Is Decay constant some kind of rate of decaying?

Question

I am in high school. So in my book, there is just the definition of decay constant. But I don’t understand the concept of it.

Answer 1

The good news is decay (as in radioactive decay) is the absolute simplest concept, it really only has a single parameter that appears in various forms, anyone of which could be called "the decay constant".

If you have an unstable particle, call it $A$, it has a decay constant. It is expressed as a probability to decay per unit time (strictly, it is a calculus limit, but that is a detail). We can pick a small unit of time and approximate:

Suppose the decay rate is $10^{-6}$ per second. That means for each second $A$ exists, it has a 1 in a million chance of decaying.

That's it. That constant never changes. If you come back in two million seconds, and the particle is still there: it still has a 1 in a million chance of decaying in the next second. The particle doesn't get 'old' and more likely to decay.

The inverse of this probability per unit time is called the lifetime:

$$ \tau = \frac 1 {10^{-6}\,{\rm s^{-1}}} = 1,000,000\,{\rm s}\approx 11.5\,{\rm days}$$

The lifetime if proportional to the half-life:

$$ t_{\frac 1 2} = \tau \times \ln{2} \approx 693,000s\approx 8\,{\rm days}$$

Different applications use lifetime or half-life, or both...depending on what is customary in the field.

So if we have a single particle $A$, there is a 50% chance that it will decay in the next 8 days. If it doesn't, and it is still there on day 9, then there is still a 50% chance that it will decay in the next 8 days.

Of course, with many materials, we're not dealing with a single atom, we have a macroscopic amount of it. Any macroscopic collection of atoms of a substance has around Avogadro's number of atoms, and that is a big number:

$$ N_A \approx 6 \times 10^{23} $$

With that many unstable atoms, the individual variations in which decay when isn't that important, and you can treat the material in bulk.

This is where 1/2 comes in handy. If we start out with 8 pounds of substance "A", in 8 days there will be very close to 4 pounds left. After another 8 days, there will be 2 pounds, and then 1, then 1/2, and so on.

The term "decay constant" could refer to anyone of these terms. The key is that it is indeed constant.