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# Half life exponential equations

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## 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.

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:**

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

**where:**

**A _{0}=** the initial activity

**A =** the activity at time, t

**N _{0} = ** 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:-

- how many you started with N
_{0}or what activity you started with, A_{0} - how quickly they decay λ
- 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 t_{½} in the question and vice versa. Not to worry - just use:-

**2.** If you need to use:

to find t or N = N_{0}e^{-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:**

**Then it's easy.**

**3.** Sometimes the question will require you to use

but it won't give you N or N_{o}. Instead it will say something like 1/3 of the atoms are left undecayed.

In this case, reorganise the equation to give:-

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

Think it through with some actual number.

So the equation can be written as:

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 10 ^{23} atoms. It emits alpha particles and has a half-life of 1620 years. How many atoms are left after 100 years?**

**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?**