Carbon dating exponential
This means that the ceiling on the usefulness of radiocarbon dating is not the number of half-lives after which the remaining concentration is too small for us to measure, but the point at which the remaining concentration cannot be distinguished from this noise.
I can do this by working from the definition of "half-life": in the given amount of time (in this case, hours.
If we assume Carbon-14 decays continuously, then $$ C(t) = C_0e^, $$ where $C_0$ is the initial size of the sample. Since it takes 5,700 years for a sample to decay to half its size, we know $$ \frac C_0 = C_0e^, $$ which means $$ \frac = e^, $$ so the value of $C_0$ is irrelevant.
Now, take the logarithm of both sides to get $$ -0.693 = -5700k, $$ from which we can derive $$ k \approx 1.22 \cdot 10^.
Carbon dioxide is distributed on a worldwide basis into various atmospheric, biospheric, and hydrospheric reservoirs on a time scale much shorter than its half-life.
Measurements have shown that in recent history, radiocarbon levels have remained relatively constant in most of the biosphere due to the metabolic processes in living organisms and the relatively rapid turnover of carbonates in surface ocean waters.