Chapter 2.4, Problem 18E

Radioactive Decay The half-life of a radioactive substance is the time H that it takes for half of the substance to change form through radioactive decay. This number does not depend on the amount with which you start. For example, carbon-14 is known to have a half-life of $\mathrm{H}=5770$ years. Thus, if you begin with 1 gram of carbon-14, then 5770 years later you will have $\frac{1}{2}$ gram of carbon-14. And if you begin with 30 grams of carbon-14, then after 5770 years there will be 15 grams left. In general, radioactive substances decay according to the formula

$A=A_0\times {0.5}^{\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$H$}\right.}$

Where H is the half-life, t is the elapsed time, $A_0$ is the amount you start with (the amount when $\mathrm{t}=0),$ and A is the amount left at time t.

a. Uranium-228 has a half-life H of 9.3 minutes. Thus, the decay function for this isotope of uranium is

$A=A_0\times {0.5}^{\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$9.3$}\right.},$

where t is measured in minutes. Suppose we start with grams of uranium-228.

i. How much uranium-228 is left after 2 minutes?

ii. How long will you have to wait until there are only 3 grams left?

b. Uranium-235 is the isotope of uranium that can be used to make nuclear bombs. It has a half-life of 713 million years. Suppose we start with 5 grams of uranium-235.

i. How much uranium-235 is left after 200 million years?

ii. How long will you have to wait until there are only 3 grams left?