Question

What is an example of an exponential decay practice problem?

Answer 1

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(t) = A_0 * (1/2)^(t/t_("1/2"))`, where

`A(t)` - the amount left after t years;
`A_0` - the initial quantity of the substance that will undergo decay;
`t_("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_("1/2") = 3000` `"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` `hours * ("1 day")/("24 hours") * ("1 year")/("365.25 days") = 0.00114` `"years"`

So, the amount left after 10 hours will be

`A(t) = 50.0 * (1/2)^(0.00114/3000) = 29.99999` `"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(t) = 50.0 * (1/2)^(100000/3000) = 0.0000000046` `"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.