Q.A radioactive sample initially contains $N_0 = 8\times10^{10}$ atoms and has a half-life of $20$ days. Find the number of atoms remaining after $60$ days.

Radioactive Decay Law – From Intuition to the Exact Statement

Imagine you have a huge hall filled with 10,000 identical, unstable atoms. Each atom has a certain probability of "breaking apart" (decaying) in the next second — but you cannot predict which atom will decay, only that, on average, a fixed fraction of them will. This is the core idea: the more atoms you have, the more decays you see per second. If you start with 10,000 atoms, you might see 100 decays in the first second. After that, only 9,900 atoms remain. In the next second, you will see roughly 99 decays — slightly fewer, because fewer atoms are left. The decay rate keeps dropping as the number of undecayed nuclei drops.

That is the intuition: the decay rate is proportional to the number of undecayed nuclei present at that instant.

The Precise Statement

Let $N(t)$ be the number of undecayed nuclei at time $t$. The rate at which they decay, $-\frac{dN}{dt}$, is proportional to $N(t)$ itself. Mathematically:

Here $\lambda$ is the decay constant — a positive constant that is different for every radioactive isotope. It tells you the probability per unit time that a given nucleus will decay. A large $\lambda$ means a fast-decaying substance; a small $\lambda$ means a slow one.

This is the Radioactive Decay Law. $N_0$ is the number of undecayed nuclei at $t = 0$. The number decreases exponentially with time.

Why Exponential?

The differential equation $-\frac{dN}{dt} = \lambda N$ is solved by separating variables:

Integrate both sides:

Exponentiate:

The logarithm of $N(t)$ falls linearly with time — that is the hallmark of exponential decay.

What This Law Tells You

• It is statistical. The law works perfectly for large numbers of nuclei. For a single nucleus, you can only talk about probability.
• The decay constant $\lambda$ is fixed. It does not change with temperature, pressure, or chemical environment (except in extremely rare cases like electron capture).
• Half-life $T_{1/2}$ is the time after which half the original nuclei remain. Set $N = N_0/2$:

A Common Mistake to Avoid

Putting It All Together

The Radioactive Decay Law is simply: the number of undecayed nuclei shrinks exponentially because the decay rate is proportional to the number present. The equation $N = N_0 e^{-\lambda t}$ is the mathematical consequence of that proportionality. Every radioactive substance has its own $\lambda$, and from it you get the half-life — the most intuitive measure of how quickly something decays.

Final result: $N(t) = N_0 e^{-\lambda t}$, with $\lambda = \frac{\ln 2}{T_{1/2}}$.

The radioactive decay law is a foundational CBSE Class 12 Physics NCERT topic under Nuclei, commonly searched as radioactive decay law formula derivation class 12 or exponential decay law important questions. This exponential decay result underlies half-life, decay constant, and activity calculations tested consistently in board exams and JEE Main/NEET physics.