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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?

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2 Answers 2

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Your mistake is here:

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.

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    $\begingroup$ Silly me. I got it. I overlooked this simple point and some time after posting the question, I came to realize it. $\endgroup$ Commented May 12, 2020 at 10:26
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    $\begingroup$ 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. $\endgroup$ Commented May 13, 2020 at 12:32
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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.

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