understand that radioactive decay is both spontaneous and random

Radioactive Decay

Learning Objective

Understand that radioactive decay is both spontaneous and random.

What is Radioactive Decay?

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. The nucleus transforms into a different nucleus (or the same element in an excited state) without any external influence.

Spontaneous Nature

The decay occurs without any external trigger. Even in a perfectly isolated environment, an unstable nucleus will eventually decay.

• Temperature, pressure, or chemical state do not affect the probability of decay.
• Only the intrinsic properties of the nucleus (its energy levels and quantum structure) determine its stability.

Random (Statistical) Nature

While the overall behaviour of a large collection of nuclei can be described by simple laws, the exact moment at which any single nucleus decays is unpredictable.

• Each nucleus has a constant probability per unit time of decaying, denoted by the decay constant \$\\lambda\$.
• The process follows a Poisson distribution for large numbers of nuclei.

Mathematical Description

The number of undecayed nuclei \$N\$ after a time \$t\$ is given by the exponential decay law:

\$N(t) = N_0 e^{-\\lambda t}\$

where:

• \$N_0\$ – initial number of nuclei at \$t = 0\$
• \$\\lambda\$ – decay constant (s⁻¹)
• \$t\$ – elapsed time (s)

Half‑Life

The half‑life \$T_{1/2}\$ is the time required for half of the original nuclei to decay. It is related to the decay constant by:

\$T_{1/2} = \\frac{\\ln 2}{\\lambda}\$

Types of Radioactive Decay

Illustrative Example

Consider a sample containing \$1.0\\times10^{6}\$ atoms of a radionuclide with a half‑life of 30 minutes. After 90 minutes (three half‑lives), the number of remaining atoms is:

\$\$N = N0 \\left(\\frac{1}{2}\\right)^{\\frac{t}{T{1/2}}}

= 1.0\\times10^{6} \\left(\\frac{1}{2}\\right)^{3}

= 1.25\\times10^{5}\$\$

This calculation shows the predictable statistical behaviour of a large ensemble, even though the exact decay time of any individual atom remains unknown.

Common Misconceptions

1. “Decay can be accelerated by heating.” – Temperature does not affect nuclear decay rates.
2. “All atoms decay at the same time.” – Decay is random; only the average behaviour follows the exponential law.
3. “Radioactive decay is a chemical reaction.” – It involves changes in the nucleus, not electron rearrangements.

Summary

• Radioactive decay is an intrinsic, spontaneous process.
• The exact moment of decay for a single nucleus is random and cannot be predicted.
• Statistical laws (exponential decay, half‑life) describe the behaviour of large numbers of nuclei.

Self‑Check Questions

1. Define the decay constant \$\\lambda\$ and explain its physical meaning.
2. If a radionuclide has a decay constant of \$2.0\\times10^{-3}\\,\\text{s}^{-1}\$, calculate its half‑life.
3. Why does temperature not affect the rate of nuclear decay?
4. Describe how you would experimentally verify that decay is random.