Activity Of A Sample

NCERT defines the activity $R$ of a radioactive sample as the number of disintegrations per unit time, and the radioactive decay law gives $R=\lambda N$, where $\lambda$ is the decay constant and $N$ is the current number of undecayed nuclei. Since $N$ itself falls exponentially, the activity also decays exponentially as $R=R_0{e}^{-\lambda t}$. The option using half-life times number is wrong because activity depends on the decay constant, not the half-life, and the units would not even be a rate. The option claiming constant activity is wrong because as nuclei are used up, fewer remain to decay, so the rate must fall. The option claiming activity increases is wrong because it directly contradicts the exponential decrease of $N$ over time. As stated in NCERT Class 12, Chapter 13 (Nuclei), the SI unit of activity is the becquerel, one disintegration per second, while the older unit is the curie. Because activity is directly proportional to the surviving population, an activity measurement can itself be used to count how many undecayed nuclei remain in a source. A plausibility check: because $R\propto N$ and $N$ shrinks with time following the same exponential law, activity must diminish steadily, which is exactly why old sources become progressively weaker.