For 1000k atoms I will have a 500k after 1 half life, and 250k after 2 half lives. But when I try to do the same with fractions something doesn’t add up. Because after “quarter life” i will have 750k atoms, but then I’ll have 562500 atoms, because of 750k multiplied by 0.75. So quarter life should be equal to 0.5 of half life, but half life doesn’t work that way. I am confused.
In: Physics
Half life is a characteristic measure of a non-linear process. It turns out that specifying three facts about a process:
1. It is exponential
2. The quantity at one time
3. The quantity at another time
fully fixes that behavior. By convention the time interval is given such that the ratio of quantities is 1:2, but it’s just convention. There’s nothing special about that particular fraction. If the half life of material was 100 years we could say the 1/3rd life was a different number and it would contain identical information.
Converting between different fractions isn’t trivial, as you’ve discovered. The math isn’t necessarily difficult, just not generally you can do without some thought.
I believe the conceptual problem is one of language. You read “half life” and consider “life” to be some quantity of time to which “half life” is 0.5 multiplied by that quantity. I.e. if half life is 50 years then quarter life must be 25, tenth life must be 5, etc.
You’re thinking “half life” is “half of a life” but it really means “life of a half” or to put it in unmistakable language “time of a halving.”
As an aside, the term “quarter life” I took to mean the time interval in which the material was reduced **to** one quarter, not **by** one quarter. I don’t know which interpretation is more common. Obviously the term half life doesn’t suffer from this confusion because both interpretations are numerically the same.
The time to reduce all of the material is infinity. We know that the time to reduce the material by half is not half of infinity. By the same thinking the time it takes to reduce by one quarter is not half of a “time of halving”.
In mathematical practice when you know the “time of halving” and want to know the “time of ___” you need to find the equation of behavior and solve it for this different fraction.
The general equation for exponential decay is:
X = A * e^(-kT)
where X is how much at time T, A is how much at time zero, and k is the decay constant.
Half life (H) is a particular time T=H where X/A is 0.5. With this information the value of k can be discovered and the full character of the decay process known at all times.
The rearrangment of the equation for k looks like:
k = -ln(X/A)/H
If the equation seems scary, don’t worry, the point is that k is a number. Just remember the value of k depends of the units of time used in T. If you use the same k and T=10 years or T=10 seconds you would get the same decay when we know they shouldn’t be. It’s another aside but it’s a classic test question mistake like half life is 10 years how much is left after 20 days and people answer one fourth.
So you want to know the time where X/A=0.75? Well go back to the equation:
X/A = 0.75 = e^-(kT)
Now we want to know what T is. Rearrange:
T = -ln(0.75)/k
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