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:
#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
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.