Why is this way of calculating mean life of radioactive atoms incorrect?

Suppose there are $$N$$ radioactive atoms and the half life of decay is $$t$$. Then after one half life the number of remaining atoms will be $$\frac{N}{2}$$. And so after each half life the number will be halved.

Which means, $$1/2$$ of the atoms will have a life of $$t$$

Half of the the remaining half or $$1/4$$ of the atoms will have a life of $$2t$$ and so on.

Then if the mean time for decay is $$\tau$$, then it should be:

$$\tau = \frac{(\frac{N}{2}t+\frac{N}{4}2t+\frac{N}{8}3t+...)}{N}$$ or $$\tau = t(\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+...)$$

But this infinite series doesn't equal to $$\frac{1}{ln2}$$. And we know that, $$\tau =\frac{t}{ln2}$$

So obviously my calculation is wrong. Why is this way of calculating the mean time for decay wrong?

Which means, 1/2 of the atoms will have a life of t

Half of the the remaining half or 1/4 of the atoms will have a life of 2t and so on.

The corrected statement is:

Which means, 1/2 of the atoms will have a life $$\le t$$

Half of the the remaining half or 1/4 of the atoms will have a life between $$t$$ and $$2t$$ and so on.

• Silly me. I got it. I overlooked this simple point and some time after posting the question, I came to realize it. Commented May 12, 2020 at 10:26
• A second, related problem, is that this statement is expressed as a discrete rule ("...after each half life the number will be halved... Half of the the remaining half or 1/4 of the atoms will have a life ..."), when the actual rule is continuous ("...at any point in time t1>0, the remaining atoms will have a half-life of t1+t"). Even with the correction you point out above, the discrete rule would not mathematically reduce to the correct single equation (exponential decay) , but would still allow many odd step-decay solutions. Commented May 13, 2020 at 12:32

Which means, $$1/2$$ of the atoms will have a life of $$t$$

is not a correct statement as some of them will have a lifetime of almost zero, some $$\frac 12t$$ some $$\frac{199}{200}t$$ etc, so you have overestimated the time that the atoms live.

To see how it should be done correctly read the answer to Mean life of radioactive substance.