We consider the phenomenon of radioactive decay which, from experimentation, we know beh...

We consider the phenomenon of radioactive decay which, from experimentation, we know behaves according to the law:

The rate at which a quantity of a radioactive isotope decays is proportional to the amount of the isotope present. The proportionality constant depends only on which radioactive isotope is used.

Carbon dating is a method of determining the time elapsed since the death of organic material. The assumptions implicit in carbon dating are that

• Carbon 14 (C-14) makes up a constant proportion of the carbon that living matter ingests on a regular basis, and

• once the matter dies, the C-14 present decays, but no new carbon is added to the matter.

Hence, by measuring the amount of C-14 still in the organic matter and comparing it to the amount of C-14 typically found in living matter, a “time since death” can be approximated. Using the decay-rate parameter you computed in Exercise 7, determine the time since death if

(a) 88% of the original C-14 is still in the material.

(b) 12% of the original C-14 is still in the material.

(c) 2% of the original C-14 is still in the material.

(d) 98% of the original C-14 is still in the material.

Remark: There has been speculation that the amount of C-14 available to living creatures has not been exactly constant over long periods (thousands of years). This makes accurate dates much trickier to determine.

Reference:

The half-life of a radioactive isotope is the amount of time it takes for a quantity of radioactive material to decay to one-half of its original amount.

(a) The half-life of Carbon 14 (C-14) is 5230 years. Determine the decay-rate parameter λ for C-14.

(b) The half-life of Iodine 131 (I-131) is 8 days. Determine the decay-rate parameter for I-131.

(c) What are the units of the decay-rate parameters in parts (a) and (b)?

(d) To determine the half-life of an isotope, we could start with 1000 atoms of the isotope and measure the amount of time it takes 500 of them to decay, or we could start with 10,000 atoms of the isotope and measure the amount of time it takes 5000 of them to decay. Will we get the same answer? Why?