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}N0 = N0 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

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.