Differential Equation Applications
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 No.
t = 0 N = No
And from the definition of half-life:
t = 5600 N = No/2
Also: dN/dt = kN
And what is it that we are looking for?
t = ? when N = 0.1No
Now, separate variables and integrate.