Here is a constructed example of the link between a physical idea and its mathematical model. An idealised radioactive sample starts with 800,000 undecayed parent nuclei and has a half-life of four minutes. The model predicts 400,000 remaining after four minutes. How many remain after eight minutes?
A tempting answer is zero: subtract another 400,000. But a half-life removes half of what remains, not the same number each time. Half of 400,000 is 200,000, so a better answer is: “After two half-lives, one quarter of the original parent nuclei remain.” These are modelled expected counts, rather than an exact prediction for every real sample.
For degree preparation, connect that reasoning to dN/dt = −λN. Here N is the number of undecayed parent nuclei, t is time, dN/dt is its rate of change, and λ is the positive decay constant. The minus sign says N decreases; it does not make the number of nuclei negative. With t in minutes, λ = ln(2)/4 ≈ 0.1733 per minute, where ln means the natural logarithm.
The solution is N(t) = N₀e^(−λt), where N₀ is the starting count and e is the base of the natural exponential. At t = 0 it gives N₀; at eight minutes it gives 800,000 × ¼ = 200,000. Because λ is measured per minute, λt has no units. Checking the starting value and units helps connect the symbols to the physical situation.
Try the next step: after twelve minutes, the model predicts 100,000 parent nuclei. If the starting count doubles to 1,600,000 while the half-life stays four minutes, the expected count after eight minutes doubles to 400,000. The fraction remaining is still one quarter.
