What is the 14C activity in decay events per minute of 1.0 g of a 3000-year-old wooden object? All living organisms contain an equilibrium concentration of radioactive 14C that gives rise to an average of 15.3 nuclear decay events per minute per gram of carbon. (14C, t1/2 = 5715 years)

Verified step by step guidance

Step 1: Understand the problem. We need to find the current activity of 14C in a 1.0 g sample of a 3000-year-old wooden object. We know the initial activity in living organisms is 15.3 decay events per minute per gram.

Step 2: Use the half-life formula to determine the decay constant (k). The half-life (t_{1/2}) of 14C is 5715 years. The decay constant k can be calculated using the formula: k = \frac{0.693}{t_{1/2}}.

Step 3: Calculate the number of half-lives that have passed in 3000 years. This can be done by dividing the age of the object by the half-life of 14C: \text{Number of half-lives} = \frac{3000}{5715}.

Step 4: Determine the remaining fraction of 14C using the formula: \text{Remaining fraction} = (0.5)^{\text{Number of half-lives}}.

Step 5: Calculate the current activity of 14C in the sample. Multiply the initial activity (15.3 decay events per minute per gram) by the remaining fraction to find the current activity in decay events per minute for the 1.0 g sample.

Key Concepts

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

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This decay occurs at a predictable rate characterized by the half-life, which is the time required for half of the radioactive atoms in a sample to decay. For carbon-14 (14C), the half-life is approximately 5715 years, meaning that after this period, half of the original amount of 14C will have decayed.

Carbon Dating

Carbon dating is a method used to determine the age of an object containing organic material by measuring the amount of carbon-14 it contains. Living organisms maintain a constant ratio of 14C to 12C while alive, but once they die, the 14C begins to decay. By comparing the remaining 14C in a sample to the known initial levels, scientists can estimate the time since the organism's death.

Activity of a Radioactive Sample

The activity of a radioactive sample refers to the number of decay events occurring per unit time, typically measured in decays per minute or becquerels. For carbon-14, the activity is directly related to the amount of 14C present in the sample. In the context of the question, the initial activity of 1.0 g of carbon in a living organism is given as 15.3 decay events per minute, which decreases over time as the 14C decays.

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