Write the corresponding exponential decay equation for the number of undecayed nuclei.

N = N₀e^(−λt), where N is the number of undecayed nuclei remaining at time t and N₀ the number present at t = 0.

Why do the activity and the number of undecayed nuclei follow the same exponential law?

The activity is directly proportional to the number of undecayed nuclei through A = λN, so both quantities fall by the same fraction in any given time.

A source of initial activity 5.0 × 10⁴ Bq has a half-life of 12 hours. Calculate its activity after 30 hours.

λ = ln2/t½ = 0.693/12 = 0.0578 h⁻¹, so λt = 0.0578 × 30 = 1.733 and A = 5.0 × 10⁴ × e^(−1.733) = 5.0 × 10⁴ × 0.177 = 8.8 × 10³ Bq.

CCEA A-Level Physics 1210

Physical Quantities, Scalars and Vectors

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The Nucleus and Radioactive Decay

Exponential decay calculations with A = A₀e^(−λt)

3 things to know
Unit A2 1

Overview

Recall questions on the decay equations for activity and for the number of undecayed nuclei, why both follow the same exponential, worked calculations of activity after a stated time and of the time to reach a stated activity, using half-lives as a shortcut, and taking logarithms to straighten a decay curve.

What you need to know

Answer each one before you reveal it.

Your recallNot started

Key term

State the exponential decay equation for activity and define every symbol.

Numbers

A source of initial activity 800 Bq has a half-life of 6.0 hours. Calculate its activity after 18 hours.

Key term

Why must the decay constant and the time be expressed in matching units in the exponential decay equation?

Questions from this subtopic

Each one with the model answer underneath.

1
State the exponential decay equation for activity and define every symbol.
Model answer
A = A₀e^(−λt), where A is the activity at time t in Bq, A₀ the initial activity in Bq, λ the decay constant in s⁻¹ and t the time in seconds.
2
Write the corresponding exponential decay equation for the number of undecayed nuclei.
Model answer
N = N₀e^(−λt), where N is the number of undecayed nuclei remaining at time t and N₀ the number present at t = 0.
3
Why do the activity and the number of undecayed nuclei follow the same exponential law?
Model answer
The activity is directly proportional to the number of undecayed nuclei through A = λN, so both quantities fall by the same fraction in any given time.
4
A source of initial activity 800 Bq has a half-life of 6.0 hours. Calculate its activity after 18 hours.
Model answer
Eighteen hours is three half-lives, so the activity is 800/2³ = 800/8 = 100 Bq.
5
A source of initial activity 5.0 × 10⁴ Bq has a half-life of 12 hours. Calculate its activity after 30 hours.
Model answer
λ = ln2/t½ = 0.693/12 = 0.0578 h⁻¹, so λt = 0.0578 × 30 = 1.733 and A = 5.0 × 10⁴ × e^(−1.733) = 5.0 × 10⁴ × 0.177 = 8.8 × 10³ Bq.
6
Why must the decay constant and the time be expressed in matching units in the exponential decay equation?
Model answer
The product λt must be a pure number for the exponential to be defined, so a decay constant in s⁻¹ requires a time in seconds and one in h⁻¹ requires a time in hours.

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