# Why does a half life exist

1. Apr 4, 2012

### Brook

Hello
I was wondering if someone could explain why a half life exist?
What attribute of radioactive matter requires that on average half of the particle decay within this so called half life period.
Why couldn't it vary?
thanks

2. Apr 4, 2012

### davenn

hi brook
welcome to PF

a halflife is just the time it takes for x ammount of the parent element to decay into the daughter element. different elements have different halflifes some from seconds to minutes others up to many millions of years

from wiki...
"Radioactive decay is a stochastic (i.e., random) process at the level of single atoms, in that, according to quantum theory, it is impossible to predict when a particular atom will decay.[1] However, the chance that a given atom will decay is constant over time. For a large number of identical atoms (of the same nuclide), the decay rate for the collection is predictable from the measured decay constant of the nuclide (or equivalently from the half-life)."

that should also answer why the halflife of a given element doesnt vary

as to why some have longer halflives than others....
again from wiki....
"Atomic nuclei consist of protons and neutrons, which attract each other through the nuclear force, while protons repel each other via the electric force due to their positive charge. These two forces compete, leading to some combinations of neutrons and protons being more stable than others. Neutrons stabilize the nucleus, because they attract each other and protons equally by the strong nuclear force, which helps offset the electrical repulsion between protons. As a result, as the number of protons increases, an increasing ratio of neutrons to protons is needed to form a stable nucleus.

However, if too many or too few neutrons are present with regard to the optimum ratio, the nucleus becomes unstable and subject to certain types of nuclear decay. Unstable isotopes decay through various radioactive decay pathways, most commonly alpha decay, beta decay, or electron capture. Many other rare types of decay, such as spontaneous fission or cluster decay are known. See radioactive decay for details."

cheers
Dave

3. Apr 4, 2012

### Brook

hmmmm, rather interesting, thanks for replying. You say "For a large number of identical atoms (of the same nuclide), the decay rate for the collection is predictable from the measured decay constant of the nuclide". However for a single atom it can't be predicted exactly.
This is weird. This is like flipping a coin a bunch of times, it works out to be 50% heads or tails....which is what bugs me. Why does it have to be that way, I mean on average. Why can't it be like the rate of eroding a mountain. Bits get washed away at varying rates and the average rate of erosion might vary from year to year and season to season.
I just find it strange that mindless atoms can behave this way, on average that is. Are there any philosophical articles on this, by chance that is, lol...

4. Apr 5, 2012

### davenn

well others say... .I personally am no atomic physics expert ;)

why does it have to be that way ?? I cant answer that maybe some one with a deeper understanding can. Maybe its as simple as .... Thats the way nature is

to quote Prof Richard Fyenman (1979) .....

Its the way nature is!
If you dont like it, go somewhere else....
To another universe, where the rules are simpler
Philosophically more pleasing, more psychologically easy

cheers
Dave

5. Apr 5, 2012

### Emilyjoint

In a way atoms are like the coins. There are only 2possible states. Coins come down either heads or tails....this is where the 50% comes from.
Atoms either have decayed or they have not yet decayed (assuming they are radioactive in the first place) and this is where the 50% comes in again.
If you were throwing dice then the fraction would be 1/6 for a number to come up.

6. Apr 5, 2012

Exponential decay can be predicted theoretically by assuming the experimentally observed facts that that on average the decay rate is directly proportional to the number of radionuclides present(for a "large" number of nuclides) and that it depends on the structure of the nuclides.
Half life is defined but we could equally define any other ratio such as third life or fifth life.There is nothing fundamental about 50% or half life,it's just that it's a convenient number to work with.

7. Apr 5, 2012

### Dickfore

Because of the properties of the exponential function, and the properties of logarithms, you can write:
$$e^{-\lambda \, t} = \left( a^{\mathrm{log}_a(e)} \right)^{-\lambda \, t} = a^{-\frac{\lambda \, t}{\ln a}}, \mathrm{log}_a(e) = \frac{1}{\ln a}$$
Then:
$$T_{1/a} \equiv \frac{\ln a}{\lambda}$$
has the meaning of the time required for an $1/a$-part of the original number of radioactive nuclei to remain undecayed (and $(1 - 1/a)$-part to decay).

When $a=2$, it is called half-life.

8. Apr 5, 2012

### sophiecentaur

It's very easy to look at these problems 'the wrong way round'.
If you first consider that an individual atom has a certain probability of decaying in the next second. Say that probability is 1/1000. This is a problem, in itself, of course - 'How do we know that?' would be a reasonable question - but just assume it has a given probability.

This implies that, for a vast number N of Identical atoms, in the next second, 1/1000 of them will decay - leaving N X999/1000. In the next second there will be fewer atoms than at the start but 1/1000 of them will decay - leaving (NX999/1000)X999/1000 (the same proportional decrease each time. After a long enough interval, the total remaining will drop to a half of what it was when you started. Whenever you started and whatever N you had initially, the time to N/2 is always the same, for a particular type of atom (isotope).

The question 'how do you know the probability?' can be answered by actually measuring the half life and then working backwards.

9. Apr 5, 2012

### Brook

to quote Prof Richard Fyenman (1979) .....

Its the way nature is!
If you dont like it, go somewhere else....
To another universe, where the rules are simpler
Philosophically more pleasing, more psychologically easy

Thought experiment....
Supposing I took 2 sets of 1000 radioactive atoms and place one set in room A and the other in room B, then on average half of them would remain in both sets after a half life that is the same for both sets because they are the same isotope.

What if I placed the atoms in 2000 rooms so that each atom is separate from each other. Is it still true that half of them would remain after the half life or will nearly all decay?
Do the atoms have to be close together in a pile for the half life property to work.

10. Apr 5, 2012

### Naty1

It doesn't have to be that way, just that way in THIS universe....The mathematics of quantum mechanics, all the stuff in the Standard Model of particle physics has the underlying mathematics...but that never answers the question 'why'..... because you can always then ask 'why this math, why not some other math'....and the only answer is that
either 'we think so' if purely theoretical or 'this math matches experimental measurements'....which is better, but not still great.....

The really interesting question is why physics follows OUR mathematical constructs...why should it???

11. Apr 5, 2012

### Brook

please elaborate on this question you are posing.....I thought the math was derived from the physics, so it has to follow it...

12. Apr 5, 2012

### sophiecentaur

Umm.
It has been found possible to make mathematical models of the Physical processes we observe. These models 'usually' work when we try to predict behaviour. It doesn't follow that either Science or Maths is subservient to the other . when the mathematical models are close to what we see, we can pat ourselves on the back and feel good about it. Maths doesn't "have to follow" reality for all cases, any more than a map of London will get you around Paris.

But what IS Maths? That's a really hard question to answer. Was it 'there' before humans discovered it or is it just a way of writing down our human thought processes?

13. Apr 5, 2012

### epenguin

It's not like find a universe where things are simpler than this. They couldn't be simpler than this.
It is just saying that of the unstable isotope all atoms have a given chance of decaying in a short time. After a fraction of them have decayed (so that the radioactivity is less) this is still true and the chance of any of the remaining ones decaying is still what it was before.

Then maybe we get into slightly complicated calculations by asking how many decay in a long time. But the atom nucleus doesn't care about this, it is just there. And then it isn't. With probability p after a second.

It isn't just radioactive disintegrations, it's lots of things. For instance chemical reactions. A molecule has a given chance of undergoing a chemical change per second say (and the reacting substance has a given half-life). Simple. Except when it hasn't. Then we say not simple - something for us to do to try and find what special is going on.

The number of glasses that accidentally break in a restaurant probably follows a similar law. The number of people dying on the other hand only within rough limits. We find periods at some times when they don't, there is old age where the probability increases, we can conclude there is a physiological cause even before we know what it is. And also young age - they have greater accident risks.

So you could say the law not holding is an indication of some complicating cause needing explanation. Mind you even when it is simple and reduces to just one figure - the probability of change or half-time or rate constant... we still hope to be able to get to an explanation or prediction of that figure too.

Last edited: Apr 5, 2012
14. Apr 5, 2012

### Brook

Very insightful responses thanks...

Any ideas on this?

15. Apr 5, 2012

### QuantumPion

If you start with 1000 atoms, after one half-life you will have 500 left. If you start with 2000 atoms, after one half-life you will have 1000 atoms left. The reason why this is true is because the chance for any atom to decay is random. The atoms don't know or care how many atoms are near by. If you have one atom then every half life it will have a 50/50 chance of decaying.

For a better experiment, physically do this: get 100 pennies. Flip them all. Remove all the ones that land tails. Then flip the ones remaining and repeat. Each flip of all the remaining pennies is the equivalent of a half-life for atoms. The only difference is flipping pennies is a discrete act that occurs all at once where as atoms decay continuously.

16. Apr 5, 2012

### Staff: Mentor

They don't even have any "memory" of how long they've lived so far. No matter how old a radioactive atom is, it has the same chance (determined by its species and the type of decay) of surviving for another second, as any other atom of the same species.

17. Apr 5, 2012

### Brook

hmmm...
So I checked out uranium and saw that for U238 the half life is 4.4 billion years while for U235 it's 700 million.

I believe lead is the highest stable atom. Which element above lead is naturally occurring and is never created by the decay of any higher atom?

18. Apr 5, 2012

### QuantumPion

All elements heavier than iron are created in the same way - supernovas. Any element with a half life long enough for there still be appreciable concentrations left on Earth would be natural. These elements can be see on the chart of the nuclides. These elements are Bismuth, Thorium, and Uranium.

19. Apr 5, 2012

### Brook

So according to wiki Thorium has a half life that is large and Bismuth's haf life is fantastically huge....

thanks....

Are there any elements with a short half life that is less the the age of the earth. If so how are they preserved from disappearing all together?

20. Apr 5, 2012

### QuantumPion

There are, such as technetium and francium. They have no stable isotopes, so they do not occur naturally. They only quantities that exist are ones created artificially in cyclotrons or nuclear reactors.