Lesson 16: Half-life part 3
Last lesson we looked at how to deal with whole numbers of half-lives. This lesson we’ll look at how to deal with fractions of a half-life.
The mathematics assumes there are a very large number of nuclei
Nuclear decay is a completely random process. It’s impossible to know when a particular nucleus will decay. But we can use mathematics to describe and predict nuclear decay as long as we have lots of nuclei.
The number of nuclei that should decay in 1 second is the chance of one nucleus decaying multiplied by the number of nuclei.
So if the chance of each nucleus decaying each second is 25% and there are are 100 nuclei then the number of nuclei that should decay in 1 second is 25% (i.e. 0.25) multiplied by 100, which equals 25 nuclei.
You'd find that on average 25 decay but most of the time it’s not exact. This is because 100 nuclei isn’t really enough for the random fluctuations to be smoothed out. The more nuclei you have the closer you’ll get to 25%.
As the nuclei decay into more stable ones, there are fewer undecayed nuclei left (though the same total). So as time passes our mathematics will get less good at modelling what actually happens.
Nuclear decay is discrete but the mathematics is continuous
Nuclear decay is ‘discrete’. In other words you either have four nuclei or five, never 4.7. Over one second none might decay.
But the mathematics is ‘continuous’. It assumes that 4.7 nuclei is a perfectly good answer. Over 1 second it might predict 0.1 decays.
Again, this doesn’t matter if we have a large number of nuclei. We normally only need answers to 3 or 4 significant figures so we never need to worry about fractions of nuclei.
We'll sometimes use small numbers for convenience
So the mathematics in this lesson assumes that we have a very large number of undecayed nuclei.
In reality we normally do. A single gram of a radioactive isotope typically consists of around a trillion trillion nuclei. We’ll often use small numbers as examples because it makes our arguments simpler to understand. But it’s important to realise that you need large numbers of nuclei for the mathematics to be a good fit to reality.
The rate of decay equation: dN/dt = -λN
The first equation we’re going to look at is the ‘rate of decay’.
- dN is a very small change in the number of undecayed nuclei
- dt is a very small change in time
- so dN/dt is the rate of change of undecayed nuclei
- the minus sign shows us that the number of undecayed nuclei decreases with time
- λ (lambda) is called the decay constant - it's the chance any nucleus has of decaying each second
- N is the number of undecayed nuclei there are left at time t
This equation says:
How fast you lose undecayed nuclei (dN/dt) at any time, t, depends on the chance of any given nucleus decaying (λ) and how many undecayed nuclei there are left at that instant (N).
The fewer you have the slower you lose them.
Remember the nuclei never disappear. They just change into more stable ones.
We can choose to start timing i.e. set t=0 whenever we like
The number of undecayed nuclei we have when we start timing is called N0. (It'll appear later on). It's just a number and doesn't change with time. So if we had 196 undecayed nuclei when we started timing, N0 would equal 196 and that would never change.
In our simulation we always start with a completely ‘clean’ sample with no decayed nuclei. In practice this is never the case, but it doesn’t matter how long the sample has been decaying for or what other nuclei are around. All you're worried about is the number of undecayed nuclei there are at the start because they're the only ones that are going to do any decaying.
A doctor might choose to ‘start timing’ when a patient was injected with a tracer, rather than when the tracer was synthesised.
The number of undecayed nuclei, N, decreases with time
‘N’ is the number of undecayed nuclei at any time, t, after we’ve started timing. The number of undecayed nuclei decreases with time. So N decreases with time. For example when t = 0 seconds, N = N0, which equals 196 undecayed nuclei.
When t = 20 seconds, N = 160 undecayed nuclei. We’ll see how to do this calculation later.
The smaller the chance of decay, λ, the slower you lose undecayed nuclei
Now ‘how fast you lose them’ (dN/dt) depends on how likely each nucleus is to decay each second, λ.
The decay constant, λ, doesn't change with time. It's specific to each isotope and depends in a very complex way on the exact makeup of the nucleus.
It’s normally expressed as a decimal fraction rather than a percentage. If each nucleus has a 1% chance of decay each second then λ equals 0.01 per second.
Two main ideas with the rate of decay equation: dN/dt = -λN
So we know that our ‘rate of decay’ equation embodies two ideas.
- ‘The fewer undecayed nuclei you have, the slower you lose them’
- ‘The smaller the chance of decay, the slower you lose them’.
For every nucleus that decays a particle is emitted that you can detect
Now there's no way to see nuclei as they decay so our rate of decay equation doesn't seem to have that much practical use.
But the key idea is that every time a nucleus decays a particle is given off. If 10 nuclei decay in 1 second then 10 beta particles (say) must have been emitted. This is what we mean by the count rate or the radioactivity, often shortened simply to activity, A.
So this equation can be rewritten as
-dN/dt = A = λN
We've shuffled the minus sign around because the count rate has to be positive.
Let's rewrite this equation again: A = -λN
So if we have 40 000 undecayed nuclei of an alpha emitter (N = 40 000) and the chance of each nucleus decaying each second is 0.002 (λ = 0.002 s-1), then the activity, i.e the number of alpha counts per second is simply 40 000 x 0.002 = 80 Bq.
Geiger counters don't catch every decay
But if you used a Geiger counter to confirm your answer it wouldn’t read 80 Bq. Most alpha emissions, say, happen inside the source so never make to the outside. And the GM tube only captures a fraction of those that do.
You also assume that the daughter nuclei are stable, otherwise they too might decay to produce more alphas or betas.
Using the Avagadro number to find N
Sometimes you won't be given N directly and you'll have to calculate it using the Avagadro number and the molar mass.
Using a graph of the number of undecayed nuclei, N, against time, t
Now let’s look at our graph of undecayed nuclei against time. The gradient of a curve is the change in the y value divided by the change in the x value.
So the gradient of the curve is dN/dt and we know that this is equal to the activity. The gradient decreases with time so the activity decreases with time.
So if we know the number of undecayed nuclei we can use the ‘rate of decay’ equation to calculate the activity.
But how can we calculate the number of undecayed nuclei, N, at any time, t?
The ‘rate of decay’ equation, dN/dt = -λN tells us the value of the slope of this curve at any point. But what we want to find is the EQUATION of the curve. In other words N as some function of N0, lambda and time.
The decay equation N = N0e-λt
It turns out that the solution to the ‘rate of decay’ equation is this
N = N0e-λt
It’s called the ‘decay equation’ and is the equation of the curve.
e is a natural number like pi. It has a value of 2.7182… and, like pi, goes on forever.
Using the decay equation to find the number of nuclei remaining
You can use the decay equation N = N0e-λt to find the value of N for any value of t if you're given λ and the number of undecayed nuclei you start off with, N0.
Using the decay equation to find the fraction of nuclei remaining
Some questions ask what fraction (or percentage) of undecayed nuclei remain after some period of time. The answer they're looking for is something like 0.3 or 65% of what you started with.
The key idea is that what you started with was N0 so if you can find e-λt then you'll end up with an answer like 0.7N0 i.e. 0.7 of what you started with remains.
Using natural logarithms to answer questions about t, i.e. how long?
The decay equation N = N0e-λt has t in it. This is fine if you're given t. In other words if the question asks something like 'How much is left after 10 hours?'.
But how do you solve for t? In other words how do you answer questions like 'How long does it take before only 20% of the original undecayed nuclei remain?'
Somehow you have to get t down as a 'normal' number rather than up as a power.
The way we do this is to take natural logarithms.
Taking natural logarithms of the decay equation we end up with
ln N = ln N0 -λt
which means λt can just be treated as normal numbers.
The activity equation A = A0e-λt
Now the activity and the number of undecayed nuclei change in the same way. For every nucleus that decays a beta particle, say, is emitted.
So we can write an equation that has exactly the same form as the decay equation, N = N0e-λt:
A = A0e-λt
This equation says that the activity at any time, t, depends on the activity when t=0 and the chance of a nucleus decaying.
Using the activity equation to find the fraction of the initial activity after a certain time
Some questions ask you to find how long it takes for the activity to drop to a particular fraction, for example 5% of its original value.
Again it doesn’t matter what the starting activity, A0, actually is. After a certain time the fraction of the activity will be the same.
We can see this mathematically
0.05A0 = A0e-λt
It should be clear that the A0s cancel so the answer is independent of A0.
Relating decay constant, λ, to half-life, t1/2
If the nuclei are likely to decay then the half-life will be short. In other words if λ is big, the half-life will be small.
There is a simple relationship between λ and half-life which can be found by the same technique as we’ve been using.
t1/2 = ln2/λ
This result can normally be quoted without proof.
Calculating the half-life from the decay constant
So in some questions you'll need to use λ but you'll only be given the half-life.
In this case you simply rearrange the equation to give
λ = ln2/t1/2
A summary of the useful equations and relationships
There are probably six equations that you need to be familiar with. Many exams will give these to you so you may not have to remember them but you should be clear where and how to use each one.
Proving an exponential relationship
One final thing you might be asked to do is to prove that a set of data shows an exponential decay.
Exponential decays all have this feature that ‘how fast you lose’ something is directly proportional to ‘how much there is left’. This implies the ‘constant ratio’ property, of which the existence of a half-life is an example.
So one thing you can do is to show that it always takes the same time to halve the activity (or number of undecayed nuclei).
But what if the value hasn’t halved within the data you have?
In this case the best thing to do is to take natural logs of the activity equation and compare the result with the equation of a straight line.
If you plot the natural log of the activity against time, you’ll end up with a straight line if the relationship is exponential.