A-Level · Physics · AQA · Mark scheme decoded
AQA A-Level Physics: Radioactive Decay and Half-Life — mark scheme explained
The short answer
Radioactive decay is a fundamental process in nuclear physics, characterized by the random nature of the decay of atomic nuclei. This section delves into the mathematical models used to describe radioactive decay, including the decay equation, activity, and half-life.
The question
A sample of a radioactive isotope has an initial activity of 100 Bq. If the half-life of the isotope is 5 years, what will be the activity after 15 years? [Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
What each mark is really for — in plain English — and the wording trap that loses it.
- S1
Identify the given values: A 0 = 100 Bq, T 1/2 = 5 years, and t = 15 years.
- S2
Calculate the decay constant using the half-life equation: λ = ln(2) / T 1/2 = ln(2) / 5 ≈ 0.1386 year -1 .
- S3
Use the activity equation to find the activity after 15 years: A(t) = A 0 e -λt = 100 × e -0.1386 × 15 ≈ 12.5 Bq.
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
Identify the given values: A 0 = 100 Bq, T 1/2 = 5 years, and t = 15 years.
- S2
Calculate the decay constant using the half-life equation: λ = ln(2) / T 1/2 = ln(2) / 5 ≈ 0.1386 year -1 .
- S3
Use the activity equation to find the activity after 15 years: A(t) = A 0 e -λt = 100 × e -0.1386 × 15 ≈ 12.5 Bq.
Final answer: 12.5 Bq
Common mistakes
- Confusing the decay constant (λ) with the half-life (T 1/2 ). — Remember that the decay constant is a measure of the probability of decay per unit time, while the half-life is the time it takes for half of the initial nuclei to decay. Use the relationship T 1/2 = ln(2) / λ to convert between them.
- Using the wrong base in logarithmic calculations. — Always use the natural logarithm (ln) when dealing with exponential decay equations. The relationship T 1/2 = ln(2) / λ uses the natural logarithm.
- Forgetting to convert units in time calculations. — Always check that the time units (e.g., seconds, years) are consistent with the decay constant. Convert units if necessary before performing calculations.
- Misinterpreting the activity equation as a linear relationship. — Remember that activity decreases exponentially with time, following the equation A(t) = A 0 e -λt . This means that the rate of decay slows down over time.
- Using the wrong initial value in decay equations. — Always use the initial number of undecayed nuclei ( N 0 ) or the initial activity ( A 0 ) in your calculations. The decay equation N(t) = N 0 e -λt and the activity equation A(t) = A 0 e -λt both require the initial value.
- Incorrectly plotting logarithmic graphs. — When using logarithmic graphs to linearize data, always plot ln(N(t)) against t . This will give you a straight line with a slope of -λ .
Where the marks go
- Full worked solution (all marking points)4 marks