A tree is chopped down and burned to produce ash.The carbon isotopes in the ash are found to have a ratio for ¹⁴C to ¹²C of 1.3 * 10^-12.Experimental tests on the ¹⁴C atoms reveal that ¹⁴C is a beta emitter with a half-life of 5730 years.At an archeological excavation,a skeleton is found next to a campfire with some wood ash.If 100 g of carbon from the ash in the campfire emits betas at a rate of 40 per hour,how old is the campfire?

Question

Quizplus · Accepted Answer

The ratio of ¹⁴C to ¹²C in the ash indicates that carbon in the tree that made the ash died about 12,000 years ago, based on the known half-life of ¹⁴C. Since the skeleton is found next to a campfire with wood ash, it must be younger than the tree that made the ash. Therefore, the age of the campfire must be less than 12,000 years. Using the formula for radioactive decay, N = N0e^(-kt), where N is the current amount of radioactive material, N0 is the initial amount, k is the decay constant, and t is time, we can find the age of the campfire. We know that 100 g of carbon emits betas at a rate of 40/hour. This means that the decay constant is k = ln(2) / (5730 years) * (40/hour) / (100 g) = 1.209 x 10^-6 / (year * g). Let's assume that all of the carbon in the campfire ash is carbon-14. Therefore, the initial amount of carbon-14 in the campfire is 100 g. We're looking for the time it takes for the amount of carbon-14 in the campfire to decay to its current level, which we'll call x: x = 100 g * e^(-1.209 x 10^-6 / (year * g) * t) We can solve for t: t = ln(x/100 g) / (-1.209 x 10^-6 / (year * g)) Plugging in x = 50 g (since the half-life of carbon-14 is 5730 years, we know that after 5730 years, half of the original carbon-14 will remain), we get: t = ln(0.5) / (-1.209 x 10^-6 / (year * g)) = 5730 years Therefore, the age of the campfire is 5730 years. However, we know that the actual age of the campfire must be less than 12,000 years based on the age of the tree that made the ash. Therefore, the best answer choice is B) 64,000 years.

A Tree Is Chopped Down and Burned to Produce Ash