[sgarrett]

I am an archaeology graduate student trying to grasp the hard science behind radiometric (specifically radiocarbon) dating.  I feel I have a general grasp of the concept of a half-life.  What is confusing me is the meanlife.  A quote from the paper I am reading will perhaps explain my confusion: "...there are several reasons why [radiocarbon dates] differ from a true age.  First, radiocarbon dates are calculated using an obsolete measurement of 14C meanlife, 8033 years, even though it is now known more accurately as 8266 years."  He moves on to explain other reasons (14C content has varied over the millenia-which I understand), without explaining what a meanlife is.  So far, Google has not been very helpful.

Is the half-life derived from the meanlife?  If so, why don't they just keep it simple and say that the known half-life is calculated using an obsolete measurement? 

FYI I do understand the difference between a radiocarbon date [statistical measurement of 14C content] and a calendar date.  That's not what I'm asking.

I would appreciate any help that you hard scientists can give me!

[sgarrett]

Is this problem in violation of rules?  I realize my subject title is not in the form of an answerable question, and I apologize for that oversight.  Or, is it outside the scope of undergrad chemistry?  If so, I may be in waaay over my head here.

I hope you see this as an interesting exercise in applied chemistry.  It will help me to understand this little corner of chemistry so that I can interpret radiocarbon dates accurately.

[MrTeo]

Is the half-life derived from the meanlife?

Not really, while the half-life is, as you know, the time needed for the decay of half the amount of substance, the mean life is the arithmetic mean of the lifetimes of the atoms decaying. This means that the two values are connected:



(where t_1/2 is the half-life and τ the mean life), but they have different derivations. This difference leads to a different form of the decay expression



while usually we find



In fact τ=λ^-1, and the first expression gives us the chance to compare the decay process with other physical or chemical exponential trends (e.g. RC and LR circuits), which use a time constant too.
To sum up you can easily verify that τ can be seen as the time needed for the decay of 1/e of the total amount of radioactive substance or the time that the whole amount would need to decay if the law wasn't exponential, but linear. Here's a mathematical explanation of what I mean:

(BONUS)
If the decay trend was linear the decay rate would be costant and equal to the decay rate at t=0, so we can find this value with the derivation of our expression:



at t=0 we have:



taking the modulus of this (the minus sign shows us the decrease of the radioactive element) and finding the time needed for N₀ atoms to decay at this rate we have:



as we wanted to show.

Anyway probably most times you'll find the decay law expressed with the decay constant (or the half life) and not with the mean life. I think that your book only wants to point out an obsolete value of the constants needed to calculate the age of something with 14C dating.

[Borek]

Carbon dating is not as simple as just looking at half life - you need tables that show how the C14/C12 ratio (which is itself a function of solar activity) changed with time. As these tables are calibrated using old value of mean life, I guess it is better to use wrong value that is incorporated in these tables, than to recalibrate whole system. I remember reading that these tables are skewed and they show increasing percentage of C14 when you go back in time, that could be artifact of the wrong half life value.

[sgarrett]

Thanks for your *delete me*  I think I get it a little better, although if I could understand all that math, I would probably be a chemist myself!  What I understand is- half-life is looking at the rate of decay of the entire sample, while meanlife looks at the individual atoms.  I am aware that it is much more complicated than simply looking at half-life.  I was just trying to get a relevant definition of the term meanlife that made sense in the context of the sentence I provided.

In fact, the entire sytem gets recalibrated every few years based on new data.  Specialists write fro the journal Radiocarbon, which is the arbiter. 

The good thing is that RCYBP (radiocarbon years before present), that is, the ratio of C14 to C12 in your particular sample, is always the same, so your point date and error range can simply be plugged into the new calibration.