Why "half-life"?
 

I knew this once upon a time, but forgot the answer... why is the decay of an element always given as the half-life? Why not just give the full time it takes to decay?

For example: Mt-268 has a half-life of just 0.07 seconds. Okay, so after .07 seconds, 50% of it exists. Why not just state when it no longer exists?

It's been a while since I've had chemistry/pre-calc (which touched on half-life/decay), and I've forgotten a lot of this stuff.. so bear with me :)

Because it never completely decays. Every 0.07 seconds, half of the remaining isotope will have decayed. Think of it like this:

You have to run 100m. You're not very fit, so you have to rest every once in a while. Let's say you take a rest when you've gone half of the remaining distance. So at first you run 50m, then 25m, then 12.5m, then 6.25m etc.

Will you ever actually reach the finish line? Well, theoretically, no. Eventually the measurements will be so small that simply breathing may move your shoe far enough to cross the finish line, but it's just an analogy :p

If you're looking for a more detailed mathematical explanation let me know.

Ahhhhhhhhhh yes, I remember now!

Half of a half of a half, etc... you never REALLY get to 0, but pretty damn close.

Thanks a ton. I tried Googling it, but I ended up with a lot of "what half-life is" as opposed to why it was measured as such.

Thanks! :D

But doesn't it eventually completely decay? I'm unsure of the specific terminology, but eventually, by repetitively dividing in half, you get to a "smallest unit" that is indivisible, and once this unit decays, it's all gone.

Same as with the running analogy, eventually, your shoe size prohibits in practice halving your run again. You reach a point where, halving the distance or not, your toes still peek over the finish line.

In a practical sense, yes. That's why they don't use Mt-268 for dating things thousands of years old. It's impossible to accurately measure how much of it is left.

Also, this isn't a very useful measurement. If you want to know how quickly something decays, then saying "it gets to 0 in 7 seconds" doesn't mean much, since it's an exponential decay, you'd have to do a lot of math in order to know how long it would take to get to, say, safe levels of radiation. By giving the half-life, it makes such calculations much simpler.

Half-life, more technically, is the period of time in which any one atom of the substance has a 50% chance of decaying.

As it happens, when dealing with large quantities of a substance (and any quantity of atoms you can weigh is 'large'), this means that pretty much exactly[1] half of it decays in that period of time.

If your isotope X had a half life of 1 day, and you had one atom, after 1 day there is a 50% chance it would be gone.

After 10 days there would be a (1-2^-10)*100% chance it would be gone, or about 99.9% chance.

There would never be a 100% chance it was gone.

Interestingly, even protons have a half life. There is a period far far far far[2] in the future when there is a greater than 50% chance that every proton in the universe will have decayed, assuming something else catastrophic doesn't happen first.

Footnotes:
[1]: Basically, the odds that >51% or <49% would be left are on the same order of magnitude as the atoms in your room spontaniously assembling into a working computer. Well, not quite... =)
[2]: No, I actually didn't use enough "far"s. Not by a long shot.

Awesome info, thanks a ton :)

I heard a great joke that went along with this at one point .. but I don't remember the whole thing..... =(. I'll tell it as best as I remember though, because I thought it was funny.


A mathematician and a theortical physicist (both are heterosexual men for this example) are brought into a room and put in a chair at one end of the room. At the other end of the room is a beautiful woman lounging seductively in the nude on a couch. The two men are told that they can have sex with her when they reach her, but that they are only allowed to move toward her by closing half the distance to her on each move.

Well, the mathematician says, "this is useless, we'll never reach her!" and he leaves. The theoretical physicist starts to take off his clothes and the coordinator says, "You do realize that you will never reach her don't you?" and he replies, "I'll get close enough."

haha....I believe the punchline you're looking for is "close enough for all practical purposes!" :-)

(used to be a data entry nerd in the physics department)

yeah yakk's got it right.
I gotta say though, that last atom that decays is damn lucky.

Or unlucky! Imagine how lonely it would be :( Better to end in a blaze of glory with half of the other protons in the universe.

Somebody got the idea of half life wrong.

It is NOT the time at which a decay will happen with a 50% probability. Half life is a chemistry term that relates to rate as opposed to chance.

Half-life is the time required for a system to change equibilrium in such a manner that only half of the concentration of "X" remains in relation to the immediate past concentration.

Rate and half-life are related as follows for nuclear (1st order) systems:

half-life = ln (2) / k

where k = the rate constant of decay

This said, if the half-life of Iodine is say, 5 minutes .. a sample of radioactive Iodine with # molecules = 1 mol initially; at time = 10 minutes we will have 0.25 mol of radioactive nucleii remaining.

Oops, sorry, you are right Sapper.

I forgot to compound my "interest" over infinitely small periods. =)

It's an important distinction that needs to be made.

besides,

http://www.ambient-tech.net/images/tfp/lim_bsc.gif