Half life exponential equations

Half life exponential equations

If we plot a graph of the number of radioactive nuclei in a sample (N) against time (t) we end up an exponential decay as shown below.

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We can go further than this. It is possible to write an equation which describes exactly how many atoms are left (and therefore what the activity is) as time passes:

It is:

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as the number of atoms is proportional to the activity we can also write

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where:

A0= the initial activity

A = the activity at time, t

N0 = the initial number of atoms in a sample

N = the number left at time t.

So these equations says that the number of atoms left, N depends on:-

  1. how many you started with N0 or what activity you started with, A0
  2. how quickly they decay λ
  3. how long you left them for, t

All pretty obvious, really! And that's it. Now all you have to do is be able to use them. So here are some tips.

There are a couple of problems/tricks with using these equations.

1. You can guarantee that if you need to use λ you'll be given t½ in the question and vice versa. Not to worry - just use:-

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2. If you need to use:

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to find t or N = N0e-λt, you have to use natural logarithms ('ln' on the calculator, not 'log')

Learn that if you take natural logarithm of both sides you get:

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Then it's easy.

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3. Sometimes the question will require you to use

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but it won't give you N or No. Instead it will say something like 1/3 of the atoms are left undecayed.

In this case, reorganise the equation to give:-

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If you've got 1/3 left that's the same as:

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Think it through with some actual number.

So the equation can be written as:

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and you can get an answer even though you don't know the number of atoms involved.

Here are a couple of worked examples to help:

Example 1

A sample of radium contains 6.64 x 1023 atoms. It emits alpha particles and has a half-life of 1620 years. How many atoms are left after 100 years?

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Example 2

A sample of wood from an old boat is found to contain 25% the number of carbon 14 nuclides as an equivalent piece from a modern sample. If the half-life of carbon 14 is taken to be 5730 years how old is the old wood?

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