Introduction

Radioactivity is a natural process where unstable atomic nuclei spontaneously transform into more stable forms by emitting particles or radiation. This transformation is called radioactive decay. It happens because certain isotopes-versions of elements with different numbers of neutrons-are inherently unstable. Unlike many physical processes that can be predicted exactly, radioactive decay is inherently probabilistic. We cannot know exactly when a particular nucleus will decay, only the probability that it will decay over a certain time interval.

Two key concepts help us understand and quantify radioactive decay: half-life and activity. Half-life tells us how long it takes for half of a sample to decay, while activity measures how many decays occur per second. These ideas are foundational in nuclear physics and critical for applications such as medical imaging, radiocarbon dating, nuclear energy, and radiation safety.

Exponential Decay Law

Imagine you have a large number of identical radioactive nuclei, say \( N_0 \) at time \( t = 0 \). Because each nucleus has a fixed probability of decaying per unit time, the number of undecayed nuclei \( N \) decreases as time passes. The rate at which nuclei decay is proportional to how many remain. Mathematically, this is expressed by the differential equation:

\[ \frac{dN}{dt} = -\lambda N \]

Here, \( \lambda \) is called the decay constant and has units of inverse time (\( s^{-1} \), for example). It quantifies the probability of decay per nucleus per second. The negative sign indicates that \( N \) decreases over time.

Solving this equation (a standard first-order differential equation) gives us the number of undecayed nuclei at any time \( t \):

\[ N = N_0 e^{-\lambda t} \]

This is known as the exponential decay law. It tells us that the number of nuclei reduces exponentially over time, never dropping abruptly but slowing as fewer nuclei remain.

Half-life and Its Derivation

The half-life of a radioactive isotope, symbolized as \( T_{1/2} \), is defined as the time taken for half of the initial number of nuclei to decay. In other words, after a time \( T_{1/2} \):

\[ N = \frac{N_0}{2} \]

Using the exponential decay law, we substitute and find \( T_{1/2} \):

\[ \frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}} \]

Dividing both sides by \( N_0 \):

\[ \frac{1}{2} = e^{-\lambda T_{1/2}} \]

Taking the natural logarithm of both sides:

\[ \ln \frac{1}{2} = -\lambda T_{1/2} \Rightarrow -\ln 2 = -\lambda T_{1/2} \]

Thus, the formula for half-life is:

\[ T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda} \]

This relationship shows that the half-life depends only on the decay constant and is independent of the initial amount or the physical or chemical state of the substance. It's a unique property of each radioactive isotope.

Activity and Its Relation to Decay

Activity (\( A \)) of a radioactive sample is defined as the number of decays occurring in the sample per unit time, typically per second. Since each decay corresponds to a nucleus disintegrating, the activity is related to how many nuclei are left and their decay probability.

The rate of change of nuclei is:

\[ \frac{dN}{dt} = -\lambda N \]

Since activity counts decays (positive quantity),

\[ A = -\frac{dN}{dt} = \lambda N \]

Activity decreases over time as nuclei decay, which follows the exponential law:

\[ A = A_0 e^{-\lambda t} \]

where \( A_0 = \lambda N_0 \) is the initial activity right at \( t = 0 \).

Units of Activity:

• Becquerel (Bq): 1 Bq = 1 decay per second (SI unit)
• Curie (Ci): 1 Ci = \(3.7 \times 10^{10}\) decays per second (historical unit)

Activity gives a practical measure of how "radioactive" a sample is at any time.

graph TD    N[Number of undecayed nuclei (N)]    L[Decay constant (λ)]    A[Activity (A = λN)]    T[Time (t)]    N -->|decreases over time| N    L --> A    N --> A    T -->|exponential decay| N    T -->|exponential decay| A

Formula Bank

Worked Examples