Q.A source contains two phosphorous radio nuclides $^{32}_{15}\text{P}$ ($T_{1/2}$ = 14.3d) and $^{33}_{15}\text{P}$ ($T_{1/2}$ = 25.3d). Initially, 10% of the decays come from $^{33}_{15}\text{P}$. How long one must wait until 90% do so?

Decay Constant — The First Meeting

Imagine you have a room full of 1000 identical, unstable nuclei. Each one is like a tiny time bomb, but with no clock — it could pop in the next second, or a million seconds from now. You cannot predict which one will go next, but you can measure how likely any single nucleus is to decay in a given interval of time.

That likelihood is the decay constant, denoted by the Greek letter $\lambda$ (lambda).

The Intuition

If $\lambda = 0.1\ \text{s}^{-1}$, it means: each nucleus has a 10% chance of decaying in the next second. Not that 10% of the nuclei will decay in exactly one second — but that the probability per unit time is 0.1.

This is a rate, not a count. It stays constant for a given isotope, no matter how many nuclei are left. Whether you have a billion nuclei or just ten, each one still "rolls the dice" with the same probability per second.

The Precise Statement

For a sample containing $N$ radioactive nuclei at time $t$, the rate at which they decay (the activity) is:

The minus sign means $N$ is decreasing. The equation says: the number of decays per second is proportional to the number of nuclei present, and $\lambda$ is the constant of proportionality.

From this, we get the exponential decay law:

where $N_0$ is the initial number of nuclei.

Connecting to Half-Life and Mean Life

The decay constant is the most fundamental of the three related quantities. The others are derived from it.

Half-life $T_{1/2}$ is the time after which half the nuclei remain. Set $N = N_0/2$:

Take natural logs:

So:

Mean life $\tau$ is the average lifetime of a nucleus. It turns out to be:

A Concrete Example

Carbon-14 has a half-life of about 5730 years. Its decay constant is:

That is a very small number — each C-14 nucleus has only a 0.012% chance of decaying in a year. That is why carbon dating works: the decay is slow enough to measure over thousands of years, but fast enough to be detectable.

Common Misconception

For example, if $\lambda = 0.1\ \text{s}^{-1}$ and $\Delta t = 0.01\ \text{s}$, about $0.1 \times 0.01 = 0.001$ (0.1%) of the nuclei decay. But over a full second, it is not 10% — because as nuclei decay, fewer are left to decay in the later part of that second. The exact fraction after one second is $1 - e^{-\lambda} \approx 9.5\%$, slightly less than 10%.

Final takeaway: The decay constant $\lambda$ is the probability per unit time that a single unstable nucleus decays. It is the fundamental rate constant of radioactive decay, and from it you get the half-life and mean life by simple formulas.

The decay constant is a core CBSE Class 12 Physics NCERT topic under Nuclei, commonly searched as decay constant formula and unit class 12 or relation between decay constant, half life, and mean life. This single constant links half-life and mean life through simple formulas that appear repeatedly in board-exam and JEE Main/NEET physics numericals.