Q.The amount of radiocarbon present after $t$ years is given by $A=A_0e^{-(\ln 2)\left(\frac{1}{5700}\right)t}$, where $A_0$ is the amount present in the living plants and animals.

Exponential Decay Half-Life: From Intuition to Precision

Imagine you have a freshly poured cup of coffee. It's hot — say 80°C. The room is at 25°C. You know the coffee cools down, but it doesn't cool at a constant rate. In the first minute, it drops a lot. In the second minute, it drops less. By the tenth minute, it's barely changing. That slowing-down pattern — where the rate of change is proportional to the amount left — is exponential decay.

Now, the half-life is a single number that captures the speed of that decay. It answers the question: How long does it take for half of whatever we're measuring to disappear?

The Intuition: A Simple Story

Suppose you have 100 radioactive atoms of a certain element. After one half-life, 50 atoms remain. After another half-life, 25 remain. After another, 12.5 (on average). Notice the pattern: each half-life, the quantity is cut in half. It doesn't matter whether you start with 1000 atoms or 10 atoms — the time to go from any number to half that number is always the same.

This is the key insight: the half-life is constant. It doesn't depend on how much you have. That's what makes it so useful. If a drug has a half-life of 4 hours in your body, then every 4 hours, half of what's left gets eliminated. After 8 hours, only a quarter remains. After 12 hours, an eighth.

The Precise Statement

Let $N_0$ be the initial quantity (atoms, drug concentration, temperature difference, etc.). Let $N(t)$ be the quantity after time $t$. Let $T_{1/2}$ be the half-life.

The defining property of half-life is:

After two half-lives:

After $n$ half-lives:

This works even for fractional half-lives. After 0.5 half-lives, you'd have about 70.7% remaining (since $\sqrt{1/2} \approx 0.707$).

The Mathematical Formula

Exponential decay follows the equation:

where $\lambda$ is the decay constant — the probability per unit time that a single atom decays. The half-life and decay constant are related by a simple formula:

Why $\log 2$? Because we want $N(T_{1/2}) = \frac{1}{2} N_0$. Plugging into the decay equation:

Cancel $N_0$:

Take natural log of both sides:

So:

And therefore:

$T_{1/2} = \frac{\log 2}{\lambda}$ …