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Computations of ages and dates

The radioactive decay of carbon-14 follows an exponential decay. A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. Symbolically, this can be expressed as the following differential equation, where N is the quantity and λ is a positive number called the decay constant:The solution to this equation is:,where C is the initial value of the particular case of radiocarbon decay, this equation is written:,where, for a given sample of carbonaceous matter:N0 = number of radiocarbon atoms at t = 0, i.e. the origin of the disintegration time,N = number of radiocarbon atoms remaining after radioactive decay during the time t,λ = radiocarbon decay or disintegration related times can be defined:

half-life: time lapsed for half the number of radiocarbon atoms in a given sample, to decay,

mean- or average-life: mean or average time each radiocarbon atom spends in a given sample until it decays.

It can be shown that:t1 / 2 = = radiocarbon half-life = 5568 years (Libby value)tavg = = radiocarbon mean- or average-life = 8033 years (Libby value)Notice that dates are customarily given in years BP which implies t(BP) = -t because the time arrow for dates runs in reverse direction from the time arrow for the corresponding ages. From these considerations and the above equation, it results:For a raw radiocarbon date:and for a raw radiocarbon age:After replacing values, the raw radiocarbon age becomes any of the following equivalent formulae:using logs base e and the average life:andusing logs base 2 and the half-life: