# Does half-life mean something can never completely decay?

Caffeine has a half-life of approximately six hours. I understand this to mean that every six hours, the amount of caffeine in the body is half of what it was six hours prior. Does that mean that caffeine never completely leaves the body? It just keeps reducing to half after a fixed time ad infintum.

I guess a broader statement of my question is: When calculating the half life of a thing, is it true that it will never reach zero? And how is this used/reconciled in a more practical setting like when calculating the amount of caffeine in the body?

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Might be better to ask at physics.stackexchange.com – DanZimm May 25 '13 at 22:52
Half-Life had a half-life of about six years, eight if you mix Half-Life with Counterstrike. – Asaf Karagila May 25 '13 at 23:04
@DanZimm He's actually asking about the pharmacological half-life, which is slightly different from the nuclear half-life. In particular, it's far less regular and predictable :) – Mike Edenfield May 26 '13 at 4:39
@MichaelEdenfield ah figured they coincided in some way (decaying of atoms/molecules) but I guess not! – DanZimm May 26 '13 at 5:35
@DanZimm They do in idealization, just the human body is complicated and non-linear. – Lucas May 26 '13 at 20:17

There is always a whole number of molecules, and at some point in time there will be zero molecules left. With time, it becomes increasingly likely that all the caffeine molecules have completely disappeared.

However, in chemistry the "concentration" is the mean, or expected, amount of molecules. This is not a whole number. This is much like the way the expected roll of a dice is $3\frac{1}{2}$ even though there is no such number on the dice - the average of your dice rolls over many rolls will be $3\frac{1}{2}$.

This expected number gets smaller and smaller but never reaches zero.

Perhaps this simulation works on you computer.

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mean with respect to multiple independent experiments? Still, for a single experiment it will become zero in finite time with probability 1. – Memming May 26 '13 at 19:48

For half-life to make sense, the assumption that the metabolism that breaks down caffeine at a speed proportional the total amount in the body, and not on concentration or other things. Then exponential decay holds. This assumption doesn't exactly hold in practice, and the exponential decay is only approximate.

And of course, the caffeine is not a continuous quantity, therefore at some point we are talking about fraction of a molecule. Continuous dynamics is a good approximation for the number of caffeine molecules, but eventually they will break down completely.