It is expressed as N(t) = N0 * (1/2)^(t/T), where N(t) is the remaining quantity at time t, N0 is the initial quantity, and T is the half-life. This formula is essential for calculating the initial activity of the sample based on the activity observed after a certain period, allowing for the determination of how much of the substance has decayed.

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