Modelling the Decay of Nuclear Medicine with dy/dx = -ky

in #steemiteducation7 years ago

Inasmuch as the differential equation dy/dx = ky models exponential growth, it follows that dy/dx = -ky models exponentially decaying processes.

A exponential decay process, which is a slowing reduction in quantity over time, follows the equation y = Ae-kx. The decay constant, which governs the rate of decay, is given by k. This family of solutions is shown in the graph below...

177_1.png
Drawn with Desmos

These curves may aptly represent for example: a radioactive decay process; the amount of charge remaining in a capacitor as current flows out; the temperature of a hot surface undergoing forced convection cooling.

In fact, let's solve a radioactive decay problem:

The half-life of a radioactive element used in the treatment of cancer is 4 hours. 12mg are produced at 10AM. Use a differential equation to find an expression for the amount present at time t, and thus the amount remaining at 4PM.

Let M be the amount (i.e. mass in mg) of the radioactive element as it changes over time t. The decrease in M over time can be expressed as:

Eq1.png

Solving this differential equation we get the general solution...

Eq2.png

We are given that the half-life of the radioactive element is 4 hours. Half-life describes the time it takes for a substance to decay half the initial amount. Thus...

Eq3.png

Therefore, our equation for modelling this radioactive decay is...

Eq4.png

At 4PM, we are 6 hours past 10AM. Thus the quantity remaining is:

Eq5.png

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