Half-life And Decay Law

Radiocarbon dating relies on the decay law from NCERT, $N=N_0{\left(\frac{1}{2}\right)}^{t/T_{1/2}}$, where the carbon-14 half-life is about 5730 years and activity is proportional to the number of undecayed nuclei. Since the measured activity is one-fourth of the living value, the fraction remaining is $\frac{1}{4}={\left(\frac{1}{2}\right)}^2$, which means two half-lives have elapsed. The age is therefore $t=2\times 5730=11460$ years. The value 5730 years is wrong because that corresponds to only one half-life, when activity would be one-half, not one-fourth. The value 17190 years is wrong because it assumes three half-lives, which would leave one-eighth of the activity. The value 2865 years is wrong because it takes only half of a single half-life, which would leave far more than a quarter of the activity remaining. As stated in NCERT Class 12, Chapter 13 (Nuclei), the fixed half-life makes such isotopic ratios reliable clocks, because carbon-14 is continuously replenished in living organisms and only begins its steady decay once the organism dies and stops exchanging carbon. Since activity is directly proportional to the number of undecayed carbon-14 nuclei, measuring the activity ratio directly counts the number of elapsed half-lives. A plausibility check: two successive halvings reduce activity as $1\to \frac{1}{2}\to \frac{1}{4}$, exactly matching the observed quarter, so an age of 11460 years is fully consistent with the decay law.