To determine the relaxation time for the atmospheric excess of Carbon-14 I downloaded the data for Fruholmen, Norway from CDIAC. Exponential decay applies to other processes besides radioactive decay. The Michaelis-Menten process published in 1903 for the action of a catalyst is one example. P. Curie and J. Danne also published the exponential laws governing radioactive decay and induced radioactivity in 1903. One can fit the excess C-14 curves to a simple model involving the rates of transmutation of N-14 and C-14, α and β respectively, and a net rate for the removal of C-14 from the atmosphere, γ,. In the model below A is the amount of N-14 and B is that of C-14 in the atmosphere. The amount of C-14 removed from the atmosphere is C.
The first three lines are the rates of change for the quantities. Note that the first equations involve only A and B and can be solved separately. The sum of the rates of change is zero which tells us that the sum is a constant and this can be used to find C. Since α is much smaller than the other two rates we can simplify the fit by assuming the exponential terms involving it are effectively constant. The equation for C-14 (B) results in a simple exponential curve. The excess C-14 in the data is based on the "modern fraction" and expressed in mills (thousandths of unity). The reference amount of C-14 in the atmosphere is taken to be the value for 1950. The data points deviate from the expected mean values since the actual number of decays are determined by a Poisson process. Since the decay involves an exponential curve one can convert it to one that is approximately linear by taking the natural log of the relative number of atoms and then smoothing it before doing the actual fit. In this case the average of a point and 21 points to each side were used to smooth the data. The resulting fit turns out quite well showing that the reference value is fairly close the the equilibrium C-14 value. I found that subtracting 0.75 from the excess C-14 values gave the least error.
The first 6 years in which there were fairly large fluctuations in the data were excluded from the fit. The relaxation time found was 15.67 years. In this simple model Δα is the difference between two quantities approximately equal to α and so B'0 is the sum of two terms which makes the it difficult to determine this rate for the formation of C-14 from the equilibrium value. The equilibrium value of C-14 in the atmosphere may be slightly off from the standard 1950 value used for reference purposes but there may also be a systematic error in data due to relatively smaller counts as time progresses.