define half-life

Radioactive Decay

Objective

To define the concept of half‑life and understand how it is used to describe radioactive decay.

What is Radioactive Decay?

Radioactive decay is a random process by which an unstable nucleus transforms into a more stable configuration, emitting particles or electromagnetic radiation. The number of undecayed nuclei, $N$, decreases exponentially with time.

Mathematical Description

The rate of decay is proportional to the number of nuclei present:

$$\frac{dN}{dt} = -\lambda N$$

where $\lambda$ is the decay constant (s\(^{-1}\)). Integrating gives the exponential law:

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

$N_0$ is the initial number of nuclei at $t = 0$.

Definition of Half‑life

The half‑life, denoted $t_{1/2}$, is the time required for half of the original nuclei to decay. Mathematically it is defined by the condition $N(t_{1/2}) = \frac{1}{2}N_0$.

Substituting into the exponential law:

$$\frac{1}{2}N_0 = N_0 e^{-\lambda t_{1/2}}$$

Solving for $t_{1/2}$ yields:

$$t_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$$

Example Calculation

1. Given a decay constant $\lambda = 2.5 \times 10^{-3}\ \text{s}^{-1}$, calculate the half‑life.
2. Use the formula $t_{1/2} = \dfrac{\ln 2}{\lambda}$.
3. $$t_{1/2} = \frac{0.693}{2.5 \times 10^{-3}\ \text{s}^{-1}} \approx 277\ \text{s}$$

Factors Influencing Half‑life

• Half‑life is an intrinsic property of a radionuclide; it does not depend on external conditions such as temperature, pressure, or chemical state.
• Different decay modes (α, β, γ) have characteristic half‑lives ranging from fractions of a second to billions of years.

Typical Half‑life \cdot alues

Radionuclide	Decay Mode	Half‑life
Carbon‑14	β‑decay	5,730 years
Iodine‑131	β‑decay	8.0 days
Uranium‑238	α‑decay	4.5 × 10⁹ years
Polonium‑212	α‑decay	0.3 µs

Key Points to Remember

• The half‑life is the time for a sample to lose half of its radioactive nuclei.
• It is related to the decay constant by $t_{1/2} = \ln 2 / \lambda$.
• Half‑life is independent of the amount of material and external conditions.
• Exponential decay means that after each successive half‑life the remaining activity is halved again.