What are half-lives? And what do they have to do with measuring the age of the solar system and predicting the effects of a morning cup of coffee? Keep on reading to find out!

If you drink two cups of coffee at 8 a.m., how much caffeine will be left in your body that night at 8 p.m.? Certainly after 12 hours it can’t be that much, right? Or could it be? Maybe even enough to mess with your sleep? I’m not going to spoil the answer, but let’s just say that after learning about the concept of a "half-life" today, you might be a little surprised.

So, what is a half-life? What’s the math behind it? And what does it have to do with the amount of caffeine left in your body at the end of the day … and even with calculating the ages of archeological artifacts and the entire solar system?

Let’s find out.

## What Is a Half-Life?

Some types of atoms do a really weird thing—they spontaneously decay into other types of atoms. A bit more precisely, some unstable isotopes of certain atoms (meaning certain versions of certain atoms that have certain, shall we say, non-standard numbers of neutrons in their nuclei) will spontaneously turn into different elements and in so doing release other particles and light along the way.

… it’s impossible to predict exactly when any individual atom in a huge pile of atoms will decay.

The spontaneous nature of this decay means that it’s impossible to predict exactly when any individual atom in a huge pile of atoms will decay. But even if we don’t know when any one particular atom will decay, we do know the overall average rate at which atoms in the pile will decay. In other words, in a big group of the same type of radioactive atoms, there is a certain amount of time—called the half-life—that will pass before more-or-less half of the atoms will have decayed. So if there are 1 billion atoms to start with, there will on average be 500 million atoms left after a time equal to their half-life.

## What Is Exponential Decay?

This type of decay—in which an average of half the members of a population disappear in a half-life of time … and then another half disappear in the next half-life … and then half of whatever is remaining disappear in the next half-life … and so on forever—is known as exponential decay. We can represent this type of decay in terms of a fairly simple formula:

N_{final} = N_{initial} • (1/2)^{t/t1/2}

In other words, if we start with a population of atoms—or, as we’ll see in a moment, anything else that decays exponentially—that we’ll call N_{initial} (that’s the initial number of those atoms), and we then multiply it by 1/2 raised to the power *t*/*t*_{1/2}, then we can calculate the number of atoms we end up with. But what are *t* and *t*_{1/2}? Well, *t* is simply the amount of time that has passed since we counted the N_{initial} atoms we began with, and *t*_{1/2} is the half-life for the decay of those atoms.