Q.Starting from the radioactive decay law $\dfrac{dN}{dt} = -\lambda N$, derive the expression $N(t) = N_0 e^{-\lambda t}$ for the number of undecayed nuclei remaining at time $t$.

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 $N = N_0/2$:

$\ln(1/2) = -\lambda T_{1/2} \quad\Rightarrow\quad T_{1/2} = \frac{\ln 2}{\lambda}$ …