Question

Carbon 14 is a radioactive isotope produced in the upper atmosphere by cosmic rays. It has a half-life measured as 5730 : 40 years. Since plants and animals absorb carbon from the atmosphere, the percentage of carbon a living organism contains that is carbon 14 is equal to the percentage of carbon 14 in the atmosphere. When an organism dies, however (or when a layer of wood is laid down as bark in a tree), it ceases to absorb carbon. Since carbon 12 (the normal form of carbon) is stable, this means that the ratio of carbon 14 to carbon 12 in an organism decays exponentially after death of that organism. This is the basis of radio-carbon dating. n organism decays exponentially after death of that Suppose that a sample of plant material from an ancient grave site is found to have a carbon 14 to carbon 12 ratio equal to only one third of the atmospheric ratio. What age would you assign this material? a. (4 pts) Suppose that an element K is to be used for radioactive dating. K occurs in two isotopes, Ki and K2 with respective half-lives t and /2, with t2>. Assuming the ratio of Ki to K2 in the atmosphere is constant and corresponds to the ratio in living organisms, find the equation describing the decay of this ratio after an organism dies (4 pts) If the atmospheric fraction of Ki is r(0) and the smallest detectable amount of K is ar(0) with a < 1, what is the maximum time into the past for which use of the isotopes of K provides a reasonable dating method? c. (4 pts) Given that use of carbon 14 for dating is useful for about 60,000 years into the past, approximately what fraction of the atmospheric concentration of carbon 14 is the minimal detectable amount? (4 pts)

Answer 1