# Examples of Half-Life Calculations in Chemistry

Explore a series of examples illustrating how to calculate decayed amounts, remaining quantities, and half-lives of radioactive substances such as Os-182, U-238, and I-131. Understand the concepts through practical applications in chemistry.

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**Example 6:-**Os-182 has a half-life of 21.5 hours. How many grams of a 10.0 gram sample would have decayed after exactly three half-lives? Solution: (1/2)3= 0.125 (the amount remaining after 3 half- lives) 10.0 g x 0.125 = 1.25 g remain 10.0 g - 1.25 g = 8.75 g have decayed Note that the length of the half-life played no role in this calculation. In addition, note that the question asked for the amount that decayed, not the amount that remaining.**Example 7:-**After 24.0 days, 2.00 milligrams of an original 128.0 milligram sample remain. What is the half-life of the sample? Solution: 2.00 mg / 128.0 mg = 0.015625 How many half-lives must have elaspsed to get to 0.015625 remaining? (1/2)n= 0.015625 n log 0.5 = log 0.015625 n = log 0.5 / log 0.015625 n = 6 24 days / 6 half-lives = 4.00 days (the length of the half-life)**Example 8:-**U-238 has a half-life of 4.46 x 109years. How much U-238 should be present in a sample 2.5 x 109years old, if 2.00 grams was present initially? Solution: (2.5 x 109) / (4.46 x 109) = 0.560 (the number of half-lves that have elapsed) (1/2)0.560= 0.678 (the decimal fraction of U-238 remaining) 2.00 g x 0.678 = 1.36 g remain**Example 9:-**How long will it take for a 40.0 gram sample of I-131 (half-life = 8.040 days) to decay to 1/100 its original mass? Solution: (1/2)n= 0.01 n log 0.5 = log 0.01 n = 6.64 6.64 x 8.040 days = 53.4 days