Singapore H2 Physics Wiki

Exponential Decay and Graphs

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Exponential Decay and Graphs

Overview

Exponential Decay and Graphs explains how radioactive quantities decrease with time and how to interpret common graph questions.

This page deepens ideas from:

Radioactive samples show predictable exponential behaviour when many nuclei are present.

Definition

Exponential decay means the quantity decreases by the same fraction over equal time intervals, rather than by the same absolute amount.

Why It Matters

Students need this idea to:

  • read half-life from decay graphs
  • distinguish true source count from measured count including background
  • decide when repeated-halving reasoning is enough and when a graph or formula is needed

Key Representations

Core Decay Equations

Number of Undecayed Nuclei

Activity

Count Rate

where:

  • , , are initial values
  • is the decay constant
  • is time

Here, refers to the source or background-corrected count rate when background radiation is present.

Shape of Exponential Decay Curves

All three graphs have the same general shape:

  • decrease rapidly at first
  • rate of decrease slows later
  • curve approaches zero
  • curve does not reach zero exactly

Graph of N Against t

Axes:

  • vertical: number of undecayed nuclei
  • horizontal: time

Features:

  • starts at
  • falls exponentially
  • after each half-life, the value halves

Example:

TimeValue

Graph of A Against t

Since:

activity is directly proportional to .

Therefore:

  • it has the same exponential shape as the graph
  • it has the same half-life
  • it starts at

Graph of Count Rate Against t

If the detector arrangement stays unchanged, the corrected source count rate follows:

So count rate also decays exponentially.

It has the same half-life as activity.

Corrected Count Rate vs Background

Measured count rate often includes background radiation.

Hence:

Important graph effect:

  • measured graph may flatten toward the background level, not zero
  • corrected graph decays toward zero

Reading Half-Life from Graphs

Method 1: Vertical Halving

Find the time taken for the quantity to reduce from:

  • 100 to 50
  • 80 to 40
  • 30 to 15

That time interval is the half-life.

Method 2: Repeat at Different Heights

Check consistency using more than one halving interval.

Repeated-Halving Reasoning

If the time equals several half-lives, use:

Number of Half-LivesRemaining Fraction
1
2
3
4

General form:

where is the number of half-lives.

Decay Constant Interpretation

From:

Meaning of :

  • larger gives a steeper graph
  • larger gives faster decay
  • larger gives shorter half-life

Using:

Logarithmic Linearisation

Taking natural logarithms:

So plotting:

  • against

gives a straight line with:

  • gradient
  • intercept

This is useful if the syllabus expects simple data analysis.

Worked Graph Examples

Example 1: Read Half-Life

Activity falls from:

  • 160 Bq at
  • 80 Bq at h

Therefore:

Example 2: Three Half-Lives

Initial count rate = 400 counts min

Half-life = 2 h

After 6 h:

  • 3 half-lives

Answer:

Example 3: Corrected Count Rate

Measured count = 72

Background = 12

Corrected source count:

Use 60 for graph analysis.

Summary

  • , , and corrected all decay exponentially
  • each halves after every half-life
  • larger means faster decay
  • subtract background count before analysis
  • exponential graphs are steep initially and then flatten

Graph View

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