Final answer:
The question asks about the time required for Carbon-14 activity in a gram of carbon from a fallen tree to decay to 1 decay per hour. The half-life of Carbon-14 is 5730 years, and the initial activity rate is not provided, which is essential to solve the problem explicitly. Theoretically, the decay rate reaching such a low level would take many half-lives, implying an answer possibly beyond d) 50,000 years.
Step-by-step explanation:
The subject of the question is Carbon-14 dating, which falls under the category of Chemistry and applies to High School education level. The premise of this question revolves around understanding the half-life of Carbon-14 (14C) to determine how long it would take for the activity in a given amount of carbon from a fallen tree to decay to a certain level. Given the half-life of 14C is approximately 5730 years, the decay constant (λ) can be calculated using the formula λ = 0.693 / t1/2.
The original decay rate of 14C in living organisms is typically around 15 disintegrations per minute (dpm) per gram of carbon. This rate decreases by half every 5730 years. To solve the question, we would need to calculate the number of half-lives required for the decay rate to reach 1.00 decay per hour from the initial rate. Since 1 dpm is equal to 60 decays per hour, we are looking at a decay rate that is dramatically reduced from the starting level. By using the relationship A = A0e-λt, where A is the activity at time t and A0 is the initial activity, we can determine the time elapsed for a decay rate to reach a given level.
However, the question appears to be hypothetical as the initial activity is not provided and decays per hour are not commonly used units for such calculations. Realistically, in typical carbon dating, one would compare the activity of the sample against the known decay rate for 14C to calculate the age of the sample. The act of translating the decay rate to decays per hour adds an unnecessary layer of complexity that isn't standard in radiocarbon dating. Nonetheless, for the purposes of answering the multiple-choice question, knowing the half-life is sufficient to approximate that such extremely reduced activity levels would likely be observable after multiple half-lives, putting us well beyond tens of thousands of years and likely even beyond the reasonably accurate range of carbon dating.
The correct option is therefore not determinable without the initial activity, but in the context of radiocarbon dating, the theoretical answer would use the concept of half-lives to estimate that it would take many half-lives, likely more than 50,000 years, to achieve such a low decay rate, where option (d) 50,000 years might be a plausible guess.