Exponential decay: The amount of a radioactive material present at time t decays at a rate proportional

to the amount present at that time. The half life h is the time at which half of the material has decayed.

The mathematical model for exponential decay is

$\displaystyle \displaystyle y(t) = y0 \frac{1}{2}^{\frac{t}{h}} $

Here y0 is the initial amount of the substance present. After $\displaystyle \frac{t}{h} $ half lives, the material has been successively halved $\displaystyle \frac{t}{h} $ times.

Hint: recall that $\displaystyle a^{b} = e^{bln(a)} $

Uranium-238 has a half life of $\displaystyle 4.5 \times 10^{9} $ years. Let A be an amount of Uranium-238. What fraction of A is the amount left after $\displaystyle 10^{9} $ years?

b. 238 grams of Uranium-238 contain Avogadro's number, approximately $\displaystyle 6 \times 10^{23} $ atoms. Using this approximation for Avogadro's number, how many atoms of 238 grams of Uranium-238 will decay in the first

year? In the second year?