Half-Life Calculator

What is Half-Life?

Half-life is the time required for a quantity to reduce to half of its initial value. In the context of radioactive decay, it's the time taken for half of the atoms in a sample of a radioactive isotope to decay. This concept is crucial in nuclear physics, radiochemistry, and medical applications involving radioactive materials.

How to Calculate Half-Life

The half-life can be calculated using the exponential decay formula. If we know the initial quantity, the quantity remaining after a certain time, and the elapsed time, we can determine the half-life.

Formula

The formula for radioactive decay is:

\[ N(t) = N_0 \cdot (1/2)^{t/t_{1/2}} \]

Where:

• \(N(t)\) is the quantity remaining after time \(t\)
• \(N_0\) is the initial quantity
• \(t\) is the elapsed time
• \(t_{1/2}\) is the half-life

Calculation Steps

To calculate the half-life, we can rearrange the formula:

1. Start with the decay formula: \[ N(t) = N_0 \cdot (1/2)^{t/t_{1/2}} \]
2. Divide both sides by \(N_0\): \[ N(t)/N_0 = (1/2)^{t/t_{1/2}} \]
3. Take the natural log of both sides: \[ \ln(N(t)/N_0) = (t/t_{1/2}) \cdot \ln(1/2) \]
4. Solve for \(t_{1/2}\): \[ t_{1/2} = \frac{-t \cdot \ln(2)}{\ln(N(t)/N_0)} \]

Example and Visual Representation

Let's calculate the half-life of a sample where:

• Initial quantity (\(N_0\)) = 1000 atoms
• Quantity after 10 hours (\(N(t)\)) = 250 atoms
• Elapsed time (\(t\)) = 10 hours

Plugging these values into our formula:

\[ t_{1/2} = \frac{-10 \cdot \ln(2)}{\ln(250/1000)} = 5 \text{ hours} \]

This graph illustrates the exponential decay of the radioactive sample:

• The blue curve shows the decay over time.
• The red dot indicates the half-life point, where the quantity has reduced to half (500 atoms) after 5 hours.
• After 10 hours (two half-lives), the quantity has reduced to 250 atoms, which is one-quarter of the initial amount.