Why is decay measured as a half life?

Factual Questions
1

Not having gone beyond high school chemistry does not often cause me much if any angst. My ability to use Google means that most of the various mundane questions I have are relatively easily and quickly answered.

There is one question to which I would like to have a more insightful answer than Google seems able to provide. This seems to be the ideal forum to get an answer even I can understand.

Question - why is the measurement of the decay of radioactive (or other) materials expressed as a half life. Why not simply indicate that the material will degrade to X value in Y amount of time? What is the significance of knowing when it will degrade to 50% of its present value?

2

but that is what the half life does! It says that in 1 year or whatever it will degrade to 50% of its value. One could introduce other sorts of life (e.g. the third life) but the half life is easily grasped.

" Why not simply indicate that the material will degrade to X value in Y amount of time? What is the significance of knowing when it will degrade to 50% of its present value?"

One could say that and occassionally does. Eg. one could say that 1 kg of plutonium will decay away to a safe level (say 2 micrograms of plutonium) in 10 billion years. However the half life is independent of the amount of material, whereas in the above example, 2 kg of plutonium would take twice as long to decay to the same level.

The half life is a useful property of the material (like say density) that is independent of the amount you are working with

3

It has to be a statistical measure because it is statistically possible for a few of the original atoms to remain undecayed for billions of years, even if half of them decay in a single half-life.

4

Decay (radioactive decay is the most obvious example, but this is true in relation to biodegradation and other forms of decay as well) follows an exponential function. Half-life is a natural expression of the function (corresponding to 69.3% of the mean lifetime.) Here’s the Wiki on half-life, which gives you the (simple) equations and explaination.

A function of “x remaining for time t” expressed as a linear or polynomial function would be an inappropriate fit, whereas ln(2)/lambda is a natural fit.

Stranger

5

Bolding mine

This confuses me.

Using your example - is the half life of 1 kg of plutonium 5 billion years?

If 2 kg takes twice as long to decay, is the half life not 10 billion years?

If this is correct, is your bolded statement true?

6

This makes sense.

Basically, a half life of measurement refers to a time when 50% of the decay has occured. This is not indicative of the same amount of time it will take for the other 50% of degradation to occur.

Is there a standard of half life measurements? Say plutonium has a half life of 5 billion years. This is obviously at the beginning of the degradation process. After 2.5 billion years, would it then have a half life of 2 billion years?

7

If your element has a half life of 1 year, then:
after 1 year, half of it will have decayed (so there will be one half left)
after a further year, half of what remains[/ will have decayed (so there will be a quarter left)
after a further year, half of what remains[/ will have decayed (so there will be one eighth left)
repeat ad inifinitum.

8

The half life tells you how long it takes for however much stuff you have to decay by half. If you start with 1kg of plutonium-239, after 24,000 years (the actual half life) you will have 500g. 24,000 years later you will have 250g. Another 24,000 years later you will have 125g, and so on.

9

Damn; I’m sure I clicked preview.

The point is that the ‘start’ is arbitrary; for any given amount of an isotope, at any given time, it can be said that after the passage of the half-life duration, half will remain; after the passage of two half-lives, one quarter will remain and so on.

10

Thanks very much for the most helpful explanations.

Ignorance takes a beating today.

11

It depends how you measure level, relative or absolute

In absolute terms 2 kg of plutomiun will take twice as long to decay down to a specified amount (say 1 g) as 1 kg would

However the half lifes of each are the same (each will lose half their mass every x years)

12

Half live is an intrinsic value of a substance; it does not change from one sample to the next. Because of this, it can be used to assist in identifying a particular radioactive isotope.

When I was working in Naval nuclear power, there were air particulate detectors in the engineroom that constantly sucked air through a continuous ribbon of filter paper which then was drawn through a detector. If the detector reached a certain threshold, an alarm sounded, all ventilation was shut off, and everybody put on gas masks. The temperature quickly reached about 120F amid the high temperature turbines and piping.

One potential cause of this alarm was naturally occurring radon gas, while the other was a horrible nuclear accident – we had to determine which it was very quickly.

To do this, we took a quick sample from the engineroom air by running a special vaccuum cleaner for a few minutes, sucking air through a disc of filter paper. We then took careful measurements of the counts coming off of the filter paper over time. If we were able to prove that the stuff on the filter paper showed a half life of 32 minutes, we could safely state that natural radon was the cause and turn on the ventilators again.

The details of which isotope we were measuring has been lost in the mists of time, but this is a definite example of application of half life to a real-life situation.

13

Besides the wikipedia article, you could try going to a half-life calculator:
www.1728.com/halflife.htm

14

I’ve got a stupid question (hey, I feel it’s a stupid question; I should know this, right?) about the fundamental reasons for a half life.

Why?? I don’t get it. I have 500g of material with a half life of 5 years. In 5 years, I’ve lost 250g of material. In 10 years, 750 grams (i.e., and additional 250 grams). Why is it not linear? Is this all probability based? Are we saying that there’s a 50% probability that in 5 years time, any given atom will have decayed? Because we’re looking at billions and billions of atoms, then, the probability of 50% is therefore a very good metric?

But then, it’s the same atom that’s not decaying every five years. That is, after 50 years of my material, I’ve got 4.441x10^13 grams of material left (how many grams does my atom – atomic weight 50, say – weigh?) But in those 50 years, the last existing atom really beat the odds, that is, for every flip of the 50-50 coin, it came up heads and didn’t decay.

15

just wanted to point out that scm1001’s post was incorrect.

This isn’t how it works.
1kg will take roughly 10 half lives (log 1000 to base 2 = 9.966) to decay to 1g.
2kg will take exactly 1 half life to decay to 1 kg.
So, 2kg will take roughly 11 half lives to decay to 1g.

16

Yes, it’s a probability thing, with the same underlying rule that governs coin tossing: what has happened in the past has no effect on what happens next. If I toss a fair coin 1000 times, and get heads each time, the chances of my getting a head on the next toss is still 50%.

17

Yes, that’s exactly what it is saying. The probability that any individual atom will decay is 50% over the half-life of the material. Over a large sample of atoms, the sum of all probabilities results in 50% (or immeasurably close to it) of the atoms decaying. An individual atom, however, may or may not decay over any given period of time. One can calculate the likelyhood of decay by comparing the period to the half-life, and as a practical matter the atom will have almost certainly decayed in five or six half-life timespans, but there is always a nonzero chance that it will have not decayed. (If you get into the quantum mechanical implications, the particle is both decayed and not decayed, until you look at it, at which time the probability waveform collapses and it is in either one state or the other, which is really freaky, but beyond the scope of this discussion.)

Stranger

18

No, no. You always, on average, lose half of the number of radioactive atoms of a particular element you started with. Lets say you started with 500 atoms of unstable element A with a half life of 5 years. In 5 years you would have 250 atoms of element A and 250 atoms of whatever element A turned into after the radioactive decay. The in another 5 years you would have half of 250 or 125 atoms left of element A. In another 5 years you would have left 62 0r 63 atoms of element A and so on. By the time you get down to just a few atoms the statistics aren’t so exact and when you get to a single atom of A there is a 50-50 chance that this will have decayed in 5 years. If it hasn’t then there is a 50-50 chance that it will have decayed in the next 5 years and so on. By the time you have gone through several half-lives there is only a small chance that the atom will still be element A. Chances are it will be some other element.

19

Others are already explaining this pretty well, so far be it from me to add anything.

Far be it, perhaps, but it be not far enough to save you people. So here I go.

Radioactive decay is a physical example of exponential decay, which is a lot like exponential growth, only in the opposite direction. That’s an important difference of course, but the point is that the two behaviors are exactly the same when you reverse the direction of time for either one of them.

I mention this because many people (I suspect) are more familiar with exponential growth than decay. A bank for example might offer savings accounts with a fixed compounding interest rate of 5%. This means that if you and I open up accounts there at the same time, regardless of the amounts we put in initially, after one year we’ll each have 5% more than we started with: the amounts will multiply by a factor of 1.05. And if we leave our money in there, after another year the two amounts will again multiply by a factor of 1.05. (This assumes we make no withdrawals or deposits.)

So the relative rate of growth is a constant: +5% per year. The absolute rate of growth varies with time, because it’s proportional to the amount present at a given moment, which keeps changing.

We don’t normally do this with savings accounts, but we could compute the doubling time of the customer’s money, analogous to the half-life of radioactive materials. For an interest rate of 5%, the doubling time is about 14.2 years. At a rival bank offering 6% rates, the doubling time would be only 11.9 years. Just like the interest rate, the doubling time is fixed; its value does not depend on the amount of money.

In principle, the decay rate of radioactive elements could also be expressed as a constant percentage per year — or per millennium, per eon, per second, or whatever unit was desired. Physicists find it more convenient to use half-lives, however. The equations are more suited to that form. (An e-folding time might be a tiny bit more convenient yet, but the half-life convention seems to have stuck.)

20

Bolivia went through some hyperinflation back in the '80’s. I make the case that you can speak of money in terms of half-lives as well. At that point, the halflife was 23 hours (purchasing power halved in terms of denomination).