Physics

Activity measures how fast nuclei decay right now, while the total number of undecayed nuclei is what's actually left in the sample. The key insight is that activity and the number of nuclei are proportional through the decay constant—if activity drops to one-third, the number of undecayed nuclei must also drop to one-third, since they decay at the same exponential rate. This is because $A = \lambda N$, where both $A$ and $N$ follow the same exponential decay curve.

Activity measures how fast nuclei decay right now, while the total number of undecayed nuclei is what's actually left in the sample. The key insight is that activity and the number of nuclei are proportional through the decay constant—if activity drops to one-third, the number of undecayed nuclei must also drop to one-third, since they decay at the same exponential rate. This is because $A = \lambda N$, where both $A$ and $N$ follow the same exponential decay curve.