Understanding Half-Life and Reading Decay Curves

CSEC Physics: The Exponential Nature of Radioactivity

Essential Understanding: Half-life is the time taken for half of the radioactive nuclei in a sample to decay. This concept allows us to predict how much radioactive material remains after any given time period. Decay curves are graphical representations of this process, showing how activity or mass decreases exponentially over time.

🔑 Key Concept: Half-Life = Time for 50% Decay

📈 Exam Focus: Reading Decay Graphs

🎯 Learning Goal: Graph Interpretation Skills

What is Half-Life?

The half-life (symbol: T½) of a radioactive substance is the time required for half of the radioactive atoms in a sample to decay. This is one of the most important concepts in nuclear physics because it allows us to:

Predict how long a radioactive material will remain dangerous
Determine the age of ancient artifacts (carbon dating)
Calculate appropriate storage times for nuclear waste
Understand the safety requirements for handling radioactive materials

⏱️ Formal Definition

Half-life is the time taken for the activity of a radioactive sample to decrease to half of its original value.

After each half-life, exactly 50% of the remaining radioactive atoms decay.

📊 Characteristic Property

Each radioactive isotope has its own unique half-life, which never changes.

Carbon-14: 5,730 years

Uranium-238: 4.5 billion years

Polonium-212: 0.3 microseconds

🔄 The Decay Pattern

After 1 half-life: 50% remaining

After 2 half-lives: 25% remaining

After 3 half-lives: 12.5% remaining

After n half-lives: (½)ⁿ of original remains

The Decay Curve

A decay curve is a graph that shows how the amount of radioactive material (or activity) decreases over time. The shape of this curve is always exponential, never linear.

Understanding the Decay Curve Shape

Watch the animation to see how the curve forms. Notice these key features:

Steep Start: In the first half-life, half the atoms decay. This creates the steepest part of the curve.

Gradual Slope: In the second half-life, only 25% remains to decay, so the curve is less steep.

Approaching Zero: The curve never actually reaches zero – it just gets closer and closer.

Reading a Decay Graph

Being able to extract information from a decay curve is an essential skill for CSEC Physics. Here is a step-by-step guide:

Identify the axes: The vertical axis (y-axis) shows activity or mass remaining, while the horizontal axis (x-axis) shows time
Find the starting point: Locate where the curve meets the y-axis – this is the initial activity (A₀) or initial mass (m₀)
Locate 50% on the y-axis: Draw a horizontal line from this point to the curve, then drop a vertical line to the time axis
Read the half-life: The time you read is the half-life of the substance
Find other half-lives: Repeat from 25%, 12.5% to verify consistency

$$A = A_0 \\left(\\frac{1}{2}\\right)^n \\quad \\text{where} \\quad n = \\frac{t}{T_{1/2}}$$

Where:

A = Activity (or mass) at time t
A₀ = Initial activity (or mass)
n = Number of half-lives elapsed
T½ = Half-life
t = Time elapsed

🎮 Interactive: Dynamic Decay Curve Simulator

Watch the decay curve form in real-time! Adjust the half-life to see how different isotopes behave.

Mass Remaining

Half-Life Markers

Activity Level

100

Mass Remaining (%)

Half-Lives Passed

Time (years)

Half-Life (years)

Half-Life: 10 years

Speed: 2x

Common Half-Lives

Different radioactive isotopes have vastly different half-lives. This is what makes them useful for different applications:

Isotope	Half-Life	Application
Carbon-14	5,730 years	Radioactive dating of organic materials
Uranium-235	704 million years	Nuclear power plants
Uranium-238	4.5 billion years	Age of Earth determination
Iodine-131	8 days	Medical thyroid treatment
Cobalt-60	5.27 years	Cancer radiation therapy
Polonium-212	0.3 microseconds	Static eliminators (very brief!)

            💡 Why Half-Lives Vary So Much: The half-life depends on how unstable the nucleus is. Some nuclei are barely unstable and take billions of years to decay. Others are so unstable that they decay almost instantly. There's no relationship between half-life and how "dangerous" the radiation is – a short half-life means more decays per second (more activity), but for a shorter time.
        

Worked Example: Reading a Decay Graph

Problem 1: Determining Half-Life from a Graph

The Graph Shows: A radioactive source with initial activity 800 Bq decays over time.

Questions:

a) What is the half-life of this source?

b) What will the activity be after 8 days?

Solution:

a) To find the half-life:

• Initial activity = 800 Bq

• 50% of 800 Bq = 400 Bq

• From the graph, activity reaches 400 Bq at t = 4 days

• Therefore, half-life T½ = 4 days

b) To find activity after 8 days:

• Number of half-lives n = 8 ÷ 4 = 2

• After 2 half-lives: (½)² = ¼ remains

• Activity = 800 × ¼ = 200 Bq

Problem 2: Finding Original Mass

Problem: After 3 half-lives, a radioactive sample has a mass of 5 g. What was the original mass?

Solution:

After 3 half-lives: (½)³ = ⅛ of original remains

Therefore: 5 g = ⅛ × original mass

Original mass = 5 g × 8 = 40 g

Summary: Key Takeaways

Half-life (T½) is the time taken for half of the radioactive nuclei in a sample to decay
The decay curve is always exponential – it never reaches zero but gets infinitely closer
Each half-life period reduces the remaining amount by exactly half, regardless of the half-life duration
To read a decay graph: Find 50% on the y-axis, trace to the curve, then to the x-axis to read the half-life
The formula A = A₀(½)ⁿ allows calculation of remaining activity after any number of half-lives
Half-life is a characteristic property that identifies each radioactive isotope and never changes