If the activity has fallen by a fraction that is not an exact power of one half, use the exponential decay equation to find the decay constant. Divide ln 2 by the decay constant to obtain the half-life.
The 2027 H2 Physics 9478 syllabus includes modelling radioactive decay with an exponential relationship. This worked example uses two activity readings and checks the result against the time needed to halve.
- Use readings for the same sample and isotope.
- Divide the later activity by the earlier activity.
- Use the natural logarithm to find the decay constant.
- Check that the resulting half-life is physically plausible.
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Set up the ratio
For a single isotope with constant decay constant λ, the activity after an elapsed time t is A = A₀ exp(−λt). Dividing by A₀ and taking natural logarithms gives ln(A/A₀) = −λt. An equivalent calculation is λ = ln(A₀/A)/t.
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Worked example | Activity drops from 800 Bq to 300 Bq
The readings are 12 hours apart. The fraction remaining is 300/800 = 0.375. The decay constant is ln(800/300) ÷ 12 ≈ 0.0817 h⁻¹. The half-life is ln 2 ÷ 0.0817 ≈ 8.48 hours. Use the unrounded decay constant in your calculator for the final step.
The sample falls below 400 Bq after 12 hours, so the half-life must be shorter than 12 hours. It stays above 200 Bq, so 12 hours is less than two half-lives. The calculated value passes both checks.
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Distinguish activity from detector readings
If the data are detector count rates instead of activities, subtract the background count rate from each reading before taking the ratio. Use comparable readings taken with the same detector arrangement, so the corrected count rate can be treated as proportional to activity.
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Frequently asked questions
Can the decay constant be negative?
For radioactive decay λ is positive. A negative result often means the ratio or the minus sign was reversed.
Do I have to use seconds?
Use a consistent unit. With elapsed time in hours, the decay constant is in h⁻¹ and the calculated half-life is in hours.