The exponential decay law and half-life

Overview

Half-life of a radioactive nuclide, plotted to turn exponential decay data into a straight line and how throwing a large number of dice models radioactive decay.

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   Define the half-life of a radioactive nuclide.

   Model answer

   The half-life is the average time taken for the number of undecayed nuclei in a sample to fall to half its starting value. It is equally the time for the activity of the sample to halve.

2. 2

   Write down the exponential decay law for the number of undecayed nuclei, defining every symbol and its unit.

   Model answer

   The law is N = N₀e^(−λt). N is the number of undecayed nuclei left at time t in s, N₀ is the number present at t = 0 and λ is the decay constant in s⁻¹.

3. 3

   Write down the exponential decay law in terms of activity.

   Model answer

   The law is A = A₀e^(−λt). A is the activity in Bq at time t, A₀ is the activity at t = 0 and λ is the decay constant in s⁻¹.

4. 4

   Why does the activity of a sample obey the same exponential law as the number of undecayed nuclei?

   Model answer

   Activity and number of nuclei are tied together by A = λN with λ constant, so the two always stand in a fixed ratio. Whatever factor N falls by, A falls by the same factor.

5. 5

   Write the decay law in its halving form and state what the exponent stands for.

   Model answer

   The law can be written N = N₀/2ˣ and A = A₀/2ˣ. Here x is the number of half-lives that have elapsed, and it need not be a whole number.

6. 6

   Derive the relationship between the decay constant and the half-life.

   Model answer

   Putting t = T½ and N = N₀/2 into N = N₀e^(−λt) gives ½ = e^(−λT½). Taking natural logarithms of both sides gives −ln2 = −λT½, so λ = ln2/T½.

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The same content, four other ways.

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