## 3. Half-Life and Mathematical Representation

As with any chemical substance, a drug will decay exponentially, meaning that it will naturally decompose over time. Specifically, it will decompose at a constant rate that is unique to that particular substance. Scientists consider the time it takes for the initial amount of the substance to decay by half as a method of measuring the decay of a substance. Consequently, the half-life of a substance is the amount of time that it takes for the initial amount of the substance to decay by half.

To understand the concept of half-life better, consider the following example of the pain-relieving drug, which was introduced in the introduction.

One dose of the pain-reliving drug is 1 pill with 400 mg of active ingredient. The half-life of the drug is 4 hours.

If you take only one dose, then every 4 hours the amount of drug in your body will:

We can represent this graphically, as in Figure 1 below.

Figure 1: A graphical representation of the decay from the example above.

As you can see, every 4 hours, the amount of drug decreases by half.

Will the amount of drug ever be zero? Explain.

In Figure 2, you can see a continuous representation of Figure 1.

Figure 2: The red curve is the continuous representation of Figure 1.

The red curve represents the amount of drug in the body measured in milligrams for any time . We need to determine what is the equation of the red curve is. To do so, we describe the decay process mathematically.

Starting at , we have:

Thus, the equation of the red curve is

Now we will generalize this process to determine , the amount of drug in the body measured in milligrams at any time for an arbitrary drug. We will denote the half-life of this substance as and denote the initial amount of the drug as mg.

In Figures 3 and 4, you can visualize the process as above in general terms.

 Figure 3: Graphical representation of decay in general terms. Figure 4: The continuous representation of Figure 3.

Again, we wish to determine the equation of the red curve. Starting at , we have:

Thus far, we have an expression for . What is ?

Let . That is, we no longer require to be an integer. Then . We substitute into , and thus

Now that we have after one dose is taken, we will consider the effect of repeating doses.