AQA A-Level Physics Paper 2, June 2023: Question 32

1 mark · Medium difficulty · Multiple Choice

Determine the initial number of nuclei in a sample from an activity against time decay curve.

Practise this question

Question

Question 32 shows a graph of activity in mega-becquerels (MBq) on the vertical axis against time in seconds on the horizontal axis. The curve starts at an initial activity of 14.0 MBq at time 0 seconds, falls to 7.0 MBq at 30 seconds, and continues downward to around 5.0 MBq at 45 seconds. Below the graph, candidates are asked to find the initial number of nuclei of X in the sample, choosing from four options: A (4.67 × 10^5), B (3.0 × 10^8), C (4.2 × 10^8), and D (6.1 × 10^8).
Question text

32 The graph shows the variation of activity with time for a sample of a nuclide X.

What was the initial number of nuclei of X in the sample?

[1 mark]

A 4.67 × 105

B 3.0 × 108

C 4.2 × 108

D 6.1 × 108

Mark scheme

Show the mark scheme Mark scheme table row for question 32 showing the correct answer as option D, with the corresponding value 6.1 × 10^8.

32 D 6.1 × 108

How to answer it

Radioactivity: Activity Graphs & Number of Nuclei

📋 What this question tests

This multiple-choice question assesses your ability to determine nuclear decay characteristics from an activity–time graph:

  • Reading initial activity ( A₀ ) and determining half-life ( T½ ) directly from experimental decay curves.
  • Applying the relationship between decay constant and half-life: λ = ln 2 / T½ .
  • Using the fundamental activity equation A = λN to calculate the number of undecayed parent nuclei.
  • Correctly handling metric unit prefixes (converting MBq to Bq ).

Question 32 Analysis

Activity vs. Time for Nuclide X [1 Mark]

✅ Correct Answer

D: 6.1 × 10⁸

Award [1 mark] for selecting option D.

💡 Key Knowledge

  • Activity definition: A = λN , where A is activity (in Bq), λ is the decay constant (in s⁻¹), and N is the number of active nuclei.
  • Half-life relationship: T½ = ln 2 / λ ≈ 0.693 / λ .
  • Direct equation: Combining the two gives N₀ = (A₀ × T½) / ln 2 .
  • SI Prefixes: 1 MBq = 1 × 10⁶ Bq .

📐 Step-by-Step Calculation

  1. Read the initial activity (A₀) from the graph:
    At t = 0 s , the curve starts at A₀ = 14.0 MBq = 14.0 × 10⁶ Bq .
  2. Find the half-life (T½):
    Half of the initial activity is 14.0 / 2 = 7.0 MBq .
    Trace horizontally from 7.0 MBq on the vertical axis to the curve, then read down to the horizontal time axis.
    The curve crosses 7.0 MBq exactly at t = 30 s .
    Therefore, T½ = 30 s .
  3. Calculate the decay constant (λ):
    λ = ln 2 / T½ = ln(2) / 30 ≈ 0.023105 s⁻¹
  4. Calculate the initial number of nuclei (N₀):
    Using A₀ = λN₀ :
    N₀ = A₀ / λ = (14.0 × 10⁶) / 0.023105 = 6.06 × 10⁸
    To 2 significant figures, N₀ = 6.1 × 10⁸ .

❌ Common Errors & Trap Options

  • Forgetting the prefix 'M': Calculating 14.0 / 0.0231 ≈ 606 (ignoring the 10⁶ multiplier).
  • Direct division trap (Option A): Evaluating 14.0 × 10⁶ / 30 = 4.67 × 10⁵ (dividing by T½ instead of λ ).
  • Using the wrong half-life point: Reading the value at the end of the graph ( t = 45 s where activity is 5.0 MBq ) instead of reading the time taken to drop to half ( 7.0 MBq ).

🧠 Exam Technique & Shortcut

For fast execution under time pressure in Section A:

  • Memorise the rearranged shortcut: N = (A × T½) / 0.693 .
  • Plug values in directly on your calculator:
    (14.0 × 10⁶ × 30) / ln 2 = 6.059... × 10⁸ .
  • This saves 30 seconds compared to finding λ separately and avoids rounding intermediate values!

Topics

Physics · Practical skills · 3.8 Nuclear physics (A-level only) · Data analysis

Question and mark scheme from the AQA A-Level Physics examination, Paper 2, June 2023. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.