Introduction

Radioactive decay is a fundamental process in nuclear physics where unstable atomic nuclei spontaneously transform into more stable configurations. This process follows precise mathematical models that allow us to predict the amount of radioactive material present at any given time.

The Mathematical Model of Radioactive Decay

The basic model of radioactive decay follows an exponential decay pattern, described by the equation:

$N_t = N_0e^{-λt}$

Where:

Half-Life: A Key Concept

The half-life ($t_{1/2}$) of a radioactive isotope is the time required for half of the initial amount to decay. This value remains constant throughout the decay process, regardless of the initial quantity.

For example, if we start with 100 atoms:

The Decay Constant

The decay constant ($λ$) is related to half-life through the equation:

$λ = \frac{\ln(2)}{t_{1/2}}$

This relationship shows that:

Units in Radioactive Decay Calculations

When performing calculations, it's crucial to maintain consistent units. The following table shows common unit combinations:

Time Unit (t)	Decay Constant (λ) Unit
seconds	s⁻¹
minutes	min⁻¹
hours	hr⁻¹
years	yr⁻¹

Note: The units chosen for time must match the units used in the decay constant.

Sample Problem

Problem: A sample contains 100 kg of Cobalt-60, which has a half-life of 5.30 years. Calculate the remaining mass after 12 years.

Solution:

Key Points to Remember