# Why does radioactive matter decay in series of half-lives?

I was studying types of nuclear decay, and I came across the concept of "half-life". And I started wondering why does nuclear matter decay in half-lives? In other words, why does the rate of nuclear decay decrease as a function of time?

Is it because as more atoms decay, the surface area of the radioactive matter decreases, and the number of atoms "exposed" decrease? Does the outer layer of atoms somehow "shield" the inner atoms and stabilize them, preventing them from decay?

Or is there another mechanism at play?

• But the rate stays the same over time (the foundation of radiocarbon dating). After $t_{1/2}$, half of what you started with is gone. At $2t_{1/2}$, you are at half of what you had after $t_{1/2}$ (so a quarter compared to $t = 0$). The rate is the same: 50% of what you have right now is gone after another $t_{1/2}$. Radioactive decay is spontaneous and statistical, there is no shielding etc. – TAR86 Jan 11 at 22:14
• The decay is an intrinsic property of an atom so it does not matter if they are in rock or a vacuum they still decay. The rate at time $t$ is proportional to the amount unreacted $dx/dt$ where $x$ is the number, and changes in time as $dx/dt=-kx$ where $k$ is a number specific to each type of atom called the rate constant or given as $\tau = 1/k$ the decay lifetime.The minus sign means the species decay. The half life is a measure of decay similar to the lifetime, the lifetime measures time to $1/e$ of the initial amount, the half life to 0.5. – porphyrin Jan 12 at 10:14

There are fewer decays because there are fewer atoms to decay

The simple reason why the number of decays (strictly, the number of decays per unit time) decreases in simple radioactive decay is because there are fewer atoms left to decay.

Nuclear decay is probabilistic. The probability of any given unstable atom decaying is constant (independent of time or the environment). So the reaction rate (the total number of decays per second) per mole is constant. If this were a chemical reaction we would call it a first-order reaction (since it isn't a chemical phenomenon we can call it a first order process but the mathematics are the same).

In any first order process there is a characteristic time when half the original material will have gone (we call this the half life of the process). So, if we measure time in half-lives at t=1 we will have, say, 1 mole of the material; by t=1 we will have 0.5 moles; at t=2 we will have 0.25 moles of material. If we count the number of emissions per second they will follow the amount of material left. So the radioactivity will decay at the same rate as the material.

We don't need to invoke any interactions between the atoms of the material to explain this.

PS it gets a little more complicated if one radioactive compound decays into another. Then the total radioactivity will depend on two competing decays with different half-lives (but the radioactive emissions may well be different as each has a characteristic energy and expensive instruments can tell them apart giving two separate emission counts). But the mathematics remains the same.

It is a general principle, not limited to nuclear chemistry, but is common for many areas, e.g. for the reaction kinetic of the 1st order.

All processes, where the value time rate is proportional to the value, have value time evolution in the form of the exponential function.

$$\frac {\mathrm{d}x}{\mathrm{d}t}= -k \cdot x$$

$$x= x_0 \cdot \exp { (-k \cdot t )}$$

Radioactive atomic kernels have constant probability of the decay over the given time interval. The actual decay rate is then for large kernel numbers proportional to this probability and the number of these kernels, described by the above equations in the differential and integral form.

When there is twice as many radioactive atoms of the same kind there is twice as high decay rate.

The shape of the exponential function shows its value decreases to a half by constant time, no matter what the value is. This constant time is the decay half life.

where

$$t_{1/2}=\frac {\ln{2}}{k}$$