Radioactive Decay Calculator

What is Radioactive Decay?

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. This process transforms the nucleus into a more stable form, often of a different chemical element. The rate of decay is characterized by the half-life, which is the time required for half of a given quantity of a radioactive isotope to decay.

Formula and Variables

The radioactive decay process is described by the exponential decay formula:

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

Where:

• \(N(t)\) is the quantity remaining after time \(t\)
• \(N_0\) is the initial quantity
• \(e\) is the base of natural logarithms (approximately 2.71828)
• \(\lambda\) is the decay constant
• \(t\) is the elapsed time

The decay constant \(\lambda\) is related to the half-life \(t_{1/2}\) by the equation:

\[\lambda = \frac{\ln(2)}{t_{1/2}}\]

Calculation Steps

To calculate the remaining amount of a radioactive substance:

1. Determine the initial amount \(N_0\)
2. Identify the half-life \(t_{1/2}\) of the substance
3. Determine the elapsed time \(t\)
4. Calculate the decay constant: \(\lambda = \frac{\ln(2)}{t_{1/2}}\)
5. Apply the decay formula: \(N(t) = N_0 \cdot e^{-\lambda t}\)
6. Simplify and solve for \(N(t)\)

Example and Visual Representation

Let's calculate the remaining amount of a radioactive substance with an initial quantity of 1000 units, a half-life of 5 years, after 10 years:

\[N(10) = 1000 \cdot e^{-\frac{\ln(2)}{5} \cdot 10} \approx 250 \text{ units}\]

This diagram illustrates the radioactive decay process. The left circle represents the initial amount of radioactive material, while the right circle shows the remaining amount after 10 years. The size difference between the circles visually demonstrates the decay that has occurred over time.