Half-Life Calculator

How to Calculate Half-Life

Half-life is a crucial concept in various scientific fields, particularly in nuclear physics and chemistry. It represents the time required for half of a given quantity of a substance to decay or transform. This calculator helps you determine half-life, elapsed time, remaining quantity, or initial quantity based on the exponential decay model.

What is the Formula?

The basic formulas used in half-life calculations are:

1. Half-Life: \[ T_{1/2} = \frac{t \ln(2)}{\ln(N_0/N)} \]

2. Elapsed Time: \[ t = T_{1/2} \frac{\ln(N_0/N)}{\ln(2)} \]

3. Remaining Quantity: \[ N = N_0 \cdot 2^{-t/T_{1/2}} \]

4. Initial Quantity: \[ N_0 = \frac{N}{2^{-t/T_{1/2}}} \]

Where:

• T₁/₂ = Half-life
• t = Elapsed time
• N₀ = Initial quantity
• N = Remaining quantity

What are the calculation steps?

1. Identify which variable you want to calculate (half-life, elapsed time, remaining quantity, or initial quantity).
2. Gather the known values for the other variables.
3. Select the appropriate formula based on your calculation goal.
4. Substitute the known values into the formula.
5. Solve the equation to find the unknown variable.

Example Calculation

Let's calculate the remaining quantity of a radioactive substance:

Given:

• Initial quantity (N₀) = 100 grams
• Half-life (T₁/₂) = 5 hours
• Elapsed time (t) = 15 hours

Step 1: Identify the formula

We're calculating the remaining quantity, so we'll use: N = N₀ · 2^(-t/T₁/₂)

Step 2: Substitute the values

N = 100 · 2^(-15/5)

Step 3: Solve the equation

N = 100 · 2^(-3) = 100 · 0.125 = 12.5 grams

Therefore, after 15 hours, 12.5 grams of the radioactive substance remain.

Diagram of Half-Life Decay

The following diagram illustrates the exponential decay process:

This diagram shows how the quantity of a substance decreases exponentially over time. The curve represents the remaining quantity, which halves after each half-life period.