Radioactive DecayJEE Main

Half-life and the exponential law — interactive Physics simulation for IIT-JEE.

Concept

Radioactive nuclei decay at random, but a large sample follows a precise exponential law. After each half-life exactly half of whatever remains is gone — so 100% → 50% → 25% → 12.5%, regardless of where you start.

Key formula

N = N_0 e^{-\lambda t} = N_0\left(\tfrac12\right)^{t/T_{1/2}}, \quad \lambda = \frac{\ln 2}{T_{1/2}}, \quad A = \lambda N

Derivation

The decay rate is proportional to the number present: $dN/dt = -\lambda N$ . Solving gives $N = N_0 e^{-\lambda t}$ .

Setting $N = N_0/2$ identifies the half-life $T_{1/2} = \ln 2/\lambda \approx 0.693/\lambda$ . The mean life $\tau = 1/\lambda = T_{1/2}/\ln2$ is longer than the half-life. Activity $A = \lambda N$ also decays exponentially.

Scenarios to explore

Radioactive Decay — Half-life and the exponential decay law.

Real-world applications

Carbon-14 dating of archaeological samples.
Medical isotopes (I-131, Tc-99m) and radiotherapy dosing.
Nuclear waste storage timescales.

JEE exam tips

After $n$ half-lives the fraction left is $(1/2)^n$ — quickest for clean multiples.
$\tau = T_{1/2}/0.693 = 1.44\,T_{1/2}$ .

Common mistakes

Halving the original amount each time instead of the current amount.
Confusing half-life $T_{1/2}$ with mean life $\tau$ ( $\tau > T_{1/2}$ ).
Forgetting activity (not just $N$ ) also decays with the same $\lambda$ .

Exam traps to avoid

Decay is statistical — you cannot predict which nucleus decays, only the average.
Half-life is independent of temperature, pressure or chemical state.