AQA A-Level Physics Paper 2, June 2023: Question 30

📋 What this question tests

This question assesses your ability to apply the law of radioactive decay quantitatively in a multiple-choice context:

Using the exponential decay equation A = A₀e^(-λt) to determine the decay constant ( λ ).
Applying exponential decay ratios or proportional indices across non-integer half-life periods.
Eliminating distractor answers derived from false linear decay approximations.

Question 30 • 1 Mark

C — 16 MBq

Mark Scheme: 1 mark awarded for selecting C.

Exponential Law: Radioactive decay follows A = A₀e^(-λt) , where A₀ is the initial activity and λ is the decay constant.
Consistent Units: If time t is given in years (y), λ can remain in year⁻¹ (y⁻¹). There is no need to convert into seconds unless calculating decay in Becquerels from total number of nuclei ( A = λN ).
Fractional Powers: Since 15 years is 1.5 × 10 years, the remaining fraction after 15 years is (A₁₀ / A₀)¹·⁵ .

Method 1: Standard Two-Step Exponential Route

Identify given values:
Initial activity: A₀ = 37 MBq at t = 0
Activity at 10 years: A₁₀ = 21 MBq at t = 10 y
Target time: t = 15 y
Calculate the decay constant ( λ ):
21 = 37 × e^(-λ × 10)
e^(-10λ) = 21 / 37 = 0.5676
-10λ = ln(0.5676) = -0.5664
λ = 0.05664 y⁻¹
Calculate activity at t = 15 y :
A₁₅ = 37 × e^(-0.05664 × 15)
A₁₅ = 37 × e^(-0.8496)
A₁₅ = 37 × 0.4276 = 15.82 MBq ≈ 16 MBq

Method 2: Ratio / Power Shortcut (Faster for Multiple Choice)

Notice the ratio of time elapsed: t₂ / t₁ = 15 / 10 = 1.5

Trap 1 (Linear Decay Assumption):
In 10 years, activity decreases by 37 - 21 = 16 MBq (1.6 MBq/year).
In 5 more years, assuming linear decay: 21 - (1.6 × 5) = 13 MBq .
This leads directly to incorrect distractor B!
Trap 2 (Adding rather than multiplying ratios):
Halving the remaining 21 MBq or misinterpreting the half-life. Note that 10 years is slightly less than one half-life (since 21 > 18.5 MBq). At 15 years, it must be slightly less than half of 37 MBq (18.5 MBq), ruling out 12 MBq (A) and 13 MBq (B) as dropping far too fast.
Rounding λ prematurely:
Rounding λ to 1 significant figure (e.g. 0.06 y⁻¹) gives 37 × e^(-0.9) = 15.0 MBq , causing hesitation between choices. Always keep full calculator precision in your memory registers.

Quick sanity check via half-life:
- Half of 37 is 18.5 MBq.
- At 10 years, activity is 21 MBq, so the half-life T½ is slightly greater than 10 years (approx. 12.2 years).
- At 15 years, only slightly more than one half-life has passed, so the activity must be just below 18.5 MBq.
- 12 MBq and 13 MBq are way too low. 17 MBq is too close to 18.5 MBq. 16 MBq is the only physically plausible value!
Units shortcut: Because both initial activity and final answers are in MBq, you do not need to convert to Bq or convert years into seconds. Keep the native units to save valuable exam time.

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