Some time ago I posted about the theoretical justification for exponential decay. In that post, I showed that you can quantify exponential decay with this equation:

where \(N(t)\) is the number of undecayed atoms at time \(t\) and \(\lambda\) is a constant representing the decay rate. If you plug in \(N(t) = \frac{1}{2}N(0)\), the condition for the half-life, you can find that

But physicists usually write the formula like this,

where \(\tau\) is called the time constant. We prefer this to using the decay rate because, as I wrote in Calculating Terminal Speed, it’s often best to write a physics question in terms of dimensionless ratios like \(\frac{t}{T}\), where \(T\) is some time characteristic of the physical system you’re studying. We could use the half-life for \(T\), but the time constant \(\tau\) is more appealing for a couple of reasons: it keeps that ugly factor of \(\ln 2\) out of the formula, and more importantly, \(\tau\) is physically meaningful because it’s the average lifetime of an individual atom …

A rather interesting question came up on Physics Stack Exchange (semi-)recently: How do we know that \({}^{14}\mathrm{C}\) decay is exponential and not linear? This question addresses something that confuses a lot of people when they’re first learning about radioactivity, namely the use of a half-life to describe the different rates at which different kinds of radioactive atoms decay. When you first hear that, say, carbon-14 has a half life of 5700 years, you might wonder why we don’t just say that it has a lifetime of twice that, or 11400 years? If half the sample is gone in the first 5700 years, won’t the other half be gone after the next 5700 years?

The model suggested by that statement is called linear decay, because the number of atoms remaining decreases linearly with time. Of course, we know from experiments that radioactive decay is not linear, it’s exponential. But you can also use simple physical reasoning to convince yourself that radioactive decay wouldn’t be described with a half-life if it were linear. Here’s a little thought experiment to show that:

1. Take a billion radioactive carbon-14 atoms, put them in a box, and …