What is an example of an exponential decay practice problem?

Dec 29, 2014

Here's an example of an exponential decay problem.

A radioactive isotope has a half-life of 3000 years. If you start with an initial mass of 50.0 g, how much will you have after
A. 10 hours;
B. 100,000 years;

So, an exponential decay function can be expressed mathematically like this:

$A \left(t\right) = {A}_{0} \cdot {\left(\frac{1}{2}\right)}^{\frac{t}{t} _ \left(\text{1/2}\right)}$, where

$A \left(t\right)$ - the amount left after t years;
${A}_{0}$ - the initial quantity of the substance that will undergo decay;
${t}_{\text{1/2}}$ - the half-life of the decaying quantity.

So, we start with 50.0 g, this represents ${A}_{0}$. Our half-life is ${t}_{\text{1/2}} = 3000$ $\text{years}$, and $t$ will be 10 hours. Since the isotope's half-life is given in years, we must convert 10 hours to years:

$10.0$ $h o u r s \cdot \left(\text{1 day")/("24 hours") * ("1 year")/("365.25 days}\right) = 0.00114$ $\text{years}$

So, the amount left after 10 hours will be

$A \left(t\right) = 50.0 \cdot {\left(\frac{1}{2}\right)}^{\frac{0.00114}{3000}} = 29.99999$ $\text{g}$ - (don't worry about sig figs, I just want to illustrate how little mass undergoes nuclear decay).

Let's set $t$ equal to 100,000 years now. The amount left will be

$A \left(t\right) = 50.0 \cdot {\left(\frac{1}{2}\right)}^{\frac{100000}{3000}} = 0.0000000046$ $\text{g}$

These values represent the two extremes of nuclear decay; after 10 hours, the amount left is, for all intended purposes, identic to the initial mass. In contrast, the amount left after 100,000 years is close to ${10}^{10}$ times smaller than the initial mass.