Q.Explain the idea of carbon dating.

Step 1. The physical basis. All living organisms continuously absorb carbon dioxide from the air; while most of this carbon is stable $^{12}_{6}C$, a tiny, nearly constant fraction ($1.3\times10^{-12}$) is the radioactive isotope $^{14}_{6}C$, with a half-life of 5730 years.

Step 2. Why the ratio stays constant while alive. Atmospheric $^{14}_{6}C$ is continuously decaying, but it is simultaneously replenished by cosmic-ray bombardment of the upper atmosphere, so the ratio of $^{14}_{6}C$ to $^{12}_{6}C$ stays essentially fixed in the atmosphere - and, through continuous absorption, in every living organism - for as long as the organism is alive.

Step 3. What changes after death. Once an organism dies, it stops absorbing new carbon, so its existing $^{14}_{6}C$ is no longer replenished and simply decays away, following the ordinary exponential law; the $^{14}_{6}C$-to-$^{12}_{6}C$ ratio in dead remains therefore steadily decreases with time since death.

Step 4. The dating calculation. Measuring a sample's present-day carbon-14 activity $R$, and knowing what its activity $R_0$ would have been while still alive (calculated from the known natural $^{14}C$ abundance and the sample's total carbon content), the elapsed time follows from $R=R_0e^{-\lambda t}$, rearranged as $t=\dfrac1\lambda\ln(R_0/R)$, using $\lambda=\ln2/T_{1/2}$ from the known half-life. …

Explain why the $^{14}C$/$^{12}C$ ratio is constant while alive and decays after death, then apply $t=\dfrac1 …

• Forgetting to explain why the ratio is constant during life (continuous atmospheric replenishment) - this is the crux of the whole method. …