# exponential decay , word problem

• January 18th 2011, 09:49 AM
Tweety
exponential decay , word problem
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

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

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

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

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

b. 238 grams of Uranium-238 contain Avogadro's number, approximately $6 \times  10^{23}$ 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?
• January 18th 2011, 09:57 AM
Unknown008
Just substitute A into your formula.

y(t) = Answer you're looking for
yo = A
t = 10^9
h = 4.5 x 10^9

b.

Take your formula again. This time,
y(1) = amount left after 1 year
yo = 6 x 10^23
t = 1
h = 4.5 x 10^9

and
y(2) = amount left after 2 years
yo = 6 x 10^23
t = 2
h = 4.5 x 10^9

Number of atoms decayed = Initial atoms - Atoms left.

EDIT: Typo in the second part of b. I put after 1 year again whilst it should be 2 years.
• January 18th 2011, 10:07 AM
Tweety
Thanks for that, life saver.

Dont know why I was thinking something else with this question....