# What is the mean life of a radioactive substance?

As far as my understanding goes, it is the average lifetime of a collection of nuclei undergoing disintegration. But doesn't each nucleus take an infinite amount of time to decay? Is that not why we use the concept of half-life? Then, shouldn't the mean life be infinite as well?

What do you mean by your question:

doesn't each nucleus take an infinite amount of time to decay?

As far as I know, this is not true. A nucleus will start in one state, and end in another "decayed state" + radiation ($$\alpha^{2+}, \beta^\pm, \gamma$$ or whatever), and this is not an infinitely long process.

A nucleus has a probability of decaying within the next time interval, say $$\delta t$$, or not. Thanks to how statistics and probability work, if we have a large number of these nuclei, they will collectively exhibit a "mean lifetime" (i.e. we are able to obtain an average time it takes for one nucleus to decay).

Perhaps you're getting confused by this formula:

$$N = N_0e^{-\lambda t} = N_0e^{-t/\tau}$$

where $$N$$ is the number of non-decayed nuclei present in your sample, and $$N_0$$ is the number of initial non-decayed nuclei.

In this case, yes it takes (in theory) an infinite amount of time for $$N$$ to reach $$0$$, though this assumes $$N$$ can vary continuously (such as taking values like $$N=0.01$$, which is non-physical - $$N$$ can only take integer values). As $$N$$ and $$N_0$$ get larger, this equation better describes the situation.

Here, $$\tau = 1/\lambda$$ is in fact the mean lifetime, and is related to the half life,$$\tau_{1/2}$$ via

$$\tau = \frac{\tau_{1/2}}{\ln 2}$$

• You are right. I did get confused with that. I understand now. So, the mean time taken for the radioactive decay per nucleus is related to the time taken for half of the total amount of substance to decay by the relation mentioned in the end of your answer? This means that time taken for a single nucleus to decay changes as the amount of substance that is decaying changes, that is, as more and more nuclei decay, the mean time taken for a single nucleus to decay increases, am I right? – Manoj Raghavendran Oct 12 '18 at 17:30
• Not quite - the half-life and mean lifetimes never change, no matter how many nuclei there are. If I have $N_0$ nuclei to start, then after time $\tau_{1/2}$ I will have $N_0/2$ nuclei left. But how long will it take to get from $N_0/2$ to $N_0/4$ nuclei? The answer is still the half-life, $\tau_{1/2}$. This is because I could arbitrarily say I've restarted my experiment at $N=N_0/2$ and call the number of particles $N_0/2=M_0$, and then a time $\tau_{1/2}$ later (a property of the nucleus itself, not the number of them) is $M_0/2=N_0/4$. – Garf Oct 12 '18 at 17:35
• – Garf Oct 12 '18 at 17:37
• I understand. Both the half-life and mean life are properties of a nucleus alone. – Manoj Raghavendran Oct 12 '18 at 17:46

The atoms of a radioactive substance are constantly disintegrating but all the atoms do not decay simultaneously. Some atoms have short life time while others have longer. Thus making the life of every atom different.Therefore it is useful to determine the mean life of the radioactive substance.

The mean life is hence determined by the ratio of the Total life of all the atoms to the Number of atoms present in the radioactive substance.

But since radioactivity is an exponential function as answer prior to me states, there will be that one nucleus that hasn't decayed making the mean life infinite. Hope this helps you.