Radioactive Decay Formula

Radioactive decay is a fundamental process by which an unstable atomic nucleus loses energy by emitting radiation. This phenomenon underpins various applications in nuclear physics, medical diagnostics, and environmental science.

Historical Background

The discovery of radioactive decay dates back to the late 19th century, with pioneers like Henri Becquerel, Marie Curie, and Ernest Rutherford contributing significantly to our understanding. This process is intrinsic to the atoms of certain elements, leading to their transformation into other elements over time.

Calculation Formula

The formula to calculate the quantity of a radioactive substance that remains after a given period is expressed as:

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

where:

• \(N(t)\) is the remaining quantity of the substance at time \(t\),
• \(N_0\) is the initial quantity of the substance,
• \(\lambda\) is the decay constant, specific to each radioactive material,
• \(t\) is the time elapsed.

Example Calculation

Suppose you have 10 grams of a radioactive substance with a decay constant of 0.693 per year. To find out how much of the substance remains after 5 years:

\[ N(t) = 10 \cdot e^{-0.693 \cdot 5} \approx 0.5 \text{ grams} \]

Importance and Usage Scenarios

Radioactive decay formulas are essential for:

• Dating archaeological artifacts through radiocarbon dating.
• Understanding the behavior of radioactive substances in nuclear reactors.
• Medical applications such as cancer treatment with radiopharmaceuticals.
• Environmental monitoring of radioactive pollutants.

Common FAQs

1. What is the decay constant?

   • The decay constant (\(\lambda\)) is a measure of the rate at which a radioactive substance decays. It is inversely proportional to the substance's half-life.

2. How is the half-life related to the decay constant?

   • The half-life of a radioactive substance is the time required for half of the substance to decay and is calculated as \(T_{1/2} = \frac{\ln(2)}{\lambda}\).

3. Can this formula predict the exact time a single atom will decay?

   • No, the formula provides the expected behavior for a large number of atoms. The decay of individual atoms is random and cannot be predicted precisely.

The radioactive decay formula is a powerful tool for understanding the stability and transformation of atomic nuclei over time, with wide-reaching implications across science and technology.