Differential Equation Applications

Radioactive Decay

Radioactive elements decay at a rate proportional to the amount of the element present. If the amount present is defined as N, write the differential equation that applies.

dN/dt = kN

Now let’s look at a problem. Most of you have probably heard of carbon-12 or carbon-14 dating used to determine how old a specimen is. Here’s how it’s done.

A piece of human bone is found at an archeological site. If 10% of the original amount of radioactive carbon-14 was present, estimate the age of the bone. The half-life of C-14 is 5600 years. (Half-life is the the time it takes for half of it to decay.)

Write down what is known.

We don’t know what the original amount is, so we’ll call it N_{o}.

t = 0 N = N_{o}

And from the definition of half-life:

t = 5600 N = N_{o}/2

Also: dN/dt = kN

And what is it that we are looking for?

t = ? when N = 0.1N_{o}

Now, separate variables and integrate.